String-Free Higher-Categorical Holography: Relative Encodings, Super-Holography, and New Capabilities in QED and QCD

String-Free Higher-Categorical Holography: Relative Encodings, Super-Holography, and New Capabilities in QED and QCD Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] 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 F:QFT →Obs, G :Enc →Obs, which select the class of observables of interest. Holography then appears as a natural equivalence F(B)≃G(E), 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( Nc )gauge theory with Nf 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 = Nc 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. I. INTRODUCTION A. Motivation 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 AdS5×S5 and N= 4 supersymmetric SU(N)Yang–Mills theory in four dimensions. Symbolically, ZIIBgµν,Φ, . . . ≡ZCFTJO,(1) 2 where the bulk partition function ZIIB as a functional of boundary values of bulk fields ( gµν, Φ , . . . )is identified with the generating functional of correlators of the dual operators O in the boundary CFT, with sources JO 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. (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 conformal (no exact CFT structure), •are not supersymmetric, •do not admit a controllable large–Nlimit, •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]. 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: 0−→ Hn(X;Z)⊗G−→ Hn(X;G)−→ TorHn−1(X;Z), G−→ 0,(2) 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 3 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: F:QFT −→ Obs,(3) where Obs is a suitable category of observables (e.g. correlation functionals, response functions, anomaly polynomials). One can also define a functor G:Enc −→ Obs (4) for the encoding category. A holographic relation between B and E in context F then becomes the existence of a natural isomorphism (or equivalence in the (∞,1) sense) α:F(B)≃ −−→ G(E).(5) This replaces the vague statement “Band Eare dual” by a precise diagram: B E F(B)G(E) H F G α ≃ (6) 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. 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: 1. non–string encodings apply in regimes where no controlled string dual is known (strong–field QED, real QCD), 2. non–string encodings can be strictly more efficient computationally (compressing four–dimensional diagrammatics into one–dimensional worldline or two–dimensional worldsheet problems), 4 3. non–string encodings naturally incorporate structures (such as anomaly coefficients, Ext/Tor 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: • 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. B. Goals of this paper 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. 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: 1. an encoding category Enc (typically a full subcategory or a closely related ( ∞, 1)–category) whose objects are encoding theories E that are simpler, lower–dimensional, or otherwise more tractable than the bulk theory B; 5 2. an encoding functor H:QFT −→ Enc,(7) which maps a bulk theory B to its encoding E = H ( B )and likewise maps morphisms in QFT to morphisms between encodings. To make any sense of a holographic equivalence between B and E = H ( B ), we must also specify a context in which that equivalence is evaluated. This is done by introducing a pair of functors F:QFT →Obs, G :Enc →Obs,(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 Band Ethen becomes the statement that there exists a natural equivalence αB:F(B)≃ −−→ G(H(B)) (9) 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 spacetime dimension (or substantially fewer degrees of freedom) than the bulk theory B, i.e. dim(E)<dim(B),(10) or more generally dimeff(E)dimeff(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]. 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. dim(E) = dim(B)+1,(11) 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 kis fixed by matching to the SU(Nc)3chiral 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=Nc. 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. 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: 1. Strong–field, rapidly varying spinor QED. We consider four–dimensional spinor QED in an arbitrary classical electromagnetic background Aµ ( x )and focus on non–perturbative electron– positron pair production in strong, time–dependent electric fields. At one loop, the exact effective action can be written in worldline form as a path integral over a 0+1D super–particle with action Swl[x, ψ;A] = ZT 0 dτ ˙x2 4+1 2ψµ˙ ψµ−ie ˙xµAµ(x(τ)) + ie ψµFµν(x(τ))ψν,(12) where xµ ( τ )are periodic bosonic coordinates and ψµ ( τ )are anti–periodic Grassmann fields [ 8 ]. Non– perturbative pair production in strong, rapidly varying fields (e.g. Sauter pulses) can be computed semiclassically 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. 2. Anomaly/WZW holography in QCD. For QCD with Nf massless quark flavours and Nc colours, the SU( Nf ) 3 chiral anomaly is captured by a six–dimensional anomaly polynomial I6 whose coefficient is proportional to Nc [ 69 ]. In the low–energy regime, the effective theory is a non–linear sigma model on SU(Nf)with a WZW term SWZW[U] = ki 240π2ZB5 Tr(U−1dU)5,(13) defined on a 5D manifold B5 with boundary M4 , where U ( x ) ∈SU ( Nf )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 Fanom ( B ) ≃G ( E )in the chiral–anomaly context, one uniquely fixes k = Nc 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. 3. Flux–tube holography in confining QCD. In confining Yang–Mills theory, the potential between a static quark–antiquark pair at large separation R is believed and observed to behave as V(R) = σR −π 12R+O(R−3),(14) where σ is the string tension and the term −π/ (12 R )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. 4. Conceptual extension to UP–RG / UP–XC encodings. Finally, we explore how renormalisation 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 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. 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. C. Main claims 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 Bto 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 αB:F(B)≃ −−→ G(H(B)) (15) in Obs for each bulk object Bin a specified class. 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 Ei and functors Hi 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. 8 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: 1. Applicability to broader classes of theories. String–based holography (AdS/CFT and its variants) 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. 2. Faithfulness to relevant structure. Different physical questions activate different kinds of structure: 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. 3. Better computational efficiency. In specific examples that we develop later in the paper, we show 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 Aµ ( 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. • 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. • 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. 9 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 Ewith the following properties: • The bulk theories B are familiar QFTs: four–dimensional spinor QED in strong, time–dependent fields; SU(Nc)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. • 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: 1. the bulk problem is intractable (or at least very difficult) in naive 4D diagrammatic or lattice terms, 2. the encoding problem is tractable in 0+1 or 1+1 dimensions, 3. and the equivalence F ( B ) ≃G ( E )is either exact (at a given loop order) or holds to controlled 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. II. CONCEPTUAL FRAMEWORK: STRING–FREE HIGHER–CATEGORICAL HOLOGRAPHY A. The ambient (∞,1)–categories 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 Z:BordS d−→ C,(17) where BordS d is a d –dimensional bordism ( ∞, d )–category with some tangential structure S (e.g. framed, oriented, spin) and C is a symmetric monoidal ( ∞, 1)–category of “targets” (e.g. VectC , 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 Z1, Z2 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]. 16 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 dimension — both are instances of the same higher–categorical pattern: F(B)αB −−−→ G(H(B)) in Obs,(41) with H:QFT →Enc, F :QFT →Obs, G :Enc →Obs.(42) The only difference is the relationship between the dimension of H ( B )and that of B , as measured by the functor dim: dim H(B)=       d0< d, compression holography, d+ 1,anomaly holography, d, self–dimensional holography (e.g. dualities). (43) 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 observables F(B)faithfully. 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→1for loop amplitudes (as in the worldline formalism), • supplemented by an anomaly holography 4 ↔ 5for chiral anomalies in the presence of pseudoscalar or axial couplings. Similarly, in QCD one may consider both flux–tube super–holography 4 → 2and anomaly holography 4 ↔ 5for chiral anomalies. The higher–categorical framework developed here accommodates such hybrid situations naturally: one simply composes or layers multiple encoding functors Hiand context functors Fi, Gi, 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. D. Enrichment and loop grading 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), 17 •inner products and adjoints (unitarity, reflection positivity), •probabilistic structure (transition probabilities, Markov kernels), •topological and homological data (Ext/Tor contributions, anomaly classes), •loop expansions (organising contributions by loop order, powers of ~or coupling constants). 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 Cconsists of: •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 ◦X,Y,Z :C(Y, Z)⊗C(X, Y )−→ C(X, Z),(44) •for each X∈Ob(C), a unit morphism uX: 1 −→ C(X, X),(45) such that the usual associativity and unitality axioms for composition hold, expressed now as commutative diagrams in V[31, 32]. Explicitly, associativity requires that for any W, X, Y, Z the diagram C(Z, W )⊗C(Y, Z)⊗C(X, Y )C(Z, W)⊗C(X, Z) C(Y, W)⊗C(X, Y )C(X, W) id⊗◦ ◦⊗id ◦ ◦ (46) commutes in V , and unitality requires that certain triangles involving uX commute. When V = Set with ⊗ = × , this reduces to the usual notion of (ordinary) category. In our applications, however, we are typically interested in Vthat carry linear, Hilbert, topological, or probabilistic structure. AV–enriched functor F:C → D between V–enriched categories is given by: •a function on objects F: Ob(C)→Ob(D), •for each pair X, Y ∈Ob(C), a morphism in V FX,Y :C(X, Y )−→ D(F(X), F(Y)) (47) 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 = VectC , 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 VectC–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, 18 • enrichment ensures that functors such as H, F, G are linear on morphisms, preserving superposition. 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 Chas: •Hilbert–space hom–objects C(X, Y ), •composition maps that are bounded bilinear maps, •unit morphisms that behave like identity operators. When 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 V=VectC×Top,(48) with tensor product ( V1, X1 ) ⊗ ( V2, X2 ) = ( V1⊗V2, X1×X2 ). A V –enriched category C then assigns to each pair of objects X, Y a pair C(X, Y ) = VX,Y , XX,Y ,(49) where VX,Y is a vector space of amplitudes and XX,Y is a topological space (or homotopy type) representing additional data such as: •homotopy classes of defects or domain walls between Xand Y, •elements of Ext/Tor groups in some homology theory [33], •topological sectors of gauge bundles or instantons. 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 Γ[Φ] = ∞ X `=0 ~`Γ(`)[Φ],(50) where Γ (`) collects all ` –loop contributions. It is natural to reflect this structure categorically by enriching over a graded monoidal category. 19 Let V be a symmetric monoidal category. A graded version of V , denoted VN , 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 (V(•)⊗W(•))(n):= M p+q=n V(p)⊗W(q),(51) and unit object given by ( 1 , 0 , 0 , . . . ). This is a standard construction in the theory of graded categories and graded tensor product [32]. AVN–enriched category Cthen has hom–objects C(X, Y ) = C(0)(X, Y ),C(1)(X, Y ),C(2)(X, Y ), . . . ,(52) where C(`) ( X, Y ) ∈ V encodes the ` –loop component of the morphism space from X to Y . Composition uses (51) and thus satisfies C(n)(X, Z)⊇X p+q=n◦(p,q)C(p)(Y, Z)×C(q)(X, Y ),(53) 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 = VectC or Hilb , so that hom–objects are graded vector spaces or graded Hilbert spaces: C(X, Y ) = M `≥0C(`)(X, Y ),(54) with C(`) ( X, Y )corresponding to ` –loop contributions. The loop expansion (50) is then a shadow of this 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 QFTN denote a loop–graded version of QFT , where objects carry graded hom–spaces and effective actions split as in (50) . An encoding functor H:QFTN−→ EncN(55) is then said to be loop–graded if it maps the ` –loop component of a given morphism to the ` –loop component of the corresponding morphism in the encoding category, i.e. H(`) X,Y :QFT(`)(X, Y )−→ Enc(`)H(X), H(Y)(56) for all objects X, Y and all `≥0, and is compatible with the graded composition structure. Similarly, context functors F:QFTN→ObsN, G :EncN→ObsN,(57) may be required to preserve loop grading, so that relative holography at each loop order is expressed as an equivalence F(`)(B)≃ −−→ G(`)H(B).(58) 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. 20 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 Hmaps QCD to an anomaly TQFT in one higher dimension. • 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. III. STRING–FREE HOLOGRAPHIC PRINCIPLE A. Formal statement 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 H:Bulk −→ Enc,(59) together with context functors F:Bulk →Obs, G :Enc →Obs,(60) 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 H:Bulk →Enc,(61) 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), 21 •or a universal encoding for a given observable context (UP–holography). The functor H also assigns to each morphism f : B1→B2 in Bulk a morphism H ( f ) : H ( B1 ) →H ( B2 ) 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 F:Bulk →Obs, G :Enc →Obs,(62) 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 Eits anomaly inflow functional. •F assigns to B its non–perturbative effective action in a specified background field; G assigns to a worldline encoding Ethe 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: α:F≃ =⇒G◦H. (63) By definition, such a natural transformation consists of: •for each bulk object B∈Bulk, an equivalence αB:F(B)≃ −−→ GH(B)(64) in Obs, • for each morphism f : B1→B2 in Bulk , a higher–homotopy–commuting diagram in Obs expressing the naturality of α, i.e. that GH(f)◦αB1≃αB2◦F(f).(65) Equivalently, we can draw the relative holography square B H(B) F(B)G(H(B)) H FG αB ≃ (66) which commutes up to coherent higher homotopies. Informally, αB 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). 22 String–free holographic principle. With this structure in place, we can now state the core principle of this paper. 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: 1. an (∞,1)–functor H:Bulk →Enc, 2. an (∞,1)–functor F:Bulk →Obs (bulk context functor), 3. an (∞,1)–functor G:Enc →Obs (encoding context functor), 4. a natural equivalence α:F≃ =⇒G◦Hin Obs, such that: •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. If such data exist, we say that the bulk theory B∈Bulk is holographically encoded by E = H ( B )in context (F, G). Several comments are important: • 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. • 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 dimeff ( H ( B )) dimeff ( 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. • 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. 23 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: 1. specify Bulk,Enc and Obs, 2. construct explicit functors H, F, G, 3. prove the existence (or at least the existence up to controlled approximations) of the natural equivalence α. In 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. B. Strings as a special case 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: 1. To show how AdS/CFT can be recast in the language of bulk categories Bulk , encoding categories Enc, and context functors F, G, with a natural equivalence F(B)≃G(E). 2. To emphasise that this string–based realisation of holography works only in particular regimes (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 AdS5×S5 with N units of five–form flux is conjecturally dual to four–dimensional N = 4 supersymmetric SU( N )Yang–Mills theory on R1,3 or on S3×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, 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). We can then define a (heuristic) encoding functor HAdS/CFT :Bulk −→ Enc,(67) which maps a string background B (such as type IIB on AdS5×S5 with flux N ) to a boundary CFT E = HAdS/CFT ( B )(such as N = 4 SU( N )SYM). The exact mathematical construction of HAdS/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: •FAdS : 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. •GCFT : Enc →Obs that maps a CFT E to its correlation functions of local operators with given sources. More concretely, let Φ( x, z )denote a bulk field (e.g. a scalar) on AdSd+1 with boundary coordinate x and radial coordinate z , and let φ0 ( x ) = limz→0z−∆− 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, Zbulk[φ0] = ZΦ|∂AdS=φ0DΦe−Sbulk[Φ],(68) is equal (up to normalisation) to the generating functional of correlation functions in the dual CFT with source φ0for an operator Oof dimension ∆: Zbulk[φ0]≡ZCFT[φ0] = exp Zddx φ0(x)O(x)CFT .(69) The corresponding n –point functions are obtained by differentiating Zbulk [ φ0 ]with respect to φ0 and then setting φ0 = 0. The map FAdS thus assigns to B the functional Zbulk [ φ0 ]and its derivatives, while GCFT assigns to E the functional ZCFT [ φ0 ]and its derivatives. The GKPW relation (69) is precisely an instance of a relative holography equivalence αB:FAdS(B)≃ −−→ GCFTHAdS/CFT(B)(70) 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 F(B)≃G(H(B)),(71) 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 AdS5×S5/N= 4 SYM duality: • The bulk description in terms of classical supergravity is valid in the limit of large ’t Hooft coupling λ = g2 YMN 1and large N , where string loop and α0 corrections are suppressed [ 38 ]. In this regime, certain protected operators and states can be computed from geodesics or classical branes in AdS. 25 • 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 α0 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: •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). In 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◦Hholds only in those contexts. Limitations of string encodings for realistic QFTs. The string–based encoding category Encstring 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. 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 Hstring :BulkSYM →Encstring,(72) where BulkSYM is a subcategory of QFTs (e.g. superconformal Yang–Mills theories) and Encstring is a subcategory of string/brane backgrounds. • For appropriate context functors FAdS, GCFT selecting protected correlators and geometric observables, there is a natural equivalence FAdS(B)≃GCFTHstring(B),(73) yielding an AdS/CFT–type holography. • However, Encstring 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 Hstring 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. 32 • identify the natural extended objects (worldlines, worldsheets, defects, boundaries, higher– codimension excitations); • identify the non–geometric structures (cohomology, anomaly classes, factorization algebras, 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. 1. 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. • 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). 2. Identifying non–geometric structures. In addition to the geometry of extended objects, one must also identify the relevant non–geometric structures that govern the observables in context F: • 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. 33 • 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. • 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. Thus, in addition to the geometry of the excitations, one must match the algebraic and cohomological structures of the problem with the encoding category. 3. Systematic procedure for selecting Enc .Putting these observations together, we can outline a systematic procedure for selecting an encoding category Enc given a bulk theory B and a context functor F: 1. Analyse the observables F ( B ) . Determine the type of observable data captured by F ( B ): 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 . 2. Identify the dominant structures in Bfor this context. Determine which degrees of freedom 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. 3. Choose a candidate encoding category Enc. Based on the identified excitations and structures, 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. • 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. • If entanglement and reconstruction dominate, choose Enc as a code or tensor–network category. 4. Construct encoding and context functors. Define H : Bulk →Enc by mapping B to an 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). 5. Prove (or test) relative holography. Show that there exists a natural equivalence αB : 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). 6. Investigate universality. If possible, show that ( E, φ )is universal in the fiber category FibF ( 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 Hrealises 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. 34 4. Strings only when worldsheets are intrinsic. Within this strategy, string and brane encodings arise only in situations where the natural extended objects and algebraic structure point towards a 2D worldsheet QFT in a higher–dimensional geometric background. More precisely, strings and branes are appropriate encodings when: • the relevant excitations of the bulk theory can be naturally organised as one–dimensional extended objects 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. 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. 5. Summary. To summarise, the selection of the encoding category Enc is not arbitrary: it is guided by the structural features of the bulk theory B and the observables in context F . By systematically analysing: •the natural extended objects (worldlines, worldsheets, defects, boundaries), • the non–geometric algebraic and cohomological structures (cohomology, anomalies, factorization algebras, code categories), 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. V. SUPER–HOLOGRAPHY IN QED: WORLDLINE (0+1D) ENCODING OF 4D QED A. Spinor QED and the worldline encoding (one loop) 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 Sspin wl [x, ψ;A] = ZT 0 dτ ˙x2 4+1 2ψµ˙ ψµ−ie ˙xµAµx(τ)+ie ψµFµνx(τ)ψν,(81) 35 and to show that the one–loop effective action in Euclidean space can be written exactly as Γ(1)[A] = −1 2Z∞ 0 dT Te−m2TZx(T)=x(0) DxZψ(T)=−ψ(0) Dψexp −Sspin wl [x, ψ;A],(82) for an arbitrary background gauge field Aµ ( x ), with Fµν = ∂µAν−∂νAµ . This worldline expression preserves the full 4–vector structure of Aµ and Fµν and is an exact rewriting of the one–loop determinant, not an approximation. Remark 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. 1. Spinor QED and the one–loop determinant. We begin with the standard action of spinor QED in Minkowski signature: SQED[¯ ψ, ψ, A] = Zd4x−1 4FµνFµν +¯ ψiγµDµ−mψ,(83) where the covariant derivative is Dµ=∂µ+ieAµ(x),(84) and we use the mostly–minus metric convention ηµν = diag (+ ,−,−,− ). Integrating out the fermionic fields (while treating Aµas a classical background) yields the effective action Γ[A]defined by eiΓ[A]=ZD¯ ψDψexp iZd4x¯ ψiγµDµ−mψ.(85) Since the fermionic path integral is Gaussian, we obtain Γ[A] = −iln det i/ D−m,/ D=γµDµ.(86) The object of interest is thus the functional determinant of the Dirac operator i/ D−m in the background Aµ(x). To put this into a form suitable for a Euclidean worldline path integral, it is convenient to Wick–rotate to Euclidean signature, x0=−ix4, and to employ the Schwinger proper–time representation. 2. Squaring the Dirac operator. A useful trick, going back to Feynman and Schwinger, is to rewrite the log–determinant of the Dirac operator in terms of a second–order differential operator. We note the identity (formally) ln det i/ D−m=1 2ln det −(/ D+im)( / D−im).(87) Equivalently, setting M=/ D+im, we have ln det i/ D−m=1 2ln det MM†,(88) 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 γµsatisfying {γµ, γν}= 2δµν , and define / DE=γµDµ, Dµ=∂µ+ieAµ(x),(89) with Euclidean indices µ= 1,2,3,4. The Euclidean Dirac operator is then DE=/ DE+m. (90) 36 The Euclidean effective action is ΓE[A] = −ln det DE.(91) We now compute D† EDE= (−/ DE+m)( / DE+m) = −D2+e 2σµνFµν +m2,(92) where D2=DµDµ, σµν := 1 2[γµ, γν], Fµν := ∂µAν−∂νAµ.(93) Let us briefly sketch the derivation of (92). One has / D2 E=γµγνDµDν=1 2{γµ, γν}DµDν+1 2[γµ, γν]DµDν=D2+1 2σµν[Dµ, Dν].(94) Using [Dµ, Dν] = ieFµν, this becomes / D2 E=D2+ie 2σµνFµν.(95) Therefore D† EDE= (−/ DE+m)( / DE+m) = −/ D2 E+m2=−D2−ie 2σµνFµν +m2.(96) Up to a convention for the factor of i (which can be absorbed into the definition of Fµν in Euclidean space), we can write this as D† EDE=−D2+e 2σµνFµν +m2,(97) which agrees with (92). The Euclidean one–loop effective action can now be written as Γ(1) E[A] = −ln det DE=−1 2ln det D† EDE=−1 2ln det −D2+e 2σµνFµν +m2.(98) We have thus reduced the problem to a determinant of a second–order operator with a scalar Laplacian −D2and a spin–coupling term e 2σµνFµν. 3. Schwinger proper–time representation. We next use Schwinger’s proper–time representation to rewrite the log–determinant as a proper–time integral. For a positive (or suitably regularised) operator M, one has the formal identity ln det M= Tr ln M=−Z∞ 0 dT TTr e−T M ,(99) where the trace is over spacetime and internal indices. Applying (99) to M = −D2 + e 2σµνFµν + m2 yields Γ(1) E[A] = 1 2Z∞ 0 dT Te−m2TTr exp −Th−D2+e 2σµνFµνi.(100) Our task is now to express the trace of the heat kernel K(T;x, y) = hx|exp −T(−D2+e 2σ·F)|yi(101) 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. 37 4. Worldline path integral representation. The operator −D2 + e 2σµνFµν acts on spinor wavefunctions. One can regard the trace in (100) as a sum over closed paths with spin degrees of freedom. Specifically, Tr e−T(−D2+e 2σ·F)=Zd4xtrspinhx|e−T(−D2+e 2σ·F)|xi,(102) where trspin denotes the trace over spinor indices. The kernel can be written as a worldline path integral over xµ ( τ )with periodic boundary conditions x (0) = x ( T ) = x0 , and over Grassmann variables ψµ ( τ ) encoding spin, with anti–periodic boundary conditions ψ(0) = −ψ(T). More concretely, one can show that hx|e−T(−D2+e 2σ·F)|xi=Zx(T)=x x(0)=xDxZψ(0)=−ψ(T)Dψexp −Sspin wl [x, ψ;A],(103) where the worldline action Sspin wl [ x, ψ ; A ]is given by (81) . The derivation proceeds by comparing the evolution operator for a relativistic spinning particle in an external electromagnetic field with the heat kernel; the Grassmann fields ψµ ( τ )satisfy {ψµ ( τ ) , ψν ( τ ) } = δµν and generate a Clifford algebra upon quantisation, reproducing the gamma matrices and spin couplings. The action (81) can be understood as follows: • The term ˙x2 4 is the usual kinetic term for a relativistic point particle in Euclidean proper time parameter τ∈[0, T]. • The term 1 2ψµ˙ ψµ 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 ˙xµAµ(x(τ)) is the coupling of the particle to the external gauge field (Wilson line). • The term ie ψµFµν ( x ( τ )) ψν reproduces the spin–field interaction e 2σµνFµν 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 x0, we obtain Γ(1) E[A] = 1 2Z∞ 0 dT Te−m2TZx(T)=x(0) DxZψ(T)=−ψ(0) Dψexp −Sspin wl [x, ψ;A],(104) 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, Γ(1)[A] = −1 2Z∞ 0 dT Te−m2TZx(T)=x(0) DxZψ(T)=−ψ(0) Dψexp −Sspin wl [x, ψ;A],(105) which is precisely (82). 5. Exactness and full vector structure. We emphasise several important features of the worldline representation (82): • The background gauge field Aµ ( x )and its field strength Fµν ( x )enter the worldline action Sspin wl [ x, ψ ; A ]with their full four–vector and tensor structure. There is no restriction to slowly varying or weak fields at this stage; the worldline representation is exact at one loop for arbitrary backgrounds. • The spinor nature of the fermion is fully captured by the Grassmann fields ψµ ( τ )and the term ie ψµFµν ( x ( τ )) ψν . Upon integrating over ψ , one reconstructs the spin factor associated with σµνFµν . • 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 τ7→ 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. 38 6. Categorical interpretation: bulk, encoding, and context. In the language of Sections II–III, the derivation above realises a concrete instance of the string–free holographic square (66) . We can identify: • Bulk category: BulkQED , a full subcategory of QFT whose objects include 4D spinor QED with background gauge fields. Our bulk theory is B= QED4. • Encoding category: Encwl , the ( ∞, 1)–category of (0 + 1)–dimensional worldline super–quantum mechanics coupled to background gauge fields Aµ ( x )and field strengths Fµν ( x ). An object of Encwl is determined by a worldline action such as Sspin wl [x, ψ;A]. • Encoding functor: Hwl : BulkQED →Encwl , which maps the Dirac QED theory B to the worldline theory E = Hwl ( B )defined by (81) . On morphisms (e.g. background–field variations or coupling deformations), Hwl maps them to corresponding deformations of the worldline action. • Context functor (bulk): F1loop : BulkQED →Obseff , which assigns to B its one–loop effective action Γ(1)[A]in an arbitrary background Aµ(x). • Context functor (encoding): G1loop : Encwl →Obseff , which assigns to a worldline theory E the corresponding worldline path integral over closed loops, interpreted as an effective action functional of Aµ(x). The Schwinger/worldline derivation shows that there is a natural equivalence αB:F1loop(B)≃ −−→ G1loopHwl(B),(106) realised explicitly by (82). In other words, in the one–loop context, Γ(1)[A] = F1loop(B)≡G1loop(E) = G1loopHwl(B),(107) with equality understood as an equality of functionals in Obseff . 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. 7. Outlook. In subsequent subsections we will exploit this worldline representation to compute non– perturbative pair production rates in strong, time–dependent fields via worldline instantons, and we will extend the encoding to two loops by incorporating bilocal photon kernels along the worldline. This will demonstrate that the worldline encoding is not merely a formal rewriting but a practically powerful tool: it compresses an infinite tower of multi–photon Feynman diagrams in four dimensions into a manageable one–dimensional (and at two loops, two–parameter) problem. This is a paradigmatic example of string–free higher–categorical super–holography in a realistic quantum field theory. B. Worldline instantons and pair production in constant fields 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. 1. Scalar QED effective action in worldline form. Consider scalar QED in Minkowski spacetime, with Lagrangian L=−1 4FµνFµν + (Dµφ)†Dµφ−m2φ†φ, (108) 39 where Dµ = ∂µ + ieAµ ( x )and Fµν = ∂µAν−∂νAµ . Integrating out the complex scalar field in the presence of a classical background Aµyields the one–loop effective action Γ(1)[A] = iln det −D2−m2=iTr ln −D2−m2.(109) As in the spinor case, we Wick rotate to Euclidean space and use Schwinger’s proper–time representation. The Euclidean one–loop effective action is Γ(1) E[A] = −ln det −D2+m2,(110) and the proper–time identity (99) gives Γ(1) E[A] = Z∞ 0 dT Te−m2TTr exp TD2.(111) The trace of the heat kernel can be represented as a worldline path integral (see e.g. [8, 57]), namely Tr eT D2=Zd4x0Zx(0)=x(T)=x0Dx(τ) exp −ZT 0 dτ ˙x2 4+ie ˙xµAµx(τ)!,(112) where xµ ( τ )are periodic paths in Euclidean spacetime with period T , and ˙x2 := ˙xµ˙xµ . The worldline action for scalar QED is therefore Sscalar wl [x;A] = ZT 0 dτ ˙x2 4+ie ˙xµAµx(τ).(113) Inserting (112) into (111) and factoring out the volume V4 = Rd4x0 of spacetime, we obtain the effective Lagrangian L(1) E= Γ(1) E/V4: L(1) E[A] = Z∞ 0 dT Te−m2TZx(0)=x(T)Dxexp −Sscalar wl [x;A].(114) This is the scalar analogue of (82) with Grassmann fields omitted. 2. Constant electric field in Euclidean space. We now specialise to a constant electric field in, say, the x3 direction. In Minkowski space, a homogeneous electric field E is described by F03 = −F30 = E , with all other components zero. In Euclidean space, after Wick rotation x0→ −ix4 , this becomes a constant imaginary magnetic field in the (x3, x4)plane, with FE 34 =iE, (115) and one convenient gauge choice is A3(x) = −iEx4, A4(x)=0, A1=A2= 0.(116) Then the worldline coupling term ie ˙xµAµbecomes ie ˙xµAµ=ie ˙x3A3(x) = ie ˙x3(−iEx4) = eE ˙x3x4.(117) Hence the scalar worldline action (113) for this background is Sscalar wl [x;E] = ZT 0 dτ 1 4˙x2 1+ ˙x2 2+ ˙x2 3+ ˙x2 4+eE ˙x3x4.(118) The path integral over the transverse directions x1, x2 factorises and yields the usual free Gaussian contributions; the nontrivial dynamics is in the ( x3, x4 )plane, where the electric field produces a coupling between ˙x3and x4. 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 xcl(τ)that extremise the worldline action (118). 40 3. Equations of motion and circular instantons. The classical equations of motion for xµ ( τ )are obtained by varying Sscalar wl [x;E]: δS δxµ(τ)= 0 ⇒¨xµ= 2ieFE µν ˙xν,(119) where FE µν is the Euclidean field strength. In our constant–field gauge (116) , the only non–vanishing components are FE 34 =iE and FE 43 =−iE. Therefore (119) becomes: ¨x3= 2ieFE 3ν˙xν= 2ieFE 34 ˙x4= 2ie(iE) ˙x4=−2eE ˙x4,(120) ¨x4= 2ieFE 4ν˙xν= 2ieFE 43 ˙x3= 2ie(−iE) ˙x3= 2eE ˙x3.(121) The x1, x2 directions are free and we may take them to be constant along the instanton (otherwise they only contribute trivial Gaussian fluctuations). Differentiating (120) with respect to τand substituting (121), we obtain ... x3=−2eE ¨x4=−4e2E2˙x3.(122) Thus ˙x3satisfies the harmonic oscillator equation with frequency 2eE: ¨ ˙x3+ (2eE)2˙x3= 0,(123) so that ˙x3 ( τ )is a linear combination of cos (2 eEτ )and sin (2 eEτ ). Similarly, one finds that ˙x4 ( τ )is a π/2–phase–shifted combination of the same frequencies. More systematically, we look for solutions of the form x3(τ) = Rcos(ωτ), x4(τ) = Rsin(ωτ),(124) representing a circle of radius Rin the (x3, x4)plane with angular frequency ω. Computing derivatives, ˙x3=−Rω sin(ωτ),˙x4=Rω cos(ωτ),¨x3=−Rω2cos(ωτ),¨x4=−Rω2sin(ωτ).(125) Substituting (124) into (120) and (121), we obtain: ¨x3=−Rω2cos(ωτ)! =−2eE ˙x4=−2eE Rω cos(ωτ),(126) ¨x4=−Rω2sin(ωτ)! = 2eE ˙x3= 2eE (−Rω sin(ωτ)).(127) These yield the same condition: ω2= 2eE ω ⇒ω= 2eE, (128) for a nontrivial solution ( ω6 = 0). Thus the classical worldline instanton is a circle of constant radius R in the (x3, x4)plane, with angular frequency fixed by the field: x3(τ) = Rcos(2eEτ), x4(τ) = Rsin(2eEτ).(129) Next, we impose the periodicity condition x (0) = x ( T ): the path must close after Euclidean proper time T. The circle parametrisation implies 2eET = 2πn, n ∈Z,(130) so that Tn=πn eE , n = 1,2,.... (131) Each integer ncorresponds to an n–fold winding of the instanton around the circle. In addition, the reparametrisation invariance of the worldline action implies a constraint on ˙x2 ( 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 ˙x2(τ) = const on shell. Here, ˙x2= ˙x2 3+ ˙x2 4=R2ω2sin2(ωτ) + cos2(ωτ)=R2ω2.(132) 41 Thus ˙x2 is indeed constant. For the tunnelling solutions of interest in pair production, it can be shown (via a stationary–phase analysis of the proper–time integral) that the on–shell condition is ˙x2 = 4 m2 , i.e. the “mass–shell” condition of the relativistic particle. This yields R2ω2= 4m2⇒R=2m ω=m eE .(133) We thus obtain the classical worldline instanton for scalar QED in a constant electric field: x(n) 3(τ) = m eE cos 2πnτ Tn, x(n) 4(τ) = m eE sin 2πnτ Tn,(134) with Tngiven by (131). 4. Classical action and Schwinger exponent. The on–shell action of the worldline instanton gives the semiclassical exponent controlling the non–perturbative pair–production rate. Plugging (133) and (131) into (118), we find the classical action for the n–th winding: S(n) wl,cl =ZTn 0 dτ ˙x2 4+eE ˙x3x4on shell .(135) We evaluate each term separately. First, ˙x2 4=1 4R2ω2=1 4·m2 (eE)2·4e2E2=m2.(136) Therefore, ZTn 0 dτ ˙x2 4=m2Tn=m2πn eE .(137) Second, for the field–dependent term: ZTn 0 dτ eE ˙x3x4=eE ZTn 0 dτ ˙x3x4.(138) Inserting the instanton ansatz, ˙x3=−Rω sin(ωτ), x4=Rsin(ωτ),(139) we obtain ˙x3x4=−R2ωsin2(ωτ),(140) so ZTn 0 dτ ˙x3x4=−R2ωZTn 0 dτ sin2(ωτ) = −R2ωTn 2.(141) Using R2ω2= 4m2and Tn=πn/(eE), we find R2ω=4m2 ω=4m2 2eE =2m2 eE ,(142) so ZTn 0 dτ ˙x3x4=−2m2 eE ·Tn 2=−m2 eE Tn=−m2 eE ·πn eE =−πnm2 e2E2.(143) Therefore, the second term contributes ZTn 0 dτ eE ˙x3x4=eE −πnm2 e2E2=−πnm2 eE .(144) 48 In the categorical language of Section IV, the bulk category BulkQED contains spinor QED with time– dependent backgrounds, the encoding category Encwl contains worldline super–QFTs with time–dependent couplings, and the context functor Fpair selects non–perturbative pair–production rates. The equivalence Fpair(B)≃GinstHwl(B)(187) is realised semiclassically by the worldline instanton solutions and their fluctuations. This is a compelling example of string–free higher–categorical super–holography in a realistic, strongly driven quantum field theory. D. Capability: 4D →1D super–holography vs. diagram explosion In 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. 1. The 4D diagrammatic expansion for pair production. Let us first recall how non–perturbative pair production appears in the standard 4D Feynman–diagrammatic approach. For definiteness, consider spinor QED in Minkowski space coupled to an external classical field Aµ(x). The effective action is Γ[A] = −iln det i/ D−m,(188) and the imaginary part of Γ[ A ]yields the rate of vacuum decay into electron–positron pairs. From a Feynman–diagrammatic viewpoint, Γ[A] = ∞ X n=1 in nAn[A],(189) where An [ A ]represents the sum of all one–loop diagrams with n external photon legs, each leg corresponding to an insertion of the background field Aµ(x). More concretely, if one expands the interaction term in the Dirac action, ¯ ψγµ(i∂µ−eAµ)ψ=¯ ψ(i/ ∂−m)ψ−e¯ ψγµAµψ, (190) the n –th order term in e in the effective action involves an n –point function of the current jµ ( x ) = ¯ ψγµψ ( x ): An[A] = (−e)nZd4x1···d4xnAµ1(x1)···Aµn(xn)hjµ1(x1)···jµn(xn)i1PI,(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 γ(k1) + γ(k2) + ···+γ(kn)→e+e−,(192) where the γ ( ki )represent background-field photons with momenta ki (obtained by Fourier transforming Aµ ( x )), and the sum runs over all n≥nmin consistent with energy–momentum conservation. In a 49 time–dependent background such as the Sauter pulse (153) , this means summing over all processes in which the electron absorbs Nquanta of frequency ωto overcome the mass gap 2m: Nω ≥2m, N = 1,2,.... (193) From a purely perturbative QED perspective, the pair–production probability is given by an infinite series of the form Pe+e−∼∞ X N=Nmin MN2,(194) where MN is the amplitude for N -photon absorption. Each MN is a sum over many Feynman diagrams involving Nexternal legs, internal Dirac propagators, and complicated spin and momentum structures. The difficulty is twofold: • 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 E0 ) 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. 2. Worldline master formula: all diagrams at once. In stark contrast, the worldline formalism expresses the entire one–loop effective action at once as a path integral over a single worldline action. For spinor QED, as derived in Subsection V A, we have in Euclidean space: Γ(1) E[A] = −1 2Z∞ 0 dT Te−m2TZx(T)=x(0) DxZψ(T)=−ψ(0) Dψexp −Sspin wl [x, ψ;A],(195) with the worldline action Sspin wl [x, ψ;A] = ZT 0 dτ ˙x2 4+1 2ψµ˙ ψµ−ie ˙xµAµx(τ)+ie ψµFµνx(τ)ψν.(196) Expanding the exponent in powers of e and Aµ , one can show that (195) reproduces all one–loop diagrams with any number of external legs: • the term linear in Aµ corresponds to a single insertion of the background field (e.g. vacuum polarisation); •terms with nfactors of Aµencode nexternal legs; • the spin factor exp ( ie RT 0dτ ψµFµνψν )generates all spin–dependent insertions, such as σµνFµν 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 Aµ. 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]. 3. From 4D diagrams to 1D ODEs in fast Sauter pulses. In the specific case of fast, strong Sauter pulses, the non–perturbative pair–production rate cannot be captured by any finite set of diagrams in the expansion (189) . Instead, as shown in Subsection VC, the worldline master formula (195) can be analysed semiclassically in terms of worldline instantons, reducing the problem to: •solving the first–order system du ds = 2γsin θ(s),dθ ds = 2 sec2u(s),(197) for (u(s), θ(s)) with periodic boundary conditions, 50 •computing the classical action S0[xinst] = m2 eE0g(γ)from the solution, •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 ( x3, x4 )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: •4D Feynman–diagram view: – infinitely many n –photon absorption processes γ + ··· + γ→e+e− contribute, with n unbounded; –each amplitude Mninvolves 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. •1D worldline–instanton view: –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); – the exponent is given by a simple functional S0 = m2 eE0g ( γ ), and the prefactor is determined by 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. 4. Complexity and scaling: qualitative comparison. Let us quantify the complexity gain at a qualitative level. Denote by Neff the effective number of background photons that can participate in the pair– production process at a given set of parameters ( E0, ω ); typically, Neff ∼mω/ ( eE0 )in the multiphoton regime. Then: • In the diagrammatic expansion, one must, in principle, sum over all N&Nmin up to Neff ; 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 C4D ∼ Neff X N=Nmin #diagrams(N)×Cint 4D,(198) where Cint 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: C1D ∼ Csolve ODE +Cfluctuations,(199) where Csolve ODE is the cost of solving the 1D ODE system (197) (typically scaling linearly with the grid size in s ) and Cfluctuations 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 Neff. 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. 51 5. Categorical formulation of the capability. From the higher–categorical perspective, the capability of 4D → 1D super–holography can be summarised as follows. In the context of non–perturbative pair production: • The bulk category BulkQED contains objects B corresponding to 4D spinor QED in external fields. •The encoding category Encwl contains objects Ecorresponding to worldline super–QFTs. • The encoding functor Hwl : BulkQED →Encwl maps B to its worldline representative E with action (196). • The context functor Fpair : BulkQED →Obspair assigns to B its non–perturbative pair–production data (exponent and prefactor). • The encoding context functor Ginst : Encwl →Obspair assigns to E the same data extracted from worldline instantons and their fluctuations. The worldline formalism and instanton analysis demonstrate a natural equivalence αB:Fpair(B)≃ −−→ GinstHwl(B),(200) which is the precise string–free holographic equivalence in this context. The crucial point is that: •Hwl is a compression functor in the sense of Subsection II C: it maps a 4D theory to a 0+1D theory; •Fpair is a context functor that selects non–perturbative observables, not all correlators; •Ginst 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. 6. Conclusion: a clear instance of string–free super–holography. To conclude, the worldline formalism in strong–field QED provides one of the clearest and most concrete examples of string–free higher–categorical super–holography: • 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. • 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. VI. TWO–LOOP SUPER–HOLOGRAPHY IN SPINOR QED A. Two–loop worldline representation with spin 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, Γ(2)[A] = 1 2Z∞ 0 dT Te−m2TZDxDψ e−Sspin wl [x,ψ;A]Ispin photon[x, ψ],(201) 52 where Sspin wl [x, ψ;A]is the worldline action (196), and Ispin photon[x, ψ] = e2ZT 0 dτ1ZT 0 dτ2Jµ(τ1)Jµ(τ2)Dx(τ1)−x(τ2),(202) with Jµ(τ) = ˙xµ(τ) + Jµ spin[ψ](τ)(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 Aµ(x). We will proceed in several steps: 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. 3. Reexpress the resulting operator traces in terms of a spinor worldline path integral and identify the worldline current Jµand photon kernel. 4. Comment on the structure of the spin part Jµ spin and on the way (201) resums all two–loop diagrams in a compact form. 1. From full QED to the two–loop effective action. We begin with the full QED action in Minkowski space, SQED[¯ ψ, ψ, A] = Zd4x−1 4FµνFµν +¯ ψi/ ∂−e/ A−mψ,(204) where now we treat Aµ as a dynamical quantum field. To define the effective action in the presence of a background field Aµ(x), we split Aµ(x) = Aµ(x) + aµ(x),(205) where aµ denotes quantum photon fluctuations and Aµ is a fixed classical background. The generating functional with background Aµis eiΓ[A]=ZDaD¯ ψDψexp iSQED[¯ ψ, ψ, A+a],(206) 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 internal photon. Expanding the action (204) to quadratic order in aµ, we have schematically SQED[¯ ψ, ψ, A+a] = SDirac[¯ ψ, ψ;A] + Sphoton[a] + Sint[¯ ψ, ψ, a;A],(207) where SDirac[¯ ψ, ψ;A] = Zd4x¯ ψ(i/ ∂−e/ A−m)ψ, (208) Sphoton[a] = Zd4x−1 4fµνfµν +Lgf[a],(209) Sint[¯ ψ, ψ, a;A] = −eZd4x¯ ψγµaµψ, (210) with fµν =∂µaν−∂νaµand Lgf a gauge–fixing term for aµ(e.g. Feynman gauge). The effective action Γ[ A ]is obtained by integrating over both ψ and aµ . In perturbation theory, one expands in powers of e and in the loop expansion (powers of ~ ). The one–loop contribution corresponds to 53 integrating out ψ with aµ 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 and one internal photon connecting two points on that loop. At order e2 , and focusing on this sector, one obtains a term of the schematic form Γ(2)[A] = i 2Tr [(SF[A]γµ)(SF[A]γν)] Dµν ,(211) where: •SF[A]is the Dirac propagator in the background A, •Dµν is the photon propagator (including gauge–fixing), •the trace is over spacetime and spinor indices. This expression encompasses all two–loop diagrams with one internal photon and one Dirac loop in the fixed background A. The factor of 1/2reflects the symmetry factor of the closed photon loop. 2. Worldline representation of the two–loop operator trace. We now wish to rewrite (211) in worldline form. The key idea, developed in various works [ 8 , 64 ], is that each propagator SF [ A ]can be represented as a line integral over a worldline action with Grassmann variables for spin, and the insertion of γµ and the background interaction combine into a worldline current Jµ(τ). Recall that, in Euclidean space, the Dirac propagator in a background Aµcan be written as SF[A](x, y)=(DE+m)G[A](x, y),(212) where DE=/ DEis the Euclidean Dirac operator and G[A]satisfies −D2+e 2σµνFµν +m2G[A](x, y) = δ(4)(x−y).(213) The scalar propagator G[A]admits a worldline representation: G[A](x, y) = Z∞ 0 dT e−m2TZx(T)=y x(0)=xDx(τ)Dψ(τ) exp −ZT 0 dτ ˙x2 4+1 2ψµ˙ ψµ−ie ˙xµAµ+ie ψµFµνψν!. (214) The insertion of ( DE + m )acts as a differential operator on the endpoints and can be represented in the worldline path integral by appropriate vertex operators involving ψµ and ˙xµ [ 64 ]. When two such propagators are connected by a photon line, one obtains, after integrating over the photon fluctuations aµ and using the Gaussian form of Sphoton [ a ], a bilocal kernel along a single worldline proportional to the photon propagator Dµν(x−y). The upshot of this construction (which involves careful normal–ordering and combinatorics, see [ 8 , 64 ]) is that the two–loop effective action in the quenched approximation can be written as Γ(2)[A] = 1 2Z∞ 0 dT Te−m2TZx(T)=x(0) DxZψ(T)=−ψ(0) Dψ e−Sspin wl [x,ψ;A]Ispin photon[x, ψ],(215) with Ispin photon[x, ψ]given by (202). We now examine the structure of Ispin photon. 3. Worldline current and photon kernel. The worldline photon factor arises from integrating out the photon fluctuations aµ in the presence of the worldline current induced by the Dirac loop. Schematically, ZDaexp iSphoton[a]−ie Zd4x aµ(x)Jµ Dirac(x)= exp −e2 2Zd4x d4y Jµ(x)Dµν(x−y)Jν(y), (216) where Jµ Dirac ( x )is the Dirac current around the loop and Dµν ( x−y )is the free photon propagator. In the worldline representation, Jµ Dirac ( x )is pulled back to the worldline as a current Jµ ( τ )supported along the trajectory: Jµ Dirac(x) = ZT 0 dτ Jµ(τ)δ(4)x−x(τ).(217) 54 Then Zd4x d4y Jµ(x)Dµν(x−y)Jν(y) = ZT 0 dτ1ZT 0 dτ2Jµ(τ1)Jν(τ2)Dµνx(τ1)−x(τ2),(218) where Dµν(x−y) = ηµνD(x−y)+(gauge–dependent terms).(219) In Feynman gauge (∂µaµ= 0,Lgf =−1 2(∂µaµ)2), we have Dµν(x−y) = ηµν D(x−y), D(x−y) = Zd4k (2π)4 eik·(x−y) k2+iε .(220) Substituting into the exponential and keeping only the term linear in the bilocal kernel (corresponding to a single photon exchange), we obtain Ispin photon[x, ψ] = e2ZT 0 dτ1ZT 0 dτ2Jµ(τ1)Jµ(τ2)Dx(τ1)−x(τ2),(221) which is precisely (202). The form of the worldline current Jµ ( τ )can be deduced by comparing the worldline representation of diagrams with and without spin. In scalar QED, the current is simply Jµ scalar(τ) = ˙xµ(τ),(222) reflecting the coupling −ie ˙xµAµ in the worldline action (113) . In spinor QED, the coupling to the background field includes the spin term ie ψµFµνψν , which gives rise to an additional contribution to the current. To leading order in the number of photon insertions, one finds Jµ(τ) = ˙xµ(τ) + Jµ spin[ψ](τ),(223) where Jµ spin contains bilinear combinations of ψ and Fµν (and possibly its derivatives), chosen so that the worldline representation reproduces the spinor Dirac trace with two photon insertions [ 8 , 64 ]. The explicit form of Jµ spin [ ψ ]is somewhat involved and not strictly needed for the general structure; it is enough to know that it is local along the worldline and linear in e. 4. 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: • Expanding the exponential e−Sspin wl in powers of the background coupling e yields worldline vertices corresponding to ˙xµAµ and ψFψ insertions along the loop; these generate the external–field interactions. • Inserting Ispin photon introduces a bilocal interaction between two points τ1, τ2 on the worldline, with kernel D ( x ( τ1 ) −x ( τ2 )) and current insertions Jµ ( τ1 )and Jµ ( τ2 ). 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. 55 5. Categorical interpretation at two loops. In the higher–categorical language developed in Sections II and III, the master formula (215) reflects the fact that the encoding category Encwl must be enriched and graded appropriately to capture higher–loop information: • The hom–objects in Encwl are no longer just single–worldline path integrals; they are graded by loop order and include functionals with bilocal kernels such as Ispin photon. • The enrichment over VectC (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. •The encoding functor Hat two loops, H(2) :Bulk(2) QED −→ Enc(2) wl ,(224) maps the two–loop sector of QED to the enriched worldline category in which Ispin photon lives. The context functors F(2) and G(2) 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. 6. Outlook for two–loop calculations in inhomogeneous fields. The master formula (215) provides a starting point for two–loop computations in inhomogeneous fields, including constant fields (where analytic results are known [ 65 , 66 ]) and slowly or rapidly varying fields (where semiclassical and numerical methods can be applied). While we will not carry out the full two–loop instanton analysis here, it is clear from the structure of (215) that: • in the semiclassical regime, the dominant contribution again comes from worldline instantons, now dressed by the photon kernel Ispin photon evaluated on the instanton trajectory; • the exponent receives a two–loop correction ∆ S(2) from Ispin photon [ xinst, ψinst ], and the prefactor receives corrections from the modified fluctuation operators (now including the bilocal interaction); • 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 τ1, τ2 from the bilocal kernel. 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. B. Semiclassical two–loop correction for fast fields 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 idea is to treat the photon–exchange functional Ispin 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 xinst ( 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 (τ1, τ2). The result has the schematic form S(1+2) eff [E]≈S0[xinst]+∆S(2)[E0, ω],(225) with S0[xinst] = m2 eE0 g(γ),(226) 56 as in (181), and ∆S(2)[E0, ω] = e2 2Ispin photon[xinst],(227) where Ispin photon[xinst] = ZT 0 dτ1ZT 0 dτ2Jµ(τ1)Jµ(τ2)ψ, xinst Dxinst(τ1)−xinst(τ2).(228) Here h···iψ, xinst 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 ) = xinst ( 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. 1. Two–loop master formula and semiclassical expansion. Recall the two–loop master formula for spinor QED in a background Aµ, derived in (215): Γ(2)[A] = 1 2Z∞ 0 dT Te−m2TZx(T)=x(0) DxZψ(T)=−ψ(0) Dψ e−Sspin wl [x,ψ;A]Ispin photon[x, ψ],(229) with Ispin photon[x, ψ] = e2ZT 0 dτ1ZT 0 dτ2Jµ(τ1)Jµ(τ2)Dx(τ1)−x(τ2).(230) The one–loop part is given by Γ(1)[A] = −1 2Z∞ 0 dT Te−m2TZDxDψ e−Sspin wl [x,ψ;A].(231) We are interested in the imaginary part of the total effective action, Γ[A]≈Γ(1)[A]+Γ(2)[A] + ··· ,(232) which yields the non–perturbative pair–production rate. In the semiclassical regime relevant for strong fields, both Γ (1) and Γ (2) are dominated by saddle–points of the worldline action Sspin wl [ x, ψ ; A ], i.e. by worldline instantons and their fluctuations. At the one–loop level, we have already seen that the exponent is given by S0 [ xinst ]; at two loops, the photon insertion Ispin photon [ x, ψ ]modifies both the exponent and the prefactor: Γ(1+2)[A]∼ZdT e−m2TN(1)[T]e−S0[T]+N(2)[T]e−S0[T],(233) where N(1) and N(2) are one– and two–loop prefactors, respectively, and S0 [ T ]is the classical action evaluated on a worldline instanton of period T . The two–loop correction can be captured semiclassically by treating Ispin photon[x, ψ]as an insertion in the one–loop saddle–point approximation: Γ(2)[A]≈1 2Z∞ 0 dT Te−m2TDIspin photon[x, ψ]E1–loop ,(234) where h·i1–loop denotes expectation value with respect to the 1D worldline measure exp ( −Sspin wl )expanded to quadratic order around the instanton. 2. Evaluating the photon functional on the instanton. We now write the fields as fluctuations around the instanton: x(τ) = xinst(τ) + η(τ), ψ(τ) = ψinst(τ) + χ(τ).(235) For the purpose of computing the exponent, one can take ψinst to be zero (the Grassmann fields enter quadratically in the free part). The one–loop effective action exponent S0 [ xinst ]is obtained by substituting x=xinst,ψ= 0 into Sspin wl and neglecting fluctuations. 57 To first approximation, the two–loop correction ∆ S(2) to the exponent arises from the photon insertion evaluated on the instanton trajectory, with ψintegrated over its quadratic fluctuations: ∆S(2)[E0, ω]≈e2 2ZT 0 dτ1ZT 0 dτ2Jµ(τ1)Jµ(τ2)ψ, xinst Dxinst(τ1)−xinst(τ2),(236) which is exactly (228) with the prefactor e2/ 2as in (227) . Here we have neglected contributions from bosonic fluctuations in x , which would contribute higher–order corrections and subleading terms in the exponent; one can systematically include them if needed. The task is thus reduced to computing the worldline current correlator Jµ(τ1)Jµ(τ2)ψ, xinst =˙xµ(τ1) ˙xµ(τ2)+ 2<˙xµ(τ1)Jspin,µ(τ2)+Jµ spin(τ1)Jspin,µ(τ2),(237) where the expectation values are taken with respect to the quadratic Grassmann action expanded around the instanton. At leading semiclassical order, it is often sufficient to approximate ˙xµ(τ1) ˙xµ(τ2)ψ, xinst ≈˙xµ inst(τ1) ˙xinst,µ(τ2),(238) and to treat the spin term as a quantum correction: Jµ(τ1)Jµ(τ2)ψ, xinst ≈˙xµ inst(τ1) ˙xinst,µ(τ2) + Jµ spin(τ1)Jspin,µ(τ2)ψ.(239) The spin correlator Jµ spin(τ1)Jspin,µ(τ2)ψcan be expressed in terms of the Grassmann propagator hψµ(τ1)ψν(τ2)i=1 2δµν signτ1−τ2,(240) for the free theory on the circle with anti–periodic boundary conditions, corrected by interaction terms proportional to Fµν . 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 Ispin photon[xinst] = ZT 0 dτ1ZT 0 dτ2K(τ1, τ2;γ),(241) with an integrand K(τ1, τ2;γ) = Jµ(τ1)Jµ(τ2)ψ, xinst Dxinst(τ1)−xinst(τ2)(242) that is fully determined by the instanton path xinst ( τ ; γ )constructed in Subsection V C and by the chosen photon propagator. 3. Effective exponent and prefactor at two loops. Combining the one–loop exponent S0 [ xinst ]and the two–loop correction ∆S(2), the total effective exponent for pair production at two loops is given by S(1+2) eff [E0, ω]≈S0[xinst]+∆S(2)[E0, ω] = m2 eE0 g(γ) + e2 2Ispin photon[xinst],(243) as in (225) and (227) . The corresponding pair–production rate (imaginary part of the effective Lagrangian) behaves roughly as =L(1+2)(E0, ω)∼ N(1+2)(γ) exp −S(1+2) eff [E0, ω],(244) 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. 64 2. Weakly gauging chiral symmetries and the anomaly. To make the anomaly manifest and to compute it systematically, one introduces background gauge fields for the chiral flavor symmetry: Lµ(x)∈su(Nf)L, Rµ(x)∈su(Nf)R,(273) and couples them minimally to the left– and right–handed quark fields: DµqL=∂µqL+igsAa µTaqL+iLµqL, DµqR=∂µqR+igsAa µTaqR+iRµqR.(274) We thus obtain an extended QCD action SQCD[A;L, R; ¯q, q] = Zd4x−1 4Ga µνGa µν + ¯qLi/ DLqL+ ¯qRi/ DRqR,(275) which is classically invariant under local transformations qL7→ UL(x)qL, qR7→ UR(x)qR, Lµ7→ ULLµU−1 L+iUL∂µU−1 L, Rµ7→ URRµU−1 R+iUR∂µU−1 R. (276) However, at the quantum level, integrating out the quark fields generates an effective action Γ[ L, R, A ] that is not invariant under these local chiral transformations. The variation of Γunder an infinitesimal SU(Nf)L×SU(Nf)Rtransformation, UL(x)≈1 + iαL(x),UR(x)≈1 + iαR(x), is given by δαΓ[L, R, A]=2πi ZM4 TrαLAL−αRAR,(277) where AL,R are local 4–forms constructed from the background field strengths FL,R (and possibly from G ). In the limit where we turn off the dynamical gluons ( Aa µ = 0) and focus on the pure flavor anomaly, the anomaly takes the well–known form [67, 69] ∂µJµ L,a =Nc 24π2µνρσ TrTaFL,µνFL,ρσ, ∂µJµ R,a =−Nc 24π2µνρσ TrTaFR,µνFR,ρσ,(278) where Ta are generators of su ( Nf ), FL,µν and FR,µν are the corresponding flavor field strengths, and Nc is the number of colors. Equation (278) can be expressed more compactly in differential–form notation. Let FL and FR be the su(Nf)–valued two–forms FL=dL +iL ∧L, FR=dR +iR ∧R. (279) Then the divergence of the left and right flavor currents is proportional to the 4–form ω4(L) = Tr(FL∧FL), ω4(R) = Tr(FR∧FR),(280) with the anomaly proportional to Nc(ω4(L)−ω4(R)). 3. The six–dimensional anomaly polynomial I6 .The four–dimensional anomaly can be encoded in a six–dimensional anomaly polynomial I6 , which is a closed 6–form built from the field strengths FL and FR and whose descent yields (278) . This is the general framework developed by Wess and Zumino [ 69 ] and refined by many authors [ 68 ]. For QCD with Nf flavors and Nc colors, the relevant anomaly polynomial for the SU(Nf)L×SU(Nf)Rchiral symmetry is I6=Nc 24π2Tr F3 L−Tr F3 R,(281) where Tr F3 L:= Tr (FL∧FL∧FL),Tr F3 R:= Tr (FR∧FR∧FR),(282) and the trace is taken in the fundamental representation of SU ( Nf ). The normalization 1 / (24 π2 )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 Gin 2ndimensions is given by the (2n+ 2)–form part of the Chern character chR(F): I2n+2(F) = hchR(F)ˆ A(T)i2n+2 ,(283) 65 where ˆ A ( T )is the ˆ A –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 chR ( F ), which, for a Weyl fermion in the fundamental representation of SU ( Nf ), is proportional to Tr F3. In QCD, each left–handed quark flavor qL,f transforms as a Weyl fermion in the fundamental representation of SU ( Nf ) L , with Nc copies (one for each color). Thus, the anomaly polynomial for the left chiral symmetry is I(L) 6=Nc·1 24π2Tr F3 L,(284) and for the right symmetry it is I(R) 6=Nc·1 24π2Tr F3 R.(285) The total anomaly polynomial, taking into account the opposite chirality of the right–handed fermions, is then I6=I(L) 6−I(R) 6, which yields (281). An important feature of (281) is the explicit factor of Nc : 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–Nclimit; instead, it scales linearly with Nc. 4. Descent and 4D anomaly from I6.The six–dimensional anomaly polynomial I6is a closed form, dI6= 0,(286) and is exact on a (2 n +1)–dimensional manifold M2n+1 when extended to a (2 n +2)–dimensional manifold B2n+2 with ∂B2n+2 = M2n+1 . In our case, we are interested in extending the 4D spacetime M4 to a 5D manifold B5and working one dimension higher to construct WZW actions and anomaly inflow terms. The descent equations proceed as follows [ 68 ]. One finds a 5–form I5 (the Chern–Simons form) such that dI5=I6.(287) Under a gauge variation of the background fields, δαA=Dα =dα + [A, α],(288) one finds δαI5=dI(1) 4(α),(289) where I(1) 4 ( α )is a 4–form valued linearly in the gauge parameter α . Integrating over B5 and using Stokes’ theorem, we find that the variation of the 5D Chern–Simons action reduces to a boundary term: δαZB5 I5=ZB5 dI(1) 4(α) = ZM4 I(1) 4(α).(290) This boundary term is precisely the anomaly in the 4D effective action. In our QCD setting, splitting I6 according to FLand FR, one obtains I5,L and I5,R, and the 4D anomaly is δαΓ[L, R]=2πi ZM4 TrαLI(1) 4,L(L)−αRI(1) 4,R(R),(291) consistent with (278) . The detailed expressions for I5,L and I(1) 4 involve standard Chern–Simons and transgression forms; they can be found in classic references [68, 69]. 5. Anomaly as an obstruction and as a holographic datum. The anomaly polynomial I6 can be understood in several equivalent ways: • As the obstruction to gauging the chiral flavor symmetry: a nonzero I6 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. 66 • As an element of a generalized cohomology group: in modern terms, I6 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. • As a boundary inflow condition for an invertible 5D TQFT: there exists a 5D topological theory whose bulk action is ∼RI5 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 I6 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, •encoding category Enc containing 5D TQFTs and WZW theories, •encoding functor Hmapping 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. 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 I6 , and showing how the requirement of anomaly holography fixes the WZW level kto be equal to the number of colors Nc. B. 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 I6(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=Ncby matching to the QCD anomaly polynomial I6. 1. Spontaneous chiral symmetry breaking and the chiral Lagrangian. As reviewed in Subsection VII A, QCD with Nfmassless quark flavours has a classical global chiral symmetry Gχ= SU(Nf)L×SU(Nf)R×U(1)B.(292) Lattice and phenomenological evidence strongly supports the expectation that, in the IR, this symmetry is spontaneously broken according to SU(Nf)L×SU(Nf)R−→ SU(Nf)V,(293) where SU ( Nf ) V is the diagonal vector subgroup. The associated Goldstone bosons are parametrised by an SU(Nf)–valued field U(x)∈SU(Nf),(294) which transforms under chiral transformations as U(x)7→ ULU(x)U† R, UL∈SU(Nf)L, UR∈SU(Nf)R.(295) Physically, U ( x )can be thought of as U ( x ) ∼exp iπa ( x ) Ta/fπ , where πa ( x )are the pseudoscalar meson fields, Taare generators of su(Nf), and fπis the pion decay constant. The leading–order IR dynamics of the Goldstone bosons is described by the chiral Lagrangian [72]: Sχ[U] = Zd4xf2 π 4TrDµUDµU†+higher–order derivative terms,(296) 67 where Dµ is a covariant derivative that can include couplings to external gauge fields (such as the electromagnetic field, or the flavor background fields Lµ and Rµ 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 Sχ [ U ]is invariant under the chiral symmetry (295) , at least in the absence of anomalies. However, to include the correct anomalous processes (such as π0→ 2 γ , 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. 2. 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 M4 be the four–dimensional spacetime manifold, and let U : M4→SU ( Nf )be the Goldstone field. Choose a five–dimensional manifold B5with boundary ∂B5=M4,(297) and extend Uto a map e U:B5→SU(Nf)such that e U|M4=U. The WZW functional is defined as Γ0[U] = i 240π2ZB5 Tre U−1de U5,(298) where Tre U−1de U5:= Tre U−1de U∧e U−1de U∧e U−1de U∧e U−1de U∧e U−1de U(299) is a closed 5–form on B5 . In local coordinates xA on B5 ( A = 1 ,..., 5) and with ABCDE the Levi–Civita symbol in five dimensions, (298) can be written as Γ0[U] = i 240π2ZB5 d5x ABCDE Tre U−1∂Ae Ue U−1∂Be Ue U−1∂Ce Ue U−1∂De Ue U−1∂Ee U.(300) A number of important remarks are in order: • The functional Γ 0 [ U ]is not manifestly local in four dimensions, since it involves an integral over B5. However, its variation under variations of Uis local and depends only on the restriction of e U to the boundary M4, due to the fact that (e U−1de U)5is a closed form: dTr( e U−1de U)5= 0. • The definition of Γ 0 [ U ]depends on the choice of extension e U : B5→SU ( Nf ). Different choices of B5and e Uthat agree on the boundary may differ by an integer multiple of 2π, as we now explain. 3. Quantisation of the WZW level and IR ambiguity. Suppose we have two different extensions e U1 : B(1) 5→SU ( Nf )and e U2 : B(2) 5→SU ( Nf )of the same boundary field U : M4→SU ( Nf ). We can glue B(1) 5 and B(2) 5 along their common boundary M4 , with reversed orientation on one of them, to obtain a closed five–dimensional manifold X5=B(1) 5∪M4B(2) 5,(301) and a map b U:X5→SU(Nf)built from e U1and e U2. The difference between the two WZW actions is Γ(1) 0[U]−Γ(2) 0[U] = i 240π2ZX5 Trb U−1db U5.(302) Now X5 is closed, and the integral is proportional to the winding number of the map b U : X5→SU ( Nf ), which lies in the homotopy group π5(SU(Nf)): π5(SU(Nf)) ∼ =Z, Nf≥3.(303) 68 This implies that ZX5 Trb U−1db U5= 2πi ·240π2·n, (304) for some integer n∈Z. Therefore Γ(1) 0[U]−Γ(2) 0[U]=2πi n. (305) Consequently, the exponentiated functional eikΓ0[U](306) is well–defined (i.e. independent of the choice of extension) provided that the level k is an integer. In other words, we are allowed to add to the action a term SWZW[U] = kΓ0[U],(307) 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(Nf). The chiral Lagrangian Sχ,eff[U] = Sχ[U] + kΓ0[U](308) is thus ambiguous up to the choice of k∈Z , with each choice defining an IR EFT that is classically invariant under SU ( Nf ) L×SU ( Nf ) R and has a well–defined path integral. Nothing in the IR effective theory alone seems to single out a particular value of k. 4. Gauged WZW term and coupling to background fields. To make contact with the anomaly polynomial I6 and to see how the WZW term reproduces the chiral anomaly, one must couple the chiral Lagrangian to the background fields Lµand Rµ. This leads to the gauged WZW functional Γ[U, L, R][11]. The basic idea is to promote the global chiral symmetry (295) to a local symmetry by introducing background connections Lµ and Rµ 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: 1. Extend U,L, and Rfrom M4to B5, choosing appropriate extensions e U, e L, e R. 2. Construct a 5D Chern–Simons action S5D [ L ]whose exterior derivative gives the Tr F3 L part of the anomaly polynomial, and similarly for S5D[R]. 3. 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 Γ[U, L, R] = Γ0[U] + ZM4 ω4(U, L, R),(309) where ω4 ( 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 δαΓ[U, L, R]=2πi ZM4 TrαLAL(L)−αRAR(R),(310) with AL,R given by the anomaly expressions in (278) . The explicit form of ω4 ( U, L, R )is somewhat lengthy and can be found in [11]; importantly, it involves terms of the form Tr LdUU−1,Tr U−1dUR,Tr(L3),Tr(R3),(311) and various combinations designed to reproduce the cohomological structure encoded in the anomaly polynomial I6. From the standpoint of the IR chiral EFT, the gauged chiral Lagrangian is then Sχ,eff[U;L, R] = Sχ[U;L, R] + kΓ[U, L, R],(312) where Sχ [ U ; L, R ]is the gauged version of (296) . Again, the integer k remains undetermined by IR considerations alone. 69 5. IR ambiguity and the need for anomaly holography. We can now summarise the IR situation: • The Goldstone bosons of spontaneous chiral symmetry breaking are described by a chiral field U(x)∈SU(Nf)and a chiral Lagrangian Sχ[U;L, R]invariant under SU(Nf)L×SU(Nf)R. • To correctly reproduce the chiral anomaly and anomalous processes, one must add a WZW term kΓ[U, L, R], with Γgiven by (309). • 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 Sχ,eff[U;L, R] = Sχ[U;L, R] + kΓ[U, L, R](313) is ambiguous up to the choice of an integer k. Physically, different values of k lead to different coefficients for anomalous processes (e.g. π0→ 2 γ ), and 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 I6 with coefficient Nc , 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 I6 , and we will show that demanding a relative holography relation F(B)≃G(H(B)) (where Bis QCD and H(B) is the anomaly TQFT plus WZW data) forces k to equal Nc . 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. C. Anomaly inflow holography as categorical equivalence 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 FanomalyQCD4=FanomalyIR EFT≃GTQFT5D,(314) and that requiring (314) to hold forces the WZW level k introduced in (308) to equal the number of colors Nc. 1. Encoding theory: 5D anomaly TQFT. As discussed in Subsection VII A, the chiral anomaly of QCD with Nfmassless quarks is captured by the six–dimensional anomaly polynomial I6=Nc 24π2Tr F3 L−Tr F3 R,(315) where FL and FR are the SU ( Nf )flavor field strengths for the left and right chiral symmetries. The anomaly polynomial I6can be viewed as the curvature of a Chern–Simons action in five dimensions: d ω5(L) = Tr F3 L, d ω5(R) = Tr F3 R,(316) where ω5 ( L )and ω5 ( R )are Chern–Simons 5–forms constructed from the background gauge potentials L and R. Explicitly, in differential–form notation, ω5(L) = TrL dL dL +3 2L3dL +3 5L5,(317) 70 with similar expression for ω5 ( R ), and dω5 ( L ) = Tr ( F3 L )follows from straightforward but involved algebra [68]. We now define a five–dimensional action functional on a 5D manifold M5 (with suitable boundary) as: S5D[L, R] = Nc 24π2ZM5ω5(L)−ω5(R),(318) as indicated in the outline. This action is purely topological (in particular, it does not depend on a 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 Encanom ⊂QFT,(319) the ( ∞, 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]. 2. Gauge variation and boundary anomaly. To see anomaly inflow explicitly, consider a 5D manifold M5 with boundary ∂M5 = M4 . Under an infinitesimal chiral gauge transformation UL ( x ) ≈ 1 + iαL ( x ), UR(x)≈1 + iαR(x), we have δL =DαL=dαL+ [L, αL], δR =DαR=dαR+ [R, αR].(320) Using the transgression formula for Chern–Simons forms [68], one finds δω5(L) = d ω(1) 4(αL, L), δω5(R) = d ω(1) 4(αR, R),(321) where the 4–forms ω(1) 4 are determined by the descent procedure applied to I6 . Substituting into (318) , we obtain δS5D[L, R] = Nc 24π2ZM5 dω(1) 4(αL, L)−ω(1) 4(αR, R) =Nc 24π2ZM4ω(1) 4(αL, L)−ω(1) 4(αR, R),(322) where we have used Stokes’ theorem and ∂M5=M4. Comparing (322) with the variation of the 4D QCD effective action Γ QCD [ L, R, A ](integrating out quarks and gluons), one finds δΓQCD[L, R, A] = −δS5D[L, R] (mod 2πi),(323) in appropriate normalisations. That is, the combined system—4D QCD on M4 plus the 5D TQFT on M5 with ∂M5 = M4 —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 BQCD ∈BulkQCD (324) and an encoding object E5D ∈Encanom (325) with action S5D [ L, R ]as in (318) ; the context functors Fanomaly and Ginflow extract, respectively, the chiral anomaly functional from the 4D theory and the boundary variation from the 5D TQFT. 3. Encoding functor and context functors. We can now identify the categorical data explicitly: •Bulk category BulkQCD: Objects include 4D QCD with Nc colors and Nf massless quarks, coupled to background chiral gauge fields (L, R)for SU(Nf)L×SU(Nf)R. •Encoding category Encanom: Objects include 5D invertible TQFTs with actions such as (318) , possibly plus boundary WZW degrees of freedom. 71 •Encoding functor H: H:BulkQCD −→ Encanom,(326) defined by mapping the QCD theory BQCD and background ( L, R )to the 5D anomaly TQFT E5D with action S5D [ 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. •Bulk context functor Fanomaly: Fanomaly :BulkQCD −→ Obsanomaly,(327) which assigns to BQCD its anomaly functional, equivalently the 6D anomaly polynomial I6(315) or a 4D inflow functional δΓQCD. •Encoding context functor G: G:Encanom −→ Obsanomaly,(328) which assigns to the 5D TQFT E5D the boundary anomaly functional given by δS5D [ L, R ]in (322) . This is equivalent data to the anomaly polynomial I6, up to coboundaries. The string–free anomaly holography condition is then the existence of a natural equivalence αBQCD :FanomalyBQCD≃ −−→ GH(BQCD),(329) expressing the equality of the QCD anomaly and the anomaly inflow from the 5D TQFT. This is the UV side of the anomaly holography. 4. IR effective theory and gauged WZW term. On the IR side, the low–energy dynamics of QCD is described by the chiral EFT with WZW term: Sχ,eff[U;L, R] = Sχ[U;L, R] + kΓ[U, L, R],(330) 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: δαΓIR[L, R] = δαkΓ[U, L, R]+δαSχ[U;L, R].(331) Since Sχ is anomaly–free (it is built from covariant derivatives and gauge–invariant traces), its variation vanishes. The only contribution comes from the WZW term: δαΓIR[L, R] = k δαΓ[U, L, R].(332) 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 ( Nf )(or, more precisely, the anomaly polynomial Tr F3 up to a normalisation). For Nf flavors and Nc colors, the IR anomaly polynomial generated by the WZW term is IIR 6=k 24π2Tr F3 L−Tr F3 R,(333) i.e. the same structure as (315) but with k instead of Nc . Therefore, if we define the IR bulk object BIR as the chiral EFT plus WZW term, the bulk context functor satisfies Fanomaly(BIR) = IIR 6=k 24π2Tr F3 L−Tr F3 R.(334) 72 Since the IR EFT must describe the same physical theory as UV QCD, anomaly matching [ 67 , 68 ] demands that the anomalies must agree: Fanomaly(BQCD) = Fanomaly(BIR),(335) which in terms of the anomaly polynomials (315) and (333) implies Nc 24π2=k 24π2⇒k=Nc.(336) This is the standard result that the WZW level k in the chiral EFT must equal the number of colors Nc in QCD [11]. In our framework, this matching is precisely the consequence of demanding that both UV and IR bulk theories share the same holographic encoding into the 5D anomaly TQFT. Standard anomaly technology. All anomaly polynomials, descent relations, Chern–Simons forms, and 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 = Nc manifest as a holographic compatibility condition between UV QCD, its IR chiral EFT, and the 5D anomaly TQFT. 5. Anomaly inflow holography as a triple equivalence. We can now summarise the anomaly holography in QCD as a triple equivalence of anomaly data: FanomalyBQCD=FanomalyBIR≃GE5D,(337) where: •BQCD is the UV QCD theory, •BIR is the IR chiral EFT with WZW level k, •E5D is the 5D anomaly TQFT with action S5D[L, R], •Fanomaly extracts the anomaly polynomial or functional from a 4D theory, •Gextracts 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 BIR embed into the same holographic diagram as BQCD. 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. 6. Categorical universality. Finally, we note that in the language of UP–holography (Subsection IV A), the encoding pair ( E5D, φ )with φ : G ( E5D ) ≃ −−→ Fanomaly ( BQCD )is in fact universal among all possible anomaly encodings. Any other ( E0, φ0 )with the same anomaly data admits a unique morphism E5D →E0 in Encanom compatible with φ, φ0 . 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. 73 D. Capability: fixing IR data without solving QCD We now highlight the concrete capability provided by anomaly inflow holography in QCD. The central point is that the higher–categorical anomaly equivalence FanomalyBQCD=FanomalyBIR≃GE5D,(338) 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 = Nc 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 π0→2γamplitude, 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. 1. Skyrmions and baryon number from the WZW level. We first consider the Skyrme model, in which baryons are described as topological solitons (Skyrmions) of the chiral field U ( x ) ∈SU ( Nf ). For simplicity, we restrict to the case Nf = 2 or Nf = 3. In either case, the topology of the Goldstone manifold satisfies π3SU(Nf)∼ =Z,(339) 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 U(x)→ 1 as |x|→∞,(340) so that the spatial slice R3can be compactified to S3, and Udefines a map U:S3→SU(Nf).(341) The winding number (degree of this map) is given by Btop[U] = 1 24π2Zd3x ijk TrU−1∂iU U−1∂jU U−1∂kU,(342) where i, j, k run over the spatial indices. This integer Btop ∈Z is the topological baryon number of the Skyrmion. It is natural to identify Btop with the physical baryon number, but strictly speaking this requires matching to the UV baryon current and anomaly structure of QCD. At the EFT level, the conserved current associated with this topological charge is Bµ(x) = 1 24π2µνρσ TrU−1∂νU U−1∂ρU U−1∂σU,(343) with baryon number Btop[U] = Zd3x B0(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: 80 so that SNG[~ X] = σTR +σ 2ZT 0 dτ ZR 0 dσ ∂τXi∂τXi+∂σXi∂σXi+O(X4),(369) where the first term σTR gives the leading area law, and the second term describes a free massless scalar theory in 1+1dimensions for each transverse coordinate Xi. More generally, the effective action is expected to include higher–derivative and self–interaction terms consistent with worldsheet symmetries (reparametrisation, residual Lorentz symmetry in static gauge, etc.) [74]. Schematically, Sws[~ X] = σZd2ξ√det h+X n≥2 λnZd2ξOn(∂X, . . . ),(370) where ξα = ( τ, σ ), On are higher–order operators built from derivatives of ~ X , and λn are low–energy couplings suppressed by powers of 1 /√σ or the UV scale. At large R and for low–energy observables (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. 3. Worldsheet Hamiltonian and ground–state energy. Let us consider the quadratic approximation to the worldsheet action for D–dimensional spacetime: S(2) ws [~ X] = σTR +σ 2ZT 0 dτ ZR 0 dσ ∂τXi∂τXi+∂σXi∂σXi, i = 1, . . . , D −2.(371) We impose Dirichlet boundary conditions at the endpoints of the string: Xi(τ, 0) = Xi(τ, R)=0,(372) since the endpoints are attached to static quark sources at fixed positions. Expanding in normal modes along the spatial direction: Xi(τ, σ) = ∞ X n=1 xi n(τ) sin nπσ R,(373) the action for each mode xi n(τ)is that of a harmonic oscillator with frequency ωn=nπ/R: S(2) ws [~ X] = σTR +σ 2 D−2 X i=1 ∞ X n=1 ZT 0 dτ ˙xi n(τ)2+nπ R2xi n(τ)2.(374) In Hamiltonian form, the worldsheet Hamiltonian is Hws(R) = σR + D−2 X i=1 ∞ X n=1 pi2 n 2σ+σ 2nπ R2(xi n)2,(375) with [xi n, pj m] = iδijδnm. The energy levels are E({Ni n};R) = σR + D−2 X i=1 ∞ X n=1 ωnNi n+1 2, ωn=nπ R,(376) where Ni n = 0 , 1 , 2 , . . . are occupation numbers. The ground–state energy is obtained by setting Ni n = 0 for all (n, i): E0(R) = σR +1 2 D−2 X i=1 ∞ X n=1 nπ R.(377) The sum in (377) is divergent and must be regularised. Using zeta–function regularisation, ∞ X n=1 n=ζ(−1) = −1 12,(378) 81 we obtain 1 2 D−2 X i=1 ∞ X n=1 nπ R=(D−2)π 2R ∞ X n=1 n=(D−2)π 2R−1 12=−π(D−2) 24R.(379) Therefore, E0(R) = σR −π(D−2) 24R+O(R−3),(380) where the O ( R−3 )terms come from higher–order terms in the worldsheet effective action and from subleading corrections to the Nambu–Goto approximation. For D = 4, (380) reproduces the Lüscher correction −π/(12R). Comparing (380) with the static potential (364), we see that V(R)≡E0(R) = σR +µ−π 12R+··· ,(381) where µ includes possible renormalisations and normal ordering constants, and the universal −π/ (12 R ) 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]. 4. Categorical encoding: Hflux , Fstatic , and Gws .We now cast the preceding discussion into the string–free holographic framework of Section III. The relevant categories and functors are: •Bulk category BulkYM: 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 Encws: 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 Hflux: Hflux :BulkYM −→ Encws,(382) mapping the 4D Yang–Mills theory with a quark–antiquark pair to an effective worldsheet theory Ews describing the confining flux tube between them. In practice, Hflux ( BYM )is specified by the string tension σ , the number of transverse fields ( D− 2), and, at higher orders, by worldsheet couplings λn. •Bulk context functor Fstatic: Fstatic :BulkYM −→ Obsstatic,(383) which assigns to the bulk theory its static quark–antiquark potential V(R)at large R: Fstatic(BYM) = {V(R)}R1/√σ.(384) •Encoding context functor Gws: Gws :Encws −→ Obsstatic,(385) which assigns to a worldsheet theory Ews the ground–state energy E0 ( R )of the string for each R , as computed from the worldsheet Hamiltonian: Gws(Ews) = {E0(R)}R.(386) 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 Fstatic(BYM)αBYM −−−−−→ GwsHflux(BYM),(387) with αBYM an isomorphism in Obsstatic 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): 82 •Bulk: 4D SU(N)YM with line defects (heavy sources), •Encoding: 2D worldsheet QFT with transverse fields, •Context: static potential and flux–tube spectrum. 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, •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 Ncand Nf, not on coupling constants. In categorical terms, this universality can be understood as follows: • The fiber category FibFstatic ( Fstatic ( BYM )) 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 Hflux. 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. C. Lüscher term and capability 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 V(R)≃σR −π 12 R+··· ,(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. 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: S(2) ws [~ X] = σTR +σ 2ZT 0 dτ ZR 0 dσ ∂τXi∂τXi+∂σXi∂σXi, i = 1, . . . , D −2.(389) We impose Dirichlet boundary conditions at the endpoints: Xi(τ, 0) = Xi(τ, R)=0,(390) corresponding to the fact that the endpoints of the flux tube are attached to fixed heavy sources. Expanding in normal modes, Xi(τ, σ) = ∞ X n=1 xi n(τ) sin nπσ R,(391) and substituting into (389), we obtain S(2) ws [~ X] = σTR +σ 2 D−2 X i=1 ∞ X n=1 ZT 0 dτ ˙xi n(τ)2+nπ R2xi n(τ)2.(392) Upon canonical quantisation, xi n and their conjugate momenta pi n satisfy [ xi n, pj m ] = iδijδnm , and the Hamiltonian for each mode is that of a harmonic oscillator with frequency ωn=nπ R.(393) The full worldsheet Hamiltonian is then Hws(R) = σR + D−2 X i=1 ∞ X n=1 pi2 n 2σ+σ 2nπ R2xi2 n,(394) leading to the energy spectrum E({Ni n};R) = σR + D−2 X i=1 ∞ X n=1 ωnNi n+1 2.(395) The ground–state energy corresponds to Ni n= 0 for all (n, i): E0(R) = σR +1 2 D−2 X i=1 ∞ X n=1 nπ R.(396) The sum in (396) diverges and must be regularised. Using zeta–function regularisation, we recall that ∞ X n=1 n=ζ(−1) = −1 12,(397) where ζ(s)is the Riemann zeta function analytically continued to s=−1. Hence 1 2 D−2 X i=1 ∞ X n=1 nπ R=(D−2)π 2R ∞ X n=1 n=(D−2)π 2R−1 12=−π(D−2) 24R.(398) Substituting back into (396), we obtain E0(R) = σR −π(D−2) 24R+O(R−3),(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 E0(R) = σR −π 12R+O(R−3).(400) Identifying V(R) = E0(R)in the IR and large–Rlimit, this reproduces (388): V(R)≃σR −π 12R+··· .(401) We emphasise that this derivation relies only on: 84 • the existence of an effective string description of the flux tube (with a well–defined string tension σ ), •the presence of D−2massless transverse modes Xion the worldsheet, •Dirichlet boundary conditions at the endpoints, 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]. 2. 4D strongly coupled → 2D free–boson QFT. The Lüscher term provides a striking example of how a strongly coupled four–dimensional gauge theory in the IR can be effectively described by a simple, weakly coupled (1 + 1)–dimensional QFT: •On the 4D side: – 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 extrapolations. •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 Xi. – 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. 3. Effective string worldsheet as non–string, defect–categorical encoding. It is important to distinguish the effective string worldsheet QFT from a fundamental string theory. In our framework: •The bulk theory BYM is a 4D confining SU(N)gauge theory, an object in BulkYM. • The encoding theory Ews is a 2D worldsheet QFT describing flux–tube fluctuations, an object in Encws, 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: 1. The worldsheet theory has a finite number of degrees of freedom and is defined only for certain ranges of R (large compared to the flux–tube thickness). It does not purport to describe the entire UV–complete theory. 2. The worldsheet couplings (e.g. σ and higher–order λn ) are determined by matching to the 4D bulk theory, not by consistency conditions as in fundamental string theory. 3. The defect 2–category D ( TYM )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. 85 4. Capability: from extended QFT structure to universal IR predictions. From a capability standpoint, the Lüscher term showcases how the string–free, higher–categorical holographic framework can produce nontrivial and universal IR predictions: • By identifying the correct encoding category Encws — the defect 2–category of worldsheet QFTs — and the correct encoding functor Hflux, 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. In the language of Section IV, the effective string worldsheet theory is a universal encoding E0 for the context Fstatic : 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. 5. Summary. The derivation of the Lüscher term from free transverse bosons on a 2D strip exemplifies the super–holographic nature of the flux–tube encoding: • A strongly coupled 4D SU( N )gauge theory in the confining phase is mapped, via the defect 2–category and the encoding functor Hflux, to a 2D worldsheet QFT. • The static potential V ( R ), a nonperturbative observable of the 4D theory, is obtained as the ground–state energy E0(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. 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. IX. BEYOND GEOMETRY: UP–RG AND DFT AS HOLOGRAPHIC ENCODINGS 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 ERG and a functor HRG :B−→ ERG,(402) with the property that any other effective field theory (EFT) reproducing the same IR observables factors uniquely through ERG 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. 86 1. Context functor for IR observables. Let Bulk be an ( ∞, 1)–category of microscopic QFTs defined with some UV regularization (e.g. lattice spacing a or momentum cutoff Λ), and let ObsIR be an (∞,1)–category of IR observables. Typical choices for ObsIR include: • the category of correlation functions of local operators at separations larger than some fixed length scale LIR, •the category of effective actions at momentum scales pΛIR, • or, more abstractly, the homotopy category of factorization algebras restricted to large–distance observables. We encode this choice as a context functor FIR :Bulk −→ ObsIR,(403) which assigns to each microscopic theory B its IR data FIR ( B ): long–distance correlation functions, critical exponents, thermodynamic functions, etc. For concreteness, imagine that FIR(B)consists of: •n–point functions of a finite set of operators {Oi(x)}at separations |xi−xj| ≥ LIR, •S–matrix elements at energies E≤EIR, •or the effective action Seff [Φ] with fields Φrestricted to low–momentum modes. 2. Fiber category of EFTs and universal encoding. Let EFT be an ( ∞, 1)–category of effective field theories with fewer degrees of freedom or lower cutoff scale, to be used as potential IR descriptions of B in the context FIR. There is a restriction or evaluation functor GIR :EFT −→ ObsIR,(404) which assigns to an EFT E its IR observables (computed within that theory). For a fixed B , we then consider the fiber category FibFIR (FIR(B)) = n(E, φ)E∈EFT, φ :GIR(E)≃ −−→ FIR(B)o,(405) whose objects are pairs ( E, φ )where E is a candidate EFT that reproduces the IR data of B via an equivalence φ in ObsIR . Morphisms between ( E1, φ1 )and ( E2, φ2 )are morphisms h : E1→E2 in EFT such that GIR(h)intertwines φ1and φ2: GIR(E1)GIR(E2) FIR(B) GIR(h) φ1φ2 (406) Definition IX.1 (Universal RG encoding (UP–RG)) . We say that B∈Bulk admits a universal RG encoding in context FIR if there exists an object ( ERG, φ0 ) ∈FibFIR ( FIR ( B )) which is initial in the ( ∞, 1)–sense: for any ( E, φ ) ∈FibFIR ( FIR ( B )), there exists a morphism h : ERG →E in EFT and a contractible space of such morphisms, making the appropriate diagram commute in ObsIR. Intuitively, ERG is the “best possible” IR effective theory for the observables in context FIR : any other EFT reproducing the same IR data is obtained by further coarse–graining or field redefinitions of ERG . The pair ( ERG, φ0 )is then a universal object, and the RG flow from B to ERG can be regarded as an encoding functor HRG :Bulk −→ EFT, HRG(B) = ERG,(407) which realises a UP–holography in the IR context: FIR(B)φ0 −−→ GIR(HRG(B)).(408) 87 3. Example: 4D → 3D finite–temperature QCD/QED (EQCD/EQED). A paradigmatic example 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 : R3×S1 β.(409) Fields obey periodic (bosons) or anti–periodic (fermions) boundary conditions in Euclidean time. The Matsubara modes with frequencies ωn = 2 πnT (or (2 n + 1) πT for fermions) of n6 = 0 are heavy at scales Tand can be integrated out to obtain an effective theory for the zero modes. Bulk and context. We take: •Bulk to include the finite–temperature 4D gauge theory (e.g. SU(N)Yang–Mills with matter), •ObsIR 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 FIR 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 A0 (the static component of the temporal gauge field), with an effective action SEQCD =Zd3x1 4Fa ijFa ij +1 2(DiA0)a(DiA0)a+1 2m2 D(Aa 0)2+λ(Aa 0Aa 0)2+···,(410) where i, j = 1 , 2 , 3, mD is the Debye mass, and the ellipsis denotes higher–dimension operators suppressed by powers of 1 /T . The parameters g2 3 (the 3D gauge coupling), m2 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 GIR to map them to static observables (e.g. static correlators and free energies). Then the pair ( ERG = EQCD, φ0 )—where φ0 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 gand 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 FIR(QED4[T]) ≃GIR(EQED3).(411) Thus, finite–temperature dimensional reduction is a clear example of UP–RG: the universal effective theory ERG (EQCD/EQED) is the initial object in the fiber category of 3D EFTs reproducing the same static IR data. 4. Example: scalar field theory and Wilsonian RG. To further illuminate the UP–RG concept, let us consider a simpler but instructive example: a real scalar field φ ( x )in d dimensions with UV cutoff Λand action S[φ] = Zddx1 2(∂φ)2+1 2m2φ2+λ 4!φ4.(412) The Wilsonian RG approach [ 49 , 50 ] constructs a family of effective actions Sk [ 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 Γk[φ]satisfies a functional RG equation such as Wetterich’s equation [13]. Suppose we are interested only in observables at scales p≤k0 for some fixed IR scale k0 . In the categorical language, we can: 88 •take Bulk to contain the UV theory at scale Λ, •define FIR to map Bto correlation functions of φfor momenta |p| ≤ k0, • take EFT to contain theories defined with UV cutoff k0 (i.e. with modes p>k0 already integrated out). The Wilsonian RG flow from Λdown to k0 , SΛ7→ Sk0 , defines an effective theory ERG with couplings ( m2 k0, λk0, . . . )that encodes all IR observables at that scale. Any other EFT E defined at k0 that reproduces the same IR observables (up to field redefinitions) can be obtained from ERG by integrating out irrelevant operators or by trivial reparametrisations. Thus, the pair ( ERG = Sk0, φ0 )is again a candidate universal encoding in the sense of Definition IX.1: it is the minimal EFT capturing the physics at scale k0 , and all other EFTs reproducing the same IR data factor through it. The functor HRG associates to each UV theory B (e.g. different bare couplings or regularizations) a universal IR encoding ERG, belonging to the same universality class. 5. Categorical and holographic interpretation. From the string–free holographic perspective, UP–RG is an instance of the general holographic pattern: FIR(B)≃GIRHRG(B),(413) 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 ERG is not only an encoding but a universal encoding: it is the initial object in the fiber category of all EFTs reproducing FIR(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. 6. Summary. In summary, UP–RG provides a categorical and holographic interpretation of the renormalization group: • For a fixed context of IR observables FIR , there exists (under appropriate conditions) a universal effective theory ERG and an encoding functor HRG such that any other EFT matching those observables factors uniquely through ERG. • 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. B. UP–XC / DFT as holography 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 EXC [ 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), 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 FDFT picks out ground–state observables that are functionals of the electron density, • the encoding functor HXC sends B to the DFT functional EXC [ 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 EXC [ ρ ]is precisely the data of the holographic encoding functor HXC , with UP–XC as a universal–property condition analogous to UP–RG. 1. 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 ˆ H[v] = ˆ T+ˆ Vee +ˆ V[v],(414) where ˆ T=−1 2 N X i=1 ∇2 i,(415) ˆ Vee =X 1≤i<j≤N 1 |ri−rj|,(416) ˆ V[v] = N X i=1 v(ri).(417) The many–body wavefunction Ψ( r1,...,rN )lives in the antisymmetric subspace of L2 (( R3 ) N )with spin degrees of freedom. The ground–state energy in the presence of vis E0[v] = inf ΨnormalisedhΨ|ˆ H[v]|Ψi.(418) The associated particle density is ρΨ(r) = NX σ1,...,σNZdr2···drN|Ψ(rσ1,r2σ2,...,rNσN)|2.(419) For the ground state, we denote ρ0(r)the corresponding ground–state density. The (first) Hohenberg–Kohn theorem [ 51 ] states that, for a given interacting Hamiltonian with fixed ˆ T and ˆ Vee and for nondegenerate ground states, the map v(r)7→ ρ0(r)(420) is one–to–one up to an additive constant in v . Thus, the density ρ0 ( r )uniquely determines the external potential v ( r ), the full ground–state wavefunction Ψ 0 , and all ground–state observables. This suggests that the density is a sufficient holographic variable for the ground–state problem, and that there exists a universal density functional F[ρ]such that E0[v] = min ρF[ρ] + Zd3r ρ(r)v(r),(421) subject to ρ≥0and Rd3r ρ(r) = N. The (second) Hohenberg–Kohn theorem and Lieb’s convex formulation [78] make this precise. Define F[ρ] = inf Ψ→ρhΨ|ˆ T+ˆ Vee|Ψi,(422) where the infimum is over all antisymmetric wavefunctions Ψyielding density ρ. Then E0[v] = inf ρF[ρ] + Zd3r ρ(r)v(r),(423) so F [ ρ ]is a universal functional (independent of v ) encoding all information about ˆ T and ˆ Vee relevant for ground–state energies and densities. 96 –worldline encodings work for generic gauge theories at one loop (and beyond in some cases), –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–Nlimits. Thus, 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. 6. Conclusion: why non–string wins. The side–by–side comparison above supports our central thesis: when holography is formulated abstractly in terms of higher–categorical encodings and context–dependent equivalences, strings are revealed as one special type of encoding among many, valuable in highly symmetric regimes but neither universal nor optimal in general. Non–string encodings: •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). 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. B. Argument that non–string encodings are more powerful 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. 1. Strong–field QED in fast backgrounds: worldlines vs. strings. In Sections V and VI, we studied non– perturbative electron–positron pair production in strong, time–dependent electric fields, with particular emphasis on Sauter pulses E ( t ) = E0sech2 ( ωt )and on one– and two–loop corrections. The key points were: • The one–loop effective action in an arbitrary background Aµ ( x )can be written exactly as a 0+1D worldline path integral with the spinor worldline action Sspin wl [x, ψ;A][Eq. (196)]. 97 • Non–perturbative pair–production exponents and prefactors are governed by worldline instantons: solutions of 1D ODE systems for xinst ( 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 Ispin photon[x, ψ], which reduces to a double integral in (τ1, τ2)plus 1D fluctuation data. 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. • 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. 2. Anomaly/WZW in QCD: TQFT and cohomology vs. strings. In Sections VII and VII C, we analysed anomaly holography in QCD with Nfmassless quarks, focusing on the chiral anomaly polynomial I6=Nc 24π2TrF3 L−TrF3 R(438) 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 S5D [ L, R ] [Eq. (318)], which define an invertible 5D TQFT. • The 4D WZW term is a boundary functional Γ[ U, L, R ][Eq. (309) ] whose variation reproduces the anomaly via descent. • 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=Nc[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 E5D ∈Encanom via the functor H. • The context functors Fanomaly and G select anomaly data from bulk and encoding, and the holographic equivalence Fanomaly ( B ) ≃G ( H ( B )) fixes IR parameters (like k ) without solving QCD. 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. 98 3. Flux–tube QCD: defect categories and effective strings vs. fundamental strings. In Section VIII, we treated confining SU( N )Yang–Mills at large distances using a defect 2–category and an effective worldsheet QFT. The key facts were: • Wilson lines and flux tubes are naturally organised into a defect 2–category D ( TYM )[ 26 , 73 ], with: –0–morphisms: vacua (e.g. the confining vacuum), –1–morphisms: line operators (Wilson lines), –2–morphisms: surface defects (flux–tube worldsheets). • The static quark–antiquark potential V ( R )at large R is encoded in the ground–state energy E0 ( R ) of a 2D worldsheet QFT for the flux tube: V(R)≡E0(R) = σR −π 12R+··· ,(439) 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 Encws , 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. 4. UP–RG and DFT: holography without geometry. In Section IX, we considered two examples of holographic encodings that are essentially non–geometric: • UP–RG: for a given IR context FIR , the RG flow produces a universal effective theory ERG (e.g. EQCD/EQED at high temperature, or an EFT at a finite scale k0 ) and a functor HRG such that any other EFT reproducing the same IR observables factors through ERG. • 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 EXC [ ρ ]). This functional is a universal encoding of the many–body problem into the space of densities. In both cases: • The encoding categories ( EFT , EncDFT ) are not geometric in the sense of AdS/CFT; they live in spaces of couplings, operators, and functionals. • The context functors FIR and FDFT select IR observables and density–dependent observables, respectively. • The universal encodings ERG and F [ ρ ]are initial objects in the relevant fiber categories, yielding UP–holographies F≃G◦Hin 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. 99 5. Abstract categorical comparison: why non–string encodings dominate. We can distil the argument into a higher–categorical statement. Let QFT be the ambient ( ∞, 1)–category of QFTs and effective theories. Let Encstring be the subcategory of encoding theories that admit a string/brane realisation in some geometric background (e.g. AdS, D–brane setups), with encoding functor Hstring defined only for those bulk theories that have known string duals. Let Encnon be the union of all non–string encoding categories we have used: Encnon =Encwl ∪Encanom ∪Encws ∪EncEFT ∪EncDFT ∪Enccode ∪··· .(440) Then: • The domain of Hstring is a strict subcategory of Bulk consisting of highly constrained theories (SUSY CFTs at large N, etc.) and a few deformations thereof. • The domain of the various non–string encoding functors Hwl , Hanom , Hflux , HRG , HXC , 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∈Encnon 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 FibF(F(B)) ∩Encstring (441) is empty or poorly understood, whereas FibF(F(B)) ∩Encnon (442) 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. 6. Conclusion: non–string higher–categorical encodings as the practical future of holography. Putting everything together, the examples in this paper show: • 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. • 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. • 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. 100 XI. CONCLUSIONS AND OUTLOOK A. 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. 1. String–free higher–categorical holography. The first central contribution is the formulation of string–free higher–categorical holography in Sections II and III. Instead of postulating “bulk–boundary dualities” in terms of strings and CFTs, we introduced: • 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 H:Bulk −→ Enc,(443) from a bulk subcategory Bulk ⊆QFT to Enc, •context functors F:Bulk →Obs, G :Enc →Obs,(444) which select the class of observables of interest. The essence of holography in this formulation is the existence of a natural equivalence α:F≃ =⇒G◦H, (445) whose components αB:F(B)≃ −−→ G(H(B)) (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)). 2. Super–holography: compression and anomaly versions. The second key conceptual contribution is the introduction of super–holography, a term we use for holographic encodings that achieve a drastic compression or extension of spacetime dimension: •Compression holography (super–holography): encodings where dim(H(B)) <dim(B)(447) (or dimeff ( H ( B )) dimeff ( B )), as in 4D → 1D worldline encodings or 4D → 2D worldsheet encodings. 101 •Anomaly holography: encodings where dim(H(B)) = dim(B)+1,(448) with H(B)an invertible TQFT, as in 4D ↔5D anomaly inflow for QCD. These two classes are both instances of the same higher–categorical pattern [Eq. (41) ], differing only in the dimension grading and the choice of encoding category. Compression holography corresponds 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 VectC , Hilb , Vect ×Top , etc.) and loop grading, ensuring that the encoding functors preserve not only observables but also loop expansions and topological data. 3. 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 Sspin wl [x, ψ;A] = ZT 0 dτ˙x2 4+1 2ψµ˙ ψµ−ie ˙xµAµ(x(τ)) + ie ψµFµν(x(τ))ψν,(449) and the exact one–loop worldline representation Γ(1)[A] = −1 2Z∞ 0 dT Te−m2TZx(T)=x(0) DxZψ(T)=−ψ(0) Dψ e−Sspin wl [x,ψ;A].(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 Ispin photon[x, ψ] = e2ZT 0 dτ1ZT 0 dτ2Jµ(τ1)Jµ(τ2)Dx(τ1)−x(τ2),(451) and showed how two–loop corrections to the exponent and prefactor can be computed from a double integral in (τ1, τ2)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 Encwl and the functor Hwl. 4. Anomaly/WZW QCD as 5D encoding: fixing IR data without solving QCD. In Sections VII 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 I6=Nc 24π2(TrF3 L−TrF3 R),(452) and the 5D Chern–Simons action S5D[L, R] = Nc 24π2ZM5ω5(L)−ω5(R).(453) 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, 102 •the WZW level kis quantised but ambiguous at the IR EFT level, and •the anomaly holography relation Fanomaly(BQCD) = Fanomaly(BIR)≃G(E5D)(454) forces k=Nc. 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. 5. 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 E0 ( 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 S(2) ws [~ X] = σTR +σ 2ZT 0 dτ ZR 0 dσ (∂τ~ X)2+ (∂σ~ X)2,(455) •the universal Lüscher term V(R)≃σR −π 12R+··· (456) arises as the Casimir energy of the transverse bosons with Dirichlet boundary conditions, • and the effective string worldsheet resides in a non–string encoding category Encws (a defect 2–category), not in a fundamental string theory in higher–dimensional spacetime. 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. 6. UP–RG and UP–XC: holographic encodings beyond spacetime. In Section IX we showed that the holographic ideas extend naturally beyond geometric encodings: • UP–RG: for a given IR context FIR , the Wilsonian RG flow produces a universal effective theory ERG and a functor HRG such that any other EFT reproducing the same IR observables factors through ERG . 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 EXC [ ρ ]) that encodes all ground–state density observables. The functional F [ ρ ]serves as a universal encoding in the density functional category EncDFT. 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. 103 7. Comparative conclusion: non–string encodings as the practical future of holography. Finally, in Section X we compared string encodings (AdS/CFT–type) to the non–string encodings we developed. We argued that: • 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. From 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. B. Future directions 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. 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 Scol[x, χ;A] = ZT 0 dτ χ†(τ)∂τ+iAa µ(x(τ))Ta˙xµχ(τ),(457) where χ(τ)carry color indices and Taare generators of su(Nc). • Background fields in non–Abelian directions: strong chromoelectric fields or color–flux backgrounds (e.g. constant Fa 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 BulkQCD,ext would include QCD with external color fields or background fields mimicking glasma configurations. 104 • The encoding category Encwl,col would consist of worldline super–QFTs with both spin and color degrees of freedom. • The context functor Fpair would select non–perturbative observables such as quark pair–production rates, Schwinger–like instabilities, and color dielectric response. Constructing encoding functors Hwl,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. 2. Fracton and topological phases: fusion categories and Ext/Tor holography. Another fertile ground for string–free holography is the domain of fracton phases and exotic topological orders [ 45 ]. These phases exhibit: •subdimensional excitations (fractons) with restricted mobility, •subextensive ground–state degeneracy depending on system geometry, •higher–rank gauge theories and subsystem symmetries. 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 Bulkfracton would include fracton Hamiltonians, higher–rank gauge theories, and their continuum limits. • The encoding category Encfusion would consist of fusion categories and their module categories, capturing fusion and braiding properties of excitations. • The observable category Obstopo 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: Ftopo(B)≃GExt/Tor(E),(458) where Ftopo extracts ground–state degeneracies, subsystem symmetry data, and fracton mobility constraints, while GExt/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. 3. More systematic UP–RG constructions for DFT and beyond. In Section IXA 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 ERG 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 ERG,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 Hrelate 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. 105 4. QFT in curved spacetimes via categorical encodings. Another natural extension is to quantum field theory in curved spacetimes and, more generally, in background geometries that are not asymptotically AdS. Traditional approaches (algebraic QFT on curved backgrounds, Hadamard states, microlocal analysis) already employ category–theoretic language (net of algebras, functors from spacetimes to algebras), but holographic aspects remain less explored outside AdS/CFT. Within our framework: • The bulk category Bulkcurved 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 Encnull 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). • 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. 5. Connections with quantum error–correcting codes. The interplay between holography and quantum error–correcting codes has been a major theme in recent work on AdS/CFT: the bulk–boundary map is often interpreted as an encoding map from a “logical” bulk Hilbert space to a larger “physical” boundary Hilbert space, with error–correcting properties reflecting entanglement wedge reconstruction. Our string–free framework suggests that one can generalise this perspective: • 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? • 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 Enccode and identifying functors H that are simultaneously holographic encodings and error–correcting codes could deepen the connection between holography, information theory, and QFT. 6. Higher–form symmetries and non–invertible defects. Recent advances in QFT have highlighted the role of higher–form symmetries, non–invertible symmetries, and their defects in classifying phases and constraining dynamics. Our defect–category approach in Section VIII can be generalised to include: •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),