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Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: [email protected]
vF : QFT → Obs, iew Holography is usually presented as a string–theoretic duality between a gravitational “bulk” and a lower–dimensional quantum field theory, epitomized by AdS/CFT. In this work we develop a string–free formulation of holography based on higher–categorical structures, in which holography is understood as a context–dependent equivalence between a bulk theory and an encoding theory in an appropriate (∞, 1)–category. Concretely, we regard quantum (and effective) field theories as objects in a higher category QFT, choose an encoding object E and a functor H : B → E from a bulk theory B, and specify a context by functors which select the class of observables of interest. Holography then appears as a natural equivalence
tn
rather than as a universal, context–independent notion. This relative, higher–categorical view generalises and sharpens the “relativity of holography” idea introduced in our earlier work [4], and incorporates both compressive “super–holography” (bulk → lower–dimensional encoding) and anomaly–inflow holography (bulk ↔ higher–dimensional invertible TQFT) within a single formalism.
We show that non–string encodings are both more general and practically more powerful than string encodings. On the QED side, we construct a string–free super–holographic encoding of four–dimensional spinor QED into a (0 + 1)–dimensional super–worldline theory, enriched by bilocal photon kernels at two loops. For strong, time–dependent electric fields (including rapidly varying Sauter pulses), the nonperturbative pair–production exponent and loop–corrected prefactors are computed from worldline instantons: solutions of one–dimensional (generally complex) classical equations of motion and their one–dimensional fluctuation operators, plus low–dimensional integrals over bilocal kernels. This replaces an intractable four–dimensional tower of multi–photon Feynman diagrams by an algorithmic, dimension–reduced computation. On the QCD side, we reformulate anomaly inflow and the Wess–Zumino–Witten term as an instance of higher–categorical holography: a four–dimensional SU(N c ) gauge theory with N f massless quarks is holographically encoded into a five–dimensional anomaly TQFT and a four–dimensional WZW functional. The holographic equivalence in the chiral–anomaly context fixes the WZW level k = N c and the baryon number of Skyrmions, without solving QCD nonperturbatively. We further interpret the effective string description of confining flux tubes in Yang–Mills theory as a two–dimensional encoding in a defect 2–category, explaining the universal Lüscher term as a super–holographic Casimir effect on the worldsheet.
Finally, we sketch how renormalisation (UP–RG) and density functional theory (UP–XC) can be recast as universal encoding problems in this framework: for a chosen context functor F , an effective theory E realises a universal property among all theories reproducing the selected observables. Throughout, we emphasise that string encodings (e.g. AdS/CFT) are special cases of our higher– categorical holographic principle, valid in particular contexts, but neither universal nor optimal. By contrast, string–free encodings—worldline theories, extended TQFTs, defect categories, and universal–property RG constructions—apply to a vastly broader range of physical situations and already yield concrete new computational capabilities in realistic QED and QCD.
The holographic principle, in its modern high-energy incarnation, is most commonly associated with the idea of a string–theoretic duality relating a gravitational theory in a (d + 1)–dimensional bulk spacetime to a non–gravitational quantum field theory (QFT) living on a d–dimensional boundary. The canonical example is the AdS/CFT correspondence, originally proposed by Maldacena [1], and further developed by Gubser, Klebanov and Polyakov [2] and Witten [3]. In its simplest form, this duality asserts an equivalence between type IIB string theory (or supergravity in an appropriate low–energy limit) on AdS 5 × S 5 and N = 4 supersymmetric SU(N ) Yang–Mills theory in four dimensions. Symbolically,
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where the bulk partition function Z IIB as a functional of boundary values of bulk fields (g µν , Φ, . . . ) is identified with the generating functional of correlators of the dual operators O in the boundary CFT, with sources J O determined by the boundary values of bulk fields via the GKPW dictionary [1–3]. In more refined language, (1) is usually understood at the level of a conjectured isomorphism of (appropriately completed) Hilbert spaces and operator algebras, and in certain regimes as an equivalence of categories of (BPS) states and line operators.
Despite its spectacular successes, the standard holographic narrative suffers from two conceptual limitations that are rarely addressed explicitly:
(i) It implicitly treats holography as absolute, i.e. as a property of a given pair of theories independent of
which observables one considers. That is, one tends to write B ≃ E where B is a “bulk” theory and E is its holographic dual, without making precise in which sense (and for which class of observables) this equivalence holds.
v(ii) It privileges string–based encodings — bulk string/brane theories and boundary CFTs — as if these
were the natural or even the only way to realise holographic equivalences.
Both points become problematic as soon as one tries to extend holography to regimes that differ from the original AdS/CFT setup: realistic QCD, strong–field QED in rapidly varying backgrounds, finite temperature and density, condensed–matter systems, and density functional theories used in materials science and chemistry. These theories typically • are not supersymmetric,
• are not conformal (no exact CFT structure), • do not admit a controllable large–N limit,
• involve complicated non–equilibrium or non–local observables.
In such regimes, the existence and usefulness of a standard string dual is either unknown or extremely doubtful.
A crucial conceptual shift was introduced in our previous work [4], where we analysed holography and quantisation through the lens of the Universal Coefficient Theorem (UCT) in algebraic topology. There we observed that:
several constructions considered as absolute until now may appear as relative, depending on individual choices of group structures needed to probe a topology [4].
tn Concretely, the UCT relates homology or cohomology with coefficients in a general abelian group G to the same (co)homology with coefficients in Z, via short exact sequences involving Tor and Ext groups:
and similarly in cohomology. The choice of coefficients G amounts to a choice of “probe” for the topology of X; different choices of G may detect or ignore torsion subgroups, thus altering the apparent content of topological invariants. In [4] it was shown that this dependence on G can be reinterpreted in the context of holography: the same quantum field theoretic configuration space can appear to have different “visible” structure depending on which coefficient groups one uses, and hence holographic “matching” of degrees of freedom is a relative statement rather than an absolute one.
In other words, the UCT indicates that topological data — and thus any holographic principle that counts or matches such data — is not an absolute property of a space or theory, but a property of a pair consisting of a space and a choice of coefficient system. This provides a mathematically precise example of what we shall call relative holography: the notion that “bulk” and “boundary” degrees of freedom can only be compared after specifying a context, i.e. what structures and equivalence relations one is allowed to use.
The first motivation of the present work is to lift this relativity of topology to a fully higher– categorical and context–dependent formulation of holography. Instead of focusing on individual groups or cohomology theories, we consider entire categories (more generally, (∞, 1)–categories) of quantum field theories and effective theories, which we denote by QFT. Objects of QFT are full theories (including their field content, interactions, and possibly their extended operator content), and morphisms are This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 3 iew F : QFT −→ Obs,
structure–preserving maps between them: embeddings, renormalisation flows, defect inclusions, etc. In this language, a bulk theory is an object B ∈ QFT, and a candidate holographic dual is another object E ∈ Enc, where Enc is an encoding subcategory or a different but related (∞, 1)–category of encoding theories.
The essential observation is that “holographic equivalence” is meaningless without specifying which observables one is comparing. Mathematically, this is encoded in a context functor:
(3)
where Obs is a suitable category of observables (e.g. correlation functionals, response functions, anomaly polynomials). One can also define a functor
This replaces the vague statement “B and E are dual” by a precise diagram:
where the functor H : QFT → Enc specifies how B is encoded in E, and α expresses the equality of observables in the chosen context. Note that there is no requirement that H be invertible or that F or G be faithful on all of QFT; they are only required to capture the prescribed class of observables.
This diagrammatic viewpoint immediately clarifies point (i) above: there is no such thing as holography in the abstract; there is only holography relative to a choice of functor F and encoding E. Different physical questions correspond to different choices of F (different context of observables); for each such context, there may be very different “best” encoding theories. For example, as we shall demonstrate in later sections, for the context of non–perturbative pair–production rates in strong, time–dependent electric fields in QED, a 0 + 1–dimensional worldline encoding is natural and computationally powerful, while for the context of chiral anomalies in QCD, a 5–dimensional invertible topological field theory and a Wess–Zumino–Witten (WZW) functional provide the correct encoding.
tn The second motivation of this work is to make explicit that string encodings are just one special class of encodings E in this general framework, and that they are generically not the minimal or most useful ones, even for problems with a clear spacetime interpretation. AdS/CFT and its variants can be recovered when Enc is chosen to be a category of string/brane theories in special geometries, and F is a functor that selects (e.g.) boundary correlators or entanglement entropies. But nothing in the formalism forces Enc to be stringy: the encoding category can be
• a worldline super–quantum mechanics (for one–loop and two–loop QED phenomena),
• an extended topological field theory with one higher spacetime dimension (for anomalies and WZW
terms),
• a defect 2–category of line and surface operators (for confining flux tubes and effective string
descriptions),
• or a category of density functionals and universal property renormalisation group flows (for DFT
and RG encodings).
In particular, the examples we will develop in QED and QCD demonstrate that:
real QCD),
diagrammatics into one–dimensional worldline or two–dimensional worldsheet problems), This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 4
refinements, and defect categories) which are awkward or opaque in a purely string/brane language.
From the point of view of category theory, strings and branes are simply concrete geometric models of higher morphisms in certain extended field theories; they are not the only such models, and they are certainly not the most general. The Universal Coefficient Theorem perspective from [4] already suggested that dualities and holographic correspondences should be understood as consequences of universal properties in homological algebra, not as primitive geometric facts. The present work extends that line of thought to a full higher–categorical, context–dependent and string–free holographic principle, and shows that it yields new capabilities in real QFT computations.
In summary, the motivation for this paper is twofold:
v• to provide a mathematically precise and higher–categorical reformulation of holography as a relative
notion, depending on choice of context and encoding, generalising and extending the topological relativity of holography found in Ref. [4];
• to demonstrate, through explicit examples in QED and QCD, that non–string higher–categorical
encodings not only subsume stringy holography as a special case but also offer concrete computational advantages in regimes where string theory has little to say.
The subsequent sections will elaborate the formal framework and then develop in detail a series of examples: worldline super–holography in spinor QED, anomaly inflow holography in QCD, defect– category holography for confining flux tubes, and universal–property holographic encodings for RG and DFT.
The primary goal of this work is to develop and demonstrate a string–free, higher–categorical formalism for holography that is both mathematically precise and practically powerful for realistic quantum field theories. In contrast to the traditional AdS/CFT paradigm, where one starts with a specific string background and then identifies a dual boundary CFT, our approach abstracts away from strings altogether and instead formulates holography directly in terms of (∞, 1)–categories of quantum field theories, encoding functors, and context functors selecting observables.
tn Higher–categorical formulation of holography. We begin by treating quantum (and effective) field theories as objects in an ambient (∞, 1)–category, which we denote by QFT. This category can be realised in several concrete ways, depending on the level of refinement:
• As an (∞, 1)–category of extended field theories in the sense of Baez and Dolan [5], where objects
are associated to (d − 1)–manifolds, 1–morphisms to d–dimensional bordisms, and higher morphisms to bordisms between bordisms.
• As an (∞, 1)–category of factorization algebras on manifolds, encoding local observables and their
OPE structure [23].
• As a more classical category enriched over topological vector spaces, where objects are Lagrangian
field theories and morphisms are suitably defined (equivalence classes of) transformations such as RG flows, duality transformations, or defect inclusions.
For the purposes of this paper, we do not commit to a single concrete model of QFT; rather, we assume that we have fixed some such model with enough structure to encode the examples we study. Crucially, theories are objects B ∈ QFT, and relations between theories are morphisms (or higher morphisms) in QFT, a viewpoint aligned with the modern functorial and higher–categorical approaches to QFT [7].
Given this ambient category, the first structural goal is to define two additional ingredients:
objects are encoding theories E that are simpler, lower–dimensional, or otherwise more tractable than the bulk theory B;
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 5 2. an encoding functor
F : QFT → Obs, G : Enc → Obs, iew
(8)
where Obs is an appropriate (∞, 1)–category of observable data: correlation functionals, effective actions, anomaly polynomials, entanglement measures, etc. The functor F specifies which observables we extract from bulk theories, and G specifies the same for encoding theories. The basic holographic relation between B and E then becomes the statement that there exists a natural equivalence
v α B : F (B) −−→ G(H(B)) in Obs. This equality is not assumed to hold for all observables, nor for all theories simultaneously; it is always relative to the choice of context functors F and G. One of the central technical goals of the paper is to make this relative holography precise and to show, through explicit examples, how the choice of context determines the appropriate encoding category and functor.
Super–holography and anomaly holography. Within this framework, we identify two important broad classes of holographic situations.
Super–holography (compressive holography).: Here the encoding theory E has strictly lower space-
time dimension (or substantially fewer degrees of freedom) than the bulk theory B, i.e.
or more generally dim eff (E) dim eff (B) in an information–theoretic sense. Examples include:
• worldline encodings of 4D QED into 0+1D worldline quantum mechanics [8, 9] for one–loop
and even two–loop processes; • effective worldsheet theories for 4D confining flux tubes in Yang–Mills theory, i.e. 4D → 2D
encodings, where the static quark potential is mapped to the ground state energy of a 2D string-like excitation [12].
tn These are instances of what we will call higher–categorical super–holography: the relevant observables of a high–dimensional interacting field theory are exactly reproduced by a much lower–dimensional encoding in a suitable context.
Anomaly holography (inflow–type holography).: Here the encoding theory E lives in one higher
spacetime dimension than the bulk, i.e.
and is typically an invertible topological field theory. This is the anomaly–inflow paradigm: anomalous variation of a d–dimensional QFT is interpreted as the boundary variation of a (d + 1)– dimensional topological term. Classical examples include:
• the Wess–Zumino–Witten (WZW) term in chiral Lagrangians, whose coefficient k is fixed by
matching to the SU(N c ) 3 chiral anomaly of QCD [11, 69]; • the parity anomaly of a single Weyl fermion in odd dimensions, compensated by a bulk
Chern–Simons term.
In our higher–categorical language, these are examples where the functor H maps a 4D QCD object to a 5D invertible TQFT (plus a 4D WZW theory) and the context functor F selects the anomaly polynomial, yielding a holographic equality F (B) ≃ G(E) that fixes non–perturbative data such as the WZW level k = N c .
A second major goal of the paper is to show explicitly that these two broad classes — super–holography and anomaly holography — are not disparate phenomena but are both instances of the same higher– categorical structure (6), distinguished only by the relative dimension of the encoding category and by the chosen context functor.
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6
Decisive capabilities in mainstream QFT. Beyond the abstract formulation, a central aim of this paper is to demonstrate that the higher–categorical, string–free holographic framework is not merely philosophically appealing, but yields decisive computational capabilities in concrete, mainstream quantum field theories. In particular, we will develop in detail the following examples:
(12)
vsemiclassically from worldline instantons, i.e. closed classical trajectories solving the Euler–Lagrange equations derived from (12). The leading exponent and loop–corrected prefactors are then determined by one–dimensional classical ODEs plus one–dimensional fluctuation determinants [9]. At two loops, a single bilocal photon insertion along the worldline adds a bilocal interaction functional; again, all two–loop diagrams are encoded in a single 0+1D object. Thus a four–dimensional, non–perturbative QED problem involving an infinite tower of multiphoton Feynman diagrams is compressed into a one–dimensional higher–categorical encoding with a finite algorithmic complexity.
colours, the SU(N f ) 3 chiral anomaly is captured by a six–dimensional anomaly polynomial I 6 whose coefficient is proportional to N c [69]. In the low–energy regime, the effective theory is a non–linear sigma model on SU (N f ) with a WZW term
tn
defined on a 5D manifold B 5 with boundary M 4 , where U (x) ∈ SU (N f ) is the chiral field. From a purely IR perspective, k is an integer that labels distinct EFTs. By constructing a 5D invertible TQFT whose boundary variation reproduces the anomaly polynomial, and by enforcing a holographic anomaly–inflow equality F anom (B) ≃ G(E) in the chiral–anomaly context, one uniquely fixes k = N c and thereby ties the IR WZW sector and the baryon number of Skyrmions to the UV quark content, without solving QCD non–perturbatively. This is an explicit higher–categorical anomaly holography example.
between a static quark–antiquark pair at large separation R is believed and observed to behave as
where σ is the string tension and the term −π/(12R) is the universal Lüscher correction [12]. From the higher–categorical viewpoint, static quarks correspond to line defects (1–morphisms) and the confining flux tube is a surface defect (2–morphism) in a defect 2–category associated to the 4D theory. In the appropriate context (long–distance static potential), one can encode this defect sector into a 1+1D worldsheet QFT whose ground state energy reproduces V (R), and in which the Lüscher term arises straightforwardly as a Casimir energy of transverse fluctuations. This is another instance of super–holography (4D → 2D encoding) where no fundamental string theory is required; instead the encoding lives in a defect 2–category of the 4D gauge theory.
malisation group flows and density functional theory (DFT) can be recast as universal encoding problems in the same higher–categorical language. Inspired by the functional renormalisation group formulation [13] and by the Hohenberg–Kohn and Kohn–Sham constructions in DFT [51, 52], we define universal–property RG (UP–RG) and universal–property exchange–correlation (UP–XC) as follows: given a context functor F that selects a class of observables (e.g. long–distance correlators or ground–state densities), an effective theory E together with a functor H is said to satisfy This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 7
UP–RG/UP–XC if it is initial in the fiber category of theories reproducing those observables. This makes precise the idea that there exists an optimal encoding theory (EFT or density functional) for a given context, and connects RG and DFT to holography as instances of higher–categorical encodings.
Main claims
v
By developing these examples in detail and within a unified categorical framework, the paper aims to demonstrate that non–string higher–categorical encodings are not only conceptually more general than string encodings, but also practically more powerful: they apply to a much wider class of physical regimes (realistic QED, QCD, DFT), they naturally accommodate structures that string theory struggles to encode (worldlines, anomalies, defects, universal properties), and they yield explicit, algorithmic computational methods (worldline instantons, extended TQFT anomaly inflow, defect Casimir energies) in situations where string–based holography has little or nothing to say.
Having introduced in Sections I A and I B the conceptual background and the general strategy of our approach, we now state the main claims of this paper in a precise and unified form. Each of these claims will be substantiated in the subsequent sections through explicit constructions, examples, and (where appropriate) rigorous proofs. Collectively, they articulate a shift from the traditional, string–centric view of holography to a broader, higher–categorical and context–dependent picture.
Claim 1: Holography is higher–categorical and context–dependent, not essentially stringy. The first main claim is that holography is most naturally and correctly formulated as a statement in higher category theory, and that it is inherently relative to a choice of context (i.e. of observables and equivalence relations), rather than as a global, context–independent duality. Mathematically, this is expressed by the existence of an ambient (∞, 1)–category QFT of quantum field theories and an encoding category Enc, together with
• a functor H : QFT → Enc sending bulk theories B to encoding theories E = H(B), as in (7),
• a pair of context functors F : QFT → Obs and G : Enc → Obs, as in (8), where Obs is a suitable (∞, 1)–category of observable data,
• a natural equivalence
tn
This is precisely the diagram (6). The essential point is that this definition does not presuppose anything about the nature of the encoding category Enc: it may be a category of string/brane theories, but it can equally well be a category of worldline theories, extended TQFTs, defect categories, density functionals, or quantum codes. In other words, the string–theoretic realisation of holography (as in AdS/CFT) is a special case where Enc is chosen to be a particular class of string backgrounds and F, G are chosen to extract specific types of correlation functions or entanglement observables [16].
From a higher–categorical perspective, holography is fundamentally about equivalences of functors between appropriate (∞, 1)–categories, not about the existence of a two–dimensional worldsheet. For example, in the framework of extended topological field theories, a fully extended d–dimensional TQFT is a symmetric monoidal functor from a bordism (∞, d)–category to a target (∞, 1)–category of linear or higher–linear objects [17]. Holography in this setting naturally appears as a statement that two such functors, when restricted to certain types of bordisms or defects, become equivalent. This holds whether or not there is any underlying string picture, and suggests that higher categories, rather than strings, are the natural home of holographic ideas.
We therefore claim that any holographic identification is always of the form (15), and that the notion of “bulk” and “boundary” must be understood as roles played by objects in QFT and Enc relative to the chosen context F, G. In particular, the same theory B may admit multiple, inequivalent encodings E i and functors H i that are holographically valid for different choices of context. This is the higher–categorical generalisation of the “relativity of holography” identified in [4] for coefficient systems and Ext/Tor corrections; here we extend it to full (∞, 1)–categorical structures.
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Claim 2: Non–string encodings are more general and structurally better adapted than string encodings. The second main claim is that, within the framework above, non–string encodings — i.e. encodings in which Enc is not a category of string or brane theories — are generically more general, more flexible, and often structurally better adapted to the physics than string encodings. This claim has several aspects:
v
requires, for its precise formulation, a number of structural assumptions: conformal symmetry, large–N limit, some amount of supersymmetry, and often special geometric properties of the bulk background. By contrast, many physically important regimes — strong–field QED in time– dependent backgrounds, real QCD at finite density, strongly correlated electron systems, density functional theory in condensed matter — do not satisfy these assumptions and have no known or practically useful string dual. Non–string encodings (worldline theories, extended TQFTs, defect categories, factorization algebras) can be defined for generic QFTs and do not rely on supersymmetry, conformality, or large–N .
worldlines of quanta, higher–form symmetries and anomalies, defect networks, code subspaces, etc. For example, anomaly problems are best captured by higher–group and cohomological structures [18], confining flux tubes by defect 2–categories [19], and holographic error–correcting codes by tensor– network categories [20]. In each case, the natural encoding category Enc is not a geometric string worldsheet, but a higher category whose objects and morphisms directly reflect the relevant structures (anomaly polynomials, defect junctions, code projections). Non–string encodings thus match the relevant structure more directly than strings, which always come with a rigid geometric interpretation.
that non–string encodings lead to strict computational simplifications. For instance:
• In strong–field, rapidly varying spinor QED, the non–perturbative pair–production rate in a
given background A µ (x) is, in principle, determined by the imaginary part of a four–dimensional functional determinant, whose Taylor expansion corresponds to an infinite tower of Feynman diagrams with arbitrary numbers of external photon insertions. In the worldline formalism, however, this entire one–loop (and even two–loop) content is encoded in a single (0 + 1)– dimensional super–worldline theory: the exponent and prefactor of the pair–production rate are extracted from worldline instantons (solutions of one–dimensional classical equations of motion) and their one–dimensional fluctuation operators. This collapses an intractable 4D combinatorial problem into a manageable 1D one.
tn • In anomaly/WZW holography for QCD, the structure of the chiral anomaly is encoded in a 5D
anomaly TQFT and a 4D WZW functional, and the WZW level and Skyrmion baryon number are fixed by an anomaly inflow condition, not by solving QCD non–perturbatively. Here, the encoding category is that of extended TQFTs and cohomological data, with no need for a string interpretation; the categorical structures (bordism, cohomology, anomaly) naturally capture the physics. rin • In confining QCD, the IR physics of static quark–antiquark potentials is efficiently encoded in a 1+1D worldsheet QFT for flux tubes, obtained from the defect 2–category, yielding the Lüscher term as a simple Casimir energy. By contrast, diagrammatic 4D calculations of such corrections are prohibitive.
These examples show that non–string encodings can reduce the complexity of concrete QFT computations, whereas string duals (when they exist at all) often serve more as conceptual guides than as practical computational tools in these regimes.
From a strictly mathematical standpoint, the superiority of non–string encodings arises because the category of string/brane theories is just one subcategory of the space of all possible encoding categories. The latter includes (and we will make explicit use of) (∞, 1)–categories of worldline theories, extended TQFTs classified by generalized cohomology theories, and defect categories with higher–morphism structures [21]. These more general encoding categories are closed under operations (such as taking defects, higher–form symmetries, or factorization structures) that naturally arise in a wide variety of QFTs, whereas string categories are tied to specific two–dimensional CFTs and their moduli.
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Claim 3: Non–string encodings already yield new capabilities in realistic QED and QCD. The third and most decisive claim is that the foregoing conceptual framework is not merely a matter of language or unification: non–string higher–categorical encodings already provide new, concrete capabilities for QED and QCD computations that are outside the practical reach of current string–based holographic methods.
To make this claim precise, we will explicitly construct, within the formalism of Section I B, several pairs (H, F ) and corresponding encodings E with the following properties:
• The bulk theories B are familiar QFTs: four–dimensional spinor QED in strong, time–dependent
fields; SU(N c ) Yang–Mills theory; QCD with massless quarks.
• The encoding theories E are lower–dimensional or higher–dimensional but non–string theories:
worldline super–quantum mechanics, anomaly TQFTs, defect worldsheet theories.
v• For a clearly specified context functor F (e.g. pair–production rates, anomaly polynomials, static
quark potentials), the observables F (B) are computed exactly or to a high degree of approximation from G(E) by means of algorithmic procedures in low dimensions (solving ODEs, computing one–dimensional determinants, evaluating finite sums).
In each case, we demonstrate that:
the bulk problem is intractable (or at least very difficult) in naive 4D diagrammatic or lattice terms,
the encoding problem is tractable in 0+1 or 1+1 dimensions,
semiclassical accuracy.
These constructions are all string–free and yet fully consistent with the higher–categorical holographic principle (15). They support the overarching thesis that holography, properly understood, is about higher– categorical encodings tailored to specific contexts, and that strings are at best one class of encodings that are valuable in some contexts but neither universal nor optimal. The rest of the paper is devoted to making these claims precise, proving them in the appropriate mathematical sense, and illustrating them through detailed QED and QCD examples.
CONCEPTUAL FRAMEWORK: STRING–FREE HIGHER–CATEGORICAL
HOLOGRAPHY
II.
tn In order to formulate holography in an intrinsically higher–categorical and string–free manner, we must first specify the ambient categorical structures in which both bulk and encoding theories live, and in which observables are valued. The aim of this subsection is to describe these ambient categories in sufficient detail to make later constructions fully precise.
The (∞, 1)–category QFT of quantum and effective field theories. The starting point of our framework is an (∞, 1)–category
QFT, (16)
whose objects are (equivalence classes of) quantum and effective field theories, and whose morphisms encode physically meaningful maps between such theories. In practice, one may choose several different models for QFT, depending on the desired level of refinement:
(1) In the context of extended topological field theories, one can take QFT to be the (∞, 1)–category of
symmetric monoidal functors
framed, oriented, spin) and C is a symmetric monoidal (∞, 1)–category of “targets” (e.g. Vect C , categories of module spectra, or higher categories). In this setting, the objects of QFT are themselves functors Z, and 1–morphisms between two such functors Z 1 , Z 2 are symmetric monoidal natural transformations, with higher morphisms given by modifications, forming an (∞, 1)–category as established in the higher–categorical formulation of the cobordism hypothesis [22].
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(2) In the context of perturbative or non–topological QFT, one may instead define QFT as an (∞, 1)–
category whose objects consist of complete data (F, S), where F is a space of fields (typically a derived stack or a locally convex topological vector space modulo gauge symmetry) and S is a local action functional modulo gauge–equivalence; morphisms are given by field redefinitions, RG flows, or factorization algebra morphisms as in Costello–Gwilliam [23]. The homotopical structure (weak equivalences, higher homotopies) reflects physically trivial deformations and gauge redundancies. In all these realizations, the key properties of QFT that we will use are: iew (3) At a more phenomenological level, one can take QFT to be the (∞, 1)–category obtained by localising
a 1–category whose objects are Lagrangian field theories and whose morphisms are equivalence classes of maps such as dualities, embeddings, or defect inclusions, with weak equivalences implementing e.g. IR equivalence or quasi–isomorphism of the corresponding factorization algebras; higher morphisms then arise from homotopies and higher coherences.
v1. The existence of a well–defined notion of objects and morphisms, with higher homotopies between
morphisms. This is the basic (∞, 1)–categorical structure.
products of theories. In the extended TQFT model (17), this is encoded in the symmetric monoidal structure of the bordism category and the target category C.
higher–categorical data of assigning objects and morphisms to manifolds of various dimensions.
We emphasise that, although the full formalisation of QFT as an (∞, 1)–category is highly nontrivial and relies on the foundations of higher category theory [24], the specific examples we consider (worldline encodings, extended TQFTs, defect categories) are sufficiently well–understood that our constructions are unambiguous and compatible with known rigorous results.
Formally, one can regard QFT as a disjoint union (or coproduct) of full subcategories
G QFT ≃ QFT
d , (18)
(19) d∈N
tn where QFT d is the full subcategory of theories defined on d–dimensional spacetimes (or whose primary spacetime dimension is d). For instance, 4D spinor QED with an external electromagnetic background belongs to QFT 4 , while a 1D worldline theory belongs to QFT 1 . The functor dim is then defined on objects by dim(B) = d if B ∈ QFT d , and on morphisms by dim(f ) = id d for any f : B 1 → B 2 with B 1 , B 2 ∈ QFT d . This makes dim a strict functor to the discrete category on N.
In some contexts, one may wish to define an effective dimension dim eff (B) that quantifies not just the spacetime dimension but also the number of active degrees of freedom (for instance, after RG flow or dimensional reduction). While such a notion is less canonical, one can model it by a functor from QFT to a poset category of “complexity measures” (e.g. number of fields, rank of factorization algebra, etc.). For the purposes of this paper, the coarse grading (18) by spacetime dimension is sufficient to distinguish the bulk and encoding theories in our examples (e.g. 4D QED vs. 1D worldline theories, 4D QCD vs. 5D anomaly TQFTs vs. 2D flux–tube worldsheet theories).
The (∞, 1)–category Obs of observables. Complementary to QFT, we must specify an (∞, 1)–category
Obs (20)
in which observables take values. Depending on the level of refinement required by a given example, Obs can be chosen in several ways:
(1) At a basic level, one can take Obs to be the homotopy category of cochain complexes of vector spaces
(dg–vector spaces), denoted by Ch(C), or its (∞, 1)–categorical enhancement. In this case, F (B) is e.g. the complex encoding correlation functions, Ward identities, or anomalies, and homotopies in Obs correspond to cohomologically trivial deformations.
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(2) For perturbative QFT, one can take Obs to be the (∞, 1)–category of factorization algebras on
spacetime [23]. The functor F assigns to a theory B its factorization algebra of local observables; morphisms in Obs correspond to local equivalences of observables and OPE structures.
(3) In the context of anomalies and topological phases, one can take Obs to be a suitable (∞, 1)–category
of generalized cohomology theories or spectra (e.g. K–theory, cobordism spectra), so that F (B) is the anomaly class or topological invariant associated to B [71]. In this setting, holography becomes a statement about the equality of cohomological invariants.
(4) In strong–field QED, one can take Obs to be the (∞, 1)–category of functionals on background
fields, or a suitable enriched category whose objects encode non–perturbative effective actions and pair–production rates as functions of the background profile.
vThe essential requirement on Obs is that it carries enough homotopical structure to distinguish physically different observable behaviours (different anomalies, different effective actions, different correlation functions), while identifying cohomologically trivial deformations. This ensures that equivalences F (B) ≃ G(E) in Obs encode genuine holographic relations rather than trivial reparameterizations.
Encoding categories Enc and subcategories of QFT. While in principle we could take Enc to be a full subcategory of QFT, in practice it is often useful to allow Enc to be a different but related (∞, 1)–category whose objects are encoding theories of a special form. For example:
• In worldline holography for QED, Enc is the (∞, 1)–category of (0 + 1)–dimensional super–quantum
mechanics with couplings to background fields, enriched over super vector spaces; the encoding functor H maps a 4D QED object to its 1D worldline encoding as in (12).
• In anomaly holography for QCD, Enc is an (∞, 1)–category of extended TQFTs in one higher
dimension, together with WZW data at the boundary; H maps a 4D QCD object to its 5D anomaly TQFT and associated WZW encoding.
• In flux–tube holography, Enc is a defect 2–category whose objects are worldsheet QFTs for flux
tubes; H embeds the defect sector of the 4D gauge theory into this 2D defect category.
In all these cases, Enc is equipped with a functor G : Enc → Obs which extracts the relevant observable data (pair–production rates, anomaly polynomials, static potentials) from the encoding theories. The conditions under which F (B) ≃ G(H(B)) define the specific holographic relations of interest.
Summary. To summarise, the ambient categories we will use throughout the paper are:
tn 1. an (∞, 1)–category QFT of quantum and effective field theories with a dimension–grading functor
dim : QFT → N capturing the primary spacetime dimension;
theories, or effective actions), where equivalence is defined up to homotopy or quasi–isomorphism;
with an encoding functor H : QFT → Enc and a context functor G : Enc → Obs.
These structures provide the categorical backbone for the string–free holographic principle developed in Section II and applied in detail to QED and QCD in later sections. They also generalise the homological perspective of Ref. [4] from coefficient systems and Ext/Tor groups to full (∞, 1)–categorical frameworks, where bulk and encoding theories are related by functors and natural equivalences in a way that is both mathematically precise and physically meaningful.
Having specified the ambient (∞, 1)–categories QFT and Obs in Subsection II A, we now introduce the central structural ingredients of our string–free holographic framework: the notions of bulk theory, encoding theory, encoding functor, and context. The interplay between these notions will lead to a precise definition of relative holography, which is the conceptual backbone of all examples developed later in the paper.
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Bulk and encoding theories as objects.
where QFT is the (∞, 1)–category of quantum and effective field theories described in Subsection II A. The terminology “bulk” is suggestive, but it is important to emphasise that it is purely a role, not an intrinsic property: the same object B may act as a bulk theory relative to one encoding, and as an encoding theory relative to another.
An encoding theory is an object
vwhere Enc is an (∞, 1)–category whose objects are theories chosen for their suitability as encodings. Typically, Enc will be a full subcategory of QFT, but in some examples it will be more convenient to regard Enc as a distinct but related (∞, 1)–category (for instance, the category of worldline theories, or that of extended TQFTs with one extra dimension). We require that there be enough structure in Enc to define an observable functor G : Enc → Obs, as explained below.
The encoding functor. The relationship between the bulk theory B and its encoding theory E is described by an encoding functor
which is an (∞, 1)–functor in the sense of higher category theory. In the simplest case, H may act as a restriction or dimensional–reduction functor: for instance, mapping a 4D QFT to a 1D worldline model that reproduces certain loop amplitudes. More generally, H may implement:
• dimensional reduction: mapping B ∈ QFT d to some E ∈ Enc d 0 with d 0 < d,
• dimensional extension: mapping B ∈ QFT d to some E ∈ Enc d+1 , e.g. a bulk anomaly TQFT
encoding boundary anomalies,
• defect restriction: mapping B to the defect sector of B, encoded as a defect k–category,
• universal encoding: mapping B to an effective theory E that satisfies a universal property with
respect to a given class of observables (as in UP–RG or UP–XC).
tn On morphisms f : B 1 → B 2 in QFT, H maps f to a morphism H(f ) : H(B 1 ) → H(B 2 ) in Enc, preserving compositions and identities up to the usual higher coherences. Physically, morphisms f might represent dualities, RG flows, or defect insertions, and H translates these bulk transformations into corresponding transformations between encodings.
A crucial feature of our approach is that H is not required to be invertible or fully faithful. Rather, H is allowed to forget or collapse information in a controlled way, provided that the information relevant for a chosen context of observables is preserved. This leads to the notion of context, to which we now turn.
Context functors and observables. The observables of interest in a given physical problem are captured by an (∞, 1)–functor
which we call a context functor. The choice of F encodes a decision about what information we consider
• assign to B its factorization algebra of local observables,
• assign to B its anomaly polynomial in some generalized cohomology theory,
• assign to B its static potential function V (R) for Wilson lines at large separation,
• assign to B its non–perturbative pair–production effective action in a strong background field.
Each such choice of F defines a different context. As emphasised in our previous work [4], topological and algebraic invariants are themselves relative to the coefficient systems or structures used to probe them; here we generalise this idea and treat the choice of F as the fundamental context–selection operation.
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Similarly, we must be able to extract observables from encoding theories. For this, we introduce a functor
For instance, if Enc is the category of worldline theories, G may map a worldline model to its effective action functional on background fields; if Enc is a defect 2–category, G may map a worldsheet theory to its ground–state energy; if Enc is an extended TQFT with one higher dimension, G may map it to its anomaly inflow functional.
Relative holography as a natural equivalence. With these ingredients in place, we can now define the notion of relative holography. For a bulk theory B ∈ QFT and its encoding E = H(B) ∈ Enc, we say that B and E are holographically equivalent in context (F, G) if there exists a natural equivalence
v
in Obs. Equivalently, there is a commutative diagram up to coherent homotopy:
(27)
where (27) is understood in the sense of (∞, 1)–categories, i.e. F (B) and G(E) are equivalent objects in Obs with higher homotopies encoding the equivalence.
Several remarks are in order:
G. Changing F (e.g. from anomaly observables to full correlator data) may break the equivalence, even if H is unchanged. Thus, holography is inherently relative to context.
complexity. For example, a four–dimensional QED theory may be mapped to a one–dimensional worldline theory E that captures only its one–loop effective action. In that context, F (B) (one– loop effective action) and G(E) (worldline effective functional) are equivalent, but higher–loop or non–perturbative aspects may require refining H and/or enlarging Enc.
tn 3. The existence of (26) often constrains parameters of E or B. For example, in anomaly holography
for QCD, requiring that the chiral anomaly data F (B) be reproduced by an encoding E (a 5D anomaly TQFT) forces the WZW level k in the IR effective theory to equal the number of colours N c . This is a typical situation where relative holography fixes otherwise undetermined parameters.
Categorical derivation of relative holography from universal properties. A more refined, and in some cases more powerful, perspective on relative holography is to view the encoding E as a universal solution to a lifting problem determined by F . Fix a bulk theory B and a context functor F : QFT → Obs. Consider the fiber Fib F (F (B)) of F over the object F (B), whose objects are pairs (E, φ) consisting of an encoding theory E ∈ Enc and an equivalence
in Obs. Morphisms in Fib F (F (B)) are morphisms h : E 1 → E 2 in Enc such that G(h) intertwines the corresponding equivalences. In many situations of interest, one can show that:
• there exists an object (E 0 , φ 0 ) ∈ Fib F (F (B)) that is initial up to higher homotopy, meaning that for
any (E, φ) in the fiber there is a unique (up to contractible choice) morphism E 0 → E compatible with φ 0 and φ;
• this initial object (E 0 , φ 0 ) gives the optimal encoding of B with respect to the context F .
When this situation occurs, we call E 0 a universal encoding of B in context F , and the corresponding functor H that selects E 0 is said to realise a universal–property holography (UP–holography). This universal viewpoint is particularly useful in the context of renormalisation group and density functional theory, where one indeed seeks optimal effective theories or functionals satisfying given constraints.
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Examples: worldline and anomaly encodings. To make the discussion less abstract, we briefly anticipate two classes of examples that will be developed in depth later in the paper:
• Worldline super–holography in QED. For 4D spinor QED, at one loop, the bulk theory B is the
Dirac QED in four dimensions with external field A µ (x), and the encoding category Enc is the category of (0 + 1)–dimensional worldline super–quantum mechanics with background couplings. The encoding functor H maps B to a worldline theory E whose action is given by (12). The context functor F extracts non–perturbative effective actions (and their imaginary parts) from B, while G computes the same quantities from worldline path integrals. The relative holography condition F (B) ≃ G(E) expresses the worldline representation of the one–loop effective action in spinor QED. At two loops, Enc is enriched by bilocal photon kernels and the same framework applies.
v• Anomaly holography in QCD. For 4D QCD with massless quarks, we choose B to be the full SU(N c )
gauge theory, Enc to be an (∞, 1)–category of 5D anomaly TQFTs plus 4D WZW theories, and H to map B to its anomaly TQFT E. The context functor F extracts the chiral anomaly data from B, whereas G extracts boundary variations from E. The equivalence F (B) ≃ G(E) implements anomaly inflow and fixes the WZW level in the IR effective theory. Again, the encoding E is non–string and lives naturally in the world of extended TQFTs and cohomology.
C.
In all these examples, the relative holography relation (26) is an explicit, calculable statement rather than a mere conjecture. This validates our higher–categorical approach and shows that it is not only conceptually appealing but also practically fruitful for nontrivial QFT computations, in line with the homological insights of Ref. [4].
Compression vs. anomaly holography category Enc, and the functors
for a bulk theory B ∈ QFT, as in (27). We also introduced a dimension grading
(31) t n
that assigns to each theory its spacetime dimension. In this subsection, we specialise this general structure to two broad and physically important classes of holographic situations:
(1) compression holography (or super–holography), where the encoding theory lives in lower spacetime
dimension than the bulk, and
(2) anomaly holography, where the encoding theory lives in higher spacetime dimension and describes
anomaly inflow via an invertible topological field theory.
We will show that both cases are instances of the same higher–categorical framework, distinguished only by the dimension grading (31) and by the nature of the encoding category Enc.
Compression holography (super–holography). We begin with the case in which the encoding theory has strictly lower spacetime dimension than the bulk. Let
be a d–dimensional bulk theory (i.e. dim(B) = d in the sense of (31)), and suppose that
is an encoding theory of dimension d 0 < d, i.e.
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The most direct interpretation of (34) is that the encoding theory E is defined on a lower–dimensional spacetime than B. However, as discussed in Subsection II A, one can also consider a refined notion of effective dimension dim eff , which measures the “information–theoretic complexity” of a theory: for instance, the number of independent degrees of freedom that contribute in a certain energy regime, or the rank of the factorization algebra of observables. In this sense, super–holography may manifest as
even when dim(H(B)) = dim(B). For the purposes of the present discussion, we focus on the geometric case (34), bearing in mind that the formalism extends to (35) as well.
Intuitively, compression holography captures situations where a high–dimensional interacting theory can, with respect to a particular context of observables, be simulated or encoded by a much lower–dimensional theory that is simpler to analyse. The canonical examples, developed in later sections, include:
v• a four–dimensional spinor QED theory whose one–loop effective action is encoded in a (0 + 1)–
dimensional worldline super–quantum mechanics;
• a four–dimensional confining Yang–Mills theory whose static quark potential at large distances is
encoded in a (1 + 1)–dimensional worldsheet QFT describing flux–tube fluctuations.
From the (∞, 1)–categorical viewpoint, the mathematical content of compression holography is the existence of a functor H with (34) and of a context functor F such that the relative holography condition (30) holds:
We emphasise that this is a nontrivial claim: it states that, even though E has fewer dimensions and (typically) fewer degrees of freedom, it suffices to reconstruct the F –observable content of B.
Anomaly holography and invertible TQFTs. In anomaly holography, the situation is almost reversed: the encoding theory E lives in one higher dimension than the bulk theory B:
t
The encoding category Enc is now typically taken to be an (∞, 1)–category of invertible topological field theories in one higher dimension. An invertible TQFT E is, roughly speaking, a symmetric monoidal functor E : Bord S d → C such that E(M ) is invertible for every manifold M (objects and morphisms assigned to closed manifolds and bordisms are invertible in C). These theories model anomaly inflow [27, 28]: the non–trivial transformation of a d–dimensional theory under a symmetry is cancelled by the boundary variation of a (d + 1)–dimensional topological action.
More concretely, consider a d–dimensional QFT B with a global symmetry group G, whose partition
where A is a background G–connection and A(g, A) is the anomaly functional capturing the lack of gauge invariance. In many cases of interest, A can be generated as the boundary variation of a (d+1)–dimensional action:
where ω d+1 is a closed (d + 1)–form whose exterior derivative encodes the anomaly polynomial in d + 2 dimensions. The (d + 1)–dimensional TQFT with action S d+1 is invertible in the sense that its partition function is a pure phase and its state spaces are one–dimensional [68, 70].
In the higher–categorical language, the anomaly data of B can be organised into an observable object F (B) ∈ Obs (for instance, an anomaly cohomology class or anomaly polynomial), while the topological field theory E yields an observable G(E) ∈ Obs given by the boundary variation of S d+1 . The anomaly inflow principle asserts that these two observables are equal:
which is precisely a relative holography condition of the form (30), but now with dim(E) = dim(B) + 1. This is what we call anomaly holography. The encoding functor H maps B to the corresponding invertible TQFT E (plus additional boundary data if necessary).
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Unified higher–categorical picture. Although compression holography and anomaly holography appear different — in the former, the encoding theory lives in lower dimension; in the latter, it lives in higher
H : QFT → Enc, iew
The only difference is the relationship between the dimension of H(B) and that of B, as measured by the functor dim:
In all cases, H maps B to some E that is better suited for computing or conceptualising the observables selected by F . The equivalence (41) asserts that the encoded observables G(E) reproduce the bulk
From a purely categorical point of view, the essential structure behind both compression and anomaly holography is the existence of adjoint or universal properties: H often has the structure of a left adjoint (or a left Kan extension) to some “forgetful” or inclusion functor, and E is initial in a fiber category of theories reproducing certain observables. In anomaly holography, the classification of invertible TQFTs by generalized cohomology [68, 70] provides a natural target for H and leads to exact anomaly inflow relations. In compression holography, worldline and worldsheet encodings can be understood as left adjoints to evaluation functors that restrict a high–dimensional theory to its loop or defect sectors, yielding effective theories in fewer dimensions.
Interplay and interpolation. An interesting and physically relevant phenomenon is that real systems may exhibit both kinds of holography simultaneously, or interpolate between them as parameters are varied. For example, in strong–field QED with time–dependent backgrounds, one can have:
• a worldline compression holography 4 → 1 for loop amplitudes (as in the worldline formalism),
• supplemented by an anomaly holography 4 ↔ 5 for chiral anomalies in the presence of pseudoscalar
or axial couplings.
tn Similarly, in QCD one may consider both flux–tube super–holography 4 → 2 and anomaly holography 4 ↔ 5 for chiral anomalies. The higher–categorical framework developed here accommodates such hybrid situations naturally: one simply composes or layers multiple encoding functors H i and context functors F i , G i , with suitable coherence conditions, to obtain a richer diagram of holographic relations.
Summary. In summary, compression holography and anomaly holography are not fundamentally different species of duality, but rather two manifestations of the same higher–categorical structure. The distinction lies only in the relative spacetime dimension of the encoding theory E compared to the bulk B, and in the choice of encoding category Enc (e.g. worldline theories vs. invertible TQFTs). Both are described by the same commutative square (41), and both play a central role in the examples we develop later: worldline super–holography in QED, anomaly/WZW holography in QCD, and defect/worldsheet holography in confining Yang–Mills theory. The next subsections will make these constructions explicit and will show, in each case, how the higher–categorical formalism leads to concrete computational advantages.
In the previous subsections we have treated QFT, Enc and Obs as abstract (∞, 1)–categories and discussed relative holography purely at the level of objects and morphisms. In practice, however, quantum field theories carry additional structures that are crucial for physics:
• linear structure (superposition of states and amplitudes), This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 17 • probabilistic structure (transition probabilities, Markov kernels), • topological and homological data (Ext/Tor contributions, anomaly classes), ed • inner products and adjoints (unitarity, reflection positivity), • loop expansions (organising contributions by loop order, powers of ~ or coupling constants). iew To faithfully encode these structures, and to ensure that holographic encodings preserve them, it is natural and often necessary to work with enriched and graded categories. The goal of this subsection is to set up these notions precisely, to explain their relevance to our string–free holographic framework, and to prepare the ground for their use in the QED and QCD examples that follow. Enriched categories: general definitions. Let (V, ⊗, 1 ) be a (symmetric) monoidal category. A V– enriched category C consists of:
v • a class of objects Ob(C), • for each pair of objects X, Y ∈ Ob(C), a hom–object C(X, Y ) ∈ V,
• for each triple X, Y, Z ∈ Ob(C), a composition morphism in V
• for each X ∈ Ob(C), a unit morphism
tn commutes in V, and unitality requires that certain triangles involving u X commute. When V = Set with ⊗ = ×, this reduces to the usual notion of (ordinary) category. In our applications, however, we are typically interested in V that carry linear, Hilbert, topological, or probabilistic structure.
• a function on objects F : Ob(C) → Ob(D),
• for each pair X, Y ∈ Ob(C), a morphism in V
that is compatible with the composition and units as per the usual enriched functor axioms [31].
Examples of enrichment relevant for QFT. We now discuss several choices of V that are physically relevant for our holographic framework.
(i) Enrichment over vector spaces (linear structure). Taking V = Vect C , the (symmetric) monoidal category of complex vector spaces with the usual tensor product, we obtain linear categories: categories whose hom–spaces are vector spaces and whose composition is bilinear. Many physically natural categories have this structure: categories of representations of a group or algebra, categories of modules over an operator algebra, and categories of states and operators in a QFT.
In our setting, QFT or Enc can be regarded as Vect C –enriched in the sense that:
• morphisms between theories (e.g. couplings, deformations) can be organised in vector spaces,
• hom–objects can encode linear combinations of amplitudes or functionals, This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 18 • enrichment ensures that functors such as H, F, G are linear on morphisms, preserving superposition. ed This is the minimal enrichment needed to reflect the linear structure of quantum mechanics. (ii) Enrichment over Hilbert spaces (unitarity). If we refine further to V = Hilb, the category of (separable) Hilbert spaces with tensor product, we can encode not only linear structure but also inner products and adjoints. A Hilb–enriched category C has: iew • Hilbert–space hom–objects C(X, Y ), • composition maps that are bounded bilinear maps, • unit morphisms that behave like identity operators.
vWhen combined with a compatible dagger structure (an involutive contravariant functor (−) † ), this yields the framework of dagger–enriched categories, which are central in axiomatic quantum mechanics and quantum information [34]. In the holographic context, insisting that the encoding functor H be Hilb– enriched and dagger–preserving ensures that unitarity (and reflection positivity in Euclidean signature) is respected by the encoding. For example, when encoding a unitary 4D QFT into a worldline or worldsheet model, the enriched structure ensures that the encoded amplitudes have the correct inner–product structure.
(iii) Enrichment over product categories (linear + topological data). To incorporate both linear and topological data, one can work with product monoidal categories such
as
with tensor product (V 1 , X 1 ) ⊗ (V 2 , X 2 ) = (V 1 ⊗ V 2 , X 1 × X 2 ). A V–enriched category C then assigns to each pair of objects X, Y a pair
where V X,Y is a vector space of amplitudes and X X,Y is a topological space (or homotopy type) representing additional data such as:
• homotopy classes of defects or domain walls between X and Y ,
• elements of Ext/Tor groups in some homology theory [33],
• topological sectors of gauge bundles or instantons.
tn Composition in C then combines the vector–space structure and the topological composition rules (e.g. group laws in cohomology). In the anomaly and TQFT examples, such enrichment allows us to treat anomaly coefficients and torsion data on the same footing as linear amplitudes.
(iv) Enrichment over probabilistic categories. For applications where probabilistic structure is explicit (e.g. in open systems, stochastic QFT, or categorical formulations of probability), one can take V to be a category of stochastic matrices or Markov kernels, such as a Markov category [35]. Objects then represent state spaces, and hom–objects represent stochastic maps or probability kernels. Enriching over such a category allows the functors F and G to encode not only amplitudes but also probabilities and conditional expectations. Although we will not fully exploit this structure in the present work, it is conceptually important: in contexts where holographic encodings are used to relate quantum and classical descriptions (e.g. in AdS/CFT interpretations of emergent classical gravity), probabilistic enrichment may be essential.
Loop grading and graded enrichment. Beyond enrichment, quantum field theories typically admit a loop expansion, in which observables are organised as formal series in powers of ~ or coupling constants. For example, the effective action in a QFT can be written as
where Γ () collects all –loop contributions. It is natural to reflect this structure categorically by enriching over a graded monoidal category.
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 19 Let V be a symmetric monoidal category. A graded version of V, denoted V N , is the category whose objects are sequences (V (0) , V (1) , V (2) , . . . ) of objects of V, and whose morphisms are sequences of morphisms in V, with tensor product defined by
and graded tensor product [32].
uses (51) and thus satisfies
where ◦ (p,q) denotes the composition map restricted to the corresponding graded components. This precisely mirrors the loop–expansion structure in perturbative QFT, where an n–loop amplitude can be written as a sum over compositions of lower–loop subdiagrams.
For our purposes, the most important choice is V = Vect C or Hilb, so that hom–objects are graded vector spaces or graded Hilbert spaces:
graded enrichment in the category of effective actions.
Loop–graded encoding functors. In the holographic setting, it is natural to require that the encoding functor H and the context functors F, G respect the loop grading. Concretely, let QFT N denote a loop–graded version of QFT, where objects carry graded hom–spaces and effective actions split as in (50). An encoding functor
n
for all objects X, Y and all ` ≥ 0, and is compatible with the graded composition structure.
Similarly, context functors
may be required to preserve loop grading, so that relative holography at each loop order is expressed as an equivalence
ep In practice, such loopwise holography is exactly what we will exploit in our examples: • At one loop ( = 1), the worldline formalism encodes the entire 4D QED effective action into a 0+1D super–worldline theory, and worldline instantons compute the non–perturbative imaginary part in strong background fields. • At two loops ( = 2), a single bilocal photon insertion along the worldline encodes all two–loop
diagrams; the same instanton trajectories and their fluctuations, together with a double integral over worldline parameters, give the two–loop corrections to the exponent and prefactor.
These are concrete instances of (58), with Enc chosen to be a worldline category enriched and graded in the appropriate way.
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Enrichment and loop grading in the examples to come. In the sections that follow, we will systematically use enrichment and loop grading to articulate how holographic encodings preserve the physically relevant structures in our QED and QCD examples:
• For worldline super–holography in spinor QED, Enc is a (0+1)–dimensional super–worldline category
enriched over super vector spaces and graded by loop order; the encoding functor H from 4D spinor QED to this worldline category preserves both the spin structure and the loop grading.
• For anomaly holography in QCD, Enc is an extended TQFT category enriched over generalized
cohomology theories (to capture anomaly classes) but with trivial loop grading (since anomalies are one–loop exact in many cases), and H maps QCD to an anomaly TQFT in one higher dimension.
v• For flux–tube holography in confining Yang–Mills theory, Enc is a defect 2–category enriched over
vector spaces or Hilbert spaces, reflecting the linear structure of defect sectors, and graded by excitation level on the worldsheet.
In each case, the enriched and graded nature of the categories ensures that the encoding functor is not only formally defined but also physically faithful: it preserves inner products, probabilities, homological data, and loop expansions, thereby making the relative holography relations both mathematically precise and physically meaningful.
STRING–FREE HOLOGRAPHIC PRINCIPLE
III.
We now assemble the ingredients introduced in Section II into a single formal statement which we refer to as the string–free holographic principle. The guiding idea is that holography need not be tied to any particular geometric realisation (such as strings propagating in an AdS background), but can be formulated abstractly as the existence of an encoding functor
together with context functors
F : Bulk → Obs, G : Enc → Obs, (59)
(60)
tn such that the observable content of a bulk theory B in context F is naturally equivalent to that of its encoding E = H(B) in context G. Here Bulk is a full subcategory of QFT containing the theories we regard as “bulk” theories for a given discussion, Enc is an encoding (∞, 1)–category, and Obs is an (∞, 1)–category of observables, all as in Section II.
Bulk and encoding data. Formally, fix:
• an (∞, 1)–category Bulk ⊆ QFT whose objects we interpret as bulk quantum (or effective) field
theories;
• an (∞, 1)–category Enc whose objects are encoding theories (worldline models, extended TQFTs,
defect theories, etc.);
• an (∞, 1)–category Obs of observables (cochain complexes, factorization algebras, generalized
cohomology classes, effective actions, etc.).
We then consider an (∞, 1)–functor
which we think of as assigning to each bulk theory B ∈ Bulk an encoding theory E = H(B) ∈ Enc. This assignment may realise:
• a dimensional reduction or defect restriction (compression holography),
• a dimensional extension to an anomaly inflow theory (anomaly holography), This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619
21
• or a universal encoding for a given observable context (UP–holography).
The functor H also assigns to each morphism f : B 1 → B 2 in Bulk a morphism H(f ) : H(B 1 ) → H(B 2 ) in Enc, preserving compositions and units up to coherent higher homotopies, as required in the definition of an (∞, 1)–functor [36, 37].
Importantly, H is not assumed to be fully faithful or essentially surjective: it may forget information, collapse degrees of freedom, or change dimension. The only requirement is that, after passing to observables via context functors F, G, the relevant information is preserved.
Context functors and observable equivalence. The notion of context is encoded in the choice of two (∞, 1)–functors
v which select the observables of interest from bulk and encoding theories, respectively. These functors are part of the data of the holographic situation and must be specified explicitly. Typical choices include:
• F assigns to a bulk theory B its anomaly polynomial or anomaly cohomology class; G assigns to an
encoding TQFT E its anomaly inflow functional.
• F assigns to B its non–perturbative effective action in a specified background field; G assigns to a
worldline encoding E the path integral that computes the same effective action.
• F assigns to B its static potential V (R) between sources; G assigns to E its ground–state energy as
a function of length, as in effective string descriptions.
We emphasise that F and G need not be unique; different physical questions or regimes correspond to different choices of context functors.
Once H, F, G are fixed, the central condition of holography is the existence of a natural equivalence in Obs:
ot By definition, such a natural transformation consists of:
in Obs,
tn • for each bulk object B ∈ Bulk, an equivalence
• for each morphism f : B 1 → B 2 in Bulk, a higher–homotopy–commuting diagram in Obs expressing
the naturality of α, i.e. that
Equivalently, we can draw the relative holography square
(66)
which commutes up to coherent higher homotopies. Informally, α B expresses the statement that “the observables of B in context F are indistinguishable from the observables of H(B) in context G”; the naturality condition ensures that this equivalence is compatible with morphisms in Bulk (e.g. with RG flows, dualities, perturbations).
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 22 With this structure in place, we can now state the core principle of
String–free holographic principle. this paper.
an (∞, 1)–functor F : Bulk → Obs (bulk context functor),
an (∞, 1)–functor G : Enc → Obs (encoding context functor),
≃ 4. a natural equivalence α : F = ⇒ G ◦ H in Obs,
vsuch that: iew Definition III.1 (String–free holographic principle). Let Bulk be a full subcategory of an (∞, 1)–category of quantum field theories QFT, let Enc be an (∞, 1)–category of encoding theories, and let Obs be an (∞, 1)–category of observables. A string–free holographic encoding consists of:
• no assumption is made that Enc is a category of string/brane theories;
• Enc may be any appropriate (∞, 1)– or higher category whose objects encode the relevant physics (worldlines, extended TQFTs, defect categories, density functionals, codes, etc.);
• the equivalence α is context–dependent: it holds for the observables selected by F, G but need not
hold for all conceivable observables.
Several comments are important:
er If such data exist, we say that the bulk theory B ∈ Bulk is holographically encoded by E = H(B) in context (F, G). • The principle is string–free in the precise sense that the definition does not mention strings, branes,
or worldsheet CFTs. String–theoretic holographies (e.g. AdS/CFT) arise as special cases where Enc is chosen to be a category of string/brane backgrounds and H encodes bulk QFTs into such backgrounds, but this is not assumed a priori.
• The principle is higher–categorical: H, F, G and α are (∞, 1)–functors and natural transformations,
and equivalence is understood up to higher homotopies. This is crucial for flexibility and for modelling gauge and homological redundancies in QFT.
tn • The principle is context–dependent: different choices of F, G (e.g. anomaly observables vs. full
correlators vs. non–perturbative rates) lead to different holographic encodings even for the same bulk theory B.
Special cases: compression, anomaly, and self–dimensional holography. Definition III.1 encompasses several familiar types of holographic behaviour as special cases, distinguished by the relationship between dim(B) and dim(H(B)) (cf. (43)) and by the nature of Enc:
• Compression (super–holography). If dim(H(B)) < dim(B) or dim eff (H(B)) dim eff (B), then H
is a compression map: a higher–dimensional bulk theory is encoded in a lower–dimensional theory that reproduces its F –observables. This is the case for worldline encodings of 4D QED into 0+1D models and for worldsheet encodings of 4D confining flux tubes into 1+1D theories.
ep • Anomaly holography. If dim(H(B)) = dim(B) + 1 and H(B) is an invertible TQFT in one higher dimension, then H encodes the anomaly of B via anomaly inflow; F extracts anomaly data and G extracts boundary variations of the TQFT. This is the case for chiral anomalies in QCD and parity anomalies in odd dimensions. • Self–dimensional holography. If dim(H(B)) = dim(B), one often recovers dualities between theories
in the same dimension (e.g. Seiberg duality, 2D conformal dualities), where H may be an auto– equivalence and F, G select particular observables.
In all these cases, the same string–free principle applies: holography is the existence of a natural equivalence of the form (63), not the presence of any specific geometric string structure.
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Proof obligations and existence statements. Definition III.1 is a formal principle, and as such it does not guarantee that an encoding functor H and context functors F, G exist for arbitrary bulk theories or arbitrary choices of observables. In concrete situations, one must:
specify Bulk, Enc and Obs,
construct explicit functors H, F, G, iew 3. prove the existence (or at least the existence up to controlled approximations) of the natural equivalence α.
vIn the worldline example for QED, the functor H is constructed from the Schwinger proper–time representation of the Dirac determinant exactly at one loop; its correctness is guaranteed by the underlying functional calculus of operators. The context functors F and G are defined by mapping theories to their effective actions in given backgrounds. The equivalence α is then essentially Schwinger’s identity for determinants.
In anomaly holography, H is constructed using the classification of anomalies via generalized cohomology and extended TQFTs; F and G are defined by mapping theories to their anomaly data and inflow functionals. The equivalence α is a consequence of anomaly matching and is often exact to all orders. In this way, Definition III.1 provides a unifying language for what, in the literature, appear as disparate constructions.
Conclusion. The string–free holographic principle encapsulated in Definition III.1 and diagram (66) is the conceptual core of this work. It emphasises that:
• holography is fundamentally a higher–categorical equivalence between functors to an observable
category,
• the encoding category Enc need not be stringy; strings and branes are just one possible realisation,
• the equivalence is always relative to a context (F, G), which expresses which part of the physics is
being encoded and compared.
In the subsequent sections we will instantiate this principle in detail for several physically rich and nontrivial cases: worldline super–holography for strong–field QED, anomaly/WZW holography for QCD, and flux–tube holography for confining Yang–Mills. Each example will reveal new computational capabilities that suggest non–string encodings are often more powerful and more widely applicable than traditional string–based holographies.
Strings as a special case t n B.
In the previous subsection we formulated the string–free holographic principle (Definition III.1) in purely higher–categorical terms, with no reference to strings or branes. We now explain how the standard string–theoretic examples, and in particular the AdS/CFT correspondence, fit into this general framework as special cases.
The purpose of this subsection is twofold:
Enc, and context functors F, G, with a natural equivalence F (B) ≃ G(E).
(large N , supersymmetry, conformality, specific operator classes), and that for many physically important systems (strong–field QED, finite–density QCD, far–from–equilibrium systems) no useful string encoding is known, whereas non–string encodings in the sense of Section II do exist.
AdS/CFT in the string–free formalism. Let us recall the prototypical AdS/CFT correspondence [38]: type IIB string theory on AdS 5 ×S 5 with N units of five–form flux is conjecturally dual to four–dimensional N = 4 supersymmetric SU(N ) Yang–Mills theory on R 1,3 or on S 3 × R. In the framework of Section II, we can identify:
• a bulk category Bulk whose objects include type IIB string backgrounds with asymptotically AdS
boundary conditions, This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 24
• an encoding category Enc whose objects include four–dimensional conformal field theories (CFTs)
with appropriate supersymmetry and gauge group,
• an observable category Obs whose objects represent correlation functions or operator algebras (e.g.
collections of n–point functions of local operators).
H AdS/CFT : Bulk −→ Enc, iew We can then define a (heuristic) encoding functor
(67)
vwhich maps a string background B (such as type IIB on AdS 5 × S 5 with flux N ) to a boundary CFT E = H AdS/CFT (B) (such as N = 4 SU(N ) SYM). The exact mathematical construction of H AdS/CFT is not presently known, but at a heuristic level it encodes the boundary limit of the bulk string / supergravity solution into data defining a conformal field theory: spectrum of operators, OPE coefficients, symmetry algebra, etc.
To define the context functors, we fix:
• F AdS : Bulk → Obs that maps a bulk string background B to the boundary limits of bulk correlation
functions of specified bulk fields, evaluated by functional derivatives of the bulk partition function with prescribed boundary sources.
• G CFT : Enc → Obs that maps a CFT E to its correlation functions of local operators with given
sources.
er More concretely, let Φ(x, z) denote a bulk field (e.g. a scalar) on AdS d+1 with boundary coordinate x and radial coordinate z, and let φ 0 (x) = lim z→0 z −∆ − Φ(x, z) be its boundary value, where ∆ − is the smaller of the two possible scaling dimensions. The AdS/CFT prescription asserts that the bulk partition function with fixed boundary data,
is equal (up to normalisation) to the generating functional of correlation functions in the dual CFT with source φ 0 for an operator O of dimension ∆:
tn The corresponding n–point functions are obtained by differentiating Z bulk [φ 0 ] with respect to φ 0 and then setting φ 0 = 0. The map F AdS thus assigns to B the functional Z bulk [φ 0 ] and its derivatives, while G CFT assigns to E the functional Z CFT [φ 0 ] and its derivatives. The GKPW relation (69) is precisely an instance of a relative holography equivalence
for the context of boundary correlators of single–trace operators. In other words, AdS/CFT can be viewed as a string–based realisation of the general pattern
with Enc chosen as a category of CFTs and Bulk as a category of string/supergravity backgrounds.
One can refine this picture by including more of the extended structure: bulk Wilson lines and branes, boundary line operators and defects, entanglement entropies, etc. Each such refinement corresponds to enlarging the observable category Obs and the context functors F, G [38, 39].
Regime of validity: large N , supersymmetry, conformality. The AdS/CFT correspondence and related string dualities are well–understood only in specific parameter regimes and for specific classes of observables. For instance, in the AdS 5 × S 5 /N = 4 SYM duality:
• The bulk description in terms of classical supergravity is valid in the limit of large ’t Hooft coupling
2 λ = g
YM N 1 and large N , where string loop and α
0 corrections are suppressed [38]. In this regime, certain protected operators and states can be computed from geodesics or classical branes in AdS.
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• On the boundary, the theory is an exactly conformal, maximally supersymmetric gauge theory.
Many explicit computations (e.g. anomalous dimensions, Wilson loop expectation values) rely on supersymmetric localisation, integrability, or perturbation theory in special limits. Outside these special regimes, the string dual is either strongly curved (so α 0 corrections are significant) or quantum gravity effects are large (so string loops are important), and no systematic control is available. Even within AdS/CFT, most explicit results focus on: iew • single–trace operators or certain BPS operators, • correlators that are protected by supersymmetry, • observables that have a direct geometric interpretation in the bulk (geodesics, minimal surfaces, classical brane embeddings).
vIn the language of Section II, this means that the context functors F, G in AdS/CFT are implicitly restricted to certain classes of observables (e.g. local correlators at large N and strong coupling) and that the equivalence F ≃ G ◦ H holds only in those contexts.
Limitations of string encodings for realistic QFTs. The string–based encoding category Enc string is thus quite specialised: it consists of CFTs with certain symmetry and large–N properties, often with additional integrability or supersymmetry structure. While this is extremely powerful in its natural regime, it becomes an obstacle when one tries to apply holography to more realistic quantum field theories.
Consider, for example:
• Strong–field QED. Four–dimensional spinor QED in a time–dependent electric field (e.g. a laser pulse)
is not conformal, not supersymmetric, and not obviously related to any known AdS background. There is no controlled string dual that directly encodes non–perturbative pair production in such fields, whereas worldline encodings (see later sections) provide a 0+1D encoding that does.
• Finite–density QCD. Realistic QCD at finite chemical potential (e.g. in neutron stars or heavy–ion
collisions) is far from conformal and plagued by sign problems; known string/brane models (e.g. top– down or bottom–up AdS/QCD) are, at best, qualitative analogues and do not yield quantitatively controlled computations for physical QCD at finite density [40].
• Generic nonequilibrium systems. Far–from–equilibrium QFTs (e.g. driven systems, quenches,
turbulence regimes) typically have no known string dual, and the observables of interest (time– dependent correlation functions, entropy production, transport in nonlinear regimes) are not easily accommodated in the standard AdS setup.
tn In all these cases, the true encoding category Enc is something other than strings: worldlines, defect categories, stochastic or dissipative QFTs, etc. The string–free holographic principle explicitly allows for such choices, whereas a string–centric view risks missing them.
Strings as one encoding among many. From the vantage point of the string–free holographic principle, we can summarise the role of strings as follows:
• In certain highly symmetric, large–N and strongly coupled regimes, there exists an encoding functor
H string : Bulk SYM → Enc string , (72)
where Bulk SYM is a subcategory of QFTs (e.g. superconformal Yang–Mills theories) and Enc string is a subcategory of string/brane backgrounds.
• For appropriate context functors F AdS , G CFT selecting protected correlators and geometric observ-
ables, there is a natural equivalence
yielding an AdS/CFT–type holography.
• However, Enc string is a small corner of all possible encoding categories; it is not the universal or
default choice. For most QFTs of physical interest, no such H string is known or useful, whereas other encodings H : Bulk → Enc do exist and provide real computational capabilities.
In this sense, strings are special cases of the encoding category Enc, not the defining ingredient of holography.
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26
Conclusion of this subsection. The lesson from this analysis is that string–based holography, and AdS/CFT in particular, fits naturally into the string–free, higher–categorical holographic principle, but only as one among many possible realisations. The encoding category Enc in AdS/CFT happens to consist of string/brane backgrounds and CFTs, and the context functors F, G happen to select correlators and geometric observables. There is nothing in the formal structure, however, that privileges strings over other encodings. In the following sections, we will exploit this flexibility to construct non–string encodings for QED and QCD, where string duals are unavailable or impractical, and we will show that these encodings yield concrete computational advantages in strong–field and strongly–coupled regimes.
Why strings are not the universal encoding
In Subsections III A and III B we presented the string–free holographic principle and showed how AdS/CFT–type string dualities fit as a particular realisation of the general pattern
with Enc chosen to be a category of string/brane backgrounds and Bulk a subcategory of QFTs admitting AdS–like completions. In this subsection, we argue that string encodings are not universal in any meaningful mathematical or physical sense, and that the space of possible encoding categories is much larger and more flexible. In particular:
• many relevant QFT structures — worldlines, higher defects, anomalies, factorization algebras,
quantum codes — have no unique or natural realisation in terms of a string worldsheet;
• many physically important regimes (strong–field QED, finite–density QCD, generic nonequilibrium
systems) have no controlled string dual;
• non–string encodings, such as worldline categories, extended TQFTs, defect 2–categories and code
categories, already provide concrete computational capabilities in these regimes.
We now substantiate these points in a more detailed and structured manner.
tn (a) Worldlines and particle–like excitations. At a very basic level, many QFT phenomena are most
naturally expressed in terms of particle worldlines rather than worldsheets: scattering amplitudes via Feynman diagrams, semiclassical instantons for tunnelling and pair production, Wilson loops for charged particles in gauge fields, etc. The worldline formalism makes this explicit by rewriting loop integrals as path integrals over particle trajectories in spacetime, with spin degrees of freedom encoded in worldline Grassmann variables. These are (0+1)–dimensional theories — super–quantum mechanics — that encode rich four–dimensional physics. There is no intrinsic necessity to “lift” these to a two–dimensional string description; the worldline category itself is the correct encoding category Enc for many one–loop (and even higher–loop) observables.
(b) Defects of various codimensions. Modern QFT makes extensive use of line, surface, and higher–
codimension defects: Wilson and ’t Hooft lines, surface operators, domain walls, junctions of defects, etc. These objects are best organised into defect n–categories, with k–morphisms representing k–dimensional defects and their junctions. While some defect configurations admit a string or brane interpretation (e.g. D–branes carrying gauge fields, or fundamental strings ending on D–branes), many do not: defects in low–dimensional condensed–matter systems, domain walls in spin systems, interfaces between different symmetry–protected phases, etc. In these cases, the natural encoding category is a defect n–category enriched over vector spaces or Hilbert spaces, not a string worldsheet CFT in a higher–dimensional background.
(c) Anomalies and invertible TQFTs. Anomalies in QFT — especially modern notions of higher–form
and mixed anomalies — are naturally classified by generalized cohomology theories, cobordism classes, and invertible topological field theories in one higher dimension. The mathematical structures involved are those of spectra, stable homotopy theory, and extended TQFTs. While some anomaly This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 27
inflow constructions can be realised in string theory (e.g. anomaly cancellation via Green–Schwarz mechanisms), the general classification and interpretation of anomalies is not string–specific: it lives in a category of invertible TQFTs and generalized cohomology classes. In our language, the encoding category Enc in anomaly holography is an (∞, 1)–category of invertible TQFTs, and the functor H maps QFTs to their anomaly inflow theories. There is no necessity to represent these TQFTs as string worldsheets; indeed, many invertible TQFTs have no known string interpretation.
view (d) Factorization algebras and local observables. The modern approach to local quantum field theory uses factorization algebras (or related structures such as vertex operator algebras) to encode local observables and their operator product expansions on manifolds. Factorization algebras are cosheaf–like assignments of cochain complexes to open subsets, satisfying a multiplicative gluing condition [42]. They form a very large and flexible (∞, 1)–category of their own, which serves as a natural target for context functors F and G. Although 2D conformal field theories can sometimes be described by vertex operator algebras and hence relate to string worldsheets, factorization algebras are defined in any dimension and are not intrinsically linked to strings. Again, the natural encoding
category for many questions is that of factorization algebras rather than the narrower class of worldsheet CFTs.
(e) Quantum codes and entanglement structures. In the context of holography and quantum information,
it has become clear that many bulk–boundary relations are best captured by quantum error– correcting codes and tensor network models. In such models, bulk degrees of freedom are encoded into boundary ones via isometric embeddings with specific entanglement structures; the relevant encoding category is one of codes and channels, or of tensor networks with certain constraints [43, 44]. Although such constructions can be drawn on hyperbolic lattices reminiscent of AdS and sometimes used as toy models for AdS/CFT, they are fundamentally code–theoretic, not string–theoretic. The worldsheet does not play a direct role; instead, one has a categorical structure of code spaces, logical and physical Hilbert spaces, and channels between them.
These examples demonstrate that the objects we wish to encode (worldlines, defects, anomalies, factorization structures, code subspaces) and the corresponding encodings often live in categories that are not naturally realised as string worldsheets or brane worldvolumes. Attempting to force them into a string language would either be unnatural or would require introducing extraneous structure that obscures the essential features.
tn
• Strong–field QED in time–dependent backgrounds. Consider four–dimensional spinor QED in the
presence of a time–dependent electric field E(t), modelling a strong laser pulse. The relevant questions concern non–perturbative pair production, vacuum instabilities, and backreaction. The theory is not conformal, not supersymmetric, and not obviously large–N . There is no known AdS–like string background whose boundary theory reproduces spinor QED in such a background with enough control to compute pair–production rates or their loop corrections. By contrast, the worldline super–holography developed later in this paper provides a well–defined encoding functor H to a (0 + 1)–dimensional worldline theory and context functors F, G that encode exactly the one–loop and two–loop effective actions relevant for pair production.
• Finite–density and finite–temperature QCD. Real–world QCD at finite chemical potential, finite
temperature, and in nontrivial phases (e.g. colour superconductivity, quark–gluon plasma, nuclear matter in neutron stars) is deeply non–conformal and exhibits complicated phase structure. Top– down string models for QCD (e.g. D–brane constructions) and bottom–up AdS/QCD models provide qualitative insights but no quantitatively controlled duals that would justify an equivalence F (B) ≃ G(H(B)) for realistic QCD observables [47]. In particular, computing the QCD phase diagram or transport coefficients at finite density via string methods remains an open problem. Non–string encodings, such as effective theories for order parameters or defects, or anomaly–based constraints, provide much more robust tools.
• Nonequilibrium and driven systems. Many condensed–matter and high–energy systems are far from
equilibrium: quenches, driven steady states, turbulence, chaotic dynamics. The observables of This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 28
interest (time–dependent correlation functions, entropy production, fluctuation theorems) are not naturally described by static AdS backgrounds. While there are some attempts to model quenches or thermalisation with Vaidya–like spacetimes and holographic entanglement entropy [46], these constructions rely on classical gravity and do not constitute controlled duals for the underlying microscopic QFTs. In such contexts, stochastic or Lindbladian encodings, or categorical formulations of nonequilibrium processes, may be more appropriate encoding categories. iew • Fracton phases and exotic topological orders. In condensed–matter physics, fracton phases and higher–rank gauge theories exhibit excitations with constrained mobility and subextensive ground– state degeneracy. These phases are best understood in terms of higher–rank tensor gauge theories, subsystem symmetries, and intricate fusion and braiding rules [45]. There is no known string worldsheet description that captures the full complexity of fracton order, and it is not clear that one exists. Encoding categories based on higher–groupoids, fusion categories with restricted mobility, or tensor network models of fractons are much better suited to describe these systems.
v In all these regimes, insisting that Enc must be a string category would either force us to work with uncontrolled approximations or leave us without any encoding at all. The string–free holographic principle, by contrast, allows us to choose the encoding category that best fits the problem at hand.
er • Worldline super–holography in QED. The worldline formalism rewrites the one–loop effective action of spinor QED in a background A µ (x) as a path integral over a (0 + 1)–dimensional worldline super–quantum mechanics of a relativistic particle coupled to A µ and F µν . At two loops, all photon corrections can be encoded by a bilocal worldline kernel. For strong, time–dependent electric fields, non–perturbative pair production can be computed semiclassically from worldline instantons and
their fluctuations, reducing a four–dimensional functional determinant to a one–dimensional ODE problem supplemented by low–dimensional integrals. There is no comparably efficient string–based method for this problem.
• Anomaly and WZW holography in QCD. The anomaly inflow picture encodes chiral anomalies of
QCD in 5D invertible TQFTs and fixes WZW term coefficients and Skyrmion quantum numbers in 4D low–energy effective theories. The relevant encoding category is that of extended TQFTs and cohomology theories, and the context functor extracts anomaly data. This yields exact, nonperturbative results (e.g. k = N c for the WZW level) with no reliance on strings.
tn • Defect/worldsheet holography for confining flux tubes. The static potential between quark sources in
confining Yang–Mills theory at large distances can be computed from an effective (1+1)–dimensional worldsheet theory of flux tubes, which arises naturally as the defect sector of the 4D gauge theory. The universal Lüscher term appears as a Casimir energy of transverse worldsheet modes. Again, this is a non–string encoding in a defect category that directly mirrors the physics.
• Quantum code encodings and emergent geometry. In holographic quantum error–correcting codes,
bulk degrees of freedom are encoded in boundary ones via isometric maps that preserve logical information against local erasures [43, 44]. The relevant category is that of code subspaces, Hilbert spaces, and channels, and context functors extract logical algebra and entanglement data. These encodings provide a rigorous and finite–dimensional setting for understanding some structural aspects of holography. They are not string–based, but rather code–based, and form a distinct encoding category.
Each of these examples is an instantiation of the string–free holographic principle with a non–string encoding category Enc and context functors F, G appropriate to the problem. They confirm that:
holography can be realised in many non–string ways that are mathematically natural;
non–string encodings yield concrete, algorithmic methods for computing physically relevant quanti-
ties;
holographic methods are both applicable and already fruitful.
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SOLUTIONS
vIV. iew
Conclusion. To conclude, while string theory provides a powerful and geometrically rich realisation of holography in certain special contexts (notably AdS/CFT), it is neither a universal nor a necessary framework for holography. Many structures at the heart of modern QFT — worldlines, defects, anomalies, factorization algebras, quantum codes — live in higher–categorical environments that are more naturally and more efficiently encoded by non–string categories. The string–free holographic principle proposed here takes this diversity as a starting point: it elevates higher–categorical encodings and context–dependent equivalences to the fundamental level, with strings understood as one particular encoding among many. The following sections will demonstrate that this perspective is not only conceptually compelling but also practically powerful in concrete examples.
We now refine the string–free holographic principle introduced in Section III by isolating a particularly important and powerful situation: when an encoding theory is not merely one possible holographic partner of a bulk theory in a given context, but is in fact universal among all such partners. This leads to what we call universal–property holography (UP–holography). The idea is directly inspired by the way universal properties (limits, colimits, adjoints, Kan extensions) are used throughout category theory to characterise mathematical constructions uniquely up to canonical equivalence [48].
Our aim in this subsection is to:
functor;
show how this notion fits into the general holographic square (66);
explain the connection with renormalisation (UP–RG) and density functional theory (UP–XC),
where one often seeks a “best possible” effective description of a many–body system with respect to a chosen class of observables.
Fiber categories of encodings. Fix an (∞, 1)–category of bulk theories Bulk, an observable (∞, 1)– category Obs, and a bulk context functor
tn action, ground–state density). We would like to consider all possible encoding theories E (living in a candidate encoding category Enc) such that E reproduces the same observable data as B in the given context, and then ask whether there exists a universal such encoding.
To this end, suppose we have chosen an encoding category Enc and an encoding context functor
as in (25). We consider the fiber category (or more properly, the homotopy fiber in the (∞, 1)–categorical sense) over the object F (B) ∈ Obs:
Objects of Fib F (F (B)) are pairs (E, φ) where E is a candidate encoding theory and φ is an equivalence in Obs between the encoded observable data G(E) and the bulk data F (B). Morphisms between two objects (E 1 , φ 1 ) and (E 2 , φ 2 ) in Fib F (F (B)) are morphisms h : E 1 → E 2 in Enc such that the following diagram commutes (up to coherent homotopy) in Obs:
(78)
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That is, φ 2 ◦ G(h) ≃ φ 1 in Obs. In this way, Fib F (F (B)) encodes the entire “moduli space” of encodings of B in the context F .
Universal encoding and initial objects. Within the fiber category Fib F (F (B)), there may exist objects that are in some sense “optimal” or “universal” among encodings. This is captured by the notion of an initial object (up to homotopy). Recall that in an ordinary category C, an object I is initial if for every object X there exists a unique morphism I → X. In an (∞, 1)–category, one says that an object I is initial if the mapping space Map(I, X) is contractible for all objects X [24], i.e. there is essentially a unique morphism from I to X up to a unique homotopy.
Definition IV.1 (Universal encoding in context F ). Let B ∈ Bulk and F : Bulk → Obs be fixed. An encoding theory E 0 ∈ Enc together with an equivalence
vis called a universal encoding of B in context F if the corresponding object (E 0 , φ 0 ) ∈ Fib F (F (B)) is initial in the (∞, 1)–categorical sense. That is, for any other (E, φ) ∈ Fib F (F (B)) there exists a morphism h : E 0 → E in Enc such that the diagram (78) commutes, and the space of such h is contractible.
Intuitively, E 0 is the “best possible” encoding of B for the observables selected by F : any other encoding E that reproduces the same observable data factors through E 0 in a unique way. This is strongly reminiscent of universal properties in other areas of category theory: for example, adjoint functors can be characterised by the initial/terminal objects in certain comma categories, and Kan extensions are defined by universal properties in functor categories [48].
UP–holography: universal–property holography. We can now define:
Definition IV.2 (UP–holography). Let B ∈ Bulk and F : Bulk → Obs be fixed. Suppose there exists a universal encoding (E 0 , φ 0 ) of B in context F in the sense of Definition IV.1. Then we say that B admits a universal–property holographic encoding (UP–holography) in context F , with universal encoding theory E 0 and encoding functor H defined on B by H(B) = E 0 .
The encoding functor H can often be extended (at least locally) to a neighbourhood of B in Bulk, giving a functor H : Bulk → Enc such that H(B) = E 0 and H behaves well with respect to morphisms (e.g. RG flows, deformations). In such cases, the relative holography equivalence F (B) ≃ G(H(B)) of Section III is sharpened by the universal property: not only is H(B) an encoding that works for the context F , but it is the canonical encoding through which all others factor.
tn UP–RG: renormalisation as universal encoding. Renormalisation group (RG) flows provide a natural arena for UP–holography. Informally, RG tells us that many microscopically different QFTs flow, at long distances, to the same effective theory, which captures all “relevant” observables at low energies [49, 50]. In our language, we can model this as follows:
• Bulk is taken to be a category of microscopic QFTs (e.g. with a UV cutoff), enriched and graded as
in Subsection II D.
• F IR : Bulk → Obs IR is a context functor that assigns to B its long–distance observables: critical
exponents, scaling limits of n–point functions, S–matrix elements below a certain scale, etc.
• Enc RG is a category of effective field theories (EFTs) with a finite number of couplings, defined at a
given scale or at a fixed point of the RG flow.
• G RG : Enc RG → Obs IR assigns to an EFT its long–distance observables.
Given a microscopic theory B ∈ Bulk, we consider the fiber category Fib F IR (F IR (B)) of all EFTs (E, φ) that reproduce the same IR observables as B:
Physically, these are all theories that are “IR–equivalent” to B in the chosen sense. In many situations, there exists an EFT E 0 with some universal property (often a fixed point or a minimal set of couplings) such that any other (E, φ) factors uniquely through (E 0 , φ 0 ). For instance:
• near a critical point, different lattice realisations of a model flow to the same continuum CFT, which
is universal for their critical behaviour;
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• in heavy–quark EFTs, the long–distance physics of different UV completions is captured by a
universal HQET or NRQCD theory.
In such cases, we can regard E 0 as a universal encoding E 0 of B in the IR context, and define H RG (B) = E 0 . This is exactly UP–holography: the RG–improved EFT is the initial object in the fiber of theories reproducing the IR observables of B.
v
UP–XC: density functional theory as universal encoding. A structurally similar situation arises in density functional theory (DFT), where the goal is to encode the physics of an interacting many–electron system in terms of a functional of the one–body density. In the Hohenberg–Kohn theorem [51], one shows that (under appropriate conditions) the ground–state density ρ(x) uniquely determines the external potential v(x) up to an additive constant, and hence the ground–state wavefunction and energy. The Kohn–Sham construction [52] then introduces an auxiliary non–interacting system whose density equals that of the interacting system, and the exchange–correlation functional E XC [ρ] encodes the difference in kinetic and interaction energies.
We can interpret this in the UP–holography language as follows:
• Bulk is the category of interacting electron QFTs in an external potential,
• F DFT : Bulk → Obs dens maps B to its ground–state density ρ(x) and possibly to a subset of
• Enc DFT is the category of density–only theories (e.g. Kohn–Sham systems plus functionals),
• G DFT : Enc DFT → Obs dens maps an encoding theory E to its output density and associated
functionals.
The Hohenberg–Kohn theorem states that for a given bulk system B, there exists (roughly) a unique density ρ B and associated universal functional F [ρ] such that all ground–state observables of interest can be expressed as functionals of ρ B . In categorical terms, this suggests that the pair (E 0 , φ 0 ) corresponding to the Kohn–Sham system with the exact exchange–correlation functional is an initial object in the fiber category of all density–only encodings reproducing the same ground–state density and energy. Thus, DFT can be viewed as a UP–holographic encoding: the density–only theory is the universal encoding for the context of ground–state observables.
tn
Summary. In summary, the notion of encoding as a universal property (UP–holography) provides a powerful and conceptually clean way to distinguish canonical holographic encodings from arbitrary ones. For a given bulk theory B and context F , the fiber category Fib F (F (B)) organises all possible encodings (E, φ); if an initial object (E 0 , φ 0 ) exists, it is the best possible encoding for that context, and any other encoding factors through it uniquely. This framework naturally unifies several familiar constructions in physics:
• RG–improved effective field theories as universal encodings for IR observables (UP–RG),
• density functionals in DFT as universal encodings for ground–state densities (UP–XC),
• and, more generally, any encoding that satisfies a universal property for a chosen class of observables.
In the remainder of the paper, we will use UP–holography as a guiding principle in designing non–string encodings, and we will show explicitly in our QED and QCD examples how universal properties arise and constrain the holographic maps H and the parameters of the encoding theories.
Universal–property holography (UP–holography), as formulated in Subsection IV A, tells us what it means for an encoding theory E to be optimal for a given bulk theory B and context F : the pair (E, φ) must be initial in the fiber category Fib F (F (B)). However, UP–holography does not, by itself, tell us how to find such encodings in practice. This subsection describes a systematic strategy for selecting the encoding category Enc based on the structural features of the physical problem at hand.
The central guiding principle is:
Select Enc to reflect the natural extended objects and algebraic structures of the problem. For each physical problem:
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• identify the natural extended objects (worldlines, worldsheets, defects, boundaries, higher–
codimension excitations); • identify the non–geometric structures (cohomology, anomaly classes, factorization alge-
bras, operator algebras, code categories) that capture the relevant observables;
• choose Enc to be an (∞, 1)– or higher category whose objects and morphisms model
these structures as directly as possible.
Strings and branes appear only when the natural structure is that of a two–dimensional extended worldsheet QFT propagating in a higher–dimensional geometric background.
We now elaborate this principle in a more precise and detailed way.
v1. Identifying natural extended objects. A first step is to examine the bulk theory B and the context functor F to identify which sorts of extended objects play a central role in the observables of interest. Concretely:
• Worldlines (1D trajectories). When one is interested in loop amplitudes of particle excitations (electrons, quarks, scalar fields) in a background, or in non–perturbative effects such as instantons in quantum mechanics or QFT, the natural extended objects are worldlines. The worldline formalism makes this explicit by rewriting determinants and loop integrals as path integrals over closed or open particle trajectories. In such cases, it is natural to take Enc to be a category of (0 + 1)–dimensional worldline theories, enriched and graded as in Subsection II D.
• Worldsheets (2D surfaces). When confining flux tubes, string–like solitons, or extended defects
stretching between sources dominate the physics, the natural extended objects are worldsheets. For example, in confining Yang–Mills theory, the field lines between a quark and an antiquark form a flux tube whose fluctuations can be modelled by a (1 + 1)–dimensional worldsheet QFT. In such contexts, it is appropriate to choose Enc as a category of worldsheet theories or defect 2–categories whose 1–morphisms are line defects and 2–morphisms are surfaces interpolating between them.
• Higher defects and boundaries. In the presence of domain walls, interfaces, junctions of defects,
or boundaries between different phases, the natural extended objects can be surfaces, volumes, or higher–codimension manifolds. Extended topological field theories and defect n–categories provide natural encoding categories where objects are boundary conditions, 1–morphisms are defects of codimension 1, 2–morphisms are junctions of such defects, and so on. For instance, the classification of topological phases and their interfaces often lives in such higher categories.
tn • No extended objects (purely local contexts). In some problems, the observables of interest are local
or quasi–local, with no clear dominance of extended objects. For instance, in certain RG or DFT contexts, one may care only about local correlators or ground–state densities. In such cases, it can be more appropriate to choose Enc to be a category of local field theories, factorization algebras, or density functionals, rather than a category of extended objects.
The key point is that extended objects of a given dimension k suggest that Enc should contain (k + 1)– dimensional worldvolume theories or k–categories of defects. This choice ensures that the encoding functor H respects the geometric and topological character of the excitations that dominate the observables F (B).
• Cohomology and anomaly classes. When the context involves anomalies, topological responses,
or classification of phases, generalized cohomology theories (e.g. cobordism, K–theory) and their Ext/Tor refinements are the natural ambient structures. The encoding category Enc should then be an (∞, 1)–category of extended TQFTs or invertible phases, with morphisms corresponding to domain walls or symmetry defects, and G : Enc → Obs extracting cohomological invariants.
• Factorization algebras and operator categories. When the context involves local operator algebraic
structure (OPEs, locality, microcausality), factorization algebras or (in 2D) vertex operator algebras form the natural target of F and G. In such cases, Enc can be chosen as a category of factorization algebras or of field theories presented via their factorization structure.
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• Quantum codes and entanglement structures. When the context focuses on entanglement properties,
code subspaces, or reconstruction of bulk operators from boundary regions, the natural non– geometric structures are those of quantum error–correcting codes and tensor network states. Here, Enc should be chosen as a category of codes and channels, with objects representing code spaces and morphisms representing encoding/decoding maps. The functor G may then extract entanglement entropies or logical operator algebras. iew • Probabilistic and stochastic structure. For nonequilibrium or open systems, stochastic processes and Markov kernels may play a central role. Then Enc may be chosen as a Markov or stochastic category, with morphisms representing probabilistic evolutions or channels rather than deterministic evolutions. Enrichment over Markov categories (see Subsection II D) is appropriate in this case.
vThus, in addition to the geometry of the excitations, one must match the algebraic and cohomological structures of the problem with the encoding category.
are they correlation functions at a given loop order, anomaly polynomials, IR effective actions, entanglement entropies, ground–state densities, etc.? This step identifies the “output type” in Obs.
or excitations of B dominate the observables: are they particle loops, extended flux tubes, topological defects, boundary modes, etc.? Also identify the symmetry and cohomology structures (global symmetries, higher–form symmetries, anomalies) that constrain these observables.
choose Enc to be an (∞, 1)– or higher category whose objects and morphisms model those directly:
• If particle loops dominate, choose Enc as a worldline category. ot • If flux tubes or string–like defects dominate, choose Enc as a worldsheet or defect 2–category. • If anomalies and topological responses dominate, choose Enc as a category of extended TQFTs and invertible phases. t n • If entanglement and reconstruction dominate, choose Enc as a code or tensor–network category.
encoding theory E = H(B) that incorporates the identified structures (e.g. the worldline path integral representation, the effective worldsheet action, the anomaly inflow TQFT). Define G : Enc → Obs by mapping E to observable data of the same type as F (B) (e.g. effective actions, anomaly functionals, potentials, densities).
≃ F (B) −−→ G(H(B)) in Obs. This may be proven exactly (as in the Schwinger worldline representation or anomaly inflow) or established in a controlled approximation (e.g. semiclassical approximation, IR limit).
Fib F (F (B)), as in Definition IV.1. If so, then Enc is not just a convenient choice but the canonical encoding category for the context F , and H realises a UP–holography.
This procedure is intentionally structural rather than ad hoc: instead of guessing an encoding theory and checking whether it happens to work, we derive the effective shape of Enc from the physical and algebraic features of the problem.
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• the relevant excitations of the bulk theory can be naturally organised as one–dimensional extended ob-
jects whose worldvolumes sweep out two–dimensional worldsheets (for strings) or higher–dimensional branes;
• the background geometry admits a meaningful notion of asymptotic boundary and a large–N or
large–radius limit in which classical gravity or string dynamics is reliable;
• the observables in context F are correlators or entanglement measures that are known (or conjectured)
to be captured by the worldsheet CFT or its brane effective theories.
v In such cases, it is natural to choose Enc to be a category of string/brane backgrounds and to define H by mapping bulk QFTs to their string embeddings. However, as discussed in Subsection III C, these conditions are restrictive and rarely satisfied in realistic QFT regimes.
By contrast, in many of the examples we consider (e.g. strong–field QED, anomaly/WZW QCD, flux–tube QCD), the natural structures point to different encodings:
• In strong–field QED, one–loop and two–loop effective actions are naturally expressed in terms of
worldlines, not worldsheets.
• In anomaly problems, the relevant data live in one higher dimension as invertible TQFTs, not as
worldsheet CFTs.
• In confining QCD, flux tubes are effective strings in the IR, but their worldsheet theories are
emergent defect theories, not fundamental string backgrounds in 10D spacetime.
In this sense, the strategic selection of Enc via structure reveals that strings are a special case, not a universal solution.
• the natural extended objects (worldlines, worldsheets, defects, boundaries),
tn • the non–geometric algebraic and cohomological structures (cohomology, anomalies, factorization
algebras, code categories), SUPER–HOLOGRAPHY IN QED: WORLDLINE (0+1D) ENCODING OF 4D QED
we can choose Enc to be a higher category that reflects these features as directly as possible. The resulting encoding functor H and observable functor G then implement a string–free holography, with the potential to be universal in the sense of UP–holography. Strings and branes appear only when the physics and mathematics genuinely demand a two–dimensional worldsheet QFT in a geometric background; in all other cases, non–string encoding categories are both more natural and more powerful.
In this subsection we develop in detail the worldline representation of the one–loop effective action of four–dimensional spinor QED in an arbitrary background gauge field. This will serve as our first concrete and fully explicit example of super–holography in the sense of Section III: a four–dimensional bulk field theory is encoded into a (0 + 1)–dimensional worldline (super–)quantum mechanics, and the one–loop observable content in a given context is captured exactly by the encoding theory.
Our main goal is to derive the spinor worldline action
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and to show that the one–loop effective action in Euclidean space can be written exactly as
(82)
for an arbitrary background gauge field A µ (x), with F µν = ∂ µ A ν − ∂ ν A µ . This worldline expression preserves the full 4–vector structure of A µ and F µν and is an exact rewriting of the one–loop determinant, not an approximation.
vRemark on methodology. For the avoidance of doubt, we emphasise that the worldline representation (82) follows from entirely standard QFT manipulations: (i) evaluating the Gaussian functional integral over fermions, (ii) rewriting the resulting ln det through the Schwinger proper-time representation, and (iii) expressing the heat kernel of a second–order differential operator as a point–particle path integral via Trotter decomposition. No Feynman-diagram resummations, dualities, or additional assumptions are introduced at any stage. The “holographic” interpretation therefore refers only to the fact that an exactly equivalent lower–dimensional representation exists, not to any new or conjectural physics.
where the covariant derivative is
and we use the mostly–minus metric convention η µν = diag(+, −, −, −). Integrating out the fermionic
tn / − m in the background The object of interest is thus the functional determinant of the Dirac operator i D A µ (x).
To put this into a form suitable for a Euclidean worldline path integral, it is convenient to Wick–rotate to Euclidean signature, x 0 = −ix 4 , and to employ the Schwinger proper–time representation.
so that the one–loop effective action depends on ln det M † M , which is a determinant of a second–order operator.
To make this explicit, consider the Euclidean version of the Dirac operator. After Wick rotation, we denote Euclidean gamma matrices by γ µ satisfying {γ µ , γ ν } = 2δ µν , and define
with Euclidean indices µ = 1, 2, 3, 4. The Euclidean Dirac operator is then
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We now compute
where
Let us briefly sketch the derivation of (92). One has
The Euclidean effective action is
(92)
(93)
(94)
(95)
(96)
Up to a convention for the factor of i (which can be absorbed into the definition of F µν in Euclidean space), we can write this as
(97)
which agrees with (92).
The Euclidean one–loop effective action can now be written as
n
where the trace is over spacetime and internal indices. Applying (99) to M = −D 2 + 2 e σ µν F µν + m 2 yields
Our task is now to express the trace of the heat kernel
as a path integral over worldline variables. The worldline representation can be obtained via several routes: canonical quantisation of relativistic particles [57, 58], or via coherent–state path integrals for spin degrees of freedom [59]. We will follow the latter approach, which leads naturally to Grassmann variables on the worldline.
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 37 4. Worldline path integral representation. The operator −D 2 + 2 e σ µν F µν acts on spinor wavefunctions. One can regard the trace in (100) as a sum over closed paths with spin degrees of freedom. Specifically,
vquantisation, reproducing the gamma matrices and spin couplings.
The action (81) can be understood as follows:
• The term 12 ψ µ ψ̇ µ is the kinetic term for Grassmann variables; after quantisation it reproduces the
anticommutation relations of gamma matrices and the spin degrees of freedom. • The term −ie ẋ µ A µ (x(τ )) is the coupling of the particle to the external gauge field (Wilson line). • The term ie ψ µ F µν (x(τ ))ψ ν reproduces the spin–field interaction 2 e σ µν F µν upon integrating over
the Grassmann variables. Boundary conditions x(0) = x(T ) ensure a closed loop in spacetime (reflecting the trace), and anti–periodic boundary conditions ψ(0) = −ψ(T ) arise from the spin structure on the circle, corresponding to fermionic degrees of freedom at one loop.
Substituting (103) into (100) and integrating over x 0 , we obtain
tn which, after Wick rotating back to Minkowski space (or adopting a unified Euclidean convention), leads to (82) up to overall factors. Restoring the minus sign from the Minkowski effective action,
which is precisely (82).
• The spinor nature of the fermion is fully captured by the Grassmann fields ψ µ (τ ) and the term
ie ψ µ F µν (x(τ ))ψ ν . Upon integrating over ψ, one reconstructs the spin factor associated with σ µν F µν . • The worldline path integral is gauge–invariant and reparametrisation–invariant up to appropriate
gauge–fixing conditions on the worldline. One can show that the path integral is invariant under worldline reparametrisations τ 7→ f (τ ) that fix the endpoints, provided one includes the appropriate ghost determinants [58].
/ − • The representation (82) transforms the original four–dimensional functional determinant ln det(i D
m) into a one–dimensional path integral over loops in spacetime and internal Grassmann variables. This is the core of the “super–holographic” compression: the one–loop physics of the 4D bulk is encoded in a 0+1D worldline theory.
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• Bulk category: Bulk QED , a full subcategory of QFT whose objects include 4D spinor QED with
background gauge fields. Our bulk theory is B = QED 4 .
• Encoding category: Enc wl , the (∞, 1)–category of (0 + 1)–dimensional worldline super–quantum
mechanics coupled to background gauge fields A µ (x) and field strengths F µν (x). An object of Enc wl
• Encoding functor: H wl : Bulk QED → Enc wl , which maps the Dirac QED theory B to the worldline
theory E = H wl (B) defined by (81). On morphisms (e.g. background–field variations or coupling deformations), H wl maps them to corresponding deformations of the worldline action.
v• Context functor (bulk): F 1loop : Bulk QED → Obs eff , which assigns to B its one–loop effective action
Γ (1) [A] in an arbitrary background A µ (x).
• Context functor (encoding): G 1loop : Enc wl → Obs eff , which assigns to a worldline theory E the
corresponding worldline path integral over closed loops, interpreted as an effective action functional of A µ (x).
The Schwinger/worldline derivation shows that there is a natural equivalence
realised explicitly by (82). In other words, in the one–loop context,
with equality understood as an equality of functionals in Obs eff . This is precisely a string–free super– holography relation: a 4D bulk QED theory is encoded in a 0+1D worldline theory for the purpose of computing one–loop effective actions.
tn
Worldline instantons and pair production in constant fields
B.
We now illustrate the power of the worldline encoding by re–deriving Schwinger’s classic result for non–perturbative pair production in a constant electric field [60] using the worldline instanton method. This provides an explicit and technically detailed realisation of the super–holographic compression 4D → 0 + 1D: the non–perturbative imaginary part of the four–dimensional one–loop effective action is obtained from classical closed trajectories of a relativistic particle in Euclidean spacetime and the Gaussian fluctuations around them.
We will focus for concreteness on scalar QED for the full derivation of the exponent and prefactor and comment on the spinor case at the end; the spinor modifications are well–known and amount to an alternating sign and a change of the worldline spin factor. The worldline representation for spinor QED was established in Subsection V A, and the derivation for the scalar case is structurally similar but technically simpler in the instanton analysis.
Lagrangian Consider scalar QED in Minkowski spacetime, with
(108)
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 39 where D µ = ∂ µ + ieA µ (x) and F µν = ∂ µ A ν − ∂ ν A µ . Integrating out the complex scalar field in the presence of a classical background A µ yields the one–loop effective action
v The trace of the heat kernel can be represented as a worldline path integral (see e.g. [8, 57]), namely ! 2 Z Z Z T ẋ T D 2 4 µ
where x µ (τ ) are periodic paths in Euclidean spacetime with period T , and ẋ 2 := ẋ µ ẋ µ . The worldline action for scalar QED is therefore
n
and one convenient gauge choice is
(117)
(118)
The path integral over the transverse directions x 1 , x 2 factorises and yields the usual free Gaussian contributions; the nontrivial dynamics is in the (x 3 , x 4 ) plane, where the electric field produces a coupling between ẋ 3 and x 4 .
Our goal is to compute the imaginary part of the Minkowski effective Lagrangian, =L (1) (E), which encodes the pair–production rate. The imaginary part arises from the nonanalytic structure of the proper–time integral in the complex T –plane, which, in the constant field case, is controlled by worldline instantons: classical closed trajectories x cl (τ ) that extremise the worldline action (118).
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The classical equations of motion for x µ (τ ) are 3. Equations of motion and circular instantons.
(119)
E where F
µν is the Euclidean field strength. In our constant–field gauge (116), the only non–vanishing E E components are F
only contribute trivial Gaussian fluctuations).
Thus ẋ 3 satisfies the harmonic oscillator equation with frequency 2eE:
so that ẋ 3 (τ ) is a linear combination of cos(2eEτ ) and sin(2eEτ ). Similarly, one finds that ẋ 4 (τ ) is a
More systematically, we look for solutions of the form
(126)
(127)
tThese yield the same condition:
the (x 3 , x 4 ) plane, with angular frequency fixed by the field:
Each integer n corresponds to an n–fold winding of the instanton around the circle.
In addition, the reparametrisation invariance of the worldline action implies a constraint on ẋ 2 (τ ). One convenient way to see this is to recall that the worldline action for a relativistic particle is proportional to its proper length, leading to the constraint ẋ 2 (τ ) = const on shell. Here,
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with T n given by (131).
(134)
Therefore,
(137)
we obtain
so t
(141)
(142)
so
(143)
Therefore, the second term contributes
(144)
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 42 Putting everything together:
v(147)
We expand the worldline around the n–th instanton:
(n)
(148)
(149)
the one–loop prefactor. The path integral over η gives
where the prime indicates that zero modes must be treated separately (e.g. translational zero modes corresponding to shifting the centre of the instanton).
For the constant–field case, the fluctuation operator factorises into transverse, longitudinal, and temporal components, and one finds that:
t• two of the four components (corresponding to free directions x 1 , x 2 ) contribute a free Gaussian
• the two components in the (x 3 , x 4 ) plane contribute a nontrivial determinant that can be evaluated
explicitly in terms of sin(eET n ),
• the translational zero mode yields an overall factor proportional to the spacetime volume and an
integral over the instanton centre, which is divided out when one computes L = Γ/V 4 .
The net result, after including the measure factors and summing over winding number n, is the well–known Schwinger formula for scalar QED in a constant field:
which exhibits the exponent (146) and a prefactor ∝ (eE) 2 /n 2 . The 1/n 2 arises from the fluctuation determinant and the integration over the proper–time parameter T near the poles at T = T n in the
For spinor QED, one finds an additional factor of 2 (for spin degrees of freedom) and an alternating sign (−1) n+1 due to Fermi statistics and the spin factor, giving
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Both formulas are recovered exactly from the worldline instanton formalism; the calculation in the spinor case is structurally identical to the scalar one, with the Grassmann path integral contributing the spin factor and modifying the fluctuation spectrum [8, 61].
Interpretational note. The constant–field results (151) and (152) are of course the classic Schwinger pair–production rates. Our use of them here is strictly as a nontrivial consistency check of the worldline formalism. The computational utility of the worldline representation emerges most clearly in time– dependent or inhomogeneous fields, where no closed diagrammatic resummation is available and the instanton method provides genuinely new calculational capability.
v• The bulk problem is four–dimensional: computing the imaginary part of the one–loop effective action
Γ (1) [A] in a uniform electric field, which in a Feynman–diagrammatic language amounts to summing an infinite series of multi–photon graphs with arbitrary numbers of external field insertions.
• The encoding problem is one–dimensional: finding classical closed trajectories x inst (τ ) in the (x 3 , x 4 )
plane and evaluating the one–dimensional fluctuation determinants around them.
• The context functor F selects the non–perturbative pair–production rate (imaginary part of the
effective action) as the observable; the encoding functor H wl maps the 4D QED theory to its worldline theory; the encoding context functor G computes the same observable from the worldline path integral.
C. ot The equivalence F (B) ≃ G(H wl (B)) in this context is not only formal but practically powerful: it replaces an intractable 4D summation problem by a manageable 1D semiclassical analysis. The exponent is given by the classical worldline action (146), and the prefactor is given by determinants of 1D fluctuation operators that can be computed with well–developed tools (e.g. zeta–function regularisation, Gel’fand–Yaglom method).
In the next subsection, we will extend this worldline super–holography to time–dependent electric fields and to higher loops, further demonstrating the capability of non–string encodings to capture non–perturbative QFT phenomena in a dimension–reduced and computationally efficient framework.
Fast time–dependent fields: Sauter pulse as a super–holographic testbed
tn We now move beyond the constant–field example of Subsection V B and consider a genuinely time– dependent electric field as a nontrivial testbed for super–holography in QED. The goal is to show that the worldline encoding developed in Subsection V A continues to provide a powerful and conceptually clear framework when the background field varies rapidly in time, and that the non–perturbative pair– production exponent and prefactor can still be obtained from a purely one–dimensional problem, albeit now with complexified Euclidean time and more intricate instanton trajectories.
We focus on the widely studied Sauter pulse,
with corresponding vector potential (in Minkowski temporal gauge A 0 = 0)
The parameter ω describes the time scale of the pulse, and E 0 its peak field strength. A useful dimensionless measure of the time dependence is the Keldysh parameter
(155)
which interpolates between the tunnelling regime (γ 1) and the multiphoton regime (γ 1). One of the virtues of the worldline instanton approach is that it captures both regimes and the crossover between them in a unified semiclassical picture [62].
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 44 1. Worldline action in a Sauter pulse background. We work again in the scalar QED setting for the instanton analysis; the spinor case will be discussed at the end. The Euclidean worldline action for a
vA 3 (t) as in (154); after analytic continuation,
needed at this step, only the analytic structure.
action in Euclidean space becomes
As in the constant–field case, the x 1 , x 2 directions are free and can be taken as constant along the instanton; the nontrivial dynamics is confined to the (x 3 , x 4 ) plane.
The Euler–Lagrange equations derived from (158) are
Then (159) yields
(161) (162)
As before, the worldline is reparametrisation–invariant and we may impose the constraint
nwith a constant along the trajectory. For tunnelling solutions relevant for pair production, the stationary– phase analysis again suggests a = 2m (mass–shell condition), as in the constant–field case.
To reduce the system (161), (162), (163) to a first–order system, it is convenient to parametrise the velocity in polar form:
(166) (167)
For generic points where neither sin θ nor cos θ vanish identically, these equations are equivalent and yield the first–order relation
Combining (164) with the constraint a = 2m, we obtain the first–order system:
which is precisely the system mentioned in the outline, with E(ix 4 ) understood via (160). This system governs the worldline instanton trajectories in Euclidean time. In general, these trajectories will be complex in x 4 for rapidly varying fields (i.e. non–small γ), but the basic structure remains as in (169).
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where prime denotes derivative with respect to τ 0 . On the other hand, from (169) we have ẋ 4 (τ ) = 2m sin θ(τ ). Equating these:
so that
(172)
(173)
(174)
tn that all dependence on the physical parameters m, ω, E 0 is captured by the single dimensionless Keldysh
The system (176), together with appropriate periodicity conditions, defines the worldline instanton problem for the Sauter pulse. It is a coupled first–order nonlinear ODE system in two variables (u(s), θ(s)), which is amenable to numerical solution by shooting methods, as we now explain.
for some integer n ≥ 1 representing the winding number. The condition on θ ensures that the worldline makes n full revolutions in the θ–coordinate over one period, consistent with a closed trajectory in the (x 3 , x 4 ) plane. In Minkowski space, this corresponds to a tunnelling trajectory connecting classically forbidden regions at different times.
To find such solutions, one typically employs a shooting method in the complex u–plane, because the Euclidean trajectory may explore complex values of x 4 for rapidly varying fields (non–small γ). The procedure is as follows [62]:
guess for the period S and the winding number n.
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46
plane (e.g. Runge–Kutta with complex arithmetic).
dimensional Newton–Raphson or shooting algorithm to minimise (|∆u|, |∆θ|). This iterative process continues until the periodicity conditions (177) are satisfied to the desired numerical accuracy.
The resulting trajectory (u(s), θ(s)) defines the worldline instanton for the given value of γ. In practice, one finds that:
v• For γ 1 (slowly varying field), the instanton stays close to the real Euclidean time axis and
resembles the circular instanton of the constant–field case.
• For γ & 1 (rapidly varying field), the instanton moves into the complex u–plane and acquires a
more intricate shape; nevertheless, it remains a periodic solution of the ODEs (176).
This behaviour reflects the transition from a tunnelling–dominated regime (quasi–static barrier) to a multiphoton–dominated regime (absorption of many quanta of frequency ω), unified by the same worldline framework.
tn The field–dependent term can be rewritten, using integration by parts and the equations of motion, into an expression involving the field strength and the path. For Sauter–profile fields, one finds that the action can be written in the compact form
(181)
where g(γ) is a dimensionless function determined by the solution of (176). Explicitly, g(γ) can be written as a contour integral along the instanton in the complex u–plane. One representative expression (see e.g. [62]) is
for a related Sauter–type background (for spatial or temporal inhomogeneities), with the integration limits determined by turning points. The precise form of g(γ) depends on conventions and on whether one studies temporal or spatial inhomogeneities, but the general structure (181) holds: the semiclassical exponent is m 2 /(eE 0 ) times a function of γ that is computed from the ODE system (176).
The behaviour of g(γ) encodes the physics of the tunnelling–multiphoton transition:
• For γ → 0 (slow field), g(γ) → π, and (181) reduces to the constant–field exponent S 0 = πm 2 /(eE 0 ).
• For γ → ∞ (fast field), g(γ) ∼ c γ for some constant c, corresponding to a scaling characteristic of
multiphoton pair production.
Thus, the function g(γ) computed from the worldline instanton ODEs captures the full non–perturbative scaling of the pair–production exponent across regimes.
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where zero modes are treated separately and lead to collective–coordinate factors.
The Gel’fand–Yaglom method provides an efficient way to compute determinants of 1D differential
interval [0, S] with suitable boundary conditions, one solves the ODE Ky(s) = 0 with initial conditions y(0) = 0, ẏ(0) = 1 and reads off det K from the value y(S). In matrix–valued cases, one solves a system of linear ODEs and computes an appropriate determinant of the fundamental solution matrix [62]. For the Sauter pulse, both K B and K F can be reduced to such 1D operators in the appropriate basis, and the determinants can be computed numerically for each γ.
Combining the exponent and prefactor, the one–loop pair–production rate in spinor QED in a Sauter pulse can be written schematically as (186)
tn
where N (γ) encodes the combined bosonic and fermionic fluctuation determinants and contains the alternating–sign structure and degeneracy factors. The precise form of N (γ) can be determined numerically using the Gel’fand–Yaglom method once the instanton trajectory (u(s), θ(s)) is known.
• The bulk problem — computing =Γ (1) [A] for spinor QED in a fast, strong time–dependent field —
is highly nontrivial in four dimensions and essentially intractable by direct diagrammatic methods (involving infinitely many multi–photon processes).
• The worldline encoding provides a 0+1D description in which the exponent and prefactor are
obtained from:
– a coupled system of one–dimensional ODEs (176) for (u(s), θ(s)) in the complex plane,
– one–dimensional linear ODEs for fluctuation operators K B , K F (Gel’fand–Yaglom method), – low–dimensional integrals (e.g. over proper time and worldline parameters).
• The entire dependence on the physical parameters is captured by the Keldysh parameter γ and
the function g(γ) computed from the ODEs; the method applies uniformly across tunnelling and multiphoton regimes.
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 48 In the categorical language of Section IV, the bulk category Bulk QED contains spinor QED with time– dependent backgrounds, the encoding category Enc wl contains worldline super–QFTs with time–dependent
theory.
Capability: 4D → 1D super–holography vs. diagram explosion
vIn the preceding subsections we developed the worldline encoding for one–loop spinor QED in arbitrary backgrounds and applied it to the two key examples of a constant electric field and a time–dependent Sauter pulse. We now step back and analyse, in detail, the computational capability that this encoding provides when compared to the standard four–dimensional Feynman–diagrammatic approach. The central point is that, in the context of non–perturbative pair production in strong, possibly rapidly varying fields, the worldline formalism implements a genuine super–holography:
• on the 4D bulk side, non–perturbative pair production is encoded in an infinite tower of multi–photon
diagrams of arbitrarily high order;
• on the 1D encoding side, the same physics is encoded in:
– a single (0 + 1)–dimensional worldline action, – a finite–dimensional problem of finding and analysing worldline instantons (solutions of a 1D
ODE system), – and low–dimensional integrals (over proper time and worldline parameters).
This is a striking reduction of computational complexity and an instructive concrete instance of the general string–free, higher–categorical holographic principle discussed earlier.
tn Feynman–diagrammatic viewpoint,
rin where A n [A] represents the sum of all one–loop diagrams with n external photon legs, each leg corre- sponding to an insertion of the background field A µ (x). More concretely, if one expands the interaction term in the Dirac action, ψ̄γ µ (i∂ µ − eA µ )ψ = ψ̄(i ∂ / − m)ψ − eψ̄γ µ A µ ψ, (190)
ep the n–th order term in e in the effective action involves an n–point function of the current j µ (x) = ψ̄γ µ ψ(x): Z A n [A] = (−e) n d 4 x 1 · · · d 4 x n A µ 1 (x 1 ) · · · A µ n (x n ) hj µ 1 (x 1 ) · · · j µ n (x n )i 1PI , (191) where the expectation is taken in the free theory. The pair production amplitude is related to the sum over amplitudes for processes of the form
where the γ(k i ) represent background-field photons with momenta k i (obtained by Fourier transforming A µ (x)), and the sum runs over all n ≥ n min consistent with energy–momentum conservation. In a This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 49
N =N min iew where M N is the amplitude for N -photon absorption. Each M N is a sum over many Feynman diagrams involving N external legs, internal Dirac propagators, and complicated spin and momentum structures.
The difficulty is twofold:
v• The number of diagrams grows rapidly with N and with the number of loops (e.g. two–loop
corrections involve inserting internal photons between different points on the loop).
• For strong fields (large E 0 ) and fast time variation (large ω), one is interested in a region of parameter
space where no finite truncation in N or in loop order is sufficient; genuinely non–perturbative physics is encoded in the behaviour of the full series.
Computationally, this leads to a diagram explosion, which makes the direct 4D diagrammatic approach essentially intractable for realistic strong–field problems beyond very special limits.
with the worldline action
ot Expanding the exponent in powers of e and A µ , one can show that (195) reproduces all one–loop diagrams with any number of external legs:
tn • the term linear in A µ corresponds to a single insertion of the background field (e.g. vacuum
polarisation);
• terms with n factors of A µ encode n external legs;
couplings.
Thus, instead of dealing with infinitely many diagrams separately, one has a single master path integral which sums them all. The worldline path integral is, in this sense, a generating functional for all one–loop processes in the background A µ .
For constant or slowly varying fields, this path integral can often be evaluated analytically (as in Subsection V B); for more complicated fields, it can be evaluated numerically by Monte Carlo methods [63] or by semiclassical approximations (instanton methods) [62].
• solving the first–order system
(197)
for (u(s), θ(s)) with periodic boundary conditions, This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 50
• computing fluctuation determinants of 1D operators around the instanton for the prefactor.
Here, u(s) is the dimensionless Euclidean time coordinate (scaled by ω), θ(s) is an angular variable encoding the direction of motion in the (x 3 , x 4 ) plane, and γ is the Keldysh parameter. The complexity of the problem is thus concentrated in the solution of a 1D ODE system and in the evaluation of 1D Gaussian integrals around the resulting trajectory.
To emphasise the contrast:
– infinitely many n–photon absorption processes γ+· · ·+γ → e + e − contribute, with n unbounded;
v– each amplitude M n involves 4D momentum integrals, spinor traces, and combinatorial sums
over diagrams;
– the non–perturbative behaviour emerges only from a delicate resummation of the entire series
and is extremely difficult to access directly.
– all n–photon processes are encoded in the single master path integral (195);
– the non–perturbative rate is dominated by one or a few classical worldline instantons, solutions
of (197);
2 m – the exponent is given by a simple functional S 0 = eE g(γ), and the prefactor is determined by
0 1D fluctuation determinants.
In numerical practice, solving (197) with a shooting method and evaluating the Gel’fand–Yaglom determinants is dramatically simpler than attempting to compute even a modest truncation of the amplitude series in the diagrammatic approach.
tn • In the diagrammatic expansion, one must, in principle, sum over all N & N min up to N eff ; the number of diagrams at each order grows combinatorially, and each diagram involves a 4D integral. Overall, the naive computational complexity scales at least as
int where C
4D denotes the cost per 4D integral and #diagrams(N ) grows factorially with N .
• In the worldline instanton method, the computational complexity is dominated by:
where C solve ODE is the cost of solving the 1D ODE system (197) (typically scaling linearly with the grid size in s) and C fluctuations is the cost of solving 1D linear ODEs for the fluctuation determinants. Both are polynomial in the number of discretisation points and do not grow combinatorially with any parameter analogous to N eff .
While this comparison is schematic, it captures the essential point: worldline super–holography converts a combinatorial 4D diagram explosion into a controlled 1D ODE problem.
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51
• The bulk category Bulk QED contains objects B corresponding to 4D spinor QED in external fields.
• The encoding category Enc wl contains objects E corresponding to worldline super–QFTs.
• The encoding functor H wl : Bulk QED → Enc wl maps B to its worldline representative E with action (196).
• The context functor F pair : Bulk QED → Obs pair assigns to B its non–perturbative pair–production
data (exponent and prefactor).
v• The encoding context functor G inst : Enc wl → Obs pair assigns to E the same data extracted from
worldline instantons and their fluctuations.
The worldline formalism and instanton analysis demonstrate a natural equivalence
• H wl is a compression functor in the sense of Subsection II C: it maps a 4D theory to a 0+1D theory;
• F pair is a context functor that selects non–perturbative observables, not all correlators;
• G inst is a context functor for the worldline theory that computes the same observables from
instantons.
Thus, in the language of the string–free holographic principle, worldline super–holography is not simply a computational trick, but a genuine higher–categorical holographic encoding for a specified context.
• It encodes the full one–loop (and, with further work, higher–loop) structure of 4D spinor QED in
arbitrary backgrounds into a 1D worldline theory.
tn • It reduces a problem that is diagrammatically intractable — especially in fast, time–dependent fields — to a tractable 1D ODE problem supplemented by low–dimensional integrals.
• It is fully compatible with the general categorical holographic framework: we can identify the bulk,
encoding, and observable categories, the encoding and context functors, and the natural equivalence relating them.
This is precisely the type of capability that justifies the string–free holographic principle: by choosing encoding categories that match the structure of the problem (worldlines, not worldsheets), we obtain not only conceptual clarity but also strong computational advantages in realistic quantum field theories.
In this subsection we extend the one–loop worldline encoding of spinor QED developed in Section V to the two–loop level, focusing on the quenched sector where a single Dirac loop interacts with an internal photon. Our goal is to derive a compact “master” worldline formula for the two–loop effective action,
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spin ψ iew with ed spin where S
wl [x, ψ; A] is the worldline action (196), and (202)
(203)
a worldline current and D(x − y) the free photon propagator in some fixed gauge (we will use Feynman gauge for concreteness). This expression encodes, in a single worldline functional, the entire set of two–loop diagrams with one Dirac loop and one internal photon exchange in an arbitrary background A µ (x).
We will proceed in several steps:
v 1. Start from the full QED path integral and integrate out the fermions in the presence of a dynamical photon field and a classical background. 2. Isolate the two–loop contribution in the quenched approximation, corresponding to a single internal
photon line connecting two points on the Dirac loop.
worldline current J µ and photon kernel.
µ and on the way (201) resums all two–loop diagrams 4. Comment on the structure of the spin part J
spin in a compact form.
background field A µ (x), we split
t
where we have suppressed gauge fixing and ghost contributions for clarity; they can be reintroduced as needed and do not affect the structure of the worldline representation for the matter loop with a single
where (207)
with f µν = ∂ µ a ν − ∂ ν a µ and L gf a gauge–fixing term for a µ (e.g. Feynman gauge).
The effective action Γ[A] is obtained by integrating over both ψ and a µ . In perturbation theory, one expands in powers of e and in the loop expansion (powers of ~). The one–loop contribution corresponds to This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 53
integrating out ψ with a µ set to zero; this is the determinant considered in Subsection V A. The two–loop contribution in the quenched approximation corresponds to diagrams with a single closed fermion loop This expression encompasses all two–loop diagrams with one internal photon and one Dirac loop in the fixed background A. The factor of 1/2 reflects the symmetry factor of the closed photon loop.
Recall that, in Euclidean space, the Dirac propagator in a background A µ can be written as
/ E is the Euclidean Dirac operator and G[A] satisfies where D E = D
(212)
(213)
The scalar propagator G[A] admits a worldline representation:
spin spin with I
photon [x, ψ] given by (202). We now examine the structure of I photon .
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0
where
(218)
(221)
which is precisely (202).
The form of the worldline current J µ (τ ) can be deduced by comparing the worldline representation of diagrams with and without spin. In scalar QED, the current is simply
reflecting the coupling −ie ẋ µ A µ in the worldline action (113). In spinor QED, the coupling to the background field includes the spin term ie ψ µ F µν ψ ν , which gives rise to an additional contribution to the current. To leading order in the number of photon insertions, one finds
µ where J
know that it is local along the worldline and linear in e.
t4. Master formula and its diagrammatic content. The master worldline formula (215) summarises the entire two–loop Dirac–loop–plus–photon sector. Let us make this explicit:
interactions.
spin • Inserting I
photon introduces a bilocal interaction between two points τ 1 , τ 2 on the worldline, with kernel D(x(τ 1 ) − x(τ 2 )) and current insertions J µ (τ 1 ) and J µ (τ 2 ). In the expansion, this corresponds to a single internal photon exchanged between two points of the fermion loop.
• Summing over all ways of distributing the background–field couplings along the loop is automatically
achieved by the path integral over x(τ ) and ψ(τ ).
From the point of view of 4D Feynman diagrams, (215) therefore generates:
• all two–loop diagrams with one closed Dirac loop,
• one internal photon connecting two points on the loop,
• and arbitrary numbers of external background–field insertions.
Instead of writing and summing a very large number of individual 4D diagrams, one works with a single worldline functional that encodes them all at once. This is the two–loop analogue of the one–loop master formula (195), now enriched by a bilocal kernel.
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• The hom–objects in Enc wl are no longer just single–worldline path integrals; they are graded by
spin loop order and include functionals with bilocal kernels such as I
photon .
• The encoding functor H at two loops,
QED −→ Enc wl , iew • The enrichment over Vect C (or Hilb) must be refined to include these bilocal structures, which can
be interpreted as 2–morphisms (or higher morphisms) in a categorical sense: the photon exchange is a “higher interaction” between worldline segments.
(224)
vspin maps the two–loop sector of QED to the enriched worldline category in which I
photon lives. The (2) (2) context functors F and G extract two–loop effective actions from the bulk and encoding, respectively, and the natural equivalence F (2) ≃ G (2) ◦ H (2) expresses two–loop worldline holography.
Thus, the two–loop worldline representation is a higher–order realisation of string–free super–holography: it compresses a large class of 4D two–loop diagrams into a compact 1D path integral with enriched structure, consistent with the loop grading and enrichment discussed in Subsection II D.
• in the semiclassical regime, the dominant contribution again comes from worldline instantons, now
spin dressed by the photon kernel I
photon evaluated on the instanton trajectory;
spin • the exponent receives a two–loop correction ∆S (2) from I
photon [x inst , ψ inst ], and the prefactor receives corrections from the modified fluctuation operators (now including the bilocal interaction); ot • the overall problem remains one–dimensional in the sense that all nontrivial dynamics takes place along the worldline parameter τ (or its rescaled versions), with additional integrals over τ 1 , τ 2 from the bilocal kernel.
tn This is a direct extension of the one–loop super–holography of Section V to the two–loop level and demonstrates that the worldline encoding naturally scales to higher loops without losing its dimensional and structural advantages.
Semiclassical two–loop correction for fast fields
B.
We now build on the two–loop worldline representation (215) derived in Subsection VI A and on the one– loop worldline instanton analysis of Subsection V C, in order to construct a semiclassical approximation to the two–loop effective action for spinor QED in strong, fast time–dependent electric fields. The central spin idea is to treat the photon–exchange functional I
photon [x, ψ] as an insertion in the one–loop worldline path integral and to evaluate its expectation value in a saddle–point expansion around the one–loop instanton trajectory x inst (τ ). In doing so, we obtain a two–loop correction to the effective exponent and prefactor which is still governed by one–dimensional dynamics, together with a two–dimensional integral over worldline proper times (τ 1 , τ 2 ).
The result has the schematic form
with
(226)
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56 as in (181), and
Here h· · · i ψ, x inst denotes the expectation value over Grassmann fluctuations ψ(τ ) around the instanton (and, if desired, over small bosonic fluctuations η(τ ), though to leading order one may set x(τ ) = x inst (τ ) in the photon kernel), and D(x − y) is the scalar photon propagator (e.g. in Feynman gauge).
We now explain this construction step by step. v 1. Two–loop master formula and semiclassical expansion. Recall the two–loop master formula for spinor QED in a background A µ , derived in (215):
with
We are interested in the imaginary part of the total effective action,
twhich yields the non–perturbative pair–production rate. In the semiclassical regime relevant for strong
worldline instantons and their fluctuations. At the one–loop level, we have already seen that the exponent spin is given by S
0 [x inst ]; at two loops, the photon insertion I photon [x, ψ] modifies both the exponent and the prefactor:
where N (1) and N (2) are one– and two–loop prefactors, respectively, and S 0 [T ] is the classical action
spin where h·i
1–loop denotes expectation value with respect to the 1D worldline measure exp(−S wl ) expanded to quadratic order around the instanton.
For the purpose of computing the exponent, one can take ψ inst to be zero (the Grassmann fields enter quadratically in the free part). The one–loop effective action exponent S 0 [x inst ] is obtained by substituting
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vAt leading semiclassical order, it is often sufficient to approximate
and to treat the spin term as a quantum correction:
(238)
(239)
er µ The spin correlator J spin (τ 1 )J spin,µ (τ 2 ) ψ can be expressed in terms of the Grassmann propagator hψ µ (τ 1 )ψ ν (τ 2 )i = 1 µν δ sign τ 1 − τ 2 , 2 (240) for the free theory on the circle with anti–periodic boundary conditions, corrected by interaction terms proportional to F µν . In constant fields, these corrections lead to the known spin–dependent factors in the Euler–Heisenberg effective Lagrangian; in Sauter pulses, they yield γ–dependent modifications of the prefactor.
The important structural point is that the entire two–loop correction ∆S (2) is encoded in the two– dimensional integral
that is fully determined by the instanton path x inst (τ ; γ) constructed in Subsection V C and by the chosen photon propagator.
(243)
as in (225) and (227). The corresponding pair–production rate (imaginary part of the effective Lagrangian) behaves roughly as
where the prefactor N (1+2) (γ) incorporates:
• the one–loop fluctuation determinants (bosonic and fermionic),
• the two–loop corrections from the modified quadratic forms around the instanton caused by the
photon insertion.
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The detailed structure of N (1+2) (γ) can again be extracted from 1D linear ODEs (Gel’fand–Yaglom method) plus the two–dimensional worldline integrals mentioned above.
From the perspective of super–holography, the crucial observation is that, even at two loops in a fast background:
• the 4D diagrammatic problem (infinite series of two–loop diagrams with arbitrary numbers of
background–field insertions) is compressed into:
– a single one–dimensional nonlinear ODE system for the instanton trajectory (already solved in
Subsection V C), – a single two–dimensional integral over worldline proper times (τ 1 , τ 2 ) for the photon kernel, – a handful of one–dimensional linear ODEs for fluctuation determinants.
v• all dependence on the physical parameters (E 0 , ω, m, e) is captured by the Keldysh parameter γ
and the worldline integrals defined above.
Thus, the worldline encoding at two loops preserves the dimensional simplification and structural compactness characteristic of one–loop super–holography, while accommodating genuine higher–loop corrections.
Algorithmic summary and outlook. For practical computations of two–loop corrections to pair production in fast fields, the worldline semiclassical approach can be summarised as the following algorithm:
Construct the one–loop instanton: For each value of γ (i.e. for each (E 0 , ω)), solve the first–
order system (197) with periodic boundary conditions to obtain the instanton trajectory x inst (τ ).
the one–loop fluctuation determinants for bosonic and fermionic modes using 1D linear ODE techniques. spin 3. Evaluate the photon functional: Compute I
photon [x inst ] via the two–dimensional integral (228), numerically if necessary. This requires:
• evaluating the worldline current correlator J µ (τ 1 )J µ (τ 2 ) ψ, x
inst (which may be approximated by its leading contribution from ẋ µ inst plus known spin corrections), • evaluating the photon propagator D x
inst (τ 1 ) − x inst (τ 2 ) for the given background and gauge.
tn 4. Assemble the two–loop exponent and prefactor: Combine S 0 [x inst ] and ∆S (2) [E 0 , ω] to ob-
From the categorical perspective, this procedure shows that the encoding functor H (2) and context functors F (2) , G (2) defined in Subsection VI A give rise to a higher–order super–holographic equivalence
with B the 4D spinor QED theory in a fast Sauter background and H (2) (B) the enriched worldline theory with photon kernel. Thus, even at two loops and beyond, the worldline formalism realises a powerful and explicit instance of string–free higher–categorical super–holography in a realistic, strongly driven quantum field theory.
We now collect and systematise the ingredients developed in Subsections V A–VI B into a concrete, implementable algorithm for computing one– and two–loop non–perturbative pair–production rates in spinor QED for strong, time–dependent background fields, with particular emphasis on rapidly varying fields such as the Sauter pulse (153). At the same time, we make explicit how this algorithm realises full–fledged, string–free, higher–loop super–holography: a four–dimensional, combinatorially explosive diagrammatic problem is compressed into an effectively one–dimensional, spin–covariant worldline computation that can be implemented numerically.
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ed • a time–dependent electric field profile E(t); we focus on E(t) = E 0 sech 2 (ωt), (246)
with vector potential A 3 (t) = −(E 0 /ω) tanh(ωt) and all other components zero. This is the Sauter pulse, but the algorithm can be adapted to other profiles.
• the mass m and charge e of the electron, and the two scales E 0 (peak field) and ω (frequency),
which combine into the Keldysh parameter
(247)
vThe goal is to compute, for given values of (E 0 , ω, m, e) or equivalently (E 0 , γ), the one– and two–loop contributions to the imaginary part of the effective Lagrangian,
(248)
which determines the non–perturbative pair–production rate in the given background.
with periodicity conditions
(249)
(250)
for some winding number n ∈ Z + . Here:
• u(s) is the dimensionless Euclidean time coordinate, u = ωx 4 (τ ), ot • s is a dimensionless proper–time parameter (essentially s ∼ (eE 0 /m)τ ), • θ(s) is an angle parametrising the velocity via
tn
Comment on novelty and methodology. The reduction of the full four–dimensional pair–production problem in a time–dependent background field to the one–dimensional nonlinear system (249) is an exact consequence of the worldline representation of the one–loop determinant, combined with the semiclassical (instanton) approximation. No diagrammatic resummation or hidden assumption enters this step. In particular, for generic rapidly varying backgrounds such as the Sauter pulse, there exists no closed– form or systematic Feynman–diagram expansion capable of reproducing the non–perturbative exponent and prefactor. The worldline formalism therefore provides genuinely new computational capability by concentrating the entire non–perturbative structure into the solution of a coupled one–dimensional boundary–value problem and its Gaussian fluctuations, rather than an infinite tower of multi–photon diagrams.
Numerical approach: shooting method. To find periodic solutions to (249), one uses a shooting method:
n.
u(s) to explore the complex plane as needed (especially for γ & 1).
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 60 4. Adjust (u 0 , S) iteratively (e.g. via Newton–Raphson) until ∆u and ∆θ are sufficiently small. ed The output of this step is the instanton trajectory x inst (τ ) = x 3,inst (τ ), x 4,inst (τ ) , (253) or equivalently (u(s), θ(s)) parametrised over s ∈ [0, S] for the chosen γ and n. For practical purposes, one–loop classical action is given by
Step 1.
The function g(γ) thus obtained reproduces known limits:
• g(γ) → π as γ → 0, recovering the constant–field Schwinger exponent,
• g(γ) ∼ c γ as γ → ∞, characteristic of the multiphoton regime. n This step yields the leading one–loop exponent S 0 for the non–perturbative rate.
spin 4. Step 3: compute the photon functional I
photon . The two–loop correction to the exponent is controlled by the worldline photon functional evaluated on the instanton, as in (227)–(228). We now outline how to compute
The algorithmic steps are:
with ∆τ = T /N . On this grid, the instanton trajectory x inst (τ ) is represented by an array of points
by:
where the first term uses numerically computed finite–difference approximations to ẋ µ inst , and the second term can be obtained from the free Grassmann correlator plus corrections determined by F µν along the instanton. For a first implementation, one may neglect the spin–spin correlator or treat it approximately, then include it systematically.
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61
using, for instance, the coordinate–space representation of the scalar propagator in Euclidean space:
with x 2 = x µ x µ and appropriate regularisation for short separations (k = `). In Minkowski space or with different gauge choices, one can adjust the expression for D accordingly.
and D k , approxi-mate
(e.g. exchanging τ 1 and τ 2 ) to reduce computational cost. spin Once I
photon [x inst ] has been computed for the given γ, the two–loop correction to the exponent is given by
and obtains a second–order differential operator K B acting on η(τ ):
(264) t n
where V B (τ ) depends on the background field and the instanton trajectory. At two loops, the bilocal photon kernel modifies V B and introduces nonlocal corrections that can be approximated by an effective local potential in the semiclassical limit.
• For the fermionic fluctuations, one writes
and obtains a first–order differential operator K F acting on χ(τ ):
(265)
(266)
where Σ µν is a representation of the spin generators. Again, the two–loop corrections from the photon kernel can be incorporated in an effective way.
The Gel’fand–Yaglom method provides a direct procedure to compute ratios of determinants of such operators [62]: one solves the homogeneous ODE Ky(τ ) = 0 with specified initial conditions and reads off the determinant from the value of y(T ), or, in the matrix–valued case, from the determinant of a fundamental solution matrix. The prefactor at two loops is then obtained by combining the relevant determinants of K B and K F (with zero modes treated separately) and multiplying by appropriate symmetry and normalisation factors.
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– The two–loop effective action involves diagrams with one Dirac loop and one photon line, with
arbitrary numbers of external legs corresponding to the background field A µ (t) (typically a superposition of plane waves for a time–dependent field).
– The number of such diagrams grows rapidly with the number of external insertions; each
diagram involves 4D momentum integrals, spinor traces, and nontrivial time–dependence.
v– The non–perturbative pair–production rate depends on an infinite resummation of these
diagrams, making direct computation essentially intractable for general E(t), especially when the field is strong and rapidly varying.
– All two–loop diagrams in the Dirac–loop–plus–photon sector are encoded in the single master
formula (215), which is a 1D path integral with a bilocal kernel.
– The leading non–perturbative behaviour is captured by a finite set of worldline instanton
trajectories (solutions of a 1D ODE system) and their fluctuations.
– The two–loop correction is given by a two–dimensional integral over (τ 1 , τ 2 ) and a handful of
1D ODE solves (for instantons and fluctuations), all of which are straightforward to implement numerically and robust to strong and fast time variation.
– The algorithm is manifestly spin–covariant: spin effects are fully encoded via the Grassmann
sector and yield the alternating signs and spin–dependent prefactors in a natural way.
From the viewpoint of the string–free holographic principle, this is an exemplary case where:
• the encoding category Enc wl (worldline super–QFTs) is structurally matched to the problem (loop
amplitudes and spin),
• the encoding functor H (2) compresses the bulk theory into a 0+1D model that retains all information
relevant to the context F pair ,
tn • the context functors F
(2) and G (2) extract the same two–loop non–perturbative observables from bulk and encoding, respectively,
• and the resulting equivalence F (2) (B) ≃ G (2) (H (2) (B)) is not merely conceptual but algorithmic
and practically useful.
• It achieves a dimensional reduction from 4D to 1D (with a 2D integral at two loops) while preserving
all spin and gauge structures relevant for pair production.
• It restructures an infinite, and in practice unmanageable, set of two–loop Feynman diagrams into a
compact functional expression that can be attacked with ODE and numerical integration methods.
• It integrates naturally into the higher–categorical framework developed in earlier sections, with
appropriate choices of Bulk, Enc, H, F , and G.
This constitutes strong evidence that non–string encodings — and in particular worldline encodings — are not only conceptually appealing but also yield real computational advantages in realistic quantum field theories, reinforcing the central thesis of this work.
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VII.
We now turn to our second major class of examples, in which holography takes the form of anomaly inflow between a four–dimensional gauge theory and a five–dimensional topological theory, rather than a dimensional reduction as in the worldline examples of Sections V–VI. In this subsection we review in six–dimensional anomaly polynomial
vof the following subsections.
where:
• G aµν is the field strength of the SU(N c ) gauge field A aµ T a ,
t
In the limit where all quarks are massless, the classical action is invariant under a global chiral symmetry group
acting as
where U L ∈ SU(N f ) L and U R ∈ SU(N f ) R act on the flavor indices, and U(1) B is the baryon number symmetry (with quarks carrying baryon number 1/N c so that baryons have baryon number 1). We will not focus on the U(1) A axial symmetry, which is broken by the QCD anomaly; our interest here is in the non–anomalous vector baryon symmetry and the potentially anomalous chiral flavor symmetries.
At the quantum level, the chiral symmetries SU(N f ) L and SU(N f ) R are anomalous when coupled to background gauge fields for these symmetries. This means that the associated currents are not conserved in the presence of background fields; the divergence of these currents is proportional to gauge–invariant polynomials in the background field strengths. This is the chiral anomaly, which can be expressed compactly through the anomaly polynomial I 6 in six dimensions.
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where A L,R are local 4–forms constructed from the background field strengths F L,R (and possibly from G). In the limit where we turn off the dynamical gluons (A aµ = 0) and focus on the pure flavor anomaly, the anomaly takes the well–known form [67, 69]
where T a are generators of su(N f ), F L,µν and F R,µν are the corresponding flavor field strengths, and N c is the number of colors.
Equation (278) can be expressed more compactly in differential–form notation. Let F L and F R be the
tn Then the divergence of the left and right flavor currents is proportional to the 4–form
with the anomaly proportional to N c (ω 4 (L) − ω 4 (R)).
(281)
where
and the trace is taken in the fundamental representation of SU(N f ). The normalization 1/(24π 2 ) is conventional and matches the anomaly in (278) when one performs the descent.
To see how (281) arises, recall that the anomaly polynomial for a chiral fermion in representation R of a gauge group G in 2n dimensions is given by the (2n + 2)–form part of the Chern character ch R (F ):
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(R) = N
c · 1
Tr F R 3 .
24π 2 iew
where Â(T ) is the –genus built from the curvature of the tangent bundle, and F is the field strength of the background gauge field. In four dimensions (n = 2), the pure gauge anomaly arises from the 6–form component of ch R (F ), which, for a Weyl fermion in the fundamental representation of SU(N f ), is
In QCD, each left–handed quark flavor q L,f transforms as a Weyl fermion in the fundamental representation of SU(N f ) L , with N c copies (one for each color). Thus, the anomaly polynomial for the left chiral symmetry is and for the right symmetry it is (285)
vI 6 (284)
The total anomaly polynomial, taking into account the opposite chirality of the right–handed fermions, is
An important feature of (281) is the explicit factor of N c : the anomaly is proportional to the number of colors. This reflects the fact that each color contributes identically to the chiral anomaly and that the anomaly is not suppressed in the large–N c limit; instead, it scales linearly with N c .
er dI 6 = 0, (286) and is exact on a (2n + 1)–dimensional manifold M 2n+1 when extended to a (2n + 2)–dimensional manifold B 2n+2 with ∂B 2n+2 = M 2n+1 . In our case, we are interested in extending the 4D spacetime M 4 to a 5D manifold B 5 and working one dimension higher to construct WZW actions and anomaly inflow terms.
The descent equations proceed as follows [68]. One finds a 5–form I 5 (the Chern–Simons form) such that
Under a gauge variation of the background fields,
(1)
none finds
(288)
(289)
where I 4 (α) is a 4–form valued linearly in the gauge parameter α. Integrating over B 5 and using Stokes’ theorem, we find that the variation of the 5D Chern–Simons action reduces to a boundary term:
This boundary term is precisely the anomaly in the 4D effective action. In our QCD setting, splitting I 6 according to F L and F R , one obtains I 5,L and I 5,R , and the 4D anomaly is
(1) consistent with (278). The detailed expressions for I
5,L and I 4 involve standard Chern–Simons and transgression forms; they can be found in classic references [68, 69].
understood in several equivalent ways:
The anomaly polynomial I 6 can be • As the obstruction to gauging the chiral flavor symmetry: a nonzero I 6 means that the partition
function of QCD cannot be made invariant under local chiral transformations without introducing additional degrees of freedom or higher–dimensional terms.
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66
• As an element of a generalized cohomology group: in modern terms, I 6 defines a class in the
appropriate group (e.g. in the cobordism group of (4 + 1)–dimensional invertible phases [70, 71]), which classifies the anomaly.
for an invertible 5D TQFT: there exists a 5D topological theory • As a boundary inflow condition
R whose bulk action is ∼ I
5 and whose boundary variation cancels the 4D anomaly, yielding an anomaly–free combined system.
The last interpretation is the one we will exploit for anomaly holography: the 4D QCD theory with chiral symmetry and anomaly polynomial I 6 is not gauge invariant by itself, but it can be realised as the boundary of a 5D invertible TQFT. This fits precisely into the string–free holographic framework, with:
• bulk category Bulk containing QCD,
v• encoding category Enc containing 5D TQFTs and WZW theories,
• encoding functor H mapping QCD to its anomaly inflow TQFT,
• context functors F, G extracting anomaly data and WZW terms,
• and a natural equivalence F (B) ≃ G(H(B)) identifying the 4D anomaly with the 5D inflow.
B.
In the next subsection we will make this correspondence explicit by constructing the Wess–Zumino– Witten term for the chiral Lagrangian, relating it to the anomaly polynomial I 6 , and showing how the requirement of anomaly holography fixes the WZW level k to be equal to the number of colors N c .
WZW term and IR ambiguity
We now move from the ultraviolet (UV) description of QCD in terms of quarks and gluons, with its chiral anomaly polynomial I 6 (281), to the infrared (IR) regime where the relevant degrees of freedom are the pseudo–Goldstone bosons of spontaneous chiral symmetry breaking. Our aim in this subsection is to introduce the chiral Lagrangian with its Wess–Zumino–Witten (WZW) term, to explain in detail how the WZW term is constructed in terms of a five–dimensional integral, and to emphasise the IR ambiguity in its normalisation: from the standpoint of the low–energy effective field theory (EFT) alone, the WZW level k appears as an integer parameter that is not fixed by IR data. In the next subsection, anomaly holography will resolve this ambiguity and fix k = N c by matching to the QCD anomaly polynomial I 6 .
t1. Spontaneous chiral symmetry breaking and the chiral Lagrangian. As reviewed in Subsection VII A, QCD with N f massless quark flavours has a classical global chiral symmetry
is spontaneously broken according to
where SU(N f ) V is the diagonal vector subgroup. The associated Goldstone bosons are parametrised by
which transforms under chiral transformations as
meson fields, T a are generators of su(N f ), and f π is the pion decay constant.
The leading–order IR dynamics of the Goldstone bosons is described by the chiral Lagrangian [72]:
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where D µ is a covariant derivative that can include couplings to external gauge fields (such as the electromagnetic field, or the flavor background fields L µ and R µ introduced earlier). The ellipsis denotes higher–order terms in the derivative and quark–mass expansion, classified by the low–energy constants of chiral perturbation theory.
The action S χ [U ] is invariant under the chiral symmetry (295), at least in the absence of anomalies. However, to include the correct anomalous processes (such as π 0 → 2γ, and more generally processes involving an odd number of Goldstone fields), one must augment the chiral action with an additional term — the Wess–Zumino–Witten term.
v2. Wess–Zumino–Witten term as a five–dimensional integral. The original Wess–Zumino term [69] was introduced as a functional Γ[U ] whose variation under a global chiral transformation reproduces the anomaly of the underlying microscopic theory. Witten [11] later showed that this term can be written in terms of a five–dimensional integral, illuminating its topological nature and its connection to higher–dimensional TQFT.
Let M 4 be the four–dimensional spacetime manifold, and let U : M 4 → SU(N f ) be the Goldstone field. Choose a five–dimensional manifold B 5 with boundary
where
is a closed 5–form on B 5 . In local coordinates x A on B 5 (A = 1, . . . , 5) and with ABCDE the Levi–Civita symbol in five dimensions, (298) can be written as
ot A number of important remarks are in order:
tn • The functional Γ 0 [U ] is not manifestly local in four dimensions, since it involves an integral over
e B 5 . However, its variation under variations of U is local and depends only on the restriction of U
e : B 5 → SU(N f ). Different choices of • The definition of Γ 0 [U ] depends on the choice of extension U
e that agree on the boundary may differ by an integer multiple of 2π, as we now explain. B 5 and U
b : X 5 → SU(N f ) built from U e 1 and U e 2 . The difference between the two WZW actions is and a map U
b : X 5 → SU(N f ), Now X 5 is closed, and the integral is proportional to the winding number of the map U which lies in the homotopy group π 5 (SU(N f )):
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 68 This implies that
Zed b −1 d U b 5 = 2πi · 240π 2 · n, Tr U (304) X 5 for some integer n ∈ Z. Therefore (1) (2) Γ 0 [U ] − Γ 0 [U ] = 2πi n.
v with k ∈ Z. This quantisation condition is analogous to the quantisation of Chern–Simons levels in three–dimensional gauge theories. From the perspective of the IR chiral EFT, the integer k is a priori undetermined: different integer values of k correspond to different effective theories that are all consistent with the symmetries and topology of the target manifold SU(N f ). The chiral Lagrangian
is thus ambiguous up to the choice of k ∈ Z, with each choice defining an IR EFT that is classically invariant under SU(N f ) L × SU(N f ) R and has a well–defined path integral. Nothing in the IR effective theory alone seems to single out a particular value of k.
The basic idea is to promote the global chiral symmetry (295) to a local symmetry by introducing background connections L µ and R µ and to modify Γ 0 [U ] by adding 4D and 5D terms so that the variation of the total WZW action under local chiral transformations reproduces the microscopic anomaly. One way to construct Γ[U, L, R] is as follows:
e , e 1. Extend U , L, and R from M 4 to B 5 , choosing appropriate extensions U L, e R. 2. Construct a 5D Chern–Simons action S 5D [L] whose exterior derivative gives the Tr F L 3 part of the
anomaly polynomial, and similarly for S 5D [R].
n3. Combine these with Γ 0 [U ] and additional 4D terms to form a gauge–invariant functional modulo
2πi.
Schematically, the gauged WZW functional can be written as
rin where ω 4 (U, L, R) is a 4–form constructed from U, L, R and their derivatives such that the total variation under local chiral transformations is Z δ α Γ[U, L, R] = 2πi Tr α L A L (L) − α R A R (R) , (310) M 4
with A L,R given by the anomaly expressions in (278). The explicit form of ω 4 (U, L, R) is somewhat lengthy and can be found in [11]; importantly, it involves terms of the form
and various combinations designed to reproduce the cohomological structure encoded in the anomaly
From the standpoint of the IR chiral EFT, the gauged chiral Lagrangian is then
where S χ [U ; L, R] is the gauged version of (296). Again, the integer k remains undetermined by IR considerations alone.
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 69 5. IR ambiguity and the need for anomaly holography. We can now summarise the IR situation: ed • The Goldstone bosons of spontaneous chiral symmetry breaking are described by a chiral field U (x) ∈ SU(N f ) and a chiral Lagrangian S χ [U ; L, R] invariant under SU(N f ) L × SU(N f ) R . • To correctly reproduce the chiral anomaly and anomalous processes, one must add a WZW term k Γ[U, L, R], with Γ given by (309). iew • Topological consistency (independence of the choice of 5D extension) requires k ∈ Z, but does not fix its value. • Thus, from the IR EFT alone, the effective action S χ,eff [U ; L, R] = S χ [U ; L, R] + k Γ[U, L, R] is ambiguous up to the choice of an integer k. (313)
v Physically, different values of k lead to different coefficients for anomalous processes (e.g. π
they affect the quantisation of Skyrmion baryon number and the spin/statistics of baryons in the Skyrme model. The microscopic QCD theory, however, has a definite anomaly polynomial I 6 with coefficient N c , and we expect that the correct low–energy effective theory must reproduce this anomaly exactly. This mismatch between the IR ambiguity and the UV anomaly data calls for a mechanism to relate them.
In the framework of the string–free holographic principle, this relation will take the form of anomaly holography: we will construct a 5D invertible TQFT whose anomaly polynomial reproduces I 6 , and we will show that demanding a relative holography relation F (B) ≃ G(H(B)) (where B is QCD and H(B) is the anomaly TQFT plus WZW data) forces k to equal N c . This is the subject of the next subsection.
In summary, the WZW term is a topological functional of the chiral field U (x), naturally expressed as a five–dimensional integral, whose presence is required to reproduce the chiral anomalies of QCD in the IR effective theory. Its normalisation, encoded in the integer level k, is quantised but not fixed by IR EFT considerations. This integer ambiguity is precisely what anomaly holography will resolve by matching to the UV anomaly polynomial.
n
We now reinterpret the chiral anomaly of QCD and the Wess–Zumino–Witten term of the chiral Lagrangian in the language of the string–free holographic principle developed in Sections III and IV. The key idea is that the four–dimensional theory (UV QCD or its IR chiral EFT) is not fully gauge invariant by itself when the chiral flavor symmetry is weakly gauged, but can be realised as the boundary of a five–dimensional invertible topological field theory whose bulk action implements anomaly inflow. This extended system is gauge invariant, and the anomaly of the boundary theory is precisely the variation of the bulk action. We will show that this structure can be expressed as a higher–categorical holographic equivalence
and that requiring (314) to hold forces the WZW level k introduced in (308) to equal the number of colors N c .
(315)
where F L and F R are the SU(N f ) flavor field strengths for the left and right chiral symmetries. The anomaly polynomial I 6 can be viewed as the curvature of a Chern–Simons action in five dimensions:
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metric) and describes an invertible TQFT in five dimensions: its partition function is a pure phase, and its state spaces are one–dimensional (or invertible objects in an appropriate target category) [70, 71]. It is natural to regard this 5D TQFT as an object in the encoding category
vthe (∞, 1)–category of anomaly TQFTs with boundary data suitable to cancel anomalies of 4D theories.
The action (318) is not gauge invariant by itself on a manifold with boundary; its variation under chiral gauge transformations produces a boundary term that matches the chiral anomaly of the 4D theory. This is the standard anomaly inflow mechanism [27].
(322)
twhere we have used Stokes’ theorem and ∂M 5 = M 4 .
Comparing (322) with the variation of the 4D QCD effective action Γ QCD [L, R, A] (integrating out quarks and gluons), one finds
in appropriate normalisations. That is, the combined system—4D QCD on M 4 plus the 5D TQFT on M 5 with ∂M 5 = M 4 —is gauge invariant. The 4D chiral anomaly is precisely cancelled by the 5D inflow.
From the viewpoint of the string–free holographic principle, this situation corresponds to a bulk QCD object
and an encoding object with action S 5D [L, R] as in (318); the context functors F anomaly and G inflow extract, respectively, the chiral anomaly functional from the 4D theory and the boundary variation from the 5D TQFT.
• Bulk category Bulk QCD : Objects include 4D QCD with N c colors and N f massless quarks, coupled
to background chiral gauge fields (L, R) for SU(N f ) L × SU(N f ) R .
• Encoding category Enc anom : Objects include 5D invertible TQFTs with actions such as (318),
possibly plus boundary WZW degrees of freedom.
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71
• Encoding functor H:
H : Bulk QCD −→ Enc anom , (326)
• Bulk context functor F anomaly :
F anomaly : Bulk QCD −→ Obs anomaly , iew defined by mapping the QCD theory B QCD and background (L, R) to the 5D anomaly TQFT E 5D with action S 5D [L, R] together with boundary WZW data. Morally, H “forgets” the dynamical gluons and quarks and retains only their anomaly structure, encoded in the 5D Chern–Simons form.
(327)
which assigns to B QCD its anomaly functional, equivalently the 6D anomaly polynomial I 6 (315) or a 4D inflow functional δΓ QCD .
v • Encoding context functor G: G : Enc anom −→ Obs anomaly , (328)
which assigns to the 5D TQFT E 5D the boundary anomaly functional given by δS 5D [L, R] in (322). This is equivalent data to the anomaly polynomial I 6 , up to coboundaries.
The string–free anomaly holography condition is then the existence of a natural equivalence
expressing the equality of the QCD anomaly and the anomaly inflow from the 5D TQFT. This is the UV side of the anomaly holography.
tn
where Γ[U, L, R] is the gauged WZW functional (309), and k ∈ Z is the WZW level. As explained in Subsection VII B, from an IR perspective alone, k is an integer that labels different IR EFTs consistent with the global and topological structure of the Goldstone manifold; there is no obvious reason to single out a particular value of k.
If we integrate out the Goldstone field U in the presence of background fields L, R, we obtain an effective action Γ IR [L, R]. Its variation under local chiral transformations is given by the variation of the gauged WZW term:
Since S χ is anomaly–free (it is built from covariant derivatives and gauge–invariant traces), its variation vanishes. The only contribution comes from the WZW term:
By construction [11, 69], the variation of Γ[U, L, R] under a local chiral transformation reproduces the anomaly associated with a single Weyl fermion in the fundamental representation of SU(N f ) (or, more precisely, the anomaly polynomial Tr F 3 up to a normalisation). For N f flavors and N c colors, the IR anomaly polynomial generated by the WZW term is
(333)
i.e. the same structure as (315) but with k instead of N c . Therefore, if we define the IR bulk object B IR as the chiral EFT plus WZW term, the bulk context functor satisfies
(334)
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vand gauged Wess–Zumino–Witten functionals used in this section follow the classic constructions of Wess–Zumino, Witten, Bardeen, and Alvarez–Gaumé without modification. Our role here is not to propose a new derivation of the chiral anomaly, but to situate the established anomaly–inflow mechanism inside the string–free holographic pattern of Section III. The resulting categorical diagram makes the equality k = N c manifest as a holographic compatibility condition between UV QCD, its IR chiral EFT, and the 5D anomaly TQFT.
where:
• B QCD is the UV QCD theory, (337)
• B IR is the IR chiral EFT with WZW level k,
• E 5D is the 5D anomaly TQFT with action S 5D [L, R], • F anomaly extracts the anomaly polynomial or functional from a 4D theory,
tn • G extracts the boundary variation of the 5D TQFT.
Equation (337) is an instance of the general holographic pattern (41), making explicit that:
• the anomaly data of UV QCD and its IR EFT are equal (anomaly matching),
• both are equal to the data provided by the 5D TQFT (anomaly inflow), • and the IR ambiguity in k is resolved by requiring that B IR embed into the same holographic
diagram as B QCD .
Thus, anomaly inflow holography provides a direct categorical mechanism to fix IR parameters, such as the WZW level, by UV anomaly data encoded in higher–dimensional TQFTs.
5D , φ) with φ : G(E 5D ) −−→ F anomaly (B QCD ) is in fact universal among all possible anomaly encodings. Any other (E 0 , φ 0 ) with the same anomaly data admits a unique morphism E 5D → E 0 in Enc anom compatible with φ, φ 0 . This means the 5D anomaly TQFT is the canonical holographic encoding for chiral anomaly data in QCD, and the requirement that the IR EFT fit into the same encoding forces its WZW level to match the UV anomaly coefficient.
In the next subsection we will discuss how this anomaly holography extends to the Skyrme model and the interpretation of baryons as Skyrmions, thereby linking the purely topological anomaly data to the quantisation of baryon number and spin.
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D.
We now highlight the concrete capability provided by anomaly inflow holography in QCD. The central point is that the higher–categorical anomaly equivalence
view discussed in Subsection VII C, does far more than provide a conceptual reinterpretation of the chiral anomaly. It fixes IR data — such as the WZW level, the baryon number of Skyrmions, and the normalisation of meson couplings to external fields — without requiring any nonperturbative solution of QCD in terms of quarks and gluons. That is, the higher–categorical structure directly constrains the IR effective field theory because the anomaly data sits in a 5D encoding category where it is easy to access and compare. In this subsection we explain in detail how:
• the WZW level k fixed by anomaly holography as k = N c determines the baryon number and
spin/statistics of Skyrmion configurations in the chiral EFT, and
• the same WZW structure determines the normalisation of couplings of Goldstone bosons (e.g. pions)
to external gauge fields, such as the π 0 → 2γ amplitude,
all without solving QCD. We also emphasise that none of this requires a string dual; the entire encoding is categorical, based on extended TQFT and cohomology.
so finite–energy field configurations U (x) at a fixed time slice t can be classified by an integer winding number. To define this, one imposes the boundary condition
so that the spatial slice R 3 can be compactified to S 3 , and U defines a map
tThe winding number (degree of this map) is given by
(341)
(342)
where i, j, k run over the spatial indices. This integer B top ∈ Z is the topological baryon number of the Skyrmion. It is natural to identify B top with the physical baryon number, but strictly speaking this
(343)
ep with baryon number Z B top [U ] = d 3 x B 0 (x). (344) Note that (343) is conserved identically (its divergence vanishes identically due to antisymmetry), reflecting its topological origin rather than a Noether symmetry.
However, as Witten showed [11], the presence of the WZW term with level k modifies both the quantisation and the identification of baryon number. The WZW term contributes an additional piece to the canonical momentum conjugate to the chiral field, and upon quantisation it leads to a shift in the baryon current and to a relation between the winding number and the baryon charge as seen by the microscopic theory. More precisely:
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 74 • The WZW term with level k introduces a coupling of the topological current (343) to the baryon
number symmetry U(1) B when one introduces a background gauge field B µ for baryon number. The gauged WZW term includes a piece of the form
with coefficient k.
• As a consequence, a Skyrmion with topological charge B top acts as a source of baryon number equal
to
vIn particular, for a unit–winding Skyrmion (B top = 1) the physical baryon number is B phys = k.
If we now impose anomaly matching and anomaly inflow holography, as in Subsection VII C, we have seen that k must equal N c . Therefore,
that it is composed of N c quarks, the identification of a single Skyrmion with B top = 1 and B phys = 1 requires that the normalization of B top and the definition of U(1) B be chosen consistently. In practice, this leads to a convention in which one rescales the topological current or the baryon gauge field so that a unit–winding Skyrmion corresponds to baryon charge one. The important point is that the factor of N c from the WZW level ensures that the Skyrmion carries the correct statistical and topological structure to represent a physical baryon.
Moreover, Witten showed [11] that the WZW term also controls the spin–statistics of Skyrmions: for odd N c , the Skyrmion behaves as a fermion, while for even N c it behaves as a boson. This can be seen by examining the transformation of the Skyrmion wavefunctional under a 2π rotation and noting that the WZW term induces a phase proportional to k = N c . Thus, anomaly holography not only fixes the normalisation of baryon number but also the fermionic nature of baryons in the Skyrme model, again without solving QCD.
tn 2. Meson couplings to external fields from the WZW term. Another important class of IR data fixed by the WZW term is the couplings of pseudo–Goldstone bosons to external gauge fields, such as the π 0 → 2γ decay amplitude. The gauged WZW functional Γ[U, L, R] contains terms linear in the Goldstone fields and quadratic (or higher order) in the external gauge fields. These terms encode anomalous processes (those that would be forbidden in a classically conserved current but are allowed by the anomaly).
For instance, in the two–flavor chiral EFT with electromagnetic coupling, one introduces an electromagnetic background A em µ and identifies it with a combination of L µ and R µ :
where Q is the electric charge matrix in flavor space (e.g. Q = diag(2/3, −1/3) for u and d quarks). Expanding the gauged WZW action to leading order in the pion fields and in A em µ , one finds a term of the form
k and the trace over flavor charges. The amplitude for π 0 → 2γ decay is then proportional to C π 0 γγ :
with the precise normalisation obtained from the detailed WZW expansion [11, 69].
By matching to the anomaly polynomial I 6 (which fixes the coefficient of Tr F L 3 − Tr F R 3 ) and by requiring k = N c , one obtains an exact prediction for the π 0 → 2γ amplitude in terms of N c and f π , which agrees with experiment for N c = 3 when phenomenological values of f π are used. This is a remarkable example of UV anomaly data fixing an IR observable.
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Similarly, other anomalous couplings of mesons to external gauge fields (e.g. η → 2γ, ω → 3π) are encoded in the WZW term and their normalisations are fixed by the same anomaly structure. From the categorical viewpoint, these couplings are part of the observable functor G(E 5D ), which extracts boundary data from the 5D anomaly TQFT and passes it to the IR EFT; the fact that they are uniquely determined by I 6 means that they are fully controlled by the holographic relation (338).
• formulated in terms of anomaly polynomials and extended TQFTs (encoding category),
• expressed as functorial relations between 4D QFTs and 5D TQFTs (encoding functor H),
• and interpreted as a categorical equivalence of anomaly data (context functors F anomaly and G).
v The anomaly polynomial I 6 lives naturally in a generalized cohomology theory (e.g. cobordism, K– theory), and the 5D anomaly inflow theory is an object in an (∞, 1)–category of invertible TQFTs. The IR WZW term is realised as boundary data in this TQFT; its level k is identified with the bulk coefficient N c via the holographic equivalence. All of this fits seamlessly into the string–free, higher–categorical framework introduced in Sections II and III.
In particular, anomaly holography demonstrates that:
and anomalous meson couplings are fixed by UV anomaly data encoded in a 5D TQFT.
This fixing is achieved without solving QCD nonperturbatively in terms of quarks and gluons.
The relevant “holographic dual” is not a string theory in AdS, but a purely topological 5D TQFT
plus a WZW boundary functional, both living in an encoding category tailored to anomalies and cohomology.
• The chiral anomaly polynomial I 6 is encoded in a 5D invertible TQFT with action S 5D [L, R].
tn • The WZW term Γ[U, L, R] in the IR EFT is the boundary realisation of this TQFT, and its level k
is quantised but, from the IR perspective alone, ambiguous.
• Requiring the categorical equivalence F anomaly (B QCD ) ≃ G(H(B QCD )), together with anomaly
matching to the IR EFT, uniquely fixes k = N c .
• This, in turn, determines the baryon number and spin/statistics of Skyrmions and the anomalous
couplings of mesons to external fields.
All of these IR data are thus fixed without solving QCD, and without invoking any string dual, by the purely categorical structure of anomaly inflow and the higher–dimensional encoding of anomaly polynomials. This is precisely the kind of capability that the string–free holographic framework is designed to expose: nontrivial, physically relevant information about strongly coupled theories can be obtained by choosing the right encoding category (5D anomaly TQFTs), the right context (anomaly observables), and the right functorial relations (H, F anomaly , G), without relying on string theory.
We now turn to confinement and flux tubes in SU(N ) Yang–Mills theory, and to the associated notion of a defect 2–category. Our goal in this subsection is to explain, in some detail, how the physics of confining strings in 3 + 1 dimensions can be organised into a higher–categorical structure in which:
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76 • 1–morphisms are line defects such as Wilson lines in various representations, ed • objects (0–morphisms) are vacua or boundary conditions of the 4D gauge theory, • 2–morphisms are surface defects (worldsheets) that interpolate between line defects, interpreted
physically as confining flux tubes stretching between static color charges.
This defect 2–category will provide the natural encoding category Enc for the static potential and flux–tube excitations, and will underlie the 4D → 2D super–holography developed in the following subsections.
v where A µ is an su(N )–valued gauge field and Tr denotes a trace in the fundamental representation. At low energies and zero temperature, this theory is believed to be in a confining phase, characterised by:
• an area law for large Wilson loops,
• a linear potential V (R) ∼ σR between static test quarks in the fundamental representation,
√ • a spectrum of flux–tube excitations with energy gap ∼ σ.
The basic object probing confinement is the Wilson loop (352):
(353)
tn
for a 4D confining string [12].
Physically, the area law and the linear potential can be understood in terms of a flux tube: the chromoelectric flux lines between a static quark and antiquark are squeezed into a string–like tube of finite thickness and constant energy density; its worldsheet sweeps out the interior of the Wilson loop. Quantum fluctuations of this flux tube are then described by an effective (1 + 1)–dimensional worldsheet QFT, which encodes the corrections to the linear potential, including the Lüscher term.
From a modern categorical perspective, Wilson lines, flux tubes, and their junctions are best organised into a defect 2–category associated to the 4D theory [26, 73]. We next describe this structure.
• Objects (0–morphisms): topological sectors of the theory or, more concretely, vacua and
boundary conditions of T on three–dimensional spatial manifolds. For pure YM in the confining phase (and neglecting θ–angle subtleties), there is a unique confining vacuum, so D(T ) has a single object v conf .
• 1–morphisms: line defects between vacua, such as Wilson lines in various representations, ’t Hooft
lines, or mixed lines. A 1–morphism L : v conf → v conf can thus be labeled by a choice of representation R (for Wilson lines) or by more complicated data for generalized line operators.
• 2–morphisms: surface defects interpolating between line defects. Given two line operators L 1 , L 2 (same endpoints), a 2–morphism S : L 1 ⇒ L 2 corresponds to a surface defect whose boundary is the union of the worldlines of L 1 and L 2 . Physically, these 2–morphisms represent the worldsheets of flux tubes stretching between static sources or more general domain walls and junctions. In particular, the confining flux tube between a quark and antiquark is a 2–morphism between two Wilson lines.
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 77 • horizontal composition of line defects (concatenating segments of worldlines), ed Composition in D(T ) is defined by:
• vertical composition of surface defects (gluing surfaces along common boundaries),
• and higher coherences encoding associativity and unit laws up to homotopy.
This 2–category is enriched over vector spaces or Hilbert spaces: the spaces of 2–morphisms between bilinear.
vmanifolds, and surface defects are images of embedded 2D manifolds. While the full construction of Z T is difficult for non–topological YM, this heuristic extended–TQFT picture guides the defect categorisation.
• A static quark at spatial position x 1 and an antiquark at x 2 as heavy external sources. Their
worldlines extend along the Euclidean time direction.
• The Wilson lines W fund (x 1 ) and W fund (x 2 ) correspond to 1–morphisms in D(T ), labelled by the
fundamental and anti–fundamental representations of SU(N ).
• A flux–tube worldsheet spanning between these two line defects corresponds to a 2–morphism
in D(T ). In Euclidean signature, its worldsheet is topologically a rectangle of spatial width R = |x 1 − x 2 | and temporal extent T .
Let us denote by L R the Wilson line in representation R along a fixed contour, and by F R,R 0 a flux–tube worldsheet connecting L R and L R 0 (here R 0 = R for a quark–antiquark pair). The set of such worldsheets, modulo equivalences, forms the 2–morphism space
tn In particular, the lowest–energy flux tube between fundamental sources defines a “ground–state 2– morphism” F fund : L fund ⇒ L fund .
The effective dynamics on the flux–tube worldsheet can be described by a (1 + 1)–dimensional QFT (effective string theory), whose field content and interactions encode the transverse fluctuations of the tube, its excitations, and its coupling to the bulk theory [12, 74]. From the defect 2–category perspective, this worldsheet theory is a label attached to the 2–morphism F fund , and its Hilbert space of states on a spatial segment of length R is the space of 2–endomorphisms of that segment.
rin 4. Worldsheet Hilbert spaces and the static potential. Given a flux–tube worldsheet theory, one can consider its Hamiltonian formulation on a cylinder of spatial circumference R. The spectrum consists of energy levels E n (R), corresponding to the ground state and excitations of the string. In the simplest Nambu–Goto approximation for a bosonic string, one finds [12] E
leading to the Lüscher term −π/(12R) in the static potential. More generally, the worldsheet QFT will have some Hamiltonian H ws (R) whose ground–state energy reproduces the static potential,
and whose excited spectrum encodes hybrid mesons and string excitations.
From the point of view of the defect 2–category, the worldsheet Hilbert space H ws (R) associated to a flux tube of length R is the 2–endomorphism space
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and the Hamiltonian is a self–adjoint operator acting on this Hilbert space. The encoding of the 4D static potential into a 2D worldsheet theory can thus be expressed as a functor
where:
• Bulk YM is the subcategory of QFT consisting of 4D SU(N ) Yang–Mills theories and their line
defects,
• Enc ws is a category of 2D worldsheet QFTs and their defect categories,
• H flux maps the 4D theory and its line defects to the 2D worldsheet theory associated to the confining
flux tube.
v5. Defect 2–category as encoding category. We can now see why the defect 2–category is the natural encoding category for flux–tube super–holography:
• The objects of interest in the 4D theory (static potentials, flux–tube spectra) are inherently
defect–dependent observables: they are defined in the presence of Wilson line defects.
• The relevant excitations are extended objects (flux tubes) whose worldvolumes are 2D surfaces and
whose dynamics is governed by a 2D QFT.
• The correct algebraic framework for such extended excitations is an (∞, 2)–category of defects, not
a category of local operators or just a worldline category.
Thus, in the language of Section IV, the encoding category Enc for confining flux tubes should be chosen as a suitable defect 2–category D(T YM ), or a subcategory thereof, and the encoding functor H flux maps bulk SU(N ) Yang–Mills to its defect structure:
where E ws is the effective worldsheet theory of the fundamental flux tube.
defect observables, while the encoding context functor G ws : Enc ws → Obs static assigns to the worldsheet
tn In later subsections we will exploit this structure to recover the Lüscher correction and discuss further IR data encoded in the worldsheet theory.
In summary, the defect 2–category framework provides a natural and mathematically coherent way to organise the confining flux tubes of SU(N ) Yang–Mills. It identifies vacua, Wilson lines, and flux–tube worldsheets as 0–, 1–, and 2–morphisms, respectively, and it provides the correct encoding category for super–holography from 4D bulk Yang–Mills to 2D flux–tube worldsheet QFTs.
Building on the defect 2–category description in Subsection VIII A, we now make explicit the effective worldsheet description of confining flux tubes and explain how it realises a 4D → 2D super–holographic encoding for the static quark potential. The main idea is that, in the IR and at large separation R between a heavy quark and antiquark, the relevant observables are governed by an effective (1 + 1)–dimensional quantum field theory living on the flux–tube worldsheet. The 4D bulk Yang–Mills theory enters only through a small number of IR parameters (string tension, low–energy couplings) and through the structure of the defect sector (line and surface operators).
We will:
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 79 2. describe the effective worldsheet action encoding the transverse fluctuations of the flux tube, ed 3. compute the ground–state energy E 0 (R) in this worldsheet theory and relate it to V (R), 4. and cast this relationship in the higher–categorical language of an encoding functor H flux and context functors F static , G ws .
The presence of these sources is implemented by inserting Wilson lines
loop (363):
C R,T
where C R,T is a rectangle of temporal length T and spatial width R, and V (R) is the static potential. In a confining phase one finds
(364)
as in (353), with σ the string tension and −π/(12R) the Lüscher term in D = 4 [12].
From the bulk viewpoint, this configuration is simply SU(N ) YM with two line defects (heavy quark and antiquark). The area law and the expansion (364) emerge from the sum over all field configurations A µ with the given boundary conditions. Our goal is to show that, in the IR and for large R, the dominant contributions to V (R) can be encoded in a worldsheet QFT describing a flux–tube connecting the sources.
tn 2. Effective worldsheet action: Nambu–Goto and beyond. The effective worldsheet description assumes √ that, at distances large compared to the intrinsic thickness of the flux tube (set by 1/ σ), one can model the tube as a fluctuating string whose worldsheet Σ is embedded in spacetime. Let (τ, σ) be worldsheet coordinates, with τ ∈ [0, T ] parametrising Euclidean time and σ ∈ [0, R] parametrising the spatial separation between the quark and antiquark. In static gauge, one identifies
and introduces (D − 2) transverse fields X i (τ, σ), i = 2, . . . , D − 1, describing fluctuations orthogonal to the (0, 1) plane. For SU(N ) YM in D = 4, there are D − 2 = 2 transverse fields, which we denote by ~ X(τ, σ).
The simplest effective action is the Euclidean Nambu–Goto action,
0
where the induced worldsheet metric is
and we have chosen the Euclidean worldsheet metric δ αβ . In the small–fluctuation regime (large R and low energies), one can expand the square root:
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so that
v(such as the ground–state energy), the leading contributions come from the quadratic part (369).
It is this quadratic part that underlies the universal Lüscher term, which we now derive explicitly as the Casimir energy of massless scalars with Dirichlet boundary conditions.
We impose Dirichlet boundary conditions at the endpoints of the string:
since the endpoints are attached to static quark sources at fixed positions. Expanding in normal modes along the spatial direction:
(374)
In Hamiltonian form, the worldsheet Hamiltonian is
(376)
where N n i = 0, 1, 2, . . . are occupation numbers. The ground–state energy is obtained by setting N n i = 0 for all (n, i):
(377)
The sum in (377) is divergent and must be regularised. Using zeta–function regularisation,
(378)
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81
where µ includes possible renormalisations and normal ordering constants, and the universal −π/(12R) term is entirely due to the Casimir energy of the transverse worldsheet fluctuations. This is the essence of the effective–string derivation of the Lüscher term [12, 74].
• Bulk category Bulk YM : Objects include SU(N ) Yang–Mills theories in 4D with heavy quark–
antiquark external sources at separation R, represented as line defects. Morphisms include changes in R, deformations of the gauge coupling, and defect fusion operations.
• Encoding category Enc ws : Objects are worldsheet QFTs defined on cylinders of circumference R (strings of length R), with a specified IR spectrum of transverse excitations. Morphisms include deformations of the worldsheet action, changes in boundary conditions, and embeddings between different worldsheet theories.
• Encoding functor H flux :
mapping the 4D Yang–Mills theory with a quark–antiquark pair to an effective worldsheet theory E ws describing the confining flux tube between them. In practice, H flux (B YM ) is specified by the string tension σ, the number of transverse fields (D − 2), and, at higher orders, by worldsheet
tn • Bulk context functor F static :
F static : Bulk YM −→ Obs static , (383)
which assigns to the bulk theory its static quark–antiquark potential V (R) at large R:
• Encoding context functor G ws :
which assigns to a worldsheet theory E ws the ground–state energy E 0 (R) of the string for each R, as computed from the worldsheet Hamiltonian:
The effective–string derivation of the static potential, in which V (R) is expressed as the ground–state energy of a worldsheet theory, precisely realises the holographic equivalence
with α B YM an isomorphism in Obs static at large R. In other words, the static potential computed from the 4D Yang–Mills theory coincides with the ground–state energy of the flux–tube worldsheet theory, at least in the IR regime where the effective string description is valid. This is a clear example of 4D → 2D compression holography (super–holography):
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• Bulk: 4D SU(N ) YM with line defects (heavy sources), • Encoding: 2D worldsheet QFT with transverse fields, • Context: static potential and flux–tube spectrum. iew 5. Universality and independence of microscopic details. A particularly striking feature of the worldsheet encoding is its universality. The leading terms in the static potential at large R—the linear term σR and the Lüscher correction −π(D − 2)/(24R)—depend only on: • the string tension σ,
v• the number of transverse massless bosons (D − 2), and are independent of the microscopic details of the 4D theory (such as the gauge coupling g, the number of colors N for sufficiently large N , or the short–distance physics). This is analogous to the way anomaly coefficients in QCD depend only on N c and N f , not on coupling constants.
In categorical terms, this universality can be understood as follows:
• The fiber category Fib F static (F static (B YM )) contains all possible worldsheet encodings (E, φ) that
reproduce the same static potential in the IR. The effective Nambu–Goto description (plus universal corrections) provides an initial object in this fiber: any other encoding that matches the IR potential must factor through it in a suitable sense.
• The low–energy effective string theory is therefore a universal encoding for the static potential
in the sense of UP–holography: it is the minimal worldsheet theory capturing the long–distance confining physics, and its parameters (σ and a small set of couplings) are determined by matching to the bulk.
Thus, the effective worldsheet QFT provides a powerful and efficient encoding of the IR defect sector of confining Yang–Mills, and the Lüscher term can be viewed as the super–holographic image of the 4D gauge dynamics under the worldsheet encoding functor H flux .
tn ot In the next subsection we will use this framework to compute the Lüscher term explicitly from the worldsheet perspective, and to show how the flux–tube super–holography fits into the broader higher–categorical and string–free holographic picture developed earlier in the paper.
We now make explicit how the effective worldsheet QFT encoding developed in Subsection VIII B reproduces the universal 1/R correction to the static potential — the Lüscher term — and how this constitutes a concrete computational capability of the 4D → 2D super–holographic framework. In particular, we will:
• derive the Lüscher term
(388)
from the Casimir energy of free transverse bosons on a 2D strip, • emphasise that this result arises from a free worldsheet QFT in two dimensions, encoding a strongly
coupled four–dimensional Yang–Mills theory in the confining phase, • and interpret the effective string worldsheet as a non–string, defect–categorical encoding in the
sense of Section IV.
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 83 1. Casimir energy of free bosons on a strip and the Lüscher term. In Subsection VIII B we considered the quadratic approximation to the worldsheet action (371) for D–dimensional spacetime:
The full worldsheet Hamiltonian is then
leading to the energy spectrum
(395)
The ground–state energy corresponds to N n i = 0 for all (n, i):
(396)
t
where ζ(s) is the Riemann zeta function analytically continued to s = −1. Hence
(397)
(398)
(399)
where the O(R −3 ) terms arise from higher–order terms in the worldsheet effective action (quantum and higher–derivative corrections). For D = 4 spacetime dimensions, this yields the Lüscher term
Identifying V (R) = E 0 (R) in the IR and large–R limit, this reproduces (388):
We emphasise that this derivation relies only on:
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 84 • the presence of D − 2 massless transverse modes X i on the worldsheet, • Dirichlet boundary conditions at the endpoints, ed • the existence of an effective string description of the flux tube (with a well–defined string tension σ),
and is independent of the microscopic details of the 4D theory. This universality of the Lüscher term has been confirmed by lattice simulations of SU(N ) Yang–Mills and other confining theories [12, 74].
v– SU(N ) Yang–Mills is nonabelian and confining at low energies; the quark–antiquark potential
is a highly nontrivial nonperturbative observable. – Lattice computations of V (R) require extensive numerical simulations and careful extrapola-
tions.
• On the 2D worldsheet side:
– The transverse fluctuations of the flux tube are described, at leading order, by a free massless
boson theory in (1 + 1) dimensions, with (D − 2) scalar fields X i . – The Lüscher term emerges as a Casimir energy of this free–boson theory on a strip of width R
with Dirichlet boundaries.
From the string–free super–holographic viewpoint, this is exactly the kind of compression one expects: a complicated 4D observable is encoded in a much simpler 2D theory whose main input is the string tension and the number of transverse modes. The free–boson worldsheet theory is not a “fundamental string” in the sense of string theory in ten dimensions; it is an effective theory in the defect category of SU(N ) Yang–Mills, describing the dynamics of flux tubes as 2–morphisms between line defects.
• The bulk theory B YM is a 4D confining SU(N ) gauge theory, an object in Bulk YM .
tn • The encoding theory E ws is a 2D worldsheet QFT describing flux–tube fluctuations, an object in
Enc ws , which is a defect 2–category (Subsection VIII A).
• The worldsheet theory is not a fundamental string theory in a higher–dimensional target; it is an
emergent effective description living in the defect category of a 4D QFT.
In particular:
ranges of R (large compared to the flux–tube thickness). It does not purport to describe the entire UV–complete theory.
ep 2. The worldsheet couplings (e.g. σ and higher–order λ n ) are determined by matching to the 4D bulk theory, not by consistency conditions as in fundamental string theory. 3. The defect 2–category D(T YM ) captures all extended objects and their interactions in 4D: vacua, line defects (Wilson lines), and 2–morphisms (flux–tube worldsheets). The worldsheet QFT is an element in this category; it is an encoding of the 4D defect sector, not a fundamental replacement.
Thus, the effective string worldsheet is a non–string encoding in our sense: it is a lower–dimensional QFT that lives in a higher–categorical defect structure and provides a holographic description of certain observables (static potential, flux–tube excitations) of the 4D theory. No appeal to 10D string theory or to AdS backgrounds is required.
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• By identifying the correct encoding category Enc ws — the defect 2–category of worldsheet QFTs √ —
and the correct encoding functor H flux , we reduce the problem of computing V (R) for R 1/ σ to the problem of computing a Casimir energy in a 2D free–boson theory.
• This reduction is robust and universal: the Lüscher term is independent of the UV details of the
4D theory and depends only on the number of transverse directions and boundary conditions.
• The calculation is straightforward and analytic: it uses well–understood tools from 2D QFT (mode
expansions, zeta–function regularisation), rather than 4D nonperturbative techniques.
v In the language of Section IV, the effective string worldsheet theory is a universal encoding E 0 for the context F static : any other IR description that reproduces the static potential in the large–R limit must factor through this encoding. The Lüscher term is therefore a universal IR datum that is determined by the structure of the encoding category and not by the microscopic details of the bulk.
er • A strongly coupled 4D SU(N ) gauge theory in the confining phase is mapped, via the defect 2–category and the encoding functor H flux , to a 2D worldsheet QFT. • The static potential V (R), a nonperturbative observable of the 4D theory, is obtained as the ground–state energy E
0 (R) of this 2D theory.
• The universal 1/R Lüscher correction arises as a Casimir effect in the 2D theory, independent of
microscopic details, and is directly calculable.
• The effective string worldsheet is a non–string, defect–categorical encoding in our string–free
holographic framework, demonstrating that higher–categorical structures and lower–dimensional QFTs can provide powerful and precise insights into confining gauge theories without invoking fundamental strings.
BEYOND GEOMETRY: UP–RG AND DFT AS HOLOGRAPHIC ENCODINGS
IX. t n In the next section, we will turn to the interplay between these defect–based encodings and other holographic structures, and discuss how the various examples (worldline QED, anomaly QCD, flux–tube YM) fit together into a unified string–free higher–categorical holographic picture.
A. UP–RG: categorical renormalization as universal encoding
In this subsection we develop the idea of universal–property renormalization group (UP–RG) within the string–free holographic framework. The central claim is that, for a given bulk theory B and a chosen IR context functor F (encoding which long–distance observables we care about), the renormalization group can be understood as providing a universal effective theory E RG and a functor
with the property that any other effective field theory (EFT) reproducing the same IR observables factors uniquely through E RG in the appropriate fiber category. This is UP–holography applied to the IR regime. We will formalize this notion first in abstract categorical terms, and then illustrate it with concrete examples such as the dimensional reduction from 4D to 3D at finite temperature (EQCD/EQED), and simple scalar models with nontrivial fixed–point structure.
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86
• the category of effective actions at momentum scales p Λ IR ,
• the category of correlation functions of local operators at separations larger than some fixed length
scale L IR , • or, more abstractly, the homotopy category of factorization algebras restricted to large–distance
observables.
We encode this choice as a context functor (403)
vF IR : Bulk −→ Obs IR ,
which assigns to each microscopic theory B its IR data F IR (B): long–distance correlation functions, critical exponents, thermodynamic functions, etc. For concreteness, imagine that F IR (B) consists of:
• n–point functions of a finite set of operators {O i (x)} at separations |x i − x j | ≥ L IR ,
• S–matrix elements at energies E ≤ E IR ,
• or the effective action S eff [Φ] with fields Φ restricted to low–momentum modes.
which assigns to an EFT E its IR observables (computed within that theory). For a fixed B, we then consider the fiber category
twhose objects are pairs (E, φ) where E is a candidate EFT that reproduces the IR data of B via an equivalence φ in Obs IR . Morphisms between (E 1 , φ 1 ) and (E 2 , φ 2 ) are morphisms h : E 1 → E 2 in EFT such that G IR (h) intertwines φ 1 and φ 2 :
(406)
F IR (B)
Definition IX.1 (Universal RG encoding (UP–RG)). We say that B ∈ Bulk admits a universal RG encoding in context F IR if there exists an object (E RG , φ 0 ) ∈ Fib F IR (F IR (B)) which is initial in the (∞, 1)–sense: for any (E, φ) ∈ Fib F IR (F IR (B)), there exists a morphism h : E RG → E in EFT and a contractible space of such morphisms, making the appropriate diagram commute in Obs IR .
ep Intuitively, E RG is the “best possible” IR effective theory for the observables in context F IR : any other EFT reproducing the same IR data is obtained by further coarse–graining or field redefinitions of E RG . The pair (E RG , φ 0 ) is then a universal object, and the RG flow from B to E RG can be regarded as an encoding functor H RG : Bulk −→ EFT, H RG (B) = E RG , (407)
which realises a UP–holography in the IR context:
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of UP–RG is provided by the dimensional reduction of finite–temperature QCD or QED to a three– dimensional effective theory [75]. Consider a four–dimensional gauge theory (QCD or QED) at temperature T , formulated in Euclidean spacetime with compact Euclidean time direction of length β = 1/T :
Bulk and context. We take: iew Fields obey periodic (bosons) or anti–periodic (fermions) boundary conditions in Euclidean time. The Matsubara modes with frequencies ω n = 2πnT (or (2n + 1)πT for fermions) of n 6 = 0 are heavy at scales T and can be integrated out to obtain an effective theory for the zero modes.
• Bulk to include the finite–temperature 4D gauge theory (e.g. SU(N ) Yang–Mills with matter),
v• Obs IR to include static, long–wavelength observables, such as:
– static correlation functions of gauge–invariant operators, – spatial string tensions, – screening masses, – the free energy density at scales T .
The context functor F IR maps the 4D theory to this collection of static IR observables.
EFT and encoding. The effective 3D theory for the static sector of hot QCD is known as EQCD (electrostatic QCD) [75]. It is a 3D SU(N ) gauge theory coupled to an adjoint scalar A 0 (the static component of the temporal gauge field), with an effective action
n
where i, j = 1, 2, 3, m D is the Debye mass, and the ellipsis denotes higher–dimension operators suppressed by powers of 1/T . The parameters g 3 2 (the 3D gauge coupling), m 2 D , λ, etc. are determined as functions of the original 4D couplings and the temperature by matching IR observables.
One can define EFT to include such 3D theories, and G IR to map them to static observables (e.g. static correlators and free energies). Then the pair (E RG = EQCD, φ 0 )—where φ 0 encodes the matching conditions—is a natural candidate for a universal encoding: any other 3D EFT reproducing the same static observables must factor through EQCD by adjusting irrelevant couplings. This is reflected in the fact that EQCD can be viewed as the minimal EFT capturing all static IR physics of hot QCD up to a given order in g and T .
The same structure arises in finite–temperature QED, leading to EQED (electrostatic QED) as the universal 3D encoding of static IR electrodynamics at high T . The matching of Debye masses and screening properties between 4D QED and EQED is a concrete instance of the equivalence
Thus, finite–temperature dimensional reduction is a clear example of UP–RG: the universal effective theory E RG (EQCD/EQED) is the initial object in the fiber category of 3D EFTs reproducing the same static IR data.
The Wilsonian RG approach [49, 50] constructs a family of effective actions S k [φ] parametrised by a sliding scale k, obtained by integrating out modes with momenta p in the shell k < p < Λ. The effective average action Γ k [φ] satisfies a functional RG equation such as Wetterich’s equation [13].
Suppose we are interested only in observables at scales p ≤ k 0 for some fixed IR scale k 0 . In the categorical language, we can:
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 88 • define F IR to map B to correlation functions of φ for momenta |p| ≤ k 0 , ed • take Bulk to contain the UV theory at scale Λ, • take EFT to contain theories defined with UV cutoff k 0 (i.e. with modes p > k 0 already integrated
out).
v
The Wilsonian RG flow from Λ down to k 0 , S Λ 7→ S k 0 , defines an effective theory E RG with couplings (m 2 k 0 , λ k 0 , . . . ) that encodes all IR observables at that scale. Any other EFT E defined at k 0 that reproduces the same IR observables (up to field redefinitions) can be obtained from E RG by integrating out irrelevant operators or by trivial reparametrisations.
Thus, the pair (E RG = S k 0 , φ 0 ) is again a candidate universal encoding in the sense of Definition IX.1: it is the minimal EFT capturing the physics at scale k 0 , and all other EFTs reproducing the same IR data factor through it. The functor H RG associates to each UV theory B (e.g. different bare couplings or regularizations) a universal IR encoding E RG , belonging to the same universality class.
but with two additional features:
• The encoding category EFT is loop–graded and enriched as discussed in Subsection II D: the loop
expansion, operator product structure, and RG flow are built into the hom–objects and morphisms.
• The encoding theory E RG is not only an encoding but a universal encoding: it is the initial object
in the fiber category of all EFTs reproducing F IR (B).
In this sense, the renormalization group flow is a categorical left adjoint (or Kan extension) that selects the “best approximation” to the bulk theory in the IR context. The RG fixed points and universal critical exponents correspond to equivalence classes of such universal encodings under EFT–morphisms.
tn • This structure is exemplified by finite–temperature dimensional reduction (EQCD/EQED) and by
simple scalar field theories, among many others.
• The RG flow, viewed categorically, is thus a special case of string–free holography: a mapping from
high–dimensional, high–energy bulk theories to lower–dimensional or lower–cutoff encoding theories that capture specific classes of observables.
In the next subsection we will extend these ideas to density functional theory (DFT), showing how UP–XC (universal–property exchange–correlation) provides another powerful example of holographic encoding in a context where geometry plays almost no role and the key structure lies instead in functional dependence on densities.
We now turn to an example of holographic encoding which is almost entirely non–geometric in the usual spacetime sense, and where the relevant structures are functional–analytic rather than local QFT fields: density functional theory (DFT). Our aim is to show that DFT — and in particular the construction of an exact exchange–correlation functional E XC [ρ] — fits naturally into the string–free holographic framework as a universal encoding in the sense of UP–holography. In brief:
• the bulk theory B is the full many–body electron QFT (or, in practice, nonrelativistic many–electron
quantum mechanics plus quantized electromagnetic field if desired), This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 89
• the encoding theory E is a density–only functional theory (DFT), specified by a universal functional
F [ρ] and its decomposition into kinetic, Hartree, and exchange–correlation parts,
• the context functor F DFT picks out ground–state observables that are functionals of the electron
density,
• the encoding functor H XC sends B to the DFT functional E XC [ρ] (plus the universal kinetic and
Hartree contributions) that encodes the many–body effects at the density level.
We will proceed in several steps: (1) recall the Hohenberg–Kohn and Kohn–Sham structures, (2) formalize DFT as a universal encoding in categorical terms, and (3) argue that E XC [ρ] is precisely the data of the holographic encoding functor H XC , with UP–XC as a universal–property condition analogous to UP–RG.
v1. Many–body electron QFT and the density. For concreteness, we work in the standard nonrelativistic many–electron setting. Consider a system of N electrons moving in an external scalar potential v(r), described by the Hamiltonian
where
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The many–body wavefunction Ψ(r 1 , . . . , r N ) lives in the antisymmetric subspace of L 2 ((R 3 ) N ) with spin degrees of freedom. The ground–state energy in the presence of v is
The associated particle density is
tn and V̂ ee and for nondegenerate ground states, the map
is one–to–one up to an additive constant in v. Thus, the density ρ 0 (r) uniquely determines the external potential v(r), the full ground–state wavefunction Ψ 0 , and all ground–state observables. This suggests
ep subject to ρ ≥ 0 and d 3 r ρ(r) = N . The (second) Hohenberg–Kohn theorem and Lieb’s convex formulation [78] make this precise. Define F [ρ] = inf hΨ| T̂ + V̂ ee |Ψi, Ψ→ρ where the infimum is over all antisymmetric wavefunctions Ψ yielding density ρ. Then
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so F [ρ] is a universal functional (independent of v) encoding all information about T̂ and V̂ ee relevant for ground–state energies and densities.
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where: iew
• T s [ρ] is the kinetic energy of the noninteracting Kohn–Sham system,
• E H [ρ] is the classical Hartree (Coulomb) energy,
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• E XC [ρ] is the exchange–correlation functional, which encodes all many–body effects beyond T s and
E H .
Thus, the ground–state energy can be written as
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Schrödinger equation with potential
where v H [ρ] and v XC [ρ] are the functional derivatives of E H and E XC , respectively [76, 77].
The key observation is that, from the standpoint of ground–state properties, the entire many–body problem is encoded in the single functional E XC [ρ]. Once E XC is known, one can solve the Kohn–Sham equations (a one–body problem) and recover ρ(r) and all observables that are functionals of ρ. In this sense, E XC [ρ] is a holographic encoding of the many–body electron QFT into a density–only theory.
tn 3. DFT as holographic encoding: bulk, encoding, and context. We now interpret DFT as an instance of the string–free holographic principle. Let:
• Bulk category Bulk el : Objects are many–body electron QFTs of the form Ĥ[v] with fixed T̂ and
V̂ ee and varying external potentials v(r). Morphisms include changes in v, adiabatic switch–on of interactions, and unitary equivalences.
• Encoding category Enc DFT : Objects are density–only theories characterised by universal func-
tionals F [ρ] (or equivalently by T s [ρ], E H [ρ], and E XC [ρ]). Morphisms are transformations of functionals (e.g. approximations, reparametrisations, or restriction to subspaces of densities).
• Context category Obs dens : Objects are ground–state observables that are functionals of the
density: ground–state energy E 0 [v], density ρ 0 (r), expectation values of operators expressible as
Define the bulk context functor
F DFT : Bulk el −→ Obs dens ,
which maps a bulk Hamiltonian Ĥ[v] to the ground–state density ρ 0 (r) and energy E 0 [v]:
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Define an encoding context functor
G DFT : Enc DFT −→ Obs dens , (431)
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91
i.e. DFT reproduces the exact density and energy. In other words, there is a natural equivalence
where the encoding functor
vH XC : Bulk el −→ Enc DFT ,
encodes the many–body Hamiltonian into the universal density functional. In particular, the exchange– correlation functional E XC [ρ] is precisely the nontrivial part of this encoding: T s and E H are known explicitly, while E XC must be constructed (or approximated) to encode the many–body correlations.
Fix the interacting Hamiltonian ( T̂ , V̂ ee ) and consider the fiber category
whose objects are candidate density–only theories E (functionals) that reproduce the exact density and energy of Ĥ[v] for all v, together with an equivalence φ in Obs dens .
Definition IX.2 (UP–XC). We say that F [ρ] (or equivalently E XC [ρ] together with T s and E H ) is a universal–property exchange–correlation encoding (UP–XC) if (F [ρ], φ 0 ) is initial in the fiber category Fib F DFT (F DFT ( Ĥ[v])): for any other encoding (E, φ), there exists a unique (up to contractible homotopy) morphism h : F [ρ] → E in Enc DFT such that
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tn Intuitively, this means that any other density–based theory reproducing the same exact ground–state observables must be obtained by further coarse–graining or trivial reparameterisation of the exact functional F [ρ]. This matches the physical intuition: the exact F [ρ] is unique and universal for a given interacting system, and all approximate functionals F approx [ρ] can be viewed as morphisms out of F [ρ] that simplify or truncate the encoding.
From this point of view, designing approximate functionals (LDA, GGA, hybrid, meta–GGA, etc.) is equivalent to constructing morphisms in Enc DFT that factor through the exact F [ρ]. The better the approximation, the closer the morphism h is to an isomorphism in the relevant region of the density space.
being:
DFT as holography is notable for
• non–geometric: the bulk and encoding theories live on the same physical space R 3 ; the dimen-
sional reduction is not spacetime but configuration–space dimensionality: from a 3N –dimensional wavefunction Ψ(r 1 , . . . , r N ) (plus spin) to a 3D density ρ(r).
• functional–analytic: the encoding is a convex functional F [ρ] on a Banach space of densities, with
well–understood mathematical properties (lower semicontinuity, convexity) [78].
• universal: the same functional F [ρ] applies to all external potentials v, and thus encodes a whole
family of bulk theories parameterised by v.
In the language of this paper, DFT is therefore a clean instance of string–free super–holography with a non–geometric encoding category. The bulk category Bulk el and the encoding category Enc DFT are linked by the encoding functor H XC , and the equivalence of context functors F DFT ≃ G DFT ◦ H XC expresses the holographic nature of the density.
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• Bulk: many–body electron QFT, with observables expressed in terms of ground–state densities and
energies.
• Encoding: density functional theory with universal functional F [ρ] and exchange–correlation
v• Context: F DFT selects observables that are functionals of ρ.
• Encoding functor: H XC maps the bulk Hamiltonian to F [ρ], with E XC [ρ] encoding the many–body
correlations.
• Universal property: the exact F [ρ] is a universal encoding in the sense of UP–holography; any other
density–only theory with the same ground–state observables factors through it.
This example demonstrates that the holographic ideas developed in this paper are not limited to geometric or stringy contexts; they also apply naturally to purely functional and operator–theoretic settings, in which the “boundary” variables are densities or other coarse–grained observables.
Side–by–side comparison ot A.
tn We now synthesise the preceding case studies — worldline QED, anomaly inflow in QCD, flux–tube worldsheet encodings, and UP–RG/UP–XC — into a systematic comparison between string encodings (in the sense of AdS/CFT and related constructions) and the non–string encodings that we have developed in this work. The objective is not to detract from the enormous achievements of string–based holography in certain regimes, but to show that, when holography is viewed through the higher–categorical and context–dependent lens, non–string encodings:
• apply in a strictly broader set of physical regimes,
ep • capture a richer variety of structures (worldlines, defects, anomalies, codes), • and yield more direct computational tools in realistic QFTs.
We will first present a schematic side–by–side table and then unpack each row in more detail, relating back to the concrete constructions in earlier sections.
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 93 String encodings (AdS/CFT–type) Non–string encodings (worldlines,
TQFT, defects, codes) Regime Supersymmetric (often maximally), con- Generic QFTs including QED in strong
formal, large N gauge theories with AdS– fields, QCD in the confining and chiral like bulk duals; controlled in special limits regimes, finite–temperature gauge theo(large N , large ’t Hooft coupling) and often ries (EQCD/EQED), density–functional with high supersymmetry [38]. systems, quantum codes; works at finite N ,
non–conformal, non–SUSY, and far–from– equilibrium regimes. Structure captured Bulk geometry (AdS, black holes), pro- Anomalies and topological data via ex-
tected operator sectors (e.g. BPS opera- tended TQFTs and anomaly polynomials; tors, stress–tensor multiplets), certain en- worldlines and loop physics (Schwinger pair tanglement and thermodynamic quanti- production, Casimir effects); defect specties (e.g. Ryu–Takayanagi entanglement tra and flux–tube dynamics; density funcentropy [39]), hydrodynamic transport in tional structures and exchange–correlation; strongly coupled plasmas [40]. quantum error–correcting code structure of
entanglement and reconstruction. Computations Gravity–based methods; classical super- Algorithmic and flexible: 0+1D ODEs and
gravity plus α 0 and loop corrections; useful fluctuation determinants in worldline QED; mainly in strongly coupled planar limits cohomology and cobordism computations in and for protected or semi–protected quan- anomaly inflow; spectral and Casimir comtities; often no direct algorithm for unpro- putations for defect/worldsheet QFTs; functected correlators at finite N . tional minimisation and convex duality in
DFT; tensor network and code–based methods for entanglement and reconstruction. Generality Narrow: a relatively small set of well– Broad: applicable to QED, QCD, EFTs,
understood dualities (e.g. AdS 5 /CFT 4 , DFT, condensed–matter systems, and quanAdS 4 /CFT 3 , some AdS 3 /CFT 2 examples) tum codes; can be constructed whenever with stringent structural requirements a suitable encoding category (worldline, (SUSY, conformality, large N ) and few rig- TQFT, defect category, functional category, orous proofs [38]. code category) and context functors can be
defined; naturally extends to non–geometric and non–local settings. ot
v
Aspect t n This table is deliberately schematic, but it captures the main qualitative differences. We now elaborate each row in detail, linking back to explicit constructions in earlier sections.
• conformal or nearly conformal, rin • supersymmetric, often maximally (N = 4 in 4D), • in the large–N limit, where planar diagrams dominate and classical gravity is reliable in the bulk [38].
In such regimes, correlators of certain operators can be computed from geodesics or minimal surfaces in the bulk, and entanglement entropies can be computed via the Ryu–Takayanagi formula [39]. However, as soon as one moves away from this regime (finite N , broken conformal symmetry, non–SUSY theories, or strongly time–dependent backgrounds), explicit string duals are either unknown or difficult to use quantitatively.
By contrast, non–string encodings we constructed are tailored to the specific structure and context:
• Worldline encodings work for QED (and other gauge theories) in arbitrary background fields,
including strong, rapidly varying fields, without requiring SUSY or conformality; the representation (195) is exact at one loop.
• Anomaly inflow encodings work for QCD and other chiral gauge theories purely in terms of anomalies
and cohomology, independent of large–N or supersymmetry.
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• Defect–category encodings (flux tubes and worldsheet QFTs) work in confining Yang–Mills for any
N ≥ 2 and for generic couplings, capturing IR flux–tube physics and static potentials.
• UP–RG encodings (EQCD/EQED) work for finite–temperature QCD/QED and other thermal
gauge theories, capturing static IR physics at high T .
• UP–XC encodings (DFT) work for generic many–electron systems, including realistic solids,
molecules, and condensed–matter systems.
In other words, string encodings inhabit a relatively small “island” in the space of QFTs (highly symmetric, large–N CFTs), whereas non–string encodings span much broader regions, including many physically relevant theories.
v3. Structure captured: what kind of data is naturally encoded? String–based encodings excel at capturing:
• geometric data: bulk metrics, curvature invariants, horizons;
• conformal data: scaling dimensions, operator product coefficients for special operators;
• certain thermodynamic and hydrodynamic data: black–hole thermodynamics, shear viscosities,
diffusion constants in strongly coupled plasmas [40];
• entanglement of spatial regions: Ryu–Takayanagi and its covariant generalisations [39].
These are all aspects of a particular “geometric” encoding category Enc string where objects are bulk string/brane backgrounds and morphisms are background deformations; context functors F, G pick out correlators and entanglement.
However, modern QFT demands the ability to encode:
• anomalies and topological phases: classified by anomaly polynomials and cobordism groups [68, 70,
71];
• worldline and loop structures: determinants, pair–production rates, Casimir energies;
• defect spectra: line and surface operators, flux tubes, domain walls, and their junctions [26, 73];
tn • density–only and functional structures: ground–state densities and functionals in DFT, convex
dualities;
• quantum error–correcting code structures: logical vs. physical operators, entanglement wedges, and
reconstruction maps. rin These structures do not have a canonical worldsheet realisation; they live in other encoding categories: • Enc anom : extended TQFTs and invertible phases,
• Enc wl : worldline super–QFTs, • Enc ws : worldsheet QFTs in defect 2–categories, • Enc DFT : categories of density functionals, • Enc code : categories of codes and channels.
Each of these encoding categories captures different structural aspects of QFT that strings do not naturally encode. The anomaly polynomial I 6 in QCD, the worldline instantons in QED, the flux–tube spectra in YM, and the DFT functional E XC [ρ] are all best understood and computed in non–string categorical frameworks.
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 95 Gravity–based computations in string
encodings rely heavily on:
• classical solutions of supergravity (solving Einstein’s equations with sources and fluxes),
• sometimes integrability (e.g. spin chains, Bethe ansatz),
• perturbative string corrections in α 0 or in the string coupling g s .
These methods are extremely powerful in certain strongly coupled regimes but are often analytically challenging, and typically have limited control over unprotected quantities at finite N . Furthermore, they tend to be highly specific to the particular string background and its symmetries.
By contrast, the non–string encodings we use are algorithmic and flexible:
– solving 1D ODEs for instanton trajectories,
v• Worldline QED: once the worldline representation (195) is known, non–perturbative pair–production
exponents and prefactors in arbitrary background fields are obtained from:
– computing 1D fluctuation determinants (Gel’fand–Yaglom), – evaluating low–dimensional integrals over proper time and worldline coordinates.
• Anomaly TQFTs: anomalies and WZW levels are computed from:
– cohomology and cobordism groups, – characteristic classes and polynomials I 6 , – Chern–Simons actions and descent equations [68, 70, 71].
• Worldsheet flux–tube QFTs: static potentials and Lüscher terms are obtained from:
– solving 2D Casimir problems for free or weakly interacting bosons, – determining worldsheet spectra from simple Hamiltonians, – matching a small set of parameters (e.g. σ) to the 4D theory.
• DFT: ground–state energies and densities for correlated electrons are obtained by:
– functional minimisation problems over ρ(r), – solving self–consistent Kohn–Sham equations,
tn – employing convex duality and functional analysis [76–78].
• Code–based encodings: entanglement and reconstruction properties are analysed via:
– linear algebra over Hilbert spaces, – graph and tensor network methods, rin – information–theoretic measures (mutual information, channel capacities). These methods are amenable to direct numerical implementation and do not require solving higher– dimensional Einstein equations or guessing string backgrounds. String encodings have so far been constructed
for:
• various AdS/CFT dualities (AdS 5 /CFT 4 , AdS 4 /CFT 3 , AdS 3 /CFT 2 ),
• some non–conformal and finite–temperature generalisations (AdS black holes, confining geometries),
• certain theories with extended supersymmetry and specific matter content.
While conjectured duals exist for a broader set of theories, rigorously established and computationally useful examples remain comparatively few.
In contrast, non–string encodings:
• apply to essentially any QFT with a well–defined set of observables in a given context:
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96 – worldline encodings work for generic gauge theories at one loop (and beyond in some cases), ed – anomaly TQFT encodings apply to any gauge theory with chiral or higher–form anomalies, – defect–category encodings apply to a wide class of QFTs with line/surface operators, – UP–RG encodings apply to any QFT with a Wilsonian RG description,
– DFT encodings apply to any many–fermion system with a well–defined ground state.
• can be “stacked” or combined: e.g. one can consider worldline encodings in the presence of anomalies,
or defect encodings at finite temperature.
• are not restricted by the existence of a smooth AdS bulk geometry or by large–N limits.
vThus, non–string encodings are strictly more general in scope: for almost any physical theory and observable context we care about, one can, in principle, identify a suitable encoding category and construct an encoding functor, while string encodings require very specific structural conditions that are rarely met in realistic systems.
• cover far more physical regimes (QED, QCD, EFTs, DFT, codes),
• match naturally to the relevant structures (worldlines, anomalies, defects, densities, codes),
• and provide concrete, algorithmic tools for computations that are otherwise intractable (pair
production, anomaly constraints, flux–tube spectra, many–body ground states).
Argument that non–string encodings are more powerful
tn B. ot In particular, the examples we developed in this paper — worldline QED, anomaly QCD, flux–tube YM, UP–RG, and UP–XC — all demonstrate how choosing the right encoding category and functors leads to genuine computational benefits, without any reliance on string backgrounds. This suggests that the future of holography, especially for realistic systems, lies not in extending string dualities to ever more complicated settings, but in systematically exploring the space of non–string encoding categories and leveraging their higher–categorical structure to extract new physics.
We now synthesise the lessons of our detailed case studies — strong–field QED, anomaly/WZW QCD, flux–tube holography in confining YM, and UP–RG/UP–XC encodings — into a global argument: from the standpoint of which encodings give us real capabilities in real QFT problems, non–string higher–categorical encodings are both strictly more general and, at present, more powerful than any string–based encodings. By “more powerful” we mean both:
• structurally richer: able to capture a wider variety of physical and mathematical structures (worldlines, anomalies, defects, densities, codes), and
• computationally superior in many regimes: providing algorithmic methods, dimensional reductions,
and universal constraints that are unavailable or much harder to realise in string frameworks.
We will organise the argument around the concrete examples developed in earlier sections and then abstract the general pattern in categorical terms.
• The one–loop effective action in an arbitrary background A µ (x) can be written exactly as a 0+1D
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• Non–perturbative pair–production exponents and prefactors are governed by worldline instantons:
solutions of 1D ODE systems for x inst (τ ) in Euclidean time, along with 1D fluctuation determinants (Gel’fand–Yaglom).
• Two–loop corrections from internal photon exchanges are encoded in a bilocal worldline kernel
spin I photon [x, ψ], which reduces to a double integral in (τ 1 , τ 2 ) plus 1D fluctuation data. iew From the string–free holographic viewpoint, this is a 4D → 1D super–holography: a complicated 4D non–perturbative QED problem (infinite series of multi–photon diagrams) is encoded in a 1D worldline super–QFT plus low–dimensional integrals. Crucially: • This worldline encoding works for generic backgrounds E(t) (no conformal or SUSY requirements), including rapidly varying strong fields.
v• There is no known string or AdS dual that reproduces these pair–production rates with comparable
control and algorithmic clarity. While string techniques can, in principle, represent QFT determinants, they do so via 2D worldsheet CFTs in higher–dimensional target spaces, which here would be an unnecessary and unwieldy overhead.
Thus, for strong–field QED in fast backgrounds, the only practical holographic encoding currently known that yields genuine computational benefits is the non–string worldline encoding. This is already a strong indication that non–string encodings can go where string encodings cannot, at least in realistic QED regimes.
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and the Wess–Zumino–Witten term in the IR chiral Lagrangian. The main structural points were:
• The anomaly is captured by a 6D characteristic class and a 5D Chern–Simons action S 5D [L, R]
[Eq. (318)], which define an invertible 5D TQFT. ot • The 4D WZW term is a boundary functional Γ[U, L, R] [Eq. (309)] whose variation reproduces the anomaly via descent.
tn • The level k in the WZW term, an a priori integer ambiguity from the IR EFT standpoint, is
uniquely fixed by anomaly matching and inflow to be k = N c [Eq. (336)].
This is a textbook example of anomaly inflow [68, 70, 71], but when placed in our categorical framework it becomes a clear instance of holography:
• The bulk QCD anomaly data and the IR chiral EFT anomaly data are both encoded in the same
5D anomaly TQFT object E 5D ∈ Enc anom via the functor H.
• The context functors F anomaly and G select anomaly data from bulk and encoding, and the
holographic equivalence F anomaly (B) ≃ G(H(B)) fixes IR parameters (like k) without solving QCD.
ep In this entire story, string theory plays no essential role: • The classification of anomalies and invertible phases is naturally formulated in terms of cohomology, cobordism, and extended TQFT, not in terms of worldsheet CFTs. • Even where string constructions exist (e.g. D–brane pictures of anomalies), they are, conceptually,
rephrasings of the underlying cohomological/TQFT structure.
Thus, for the problem of fixing IR anomaly data in QCD, the relevant holographic encoding is purely categorical (TQFT + cohomology), and the computational capability (fixing WZW levels, Skyrmion baryon numbers, anomalous couplings) comes from this non–string encoding, not from any string dual.
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• Wilson lines and flux tubes are naturally organised into a defect 2–category D(T YM ) [26, 73], with:
– 1–morphisms: line operators (Wilson lines), – 2–morphisms: surface defects (flux–tube worldsheets). iew – 0–morphisms: vacua (e.g. the confining vacuum), • The static quark–antiquark potential V (R) at large R is encoded in the ground–state energy E 0 (R)
of a 2D worldsheet QFT for the flux tube:
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v
where the Lüscher term −π/(12R) arises as a Casimir energy of free transverse bosons on a strip.
• The worldsheet theory is an emergent effective theory in the defect category; it is not a fundamental
string theory in 10D, but a lower–dimensional QFT describing a specific defect sector of a 4D theory.
One might be tempted to identify this effective string with a fundamental string in a string dual of QCD, but such a fundamental dual is neither known nor needed here. Instead:
• The worldsheet theory arises directly from the defect structure of 4D YM as an object in Enc ws ,
the encoding category of worldsheet QFTs.
• The holographic encoding is 4D → 2D via defect categories and effective QFT, not via a fundamental
string propagating in a higher–dimensional spacetime.
The computational payoff is clear: we can compute universal IR features of the static potential (Lüscher term) and flux–tube excitation spectra using 2D worldsheet methods, sidestepping the need for a full string dual of YM. The encoding is defect–categorical and non–string, and it directly yields capabilities (e.g. IR asymptotics of V (R)) that string theory for QCD has not yet delivered in a quantitatively controlled way.
tn • UP–RG: for a given IR context F IR , the RG flow produces a universal effective theory E RG (e.g.
EQCD/EQED at high temperature, or an EFT at a finite scale k 0 ) and a functor H RG such that any other EFT reproducing the same IR observables factors through E RG .
• UP–XC / DFT: for many–electron systems, the Hohenberg–Kohn and Kohn–Sham theorems
guarantee that all ground–state observables that are functionals of the density ρ(r) can be encoded in a universal functional F [ρ] (and in particular in the exchange–correlation functional E XC [ρ]). This functional is a universal encoding of the many–body problem into the space of densities.
In both cases:
• The encoding categories (EFT, Enc DFT ) are not geometric in the sense of AdS/CFT; they live in
spaces of couplings, operators, and functionals.
ep • The context functors F IR and F DFT select IR observables and density–dependent observables, respectively. • The universal encodings E RG and F [ρ] are initial objects in the relevant fiber categories, yielding UP–holographies F ≃ G ◦ H in each case.
There is no string picture here at all. Attempting to force one (e.g. by imagining some AdS dual of a condensed–matter system or of electrons in a solid) does not presently offer a competitive computational or conceptual advantage compared to the direct functional and RG encodings. Yet, these non–string encodings are responsible for much of our practical ability to compute properties of real materials, nuclear matter, and finite–temperature QFTs.
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Then:
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• The domain of H string is a strict subcategory of Bulk consisting of highly constrained theories (SUSY
CFTs at large N , etc.) and a few deformations thereof.
v• The domain of the various non–string encoding functors H wl , H anom , H flux , H RG , H XC , etc.
collectively covers a much larger class of bulk theories, including realistic QED, QCD, EFTs, and many–body systems.
• For many contexts F (strong–field observables, anomalies, defect observables, IR observables, density
observables), there exist non–string encodings E ∈ Enc non with F (B) ≃ G(E) that are manifestly useful and calculable, while no practical string encodings are currently known.
Formally, for most physical theories B ∈ QFT and contexts F , the set
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er Fib F (F (B)) ∩ Enc string is empty or poorly understood, whereas contains explicit and computationally useful encodings. This is the precise sense in which non–string encodings are strictly more general and currently more powerful than string encodings.
• Strong–field QED in fast backgrounds: only worldline (non–string) encodings provide practical,
algorithmic methods to compute non–perturbative pair production and higher–loop corrections.
tn • Anomaly/WZW in QCD: extended TQFT and cohomological encodings fix IR data (WZW levels,
Skyrmion baryon numbers, anomalous couplings) without solving QCD and without any appeal to strings.
• Flux–tube QCD: defect–category and worldsheet encodings yield universal static potentials and
flux–tube spectra using 2D QFTs; no fundamental string dual of YM with comparable predictive power exists. rin • UP–RG and DFT: holographic encodings in the spaces of couplings and densities provide universal IR and many–body encodings that drive much of our practical understanding of condensed matter, nuclear matter, and finite–temperature QFT, again without strings.
Therefore, from the standpoint of which encodings give us real capabilities in real QFT problems, non–string higher–categorical encodings are empirically and conceptually superior:
• They apply to more theories (QED, QCD, EFTs, DFT, codes).
• They capture more kinds of structure (worldlines, anomalies, defects, densities, codes).
• They yield more direct computational tools (ODEs, cohomology, Casimir problems, functional
optimisation).
This does not diminish the conceptual elegance or deep insights provided by string encodings in their domain of applicability. Rather, it suggests that the natural extension of holographic ideas to the full breadth of modern physics must be built on non–string higher–categorical encodings, with strings recognised as one particularly geometric instance rather than as the foundation of all holography.
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 100 CONCLUSIONS AND OUTLOOK A. ed XI.
Summary of contributions
In this work we have developed a unified, string–free, higher–categorical framework for holography and demonstrated that it is not only conceptually coherent but also practically powerful across a wide range of quantum field theories and contexts. Here we summarise the main contributions, organising them according to the conceptual advances and the concrete computational capabilities they enable.
v• an ambient (∞, 1)–category QFT of quantum and effective field theories, equipped with a dimension
functor dim : QFT → N [Eq. (31)],
• an observable (∞, 1)–category Obs, which may consist of factorization algebras, cochain complexes,
anomaly classes, or other structured observable objects,
• one or more encoding (∞, 1)–categories Enc, whose objects are encoding theories (worldline QFTs,
extended TQFTs, worldsheet theories, DFT functionals, etc.),
• an encoding functor
from a bulk subcategory Bulk ⊆ QFT to Enc,
• context functors
which select the class of observables of interest.
twhose components
(445)
(446)
express the equality (up to higher homotopy) of bulk and encoding observables in the chosen context. This shifts the focus from strings and AdS geometries to higher–categorical structures and context–dependent equivalences, revealing holography as a general phenomenon of functorial encoding rather than a specific property of string theory.
A crucial aspect of this formulation is its relativity: holography is always relative to a chosen context (F, G) and an encoding category Enc; different contexts lead naturally to different encodings. This was made precise in the notion of relative holography [Eq. (66)] and in the classification of compression vs. anomaly holography in Subsection II C, based on the relation between dim(B) and dim(H(B)).
• Compression holography (super–holography): encodings where
(or dim eff (H(B)) dim eff (B)), as in 4D → 1D worldline encodings or 4D → 2D worldsheet encodings.
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101
• Anomaly holography: encodings where
to worldline and worldsheet encodings of loop and defect physics in lower dimensions, while anomaly holography corresponds to extended TQFT encodings of anomalies in higher dimensions.
In Subsection II D we further refined this picture by incorporating enrichment (over Vect C , Hilb, Vect × Top, etc.) and loop grading, ensuring that the encoding functors preserve not only observables but also loop expansions and topological data.
v3. New computational capabilities in fast, strong–field QED via 1D worldline encoding. A central technical and conceptual achievement is the demonstration, in Sections V and VI, that the worldline encoding of spinor QED provides a genuine super–holographic compression from 4D to 1D with concrete computational benefits:
• We derived the spinor worldline action
er and the exact one–loop worldline representation Z Z Z spin 1 ∞ dT −m 2 T (1) Γ [A] = − e Dx Dψ e −S wl [x,ψ;A] . 2 0 T x(T )=x(0) ψ(T )=−ψ(0)
(449)
(450)
• We showed that non–perturbative pair–production rates in constant and time–dependent electric
fields can be computed semiclassically from worldline instantons (solutions of 1D ODEs) and their fluctuations, transforming a 4D non–perturbative QED problem into a 1D instanton and fluctuation problem.
• We extended this to two loops by introducing a bilocal worldline photon insertion
(451)
tn integral in (τ 1 , τ 2 ) plus 1D fluctuation determinants.
These results exemplify the 4D → 1D super–holography: all one– and two–loop, and in principle higher–loop, effects in a wide class of background fields are encoded in a lower–dimensional worldline theory, with algorithmic methods (ODEs, determinant calculations) replacing 4D Feynman diagram resummations. Importantly, this is achieved without any string dual, purely through the non–string encoding category Enc wl and the functor H wl .
and VII C we developed a second super–holographic example: anomaly inflow holography for QCD, where a 4D gauge theory is encoded in a 5D anomaly TQFT via the anomaly polynomial
(452)
and the 5D Chern–Simons action
We showed that:
• the WZW term Γ[U, L, R] in the IR chiral EFT is naturally understood as boundary data of this
5D TQFT, This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 102
• the WZW level k is quantised but ambiguous at the IR EFT level, and Thus, IR data—WZW levels, Skyrmion baryon numbers, anomalous couplings to external fields—are fixed exactly by a 5D extended TQFT encoding and anomaly matching, without solving QCD. This is a concrete demonstration of how higher–dimensional non–string encodings (extended TQFTs) can yield nontrivial IR information about strongly coupled gauge theories.
v5. Flux–tube YM as 2D encoding: Lüscher term and defect–categorical structure. In Section VIII we exhibited a 4D → 2D super–holography in confining SU(N ) Yang–Mills, using the defect 2–category of vacua, line defects, and flux–tube surfaces. We showed that:
• the static quark–antiquark potential V (R) at large R is encoded in the ground–state energy E 0 (R)
of an effective worldsheet QFT, • in the simplest approximation this worldsheet QFT is a free massless boson theory in 1+1 dimensions,
with action
(456)
• and the effective string worldsheet resides in a non–string encoding category Enc ws (a defect
2–category), not in a fundamental string theory in higher–dimensional spacetime.
tn This provides a dimensionally reduced, analytically tractable description of the IR behaviour of the static potential, again without invoking any fundamental string dual of Yang–Mills. It is a clean example of super–holography via defect categories and effective 2D QFT.
• UP–RG: for a given IR context F IR , the Wilsonian RG flow produces a universal effective theory
E RG and a functor H RG such that any other EFT reproducing the same IR observables factors through E RG . This was illustrated by finite–temperature dimensional reduction (EQCD/EQED) and scalar field theory.
• UP–XC / DFT: for many–electron systems, the Hohenberg–Kohn and Kohn–Sham constructions
yield a universal functional F [ρ] (and in particular an exchange–correlation functional E XC [ρ]) that encodes all ground–state density observables. The functional F [ρ] serves as a universal encoding in the density functional category Enc DFT .
These examples show that holography need not involve spacetime dimension at all: it can be an encoding from a high–dimensional configuration space (many–body wavefunctions) to a lower–dimensional functional space (densities), governed by universal properties in convex functional analysis. The categorical language of UP–holography provides a unifying framework for these phenomena.
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• string encodings are powerful but restricted to highly symmetric, large–N regimes with AdS–like
bulk geometry,
• non–string encodings (worldlines, TQFTs, defect categories, RG/DFT functionals, codes) apply to
a much wider class of QFTs and observables,
• and, in many physically relevant problems (strong–field QED, real QCD, finite–T EFTs, many–
electron systems), only non–string encodings currently provide practical computational benefits.
vFrom the standpoint of which encodings yield real capabilities in real QFT problems, non–string higher– categorical encodings are thus strictly more general and currently more powerful than string–based encodings. Strings remain an elegant and important special case, but the conceptual and computational centre of gravity in holography shifts, in our framework, to the broader landscape of non–string encoding categories and their functorial relations.
In the next subsections we will discuss open directions and potential applications of this framework, including its extension to nonequilibrium systems, quantum information settings, and the interplay between different encodings (e.g. worldlines and defects, anomalies and RG), which may yield yet more powerful hybrid holographic methods.
The string–free higher–categorical holographic framework developed in this work opens up a broad range of directions for further research. In this subsection we outline a non–exhaustive list of promising avenues, organised according to physical regimes and mathematical structures. Our guiding principle throughout will be the same: for each class of problems, identify the natural encoding category (worldlines, TQFTs, defect categories, functional categories, code categories), define the appropriate context functors, and construct encoding functors realising UP–holography. In many cases, this entails extending and combining the tools (worldline instantons, anomaly TQFTs, defect 2–categories, UP–RG/UP–XC) that we have already developed.
tn 1. Non–Abelian strong–field QCD: worldlines with color. A natural extension of our worldline analysis is to non–Abelian gauge theories such as QCD in strong external fields. While the Abelian worldline formalism is well–developed and we have exploited it to full effect in Sections V and VI, the non–Abelian case introduces additional structure:
• Color degrees of freedom on the worldline: in non–Abelian cases, the worldline action must be
augmented by additional variables (e.g. Grassmann or bosonic color fields) that transform in the appropriate representation of the gauge group, with couplings such as
where χ(τ ) carry color indices and T a are generators of su(N c ).
ep • Background fields in non–Abelian directions: strong chromoelectric fields or color–flux backgrounds a (e.g. constant F µν in a Cartan subalgebra) can be treated in a worldline instanton formalism, but the structure of the instanton equations becomes richer due to color precession and non–commutativity of field strengths. A detailed exploration of non–Abelian worldline instantons in rapidly varying chromoelectric backgrounds (e.g. time–dependent color fields in heavy–ion collisions) is a natural next step. From the holographic perspective:
• The bulk category Bulk QCD,ext would include QCD with external color fields or background fields
mimicking glasma configurations.
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• The encoding category Enc wl,col would consist of worldline super–QFTs with both spin and color
degrees of freedom.
• The context functor F pair would select non–perturbative observables such as quark pair–production
rates, Schwinger–like instabilities, and color dielectric response.
Constructing encoding functors H wl,col and establishing their semiclassical instanton structure could yield new insights into non–perturbative particle production and transport in non–Abelian fields, in regimes where traditional lattice or diagrammatic methods are difficult to employ.
• subdimensional excitations (fractons) with restricted mobility, • higher–rank gauge theories and subsystem symmetries.
v • subextensive ground–state degeneracy depending on system geometry, The mathematical structures underlying these phenomena involve higher–rank tensor gauge fields, fusion categories with restricted mobility, and intricate Ext/Tor data in group and cohomology theories. In this context:
• The bulk category Bulk fracton would include fracton Hamiltonians, higher–rank gauge theories, and
their continuum limits.
• The encoding category Enc fusion would consist of fusion categories and their module categories,
capturing fusion and braiding properties of excitations.
• The observable category Obs topo could be a category of topological response functions, degeneracies,
and higher–form symmetry charges.
An “Ext/Tor holography” in this setting would be a categorical relation:
where F topo extracts ground–state degeneracies, subsystem symmetry data, and fracton mobility constraints, while G Ext/Tor computes corresponding invariants from Ext/Tor groups associated to fusion categories. This would generalise the anomaly holography of Section VII to more exotic phases, with the TQFT replaced by a higher–categorical topological order and its cohomological invariants.
tn 3. More systematic UP–RG constructions for DFT and beyond. In Section IX A we sketched how UP–RG can be formulated as a universal–property encoding for IR observables, and in Section IX B we treated DFT as a universal encoding of many–body systems into the space of densities. There is substantial scope to refine and extend these ideas:
• UP–RG beyond perturbation theory: For strongly coupled QFTs and nontrivial fixed points (e.g.
Wilson–Fisher fixed point in 3D, non–Fermi liquids), one can attempt to characterise E RG as an object in an (∞, 1)–category of CFTs or EFTs with special universal properties, and to define RG flows as morphisms with universal mapping properties (e.g. left adjoints or Kan extensions).
• UP–RG + DFT: One can combine RG ideas with DFT in multiscale problems: for instance,
constructing density functionals at various scales (coarse–grained densities, cluster densities) and treating them as intermediate encodings E RG,k in a chain of UP–RG encodings. This could help formalise and systematise multi–scale DFT and embedding methods.
• Beyond DFT: Other functional theories (e.g. current–density functional theory, time–dependent
DFT, reduced density–matrix functional theory) can be treated as encoding categories with universal properties. One can ask: for which contexts (observables) are these functionals universal encodings, and how do their encoding functors H relate to one another?
Developing a systematic “taxonomy” of UP–RG and UP–XC encodings across QFT, condensed matter, and quantum chemistry would provide a more unified view of effective theories and functionals as holographic encodings in non–geometric categories.
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Within our framework: iew • The bulk category Bulk curved would consist of QFTs defined on globally hyperbolic spacetimes (M, g), with morphisms given by embeddings of spacetimes and associated pushforwards/pullbacks of fields and algebras. • The encoding category Enc null could be defined by theories living on null surfaces, horizons, or causal boundaries (e.g. light–front quantisation, BMS–like symmetries at null infinity, horizon algebras).
v• The context functors F and G could select observables such as response functions, Unruh radiation,
Hawking radiation, or modular Hamiltonians for wedge regions.
An anomaly–like holography might relate bulk stress–tensor anomalies to boundary fluxes on horizons or null infinity. Worldline encodings in curved backgrounds (with geodesic worldlines and curvature couplings) could provide another route to curved–space holography, using instantons and fluctuation determinants adapted to curved geometries. Developing explicit encodings for QFT in cosmological backgrounds, black hole spacetimes, or non–trivial topologies could offer a string–free alternative or complement to AdS/CFT in gravitational settings.
• For any holographic encoding H : Bulk → Enc, one can ask whether H extends to a code structure:
is there an interpretation of H as an isometric embedding between Hilbert spaces with known error–correcting properties?
• For worldline and worldsheet encodings, one can ask whether the mapping of bulk states to
worldline/worldsheet states defines a code: e.g. is the information about certain bulk observables robust under restricted perturbations of the encoding theory?
tn • For DFT, one can interpret the map from wavefunctions to densities as an encoding that discards
some information; one can ask whether there is a code–theoretic characterisation of the lost information (e.g. what kinds of “errors” in the full wavefunction are invisible at the density level).
Developing a categorical description of quantum codes as encoding categories Enc code and identifying functors H that are simultaneously holographic encodings and error–correcting codes could deepen the connection between holography, information theory, and QFT.
• higher–form symmetries (e.g. 1–form center symmetries in Yang–Mills),
• non–invertible symmetry defects (e.g. condensation defects, topological line operators with fusion
rules not forming a group),
• categorical symmetries realised as higher–group or fusion–category actions on the defect categories.
In such settings, holographic encodings might relate bulk symmetry structures to boundary or defect data in higher dimensions, leading to new anomaly inflow relations for higher–form and non–invertible symmetries. For example, one can envisage:
• encoding a 4D theory with a non–invertible symmetry into a 5D TQFT with extended defect
structure (higher–categorical anomaly inflow), This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619
106
• encoding a QFT with higher–form symmetries into a topological theory on extended defects (e.g.
surfaces or volumes) that capture the symmetry action.
Such generalisations would require combining the anomaly TQFT techniques of Section VII with the defect 2–category structures of Section VIII in a higher–categorical setting.
• a site (C, J) (a category with a Grothendieck topology) can be used to encode what counts as “local
data”,
• sheaves F on this site encode observables that can be patched together from local information,
v • global sections of F represent “omniscient” knowledge in that context, • and presheaves that fail to satisfy the sheaf condition represent contexts where local data cannot be
consistently glued.
One can envision:
• building higher topoi of QFTs or observables, where different encodings correspond to different sites
and Grothendieck topologies,
• formalising relative omniscience and the limitations of encodings (e.g. due to anomalies or contextu-
ality) in terms of sheaf cohomology and obstructions,
• relating UP–holography to adjunctions and Kan extensions in higher topos theory, where universal
encodings are characterised as initial or terminal objects in appropriate higher categories of sheaves or stacks.
This direction is less tied to specific physical computations but could provide a deeply structural understanding of why holographic encodings exist, what their limitations are, and how different encodings are related by higher–categorical transformations.
tn In conclusion, the string–free higher–categorical holographic framework we have developed is not an endpoint but a starting point. It suggests that the future of holography in physics will be shaped not by a single geometric paradigm (strings in AdS), but by a rich landscape of encoding categories — worldlines, TQFTs, defects, functionals, codes, and higher topoi — and by the functorial relations that connect them to bulk theories in context–dependent ways. Exploring this landscape systematically promises both new conceptual insights and new practical tools for tackling some of the most challenging problems in quantum field theory and beyond.
At a conceptual level, the main message of this work is that holography, when stripped of its contingent historical association with string theory and AdS/CFT, is not a statement that “the world is a string in AdS” but rather a family of context–dependent higher–categorical equivalences between bulk theories and encoding categories. These equivalences take the schematic form
where Bulk is a subcategory of an ambient (∞, 1)–category of QFTs, Enc is an encoding category, H : Bulk → Enc is an encoding functor, and F, G are context functors selecting observables. The equivalence (459) can be as mundane as equality of one–loop effective actions or as profound as the matching of anomaly polynomials and topological phases. The point is that holography is a relationship between functors and categories, not a metaphysical claim about the “true dimensionality” or stringiness of the world.
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In particular:
• The duality is always invoked in a specific context: correlators of local operators, entanglement
entropies of spatial regions, Wilson loops, etc. Each such context corresponds to a specific choice of F and G.
v• In many cases, the equivalence is proven or strongly supported only in certain limits (large N , large
coupling, protected sectors), so that (459) holds for a restricted set of observables and regimes.
• Different encodings (e.g. different bulk backgrounds, different boundary CFT formulations) can
coexist and be useful for different contexts, emphasising the relativity rather than absoluteness of the holographic description.
The higher–categorical formalism makes this relativity explicit: a holographic equivalence is always relative to a choice of context functors F, G and encoding functor H. There is no single, context– independent “bulk” that is the true image of a boundary theory; instead, there is a family of encodings into different categories (worldlines, TQFTs, defects, functionals, codes), each appropriate for different questions.
• topological invariants (homology/cohomology groups) depend on the choice of coefficient groups,
• the UCT expresses this dependence as
tn • Ext and Tor groups capture torsion and extension data that can be invisible in one coefficient
system and visible in another.
In that setting, the relativity of topology arises because different choices of coefficients G lead to different apparent invariants; what is seen as “topological” depends on which probes (coefficient groups) one uses. The lesson for holography is that even in something as seemingly objective as topology, the invariants are not absolute; they are relative to a choice of context.
In the present paper, we have upgraded this idea from cohomology to full QFT and encoding categories:
• The role of “coefficient group” is played by the encoding category Enc and context functors F, G:
different choices of (Enc, F, G) reveal or obscure different structures in the bulk theory.
• The role of Ext/Tor data in topology is played by anomaly polynomials, fusion rules, worldline
spectra, defect categories, and functional structures (DFT, RG flows), all of which may be invisible in one encoding and explicit in another.
• Just as the UCT gives exact sequences relating cohomology groups with different coefficients, our
higher–categorical holography provides equivalences and universal properties relating bulk theories and different encoding theories in various contexts.
In this sense, the present work can be seen as a “categorified” and “capability–rich” generalisation of the UCT–based relativity of holography: we have moved from the level of topological invariants and Ext/Tor groups to the level of full QFTs, worldlines, TQFTs, defects, and functionals, all organised within an (∞, 1)–categorical and UP–holographic framework.
This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=6366619 108 The examples we have studied
suggest a philosophical reinterpretation of holography:
• Holography as encoding: A holographic relation is a functorial encoding of one theory into another,
relative to a context of observables. The bulk is not “more real” than the encoding, nor vice versa; each is a representation of the same observable content in a given context. iew • No privileged geometry: There is nothing in the formalism that privileges a geometric string background or an AdS geometry. Encodings into worldline QFTs, TQFTs, defect categories, DFT functionals, or quantum codes are on the same footing as encodings into string/brane backgrounds.
v• Emergence as functoriality: “Emergence” of geometry or of other structures (e.g. entanglement
wedges, flux–tube worldsheets, density functionals) is, in our framework, a statement about the behaviour of encoding functors and the existence of universal objects in fiber categories. Emergence is thus a categorical property, not a metaphysical one.
This perspective also clarifies why some holographic constructions are more useful than others: an encoding H : Bulk → Enc is valuable to the extent that it makes certain structures manifest, simplifies computations, or clarifies universal properties. In this paper we have emphasised capability as the key metric: does the encoding allow us to compute or constrain nontrivial observables that are otherwise difficult to access?
• worldline QED: reduction of 4D multi–loop non–perturbative problems to 1D instanton and
determinant problems;
• anomaly QCD: exact fixing of IR anomaly data (WZW levels, Skyrmion properties) from 5D TQFT
encodings;
• flux–tube YM: derivation of universal static potentials and Lüscher corrections from 2D worldsheet
QFTs; ot • UP–RG: systematic mapping of microscopic theories to universal EFTs (EQCD, EQED, etc.) for IR observables;
tn • UP–XC / DFT: encoding of many–body electron QFT into universal density functionals with
practical algorithms.
By contrast, string encodings are powerful in specific regimes but have not yet provided comparable capabilities for these problems. This suggests a philosophical shift: instead of seeking a single, universal, string–based duality for all physics, we should focus on the plurality of encoding categories and on the usefulness of their associated functors in different contexts. Holography then becomes a toolbox of encodings, each with its domain of validity and computational advantages, rather than a monolithic ontological claim.
• The relativity of topology under different coefficient choices becomes the relativity of holography
under different contexts and encoding categories.
• The Ext/Tor corrections that capture torsion and hidden structure in cohomology are paralleled
by anomaly polynomials, fusion rules, worldline instanton spectra, and functional encodings that become visible only in certain encodings.
• The UCT’s exact sequences are reflected by the universal–property constructions (UP–RG, UP–XC),
where initial objects in fiber categories play the role of universal encodings.
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• The author declares no conflicts of interest
v• There is no data associated to this manuscript iew
In short, the 2014 paper initiated a conceptual shift from absolute dualities to relative, context– dependent structures; the present paper extends this shift to full QFTs and shows that it not only makes philosophical sense but also yields nontrivial computational results in QED, QCD, YM, EFTs, and many–body systems. In doing so, it suggests a broader and more flexible vision of holography: one in which strings, AdS geometries, worldlines, TQFTs, defect categories, DFT functionals, and quantum codes all appear as different encoding categories in a single higher–categorical landscape, and in which the choice of encoding is guided not by metaphysical commitments but by the concrete capabilities it affords.
tn
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