From Gauge Invariance to Categorical Coherence: A Coherence-First Framework for the Strong-CP Problem, the Higgs Mechanism, Emergent Symmetry, and Condensed-Matter Phenomena
Abstract
This work proposes a “coherence-first” reformulation of gauge theory in which the fundamental organizing principle is not gauge invariance itself, but the requirement that many local descriptions can be glued into a globally consistent physical account. In the standard symmetry-first paradigm, one begins by postulating a symmetry group and then introduces gauge fields as compensators that preserve local invariance. Here the logic is inverted: the primary requirement is global consistency of local descriptions, and what is traditionally called “gauge symmetry” is reinterpreted as the first visible shadow of a deeper hierarchical coherence structure. The core conceptual shift is that physical consistency is governed by a ladder of compatibility conditions. At the lowest rung, ordinary gauge connections enforce comparability of local frames; at the next rung, higher-form fields enforce comparability of those comparisons; and so on, producing a hierarchy that naturally belongs to categorical and higher-categorical geometry. In this view, curvature is not merely a field strength but the measurable residue of “non-glueability”: the obstruction that cannot be removed by reparametrizing local descriptions. The paper formalizes this hierarchy in a concrete field-theoretic framework and develops a variational principle that drives systems toward maximal coherence across multiple levels. A central technical element is the introduction of a structure field that dynamically selects the local “symmetry type” (more precisely, the local grammar of gluing). Instead of assuming a fixed gauge structure everywhere, the theory allows the type of redundancy to vary over spacetime, enabling a unified description of symmetry emergence, symmetry-type crossovers, and intermediate “proto-gauge” regimes in which familiar gauge notions are only approximately realized. These proto-gauge regimes are treated as physically meaningful, controlled departures from strict first-level gauge invariance that remain consistent because higher coherence conditions continue to hold. Within this coherence-first setting, the manuscript develops several connected contributions: 1) A variational principle of maximal coherence The paper formulates an action principle in which the fundamental “energy” is a measure of coherence defect across multiple categorical levels. Ordinary Yang–Mills theory emerges as a limiting corner in which only the first level of coherence is active and the higher coherence structure is frozen or trivial. More general stationary points correspond to higher-gauge dynamics in which multiple compensator fields coexist and exchange “charge” between levels through generalized Bianchi/Ward-type identities. This establishes a single conceptual and technical framework that unifies standard gauge dynamics, higher-form extensions, and topological sectors. 2) Two coherence-based mechanisms for the strong-CP problem The manuscript presents two routes by which the effective strong-CP angle becomes unobservable without relying on an imposed Peccei–Quinn global symmetry or requiring an ultralight axion. Cohomological trivialization route: in an extended coherence complex, the CP-odd density that controls strong-CP sensitivity becomes exact in the extended descent structure, so the would-be CP-violating dependence is reduced to a boundary artifact (or eliminated under appropriate conditions). The physical interpretation is that the strong-CP sensitivity is a residue of incomplete gluing at lower coherence level and can disappear once higher coherence is properly accounted for. Structural relaxion route: the structure field couples in such a way that minimizing the total coherence defect dynamically drives the effective strong-CP angle to zero. Importantly, the field responsible is not postulated as a conventional axion with an assumed shift symmetry; it is a structural field controlling the local coherence grammar. Its mass and couplings are set by coherence stiffness parameters rather than by an axion decay constant. These mechanisms are framed to be compatible with standard anomaly physics and are accompanied by concrete diagnostic directions (for example, via lattice-style toy models that incorporate the extended coherence sector and test flattening of the strong-CP dependence as coherence penalties are strengthened). 3) Higgs mechanism reinterpreted as a coherence lift A major reinterpretation offered here is that the Higgs phenomenon can be understood as a coherence transmutation rather than simply “spontaneous symmetry breaking.” In the coherence-first view, a condensate reorganizes where consistency lives in the hierarchy: strict first-level coherence is traded for controlled higher-level coherence. Vector-boson masses acquire a geometric meaning as the energetic cost of maintaining higher coherence; the Higgs mass becomes a stiffness parameter of the higher-coherence condition. This perspective is designed to preserve consistency constraints (including generalized Ward identities) and to clarify why gauge–Higgs systems remain coherent even when the usual “symmetry breaking” narrative is conceptually misleading (because gauge redundancy is not an observable symmetry). 4) Emergent symmetry and proto-gauge regimes The manuscript formalizes emergent symmetry as the limit of near-perfect coherence: when coherence defects are small, approximate conservation laws and selection rules become quantitatively controlled by the size of the defect. This makes the common statement “symmetry emerges in the infrared” more operational: it becomes a statement about how quickly coherence defects decay under coarse-graining. The paper also introduces “proto-gauge phases,” transitional regimes in which familiar gauge closures fail slightly, yet the overall higher coherence remains intact. These regimes are argued to be common across both high-energy and condensed-matter settings and to admit measurable signatures. 5) Condensed-matter applications and predictions A distinctive feature of the manuscript is that it does not treat higher coherence as purely philosophical. It exports the formalism to several condensed-matter contexts where “gluing problems” and emergent gauge structures are already physically central: Topological insulators / axion electrodynamics: the coherence-first approach reframes bulk axion-like terms as artifacts of extended exactness while emphasizing quantized interfacial responses. Spatial textures of the structure field act as programmable domain walls where quantized transport steps and protected channels can arise. U(1) to Z2 spin-liquid crossovers: the structure field controls the effective gauge type, producing a controlled crossover between gapless and gapped gauge regimes, with predictions for spectral reweighting and domain-wall bound modes. Fracton elasticity duality: the higher-form coherence language is connected to defect compatibility and constrained mobility. Tuning coherence stiffness provides a principled route to mobility crossovers and defect suppression patterns. Hydrodynamic electron flow: the manuscript proposes a new collective excitation (“naturon”) associated with coherence fluctuations between microscopic and effective transport descriptions, along with signatures in nonlocal transport patterns and tunable response inversions. Across these examples, the emphasis is on testable signatures (spectral features, quantized steps, crossover scales, interface modes) that would be difficult to motivate from a fixed-symmetry viewpoint. 6) Discrete/numerical pathway To bridge formalism and data, the paper provides a discrete exterior calculus and lattice-style formulation in which the relevant coherence constraints and generalized Bianchi identities hold exactly at finite discretization. This is intended as an algorithmic route toward simulation of symmetry-type dynamics, domain walls in the structure field, and quantitative tests of the proposed coherence-based mechanisms in controlled models. Intended audience and positioning This manuscript is written as a foundational and unifying research document intended for readers interested in: gauge theory and higher gauge theory, categorical and higher-categorical structures in physics, anomalies and topological response, emergent symmetry and effective field theory, strongly correlated matter, topological phases, and fracton physics, computational formulations of generalized gauge structures. While the paper touches multiple domains, the unifying thread is operational: physical consistency is treated as coherent gluing of local descriptions, and the hierarchy of compensators and residual obstructions is used as the common language across high-energy and condensed-matter contexts.
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From Gauge Invariance to Categorical Coherence: A Coherence-First Framework for the Strong-CP Problem, the Higgs Mechanism, Emergent Symmetry, and Condensed-Matter Phenomena Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We propose a coherence-first reformulation of gauge theory in which categorical and highercategorical coherence, rather than gauge invariance, defines the fundamental organizing principle of physical interactions. In this framework, local descriptions of fields are glued together not by postulated symmetries but by hierarchies of coherence conditions: 1-coherence corresponds to conventional gauge connections, 2-coherence to higher-form fields, and higher levels to generalized associativity of local transformations. Gauge invariance then appears as the first visible shadow of a deeper coherence structure, and may be relaxed provided higher-order coherence is preserved. We construct a variational principle of maximal coherence, whose stationary points reproduce Yang–Mills theory as the 1-coherent limit and generate dynamical equations for the higher-coherence fields ( A, B, Φ). Within this setting we obtain two categorical mechanisms that resolve the strong-CP problem—either by rendering F∧F exact in the extended complex or by a structural relaxion Φ that dynamically cancels the QCD θ –term. The Higgs mechanism is reinterpreted as a coherence transmutation from 1to 2-coherence: the gauge-boson mass measures the energy of maintaining higher-level coherence, while the Higgs mass quantifies its stiffness, providing a geometric explanation for its radiative stability. Functorially, the Higgs field acts as the internal “quantization” map of gauge theory, converting continuous redundancy into discrete, quantized excitations. Beyond particle physics, the same coherence hierarchy governs emergent symmetry and proto-gauge phases in condensed matter. Applications to topological insulators, U(1) ↔Z2 spin-liquid transitions, fracton elasticity, and hydrodynamic electron flow yield concrete predictions: dynamically tunable axion responses, quantized Hall and thermal steps at coherence domain walls, and new collective modes (“naturons”) arising from fluctuations of the coherence field. We conclude that categorical coherence subsumes gauge symmetry as the fundamental requirement of physical consistency, providing a unified geometric language that links the Standard Model, quantum anomalies, and emergent gauge phenomena in matter. I. INTRODUCTION AND SUMMARY OF RESULTS A. Motivation Why the symmetry-first dogma is powerful. For nearly a century, the modern language of fundamental interactions has been built on the symmetry-first dogma: begin with a global invariance, promote it to a local one, and introduce a gauge connection to repair the failure of naive derivatives to respect the local symmetry. This paradigm explains the astonishing empirical success of the Standard Model of particle physics, where non-abelian Yang–Mills gauge theories with compact structure groups reproduce the strong and electroweak interactions with quantitative precision. The gauge principle also organizes anomalies, selection rules, and topological responses into a coherent algebraic-geometric framework. In condensed matter, the same idea underlies Berry connections, Chern numbers, and axion electrodynamics in topological phases, as well as a growing dictionary relating emergent gauge structures to long-wavelength constraints in quantum matter [2, 9, 141, 148, 271]. Why it is limited. At the same time, the symmetry-first viewpoint smuggles in a conceptual tension: gauge symmetry is not itself observable. Elitzur’s theorem formalizes the point that local gauge invariance cannot be measured directly; only gauge-invariant operators and curvatures carry physical content [ 277 ]. More strongly, a broad body of work argues that gauge symmetry may emerge in suitable infrared limits rather than being fundamental [ 4 , 5 ]. The practical meaning of this tension is clear: we routinely treat a redundancy of description as if it were a primitive law of nature and then interpret forces as the “price” of maintaining that redundancy. But if redundancy can emerge or even change its type, then it cannot be the most basic organizing principle. This motivates a deeper organizing idea that explains when and why gauge descriptions appear and how they can change across scales and regimes. Coherence before symmetry. In this work we develop a coherence-first perspective. The fundamental problem, in our view, is not to enforce invariance, but to glue many local descriptions into a single, globally consistent account. Concretely, fields are specified patchwise; overlaps require transition data; triple overlaps require compatibility of those transitions; and so on in a hierarchy. Gauge connections,
2 higher-form fields, and their curvatures are the compensators and residuals of this gluing problem: • at level 1, a connection A implements 1-coherence (comparability of local frames), with curvature Fmeasuring the remaining obstruction; • at level 2, a 2-form B implements 2-coherence (comparability of comparisons), with curvature H measuring the failure of associativity of local transformations; •higher levels continue this pattern, encoding the coherence of the whole hierarchy of comparisons. In this hierarchy, gauge symmetry appears as the first visible shadow of a deeper, categorical notion of coherence: it is the statement that different local frames produce equivalent physics because there exist compensators that glue them consistently. Crucially, it follows that strict gauge invariance need not be preserved provided that a suitable higher coherence survives: a failure of 1-coherence may be harmless (or even physically meaningful) if repaired by an appropriate 2-coherence, and similarly for higher levels. The practical motivation from physics. A coherence-first stance better matches what we observe across disparate domains: 1. Standard Model consistency without reifying redundancy. Gauge transformations are redundancies; observables are curvatures and Wilson operators. Anomalies signal failures of 1coherence; their cancellation via inflow or Green–Schwarz-like terms restores 2-coherence without requiring literal invariance of each sector, aligning with the modern view of generalized symmetries and their interplay with topology [5, 141]. 2. Topological matter and axion electrodynamics. Quantized responses such as magnetoelectric coupling in topological insulators reflect global obstructions to gluing local Bloch frames; Berry connections and Chern–Simons terms are the bookkeeping of that obstruction [ 148 ]. The quantization is not a postulate of symmetry; it is the residue of nontrivial gluing data. 3. Fractons and constrained dynamics. Fracton phases exhibit subdimensional mobility and rigid conservation laws that fit naturally into higher-form and higher-rank gauge descriptions, where the dominant physics is coherence of constraints rather than invariance under a fixed group action. Their phenomenology is most economically expressed as a hierarchy of compatibility conditions [ 8 ]. 4. Hydrodynamic crossovers. In ultra-clean conductors the crossover from ballistic to hydrodynamic flow exposes an interface between two valid local descriptions (kinetic and hydrodynamic). The measurable anomalies in nonlocal transport are governed by how well these descriptions cohere under coarse-graining, not by any microscopic gauge invariance [9]. From emergent symmetry to proto-gauge structure. In many of these settings, one encounters transitional regimes where familiar gauge descriptions only approximately apply. We will call these proto-gauge phases: the compensators exist and behave gauge-like, but the algebraic rules that define the redundancy do not close strictly. In the coherence hierarchy this means 1-coherence is imperfect while 2-coherence (and perhaps higher) already operates to keep the physics well-defined. Such phases are ubiquitous in deconfined criticality, emergent electrodynamics in spin liquids, and the onset of axion responses in topological materials, and they are most transparently described by categorical—not purely group-theoretic—data [4, 5, 141]. Reframing two canonical puzzles. Thinking coherence-first reshapes two canonical stories in particle physics: •Strong–CP problem. The θ -term multiplies a global obstruction class ( F∧F ). In a highercoherent extension of the gauge complex, this class can either become exact (cohomology trivialization) or be dynamically neutralized by a structural field that minimizes the total coherence defect (structural relaxion). In both cases, observables become insensitive to θ without introducing a new global Peccei–Quinn symmetry, in line with the modern understanding that what must vanish is not “symmetry breaking” but an obstruction in the extended complex [5]. •Higgs mechanism. The conventional picture says that a scalar vacuum expectation value (vev) breaks a gauge symmetry and gives mass to vector bosons. The coherence-first picture instead says: a Higgs vev lifts the system from 1-coherence to 2-coherence, allowing local frames to compose consistently up to a controlled higher morphism. The gauge-boson mass is then the energetic cost of maintaining this higher-level coherence; the Higgs mass measures the stiffness of the higher-coherence condition. This reframing helps explain why the gauge sector remains consistent and why aspects of the Higgs mass can be protected geometrically.
3 What we do in this paper (overview). Building on these observations, we formulate a variational principle of maximal coherence in which an action functional penalizes the total coherence defect across levels (schematically, kFk2 + kHk2 + kD Φ k2 ), and a structure field Φselects the local rules (the type of redundancy itself) in space and time. The stationary points reproduce Yang–Mills in the 1-coherent limit and generate consistent dynamics when higher-coherence data are present. We then apply this to: 1. the strong–CP problem , showing two categorical mechanisms that make θ unobservable while preserving known anomaly physics; 2. the Higgs mechanism , reinterpreting gauge-boson mass as the energy of 2-coherence and arguing for a geometric protection of the Higgs stiffness; 3. emergent symmetry , explaining symmetry as the limit of perfect coherence and identifying proto-gauge regimes where gauge/matter distinctions blur; 4. condensed-matter predictions , including quantized steps at coherence domain walls in topological insulators, bound channels and spectral reweighting at U(1) ↔Z2 spin-liquid interfaces, fracton mobility crossovers, and a new collective naturon mode in hydrodynamic electron flow. Throughout, we emphasize falsifiable consequences and computational schemes (continuum and discrete) that turn the coherence-first perspective into testable physics. B. Core Thesis Physics as coherent gluing of local descriptions. Our starting point is a shift in emphasis: physics is fundamentally about gluing many consistent local descriptions into a single global account. Local observers (or patches) each carry valid descriptions of degrees of freedom; the problem is to compare and reconcile them on overlaps, and to ensure that multi-overlap compositions agree in a hierarchy of compatibility conditions. This perspective is older than the symmetry-first dogma—it is the geometric heart of fiber bundles, connections, and curvature—but it becomes truly structural once one organizes the gluing data categorically, with objects (local models), morphisms (changes of frame), and higher morphisms (relations between changes of frame) supplying the full ladder of coherence conditions [ 11 , 305 ]. In this ladder, coherence—not invariance—is primary; invariance emerges in special limits where the compensators succeed perfectly at every level [4, 5, 15]. Gauge fields as compensators of coherence defects. In the coherence-first view, a gauge connection is not the manifestation of a metaphysical symmetry; it is a compensator that repairs the failure of naive comparisons to glue across overlaps. At level 1 (“1-coherence”), a one-form connection A specifies how to compare local frames, and the corresponding curvature F measures the residual misfit. At level 2 (“2-coherence”), a two-form field B repairs failures in how comparisons themselves compose (associativity up to a controlled correction), with H measuring the remaining defect [ 276 , 305 ]. Seen this way, the chain “global invariance ⇒ local invariance ⇒ gauge field” is replaced by “local descriptions ⇒ need to glue ⇒compensators that implement coherence.” Curvatures as residual obstructions. Curvature is the geometric shadow of what cannot be glued away. The familiar statement that parallel transport around a loop fails to return to identity is precisely the measure of a global obstruction in the cohomology of the relevant complex [ 276 ]. In higher gauge theory, the same logic persists: the two-form curvature H is the obstruction to strict associativity of transition data, and its nontrivial classes record global defects of coherence (e.g., quantized charges, anomalies, or topological responses) [ 305 ]. Quantization is therefore not postulated by symmetry; it reflects the discrete nature of global obstructions in the gluing problem. Higher coherence can supersede strict gauge invariance. Because coherence—not literal invariance—is fundamental, a theory can remain perfectly consistent when strict level-1 invariance is relaxed, provided an appropriate level-2 (or higher) compensator restores overall coherence. This phenomenon is already familiar in anomaly inflow and the Green–Schwarz mechanism: a naively anomalous current (failure of 1-coherence) is rendered consistent by adding a higher-form topological term that supplies the missing 2-coherence [ 5 , 157 , 309 ]. Categorically, this is expressed by replacing a single structure group with a 2-group (a crossed module) and a 2-connection ( A, B )whose curvatures obey coupled Bianchi identities; physical consistency is encoded in the closure of this higher complex, not in the invariance of any single piece [11, 305].
4 Emergent symmetry as the limit of perfect coherence. Symmetry appears, in this framework, as the limit in which the hierarchy of compensators achieves perfect gluing: the residual obstructions vanish ( F = 0, H = 0, . . . ), and the freedom to change local frames without affecting observables can be summarized by a group action. In less symmetric situations—intermediate or “proto-gauge” phases—the compensators exist but do not close strictly; nevertheless, the higher complex still coheres and the physics remains well-defined. This explains why gauge symmetry can emerge in infrared limits and why it may change its type across regimes, a theme that has appeared repeatedly in both high-energy and condensed-matter contexts [4, 5, 15]. Higgs as a coherence lift (1 → 2), not merely symmetry breaking. The Higgs mechanism is traditionally described as spontaneous breaking of a gauge symmetry, giving mass to gauge bosons [16, 17, 164, 166]. From the coherence-first perspective, a scalar condensate does not break consistency; it relocates it: strict 1-coherence is traded for a controlled 2-coherence supplied by the Higgs sector. Concretely, the vacuum organizes into a higher bundle whose 2-morphism data ensure that local frames compose consistently up to a correction carried by the condensate. The gauge-boson mass then measures the energetic cost of maintaining this higher-level coherence; the Higgs mass measures the stiffness of the 2-coherence condition. This reframing clarifies why the combined gauge–Higgs system remains anomaly-free and consistent, and it motivates reconsidering “naturalness” as a question about the renormalization of coherence stiffness rather than of an unconstrained scalar mass. Summary of the thesis. Putting these points together yields the paper’s core thesis: (i) physics is fundamentally the problem of coherently gluing local descriptions; (ii) gauge fields are the compensators that implement this gluing, while curvatures are the residual obstructions that cannot be removed; (iii) higher-categorical coherence can supersede strict gauge invariance, so that relaxing 1-coherence is consistent if 2-coherence (and beyond) is preserved; (iv) symmetry is an emergent limit of perfect coherence, not a primitive axiom; and (v) the Higgs mechanism is best understood as a coherence lift from level 1 to level 2, rather than as a mere breaking of symmetry. In the remainder of the paper we convert this thesis into precise field equations, show how it reorganizes canonical problems (strong–CP, Higgs), and demonstrate how it leads to concrete, testable predictions in condensed matter, where transitional and proto-gauge regimes abound. C. Main Results We now summarize our principal contributions. Each point below is stated precisely in later sections, but we record here the conceptual content, the mathematical statement, and the falsifiable consequences. To orient the reader, we use 1-coherence to denote the ordinary gauge-connection level with curvature F , and 2-coherence to denote the next categorical level with a 2-form field B and curvature H (cf. higher gauge theory and 2-groups). Where relevant, we indicate connections to existing formalisms such as BF/Chern–Simons and AKSZ-type topological actions [20–22]. •A coherence variational principle. We formulate an action functional that penalizes the total coherence defect across categorical levels, S[A, B, Φ] = Z1 2g2kFΦk2+1 2h2kHΦk2+κ 2kDΦk2+V(Φ) + Stop[A, B; Φ], where FΦ = dA + A∧A−tΦ ( B )and HΦ = dB + A .ΦB , the field Φselects the local structure maps ( tΦ, .Φ ), and Stop contains allowed topological couplings (BF/CS, WZ, AKSZ-type terms) [20, 22–25]. We prove: 1. For fixed Φsuch that tΦ≡ 0and .Φ trivial, the stationary equations reduce to Yang–Mills at level 1. 2. For nontrivial ( tΦ, .Φ ), the stationary equations are those of a generalized higher-gauge theory at level 2, with coupled Bianchi identities and modified Ward relations (charge exchange between the 1and 2-form sectors). 3. The Φ-equation (a wave-map with potential and source terms from FΦ, HΦ ) drives adaptive symmetry-type dynamics: different gauge types are local minima of V (Φ), and spatial textures of Φcreate domain walls with protected responses (quantized steps, chiral edge channels). This establishes a single variational principle from which ordinary gauge theory and its highercategorical extensions descend in limits [20–22].
5 •Strong–CP via categorical coherence: two mechanisms rendering θunobservable. We identify two distinct, categorical routes to the empirical invisibility of the QCD θ -term, compatible with standard anomaly physics and large-Ninsights [27, 151]: 1. Exactness in the extended complex: in the higher-coherence complex, the Pontryagin density becomes a total derivative, tr ( F∧F ) = d Ξ( A, B ; Φ) , so that Rθtr ( F∧F )reduces to a boundary term (or vanishes under appropriate boundary conditions). Physically, the θ-dependence is a pure gluing artifact once 2-coherence is enforced. 2. Structural relaxion: allow a mixed topological coupling Smix [Φ; A ] ∼Rhω3 (Φ) , d CS ( A ) i . Minimizing the coherence functional pins the effective angle to zero, θeff = θ + αhω3 (Φ) i → 0 , without postulating a global Peccei–Quinn symmetry [ 29 , 30 , 153 ]. Here Φis a structural field (no mandatory ultra-light axion), and its mass/couplings are set by coherence stiffnesses rather than a decay constant. Both mechanisms are testable: they predict vanishing neutron EDM, preserve nonzero topological susceptibility at θ = 0, and imply specific highT scaling of χt ( T ); lattice toy models with a 2-form sector and a coherence penalty should exhibit flattened θ-dependence as the penalty is increased. •Higgs mechanism as a coherence lift (1 →2): masses from higher-level consistency, not mere breaking. We recast the Higgs effect as relocation of coherence: a scalar condensate trades strict 1-coherence for controlled 2-coherence. Concretely, the vacuum organizes so that F = tΦ ( Bφ ) , with Bφ the 2-morphism data carried by the condensate. Expanding the coherence functional around this vacuum yields: 1. aProca mass for the vector bosons, m2 A∼g2v2 , interpretable as the energy cost of maintaining the lifted (2-)coherence; 2. aHiggs mass that measures the stiffness of 2-coherence. We argue that radiative corrections renormalize this stiffness in a constrained (“soft-natural”) manner because higher Bianchi identities tie it to protected gauge-sector data, echoing Stueckelberg-type protection mechanisms [31, 32, 167]. This coherence-lift picture preserves gauge consistency and offers a geometric rationale for the observed stability of the electroweak scale once the full higher complex is taken into account. •Emergent symmetry: symmetry as the shadow of coherence; proto-gauge phases as partial coherence. We formalize emergent symmetry as the perfect-coherence limit ( F = H = 0), and we define proto-gauge phases where 1-coherence fails (slightly) but 2-coherence already operates to keep physics consistent. This clarifies: –why approximate gauge invariance can arise in the IR without being fundamental; –why selection rules and conservation laws can hold parametrically in crossover regimes; –how deconfined quantum criticality fits naturally into a higher-coherence landscape [34]. The framework predicts concrete signatures in these intermediate regimes: coexisting soft gauge-like modes, partial quantization, and selection-rule violations that scale with the measured coherence defect. •Condensed-matter predictions: spin liquids, axion electrodynamics, fracton elasticity, hydrodynamic electrons. We export the formalism to materials and simulators, obtaining specific, falsifiable predictions: 1. U(1) ↔Z2 spin liquids: the structure field Φselects the gauge type; at domain walls, the theory predicts paired bound modes—a gauge “photon”-derived channel and a naturon (coherence) mode—with characteristic dispersions and polarizations; the lowT specific heat crosses over from T3to activated behavior as mγ(Φ) opens; see Refs. [38, 211] for context. 2. Topological insulators / axion electrodynamics: treating inter-patch Berry data categorically, F∧F becomes exact in the extended complex so the bulk axion term is inert while surface Hall and thermal responses are quantized. Spatial textures of Φyield quantized steps at coherence walls and tunable magneto-electric responses [198, 199].
6 3. Fracton elasticity duality: the 2-form curvature H maps to disclination density; a coherence penalty kHΦk2 produces an elastic energy interpolating between ordinary elasticity and tensorgauge elasticity, predicting a mobility crossover and defect suppression patterns [39]. 4. Hydrodynamic electrons: modeling the translation between kinetic and hydrodynamic descriptions by a “naturality” connection η , the curvature K produces a new collective naturon mode at intermediate frequencies, a specific inversion of nonlocal resistance patterns, and quantized pumping under cyclic drives. These are testable in graphene and delafossites [227, 229]. Together these results demonstrate that higher-categorical coherence is not only conceptually clarifying but also predictively powerful: it yields calculations and signatures that would be opaque or untouchable in a symmetry-first, fixed-group framework. D. Roadmap Organization and guiding logic. The remainder of the paper converts the coherence-first thesis into precise field theory, derives equations of motion from a single variational principle, and develops applications in both high-energy and condensed-matter contexts, with a parallel discrete formulation suited for numerics. For clarity, we outline the flow and what each section establishes or predicts. Sec. 2: Concepts and formalism (coherence hierarchy, higher gauge, structure field Φ). We formalize the hierarchy of coherence levels and the attendant compensators: (1) 1-coherence is implemented by a connection A with curvature F ; (2) 2-coherence is implemented by a 2-form B with curvature H ; higher levels follow the same pattern via higher-form data. We introduce crossed modules/2-groups and 2-connections ( A, B ), together with F = dA + A∧A−tΦ ( B )and H = dB + A .ΦB , and state the coupled Bianchi identities and Ward relations that encode global consistency of the higher complex. A central new ingredient is the structure field Φ: a map into the moduli of allowed structure maps ( tΦ, .Φ ) that selects the type of local redundancy (the “symmetry type”) in space and time. We give concrete exemplars (U(1) ↔Z2 and a color-sector extension) and discuss how quantization of responses reflects global obstruction classes in this extended complex. Sec. 3: Variational principle of maximal coherence and equations of motion. We define a single action functional that penalizes total coherence defect across levels, schematically kFΦk2 + kHΦk2 + kD Φ k2 , augmented by admissible topological couplings. Varying A , B , and Φyields: (i) Yang–Mills at fixed and trivial ( tΦ, .Φ ); (ii) generalized higher-gauge dynamics otherwise, with explicit 1-to-2form charge exchange in the Noether data; (iii) a sourced wave-map equation for Φthat drives adaptive flow between symmetry types. We analyze boundary conditions, energy-momentum, and the role of topological terms (BF/Chern–Simons/Wess–Zumino/AKSZ-type) in quantized responses and anomaly inflow. This section establishes well-posedness and clarifies how coherence, not strict invariance, ensures consistency. Sec. 4: Gauge vs. Coherence—why higher coherence surpasses strict gauge invariance. Here we separate two notions often conflated: gauge invariance (redundancy of description at level 1) and coherence (closure of the full higher complex). We show by example that a controlled failure of 1-coherence can be harmless if compensated by 2-coherence, as in anomaly inflow/Green–Schwarz mechanisms and Stueckelberg-like completions. We formulate sufficient criteria (in terms of the higher Bianchi identities and conserved currents) under which relaxing level-1 invariance leaves physical observables gauge-independent and unitary, thus making precise in what sense “higher coherence supersedes strict gauge invariance.” Sec. 5: Applications—(i) strong–CP, (ii) Higgs reinterpretation, (iii) functorial relation between quantization and Higgs, (iv) emergent symmetry. We present two categorical mechanisms that render the QCD θ -angle unobservable: (A) cohomology trivialization of tr ( F∧F )in the extended complex, and (B) a structural relaxion in which Φdynamically cancels θ without introducing a global Peccei–Quinn symmetry. We then recast the Higgs mechanism as a coherence lift (1 → 2): the gauge-boson mass measures the energy of maintaining lifted coherence; the Higgs mass measures its stiffness, suggesting asoft-naturalness rationale for radiative stability. We make explicit the functorial parallel between geometric quantization (line bundles → gerbes) and the Higgs lift (1-bundles → 2-bundles), arguing that the Higgs field acts as an internal quantization functor for gauge degrees of freedom. Finally, we formalize emergent symmetry as the perfect-coherence limit and characterize proto-gauge phases as partially coherent regimes, with clear experimental and simulation signatures. Sec. 6: Condensed-matter calculations & predictions. We port the formalism to materials and programmable platforms:
7 • U(1) ↔Z2 spin liquids: Φcontrols the gauge type; we compute photon gaps, vison masses, and the spectra of paired domain-wall modes (gauge and coherence/naturon channels), together with specific-heat and neutron-scattering reweighting. • Topological insulators/axion electrodynamics: treating inter-patch Berry data categorically renders the bulk axion term inert (exact in the extended complex) while quantizing surface responses; Φ-textures produce quantized steps at coherence walls and tunable magneto-electric couplings. • Fracton elasticity: we map H to disclination density and derive an elastic energy that interpolates between ordinary and tensor-gauge elasticity, predicting mobility crossovers and defect suppression. • Hydrodynamic electrons: introducing a “naturality” connection η between kinetic and hydrodynamic lenses, we compute its curvature K and predict a distinct naturon resonance in microwave conductivity, geometry-dependent inversions of nonlocal resistance, and quantized pumping under cyclic drives. Each case yields concrete, falsifiable signatures (gaps, steps, exponents, bound dispersions) not accessible in a fixed-group symmetry-first treatment. Sec. 7: Numerics and lattice/DEC formulation for coherence fields. We present a discrete exterior calculus (DEC) and lattice-gauge realization of the higher complex suited for Monte Carlo and HMC. Links carry G0 variables U` , faces carry G1 variables Wf , and sites carry Φ p . We construct exact lattice curvatures FΦ ( f )and HΦ ( τ )obeying discrete Bianchi identities and implement update schemes (link heat-bath/Metropolis; surface-worm moves on faces; HMC for Φ), alongside observables (Wilson loops/surfaces, curvature densities, Φstructure factors) and finite-size scaling collapses. This section supplies the algorithmic bridge from formalism to data, drawing on established DEC and lattice-gauge methods [42, 44, 46, 235, 251]. Sec. 8: Discussion and broader implications. We synthesize the conceptual shift—coherence subsumes symmetry—and the technical payoffs: a unified language for anomalies, Higgs, strong–CP, emergent gauge structures, and condensed-matter responses. We outline open problems (coherence RG and fixed points, reflection positivity for nonabelian 2-gauge discretizations, classification of Φ-flows under anomaly constraints, gravitational extensions), and comment on links to quantum information (entanglement as microscopic coherence) and spacetime geometry as the coherence of local frames. Appendices. Appendix A derives the Φ-modified Bianchi identities and the full Euler–Lagrange system. Appendix B details crossed-module examples (U(1) ↔Z2 , color-sector extensions) and the two strong–CP mechanisms. Appendix C gives DEC/lattice discretization proofs and update algorithms. Appendix D presents linear-response calculations (thermal/charge Hall steps, naturon spectral weight). Appendix E develops the necessary cohomology (with local coefficients), Künneth decompositions for H4 ( BG0×M ), and relations to SPT/anomaly inflow classifications. II. CONCEPTUAL FRAMEWORK: FROM SYMMETRY TO COHERENCE A. Local descriptions, gluing, and redundancy Local observers and the need to glue. Physical descriptions are local in two independent senses. First, they are specified on open sets {Ui} covering spacetime M ; second, they rely on local frames (gauges, bases) chosen by observers on each Ui . Consistency demands that these local pictures be glued across overlaps Uij := Ui∩Uj , and that glueings themselves be compatible on triple overlaps Uijk := Ui∩Uj∩Uk . In differential-geometric terms, this is the content of principal bundles, local trivializations, transition functions, and the associated Cech cocycle conditions [48, 49, 67, 68, 317]. Two lenses F (microscopic) vs. G (effective). Frequently there exist two faithful local descriptions of the same physics: a microscopic lens F (e.g. kinetic or parton/BdG) and an effective lens G (e.g. hydrodynamic or emergent gauge). Formally, one may view F and G as functors from a site of local models Cloc (objects: patches Ui with local fields; morphisms: restriction/flow maps f : Ui→Uj ) into appropriate categories of state spaces or observables. A comparison (or translation) between the two lenses is a family of maps {ηU : F ( U ) →G ( U ) }U⊂M ; naturality of the translation requires that for every morphism f:U→Vone has G(f)◦ηU=ηV◦F(f).(1)
8 Equation (1) is a categorical formulation of coherence between lenses. Its failure, measured by ∆(f) := G(f)◦ηU−ηV◦F(f),(2) is a naturality defect living on morphisms f . As we will see, compensating such defects gives rise to connections—the gluing instructions that make local frames comparable. Principal bundles, local frames, and transition functions. Let G be a (matrix) Lie group and π : P→M a principal G -bundle. A local frame on Ui is a section si : Ui→P . On overlaps Uij two frames are related by a transition function gij :Uij →Gdefined by sj(x) = si(x)·gij(x), x ∈Uij.(3) Choosing a good cover ensures that the gij satisfy the Cech cocycle condition on triple overlaps: gij(x)gjk(x)gki(x) = efor all x∈Uijk.(4) Two choices of frames {si} and {s0 i} are related by a family of gauge transformations hi : Ui→G via s0 i=si·hi, which induce the usual coboundary action on transitions, g0 ij =h−1 igij hj.(5) Thus isomorphism classes of principal bundles are captured by Cech 1-cocycles gij modulo 0-cochains hi [48, 49]. Connections as gluing instructions; curvature as residual mismatch. A principal connection is a g -valued one-form ω∈ Ω 1 ( P, g )that is G -equivariant and reproduces generators on vertical vectors [ 67 , 68 ]. Pulling back by local frames gives the familiar local gauge potentials Ai := siω∈ Ω 1 ( Ui,g ). Equation (3) implies the gluing law for connections on overlaps: Aj=g−1 ij Aigij +g−1 ij dgij ,on Uij.(6) The curvature 2-form F := dω + 1 2 [ ω, ω ] ∈ Ω 2 ( P, g )pulls back to Fi = dAi + Ai∧Ai , which transforms covariantly: Fj=g−1 ij Figij ,on Uij.(7) Equations (6) – (7) encode precisely how A compensates the mismatch of local frames so that F becomes a globally-defined adjoint tensor. The Bianchi identity DF = 0 follows from D2 = [ F, · ]and expresses consistency of the gluing data across triple overlaps [67, 317]. Obstructions and Chern classes via Cech–de Rham. For abelian G = U (1), the topological class of P is captured by the first Chern class c1 ( P ) ∈H2 ( M, Z )obtained by exponentiating the gij into a Cech 1-cocycle and using the boundary map to ˇ H2 ( M, Z )[ 49 , 317 ]. In the presence of a connection, the Chern–Weil representative is i 2πF , whose de Rham class maps to c1 ( P )under the de Rham isomorphism. The Cech–de Rham double complex organizes these facts: on Uij , Aj−Ai = dlog gij ; on Uijk , log gij + log gjk + log gki ∈ 2 πi Z ; the total differential packages local data into a global class [ 317 ]. More generally, U (1)-bundles with connection are classified by degree-2Deligne/Beilinson cohomology, and their higher analogs (gerbes) classify B-fields with curvature H[176, 316]. Example: Dirac monopole as a gluing obstruction. Let M = S2 with cover UN = S2\ {south} , US = S2\ {north} . For G = U (1) take transition gNS ( ϕ ) = einϕ on the equator (angle ϕ ). Define AN = n 2 (1 −cos θ ) dϕ , AS = −n 2 (1 + cos θ ) dϕ . Then AS = AN + dϕ n on UNS and F = dAN = dAS = n 2sin θ dθ ∧dϕ is globally defined. The integral i 2πRS2F = n gives c1 = n∈Z , measuring the obstruction to a single global frame—the textbook Dirac monopole [54, 55]. Gauge transformations as changes of local frames (redundancy). A gauge transformation is a vertical automorphism of P covering the identity on M ; locally it appears as Ai7→ h−1 iAihi + h−1 idhi . Physically, this is a change of frame: A contains redundant information needed to glue local descriptions; the redundancy is benign because all observables are functions of the globally defined F and holonomies (Wilson loops/surfaces) [ 68 ]. In particular, for abelian G one may compute Berry holonomies along loops in parameter space; the corresponding Chern number counts the obstruction to globally defining eigenstates—a direct, measurable incarnation of the gluing problem [55].
9 From Cech data to the emergence of a connection. Return to two lenses F and G on a fixed cover {Ui}. Given local comparison maps ηi:F(Ui)→G(Ui), define on overlaps ∆ij := G(ιij)◦ηi−ηj◦F(ιij),(8) where ιij : Uij ,→Ui is the inclusion. The collection { ∆ ij} is a Cech 1-cochain valued in the sheaf of local morphisms between the two lenses. If there exist morphisms ai on Ui such that ∆ ij = aj−ai (i.e. { ∆ ij} is a coboundary), then redefining η0 i := ηi−ai makes the family {η0 i} natural (coherent) to first order. This is the abstract analog of (6) : the ai play the role of a connection that compensates the mismatch of the two descriptions. On triple overlaps, the residual ∆ ij + ∆ jk + ∆ ki measures a curvature-like obstruction: it vanishes iff the descent data for the comparison glue on Uijk . In this way, any attempt to compare local lenses forces the introduction of compensators whose transformation law matches that of a connection, and whose residual failure to trivialize the descent is a curvature class. Physical content vs. redundancy: Wilson data and observables. Because connections arise to implement gluing, the physical information they carry is encoded in holonomies (Wilson lines and surfaces) and in local curvature densities. For a loop γ⊂Ui, Wγ(A) = tr PexpZγ A,(9) is invariant under re-trivializations (gauge) up to conjugation, and so descends to a global observable. Likewise, for higher-form data one considers Wilson surfaces exp ( iRΣB )subject to the appropriate gauge structure of gerbes [176, 316]. These are the operational probes of the gluing obstruction. Redundancy across scales: renormalization as controlled re-trivialization. Coarse graining relates F to G through scale transformations. From this viewpoint, the renormalization group is a functorial procedure that replaces a microscopic atlas by an effective one while preserving coherence as much as possible [ 245 ]. The emergent gauge structure in the IR is the residual of this process: it is the minimal compensator required to glue the effective charts. In partially coherent crossovers, the comparison fails to be strictly natural; the resulting defects are then sources for the compensators, leading to measurable “anomalies” in transport or response. Summary and outlook within §II. We have made precise the claim that gauge transformations are changes of local frame and that connections are gluing instructions, with curvature the residual obstruction that cannot be gauged away. We have also shown, abstractly, how attempting to compare two legitimate local lenses forces the introduction of compensators with the same descent properties as gauge fields. The next subsections elevate this to the hierarchical setting of higher gauge theory (connections ( A, B )and curvatures ( F, H )) and then introduce a structure field Φthat selects the local rules of gluing, opening the door to adaptive, higher-categorical coherence. B. Coherence hierarchy (categorical levels) Overview. In §II A we emphasized that the basic problem is to glue local descriptions coherently. Mathematically, this amounts to building consistent descent data on multiple overlaps and organizing the resulting compatibility conditions into a hierarchy. Physically, compensators (connections and their higherform generalizations) implement coherence at each level, while curvatures quantify residual obstructions that cannot be glued away. We now formalize three successive layers: 1. 1-coherence: morphisms compose strictly; the compensator is a 1-form connection A with curvature F. 2. 2-coherence: associativity holds only up to a controlled correction; the compensator is a 2-form B with curvature Horganized by a crossed module/strict 2-group [59–61]. 3. Higher levels: coherence of coherence; this is governed by L∞ -algebras and their higher connections and curvatures [63, 64, 177]. We supply explicit descent equations, gauge structures, and Bianchi identities at 1and 2-level, and then record the compact general form at all levels via the L∞Maurer–Cartan formalism.
16 Rank stratification and orbit types. The rank of tis an Aut0-invariant. Let M(r):= {[t, .]∈ M2-grp |rank(t) = r}(57) be the rankr stratum. The Whitney conditions hold for the natural algebraic stratification (hence, for generic paths Φ( γ ), limits are well-defined) [ 79 ]. Different strata correspond to qualitatively distinct gauge types (e.g. t= 0 gives ordinary Yang–Mills, rank(t) = dim g1yields maximal “2-screening”). Local lifts and coordinates. Given a chart U⊂ M2-grp with slice σ : U→ SP , a structure field Φ : M→Ulifts to (tΦ, .Φ) := σ◦Φ. Variations δΦact via differentials (dt)Φ·δΦ∈Hom(g1,g0),(d.)Φ·δΦ∈Bil(g0×g1,g1).(58) Deformations are controlled by the cohomology of the dgLa governing crossed-module deformations (Gerstenhaber/Nijenhuis–Richardson), ensuring that (54) are preserved to first order [80–82]. Kinematics with variable structure maps Curvatures and Bianchi identities with Φ-dependence. Let ( A, B )be a 2-connection for the pointwise crossed module g1 tΦ(x) −−−→ g0, .Φ(x). Define FΦ:= dA +A∧A−tΦ(B), HΦ:= dB +A .ΦB. (59) Because tΦ and .Φ now vary with x , their differentials contribute to the Bianchi identities. A direct calculation yields DAFΦ+tΦ(HΦ) = (dt)Φ(DΦ) ∧B, (60) DAHΦ=FΦ.ΦB+d . Φ(DΦ) ·(A, B),(61) where D Φis the ordinary differential of Φ(a TM2-grp -valued 1-form on M ) and ( dt ) Φ ( D Φ) ∧B denotes the obvious contraction of a 1-form with values in Hom ( g1,g0 )against the g1 -valued 2-form B .Sketch of derivation. Apply d to (59) , use d ( A∧A )=[ dA, A ], and record d ( tΦ ( B )) = ( dt ) Φ ( D Φ) ∧B + tΦ ( dB ) (chain rule on the bundle Hom ( g1,g0 )pulled back by Φ); similarly for the action term in HΦ . The Peiffer identities guarantee cancellations that leave (60)–(61). When Φis constant, the extra terms vanish and we recover (33). Gauge transformations. Infinitesimal 2-gauge transformations ( , λ )act as before on ( A, B )and induce the natural pushforward action on FΦ, HΦ . The new terms in (60) – (61) transform covariantly because dt and d. are tensors on M2-grp and D Φis a geometric 1-form on M . Hence, the variable-structure Bianchi identities are 2-gauge covariant. Kinetic terms, covariant derivatives, and variations Metric and covariant derivative on M2-grp .Equip M2-grp with a (slice-wise) Riemannian metric GΦ induced from an Aut0 -invariant inner product on SP . Define the kinetic term for Φand the covariant derivative DµΦusing the Levi–Civita connection ∇Mof GΦ: LΦ=κ 2GΦ(DµΦ, DµΦ) −V(Φ), DµΦ := ∇M ∂µΦ.(62) On overlaps of slices, GΦ and ∇M are related by the slice transition functions; Palais’ slice theorem implies smoothness of these assignments [ 77 ]. The Euler–Lagrange equation for Φis a wave-map (harmonic-map) equation with sources from the (A, B)-sector: κMΦa+κΓabc(Φ) DµΦbDµΦc+∂V ∂Φa=Sa[A, B; Φ],(63) where the source Sais computed below and Γare the Christoffel symbols of GΦ[83, 84].
17 Variation of the gauge-sector energy w.r.t. Φ.Let Lgauge =1 2g2hFΦ, FΦi+1 2h2hHΦ, HΦi+Ltop(Φ; A, B), with h·,·i denoting inner products on g0and g1-valued forms. Then δΦLgauge =−1 g2FΦ,(dt)Φ(δΦ) ∧B+1 h2HΦ,(d.)Φ(δΦ) ·(A, B)+δΦLtop.(64) Thus the source in (85) takes the explicit form Sa[A, B; Φ] = 1 g2DFΦ,∂tΦ ∂Φa∧BE−1 h2DHΦ,∂.Φ ∂Φa·(A, B)E−∂Ltop ∂Φa.(65) These terms encode the feedback: curvature defects source motion of Φ, so that the system can dynamically reorganize its local redundancy to reduce the total coherence defect. Domain walls, rank jumps, and interfacial responses Rank-jump walls. Let Σ ⊂M be a smooth hypersurface along which Φcrosses from stratum M(r−) to M(r+). In a thin-wall limit, (dt)Φ(DΦ) localizes on Σas a distribution valued in Hom(g1,g0): (dt)Φ(DΦ) thin wall −−−−−−→ ∆t n[δΣ,(66) where ∆ t := tΦ+−tΦ− , n[ is the conormal 1-form, and δΣ is the Dirac distribution on Σ. Inserting (66) into (60) shows that DAFΦ acquires a localized source proportional to ∆ t∧B . Integrating across a Gaussian pillbox yields the jump condition FΦΣ=−∆t∧BΣ,(67) a precise higher-gauge analog of Maxwell’s boundary conditions with an interfacial polarization induced by the change of structure map. Physically, (67) predicts domain-wall channels (bound modes) and quantized steps of response when ∆ t corresponds to a change of a topological class (e.g. a change in an associated ω3(Φ) coupling). Whitney stratification and regularity. By Whitney’s conditions, generic level sets of Φmeet strata transversely and domain walls inherit well-defined tangent/normal data; Ehresmann’s fibration theorem implies that away from Σthe field theory defines locally trivial families of 2-bundles [ 79 ]. This underlies the robustness of interfacial observables: they depend only on the homotopy class of the path Φtakes across the stratification. Topological interfacial terms. Let Ltop include a mixed topological coupling Ltop ⊃2πi Dω3(Φ), d CS(A)E,(68) with ω3a closed 3-form on M2-grp. Then ∂µDω3(Φ),CS(A)E=D(ιDµΦω3)(Φ),CS(A)E+Dω3(Φ),tr(F∧F)Eµ, so that across a domain wall one picks up a quantized surface term proportional to RΣω3 (Φ + ) −ω3 (Φ − ) ∧ CS ( A ), i.e. a quantized Hall/thermal step in direct correspondence with the Φ-jump. This is the categorical generalization of axion domain-wall responses and anyon-condensation interfaces. Discrete picture and computational use Site fields and update moves. On a cell complex K , place Φ p∈ M2-grp on sites p . Local lifts σp define ( tΦp, .Φp )used to compute face/3-cell curvatures F ( f ) ,H ( τ )as in §II C. A change Φ p→ Φ 0 p updates U`, Wf on incident cells through the induced ∆ t, ∆ . , implementing the discrete analogs of (60) – (67) . This makes it straightforward to simulate adaptive gauge types and domain-wall physics in Monte Carlo/HMC schemes (details in §VI F).
18 Summary. The structure field Φupgrades higher gauge theory from a fixed kinematical backdrop to a dynamical geometry of local redundancy. It provides the minimal and natural way to encode symmetry-type transitions, to quantify the feedback between curvature and the grammar of gluing, and to compute robust interfacial phenomena when the rank or orbit type of tΦchanges across spacetime. E. Emergent symmetry as perfect coherence Theme. The coherence-first viewpoint developed in §II A–II D suggests a precise rephrasing of what “having a symmetry” means. Rather than postulating an a priori invariance group, we define symmetry as the zero-curvature (perfect-coherence) limit of the higher gluing problem: when the compensators at each categorical level trivialize the descent data, changes of local frames become cost-free and organize into an effective invariance. Conversely, when coherence is only partial (nonzero curvatures), the same frame changes are approximate redundancies, producing proto-gauge regimes with nearly conserved currents and nearly quantized responses. On this basis, “symmetry breaking” is not a literal violation but a re-organization of coherence (often a lift from 1to 2-coherence, cf. §II B and §II C). Perfect coherence and emergent invariance Flat 2-bundles and transport 2-functors. Let ( A, B )be a 2-connection for the pointwise crossed module g1 tΦ −→ g0, .Φ selected by the structure field Φas in §II D, with curvatures ( FΦ, HΦ )from (59) . Define perfect coherence at level ≤2by FΦ= 0, HΦ= 0.(69) Then parallel transport defines a smooth 2-functor HolA,B : Π 2 ( M ) → GΦ (§II C) that sends any thinhomotopic loop or bigon to the identity (non-abelian Stokes, Eq. (45) ). All Wilson lines/surfaces reduce to global holonomies depending only on homotopy classes of noncontractible cycles. Proposition II.1 (Flatness ⇒emergent invariance).Assume (69). Then: (i) Variations by infinitesimal 2-gauge parameters ( , λ )(§II C) leave all gauge-invariant functionals built from Wilson lines/surfaces invariant. (ii) The space of classical configurations modulo 2-gauge, Cphys = { ( A, B ) |FΦ = HΦ = 0 }/G(2) , is naturally equivalent to Hom Π 2 ( M ) ,GΦ/∼ (equivalence by whiskering), i.e. to representations of the fundamental 2-groupoid. In particular, changes of local frames cost no action: they form an emergent invariance. Proof. (i) follows because under ( , λ ), the curvatures transform covariantly (38) – (39) , and the constraints (69) annihilate all variations of Wilson data (they depend only on FΦ, HΦ via the non-abelian Stokes theorem). (ii) is the standard classification of flat (higher) connections by transport functors [ 96 , 240 ]: 2-gauge transformations act by natural equivalences (whiskering) on functors, and flatness ensures thin-homotopy invariance, hence descent to Π2(M). Emergent global vs. gauge symmetry. In the flat case, two kinds of invariance appear: (i) gauge redundancies (2-gauge) reflecting frame choices; (ii) global symmetries acting on matter that commute with transport. Both arise because perfect coherence makes the action independent of local frame choices; the effective global symmetry is the subgroup of automorphisms of the flat transport 2-functor that acts trivially on superselection sectors. Thus, symmetry is not an axiom but the shadow of perfect coherence. Proto-gauge regimes: small coherence defects Nearly-flat expansion and approximate Ward identities. Let the action for the coherence sector be S[A, B, Φ] = Z1 2g2kFΦk2+1 2h2kHΦk2+κ 2kDΦk2+V(Φ) + Ltop(Φ; A, B),
19 as in §II D. Under an infinitesimal 2-gauge variation ( , λ ), using (38) – (39) and integrating by parts one obtains a Noether-type identity (Noether II [89, 90, 269]) δS δA, DA−t(λ)+δS δB , DAλ+ .ΦB | {z } ≡div J,λ =1 g2hFΦ, t(DAλ)i− 1 h2hHΦ, FΦ.Φλi | {z } ∝ kFΦk2,kHΦk2 +··· ,(70) where the ellipsis collects Φ-variation pieces proportional to ( dt ) Φ ( D Φ) and ( d. ) Φ ( D Φ) (cf. (60) – (61) ). On-shell ( δS/δA = δS/δB = δS/δ Φ=0), the divergence of the would-be Noether current is suppressed by the curvature defects. Thus, in proto-gauge regions where kFΦk,kHΦk,kD Φ k are small, currents are approximately conserved and selection rules hold up to calculable corrections. This is the precise content of “emergent symmetry” in the IR: it is nothing but near-perfect coherence [91]. RG viewpoint. In a Wilsonian picture, operators that violate level-1 invariance but preserve higher coherence often have positive RG dimension and become irrelevant. Equation (70) shows that their effects are tied to curvature defects; as coarse-graining lowers these defects, the symmetry sharpenes. Concrete lattice and continuum studies find emergent O ( N )or SO (5) invariances at deconfined critical points [92, 93], precisely consistent with a flow toward perfect coherence. Breaking as re-organization of coherence From 1to 2-coherence. Consider a one-parameter family of crossed modules selected by Φ( s )such that tΦ(0) = 0 (ordinary Yang–Mills) and tΦ(1) injective. For small s the action reduces to Yang–Mills; for large s , the fake curvature FΦ is constrained to vanish and A becomes massive via the tΦ ( B )term (Stueckelberg-like completion). No invariance is violated: the location of coherence changes. The Higgs mechanism of §II B is exactly such a transition (trading strict 1-coherence for controlled 2-coherence). From the standpoint of Ward identities, the conserved current is redefined by 2-gauge data, not destroyed. Anomalies and inflow as higher coherence. Quantum anomalies are failures of level-1 invariance constrained by the Wess–Zumino consistency conditions [ 94 ]. They are cured by inflow from higherdimensional topological terms (e.g. Chern–Simons/Wess–Zumino) [ 95 ]. In our language, this is the statement that a level-1 coherence defect can be absorbed into level-2 transport data so that the total higher complex closes: precisely the second Bianchi identity in (33) or its Φ-dependent extension (61) . Thus, “anomaly cancellation” is restoration of coherence, not restoration of a broken invariance at a single level. Mathematical characterization: symmetry as automorphisms of coherent descent Automorphism 2-group. Let HolA,B be the transport 2-functor of a perfectly coherent configuration (69) . Define its automorphism 2-group Aut ( HolA,B )as invertible 2-natural transformations of HolA,B . This 2-group acts on all Wilson data and leaves correlation functions invariant. The ordinary global symmetry group is recovered as the π0 of this 2-group (isomorphism classes of automorphisms), while higher symmetries (1-form, 2-form) are encoded in π1, π2 . Thus emergent generalized symmetries arise automatically in the perfect-coherence limit (cf. generalized symmetries as functor automorphisms). Flatness with nontrivial holonomy. Even with FΦ = HΦ = 0, nontrivial global sectors can persist (flat but nontrivial holonomies). These produce discrete superselection sectors; the emergent symmetry acts within sectors but can permute them when allowed by boundary conditions. This matches the physics of topological phases and deconfined gauge theories: perfect local coherence, yet nontrivial global responses. Examples and consequences Topological interface quantization. Across a domain wall where Φchanges rank or orbit type (§II D), the jump condition (67) forces a surface channel and a quantized response step controlled by the change of a class ω3 (Φ) in the mixed topological term (68) . On either side, coherence is perfect; the step is the global relic of different perfect-coherence grammars.
20 Deconfined criticality and emergent SO (5).At the Néel–VBS transition, numerical evidence supports an emergent SO (5) rotating order parameters [ 92 , 93 ]. In our framework, this reflects a near-flat 2-connection with small curvature defects, so the automorphism 2-group of transport is enlarged approximately to SO (5). Deviations from perfect scaling correspond to the defect terms in (70) ; their size quantifies the “distance from perfect coherence.” Summary. Symmetry is what perfect coherence looks like. When curvature defects vanish, frame changes become cost-free and organize into emergent generalized symmetries. When defects are small, one obtains controlled proto-gauge regimes with approximate conservation laws and selection rules. Transitions between “symmetric” and “broken” phases are best seen as re-organizations of where coherence lives in the hierarchy, not as literal violations. III. VARIATIONAL PRINCIPLE OF MAXIMAL COHERENCE A. Coherence functional and action Set-up and conventions. Let ( M, g )be an oriented pseudo-Riemannian d -manifold with metric signature ( −, + ,..., +) (Euclidean signature is obtained by Wick rotation). We denote by ∗ the Hodge operator and by hh·,·ii the L2-pairing of g-valued forms: hhX, Y ii := ZMhX∧∗Yi, where h·,·i is an Ad -invariant nondegenerate bilinear form on the relevant Lie algebra (or module), extended to forms by wedge product. Our sign and normalization conventions for the Hodge dual and L2 -pairings follow [ 98 ]. We work with a structure field Φ : M→ M2-grp as in §II D, which selects at each point the crossed-module data (g1 tΦ −→ g0, .Φ), and with a 2-connection (A, B)and its curvatures FΦ=dA +A∧A−tΦ(B), HΦ=dB +A .ΦB, (71) as defined in (59) . We assume that h·,·i0 on g0 and h·,·i1 on g1 are chosen invariantly so that the adjoints t† Φ:g0→g1and (.Φ)†are well-defined by htΦ(Y), Xi0=hY, t† Φ(X)i1,(72) hX .ΦU, V i1=hX, (.Φ)†(U, V )i0,(73) for X∈g0,U, V ∈g1(the latter understood as a bilinear map valued in g0). The coherence functional. The coherence functional is the L2-norm of the total defect across levels: C[A, B, Φ] := ZMkFΦk2+kHΦk2+kDΦk2,kFΦk2:= hFΦ∧∗FΦi0,kHΦk2:= hHΦ∧∗HΦi1, (74) where D Φis the covariant differential on M2-grp as in (62) . Perfect coherence at levels ≤ 2is attained when FΦ=HΦ= 0 and Φis constant (or a geodesic with vanishing kinetic term). Action with kinetic, potential, topological and matter couplings. The variational principle of maximal coherence is encoded in the action S[A, B, Φ; Ψ] = ZM1 2g2hFΦ∧∗FΦi0+1 2h2hHΦ∧∗HΦi1+κ 2GΦ(DΦ, DΦ) volg+V(Φ) volg +Stop[Φ; A, B] + Smatter[Φ; A, B; Ψ] . (75) with positive couplings g2, h2, κ > 0, a Φ-dependent metric GΦ on M2-grp (slice-wise; cf. §II D), and a potential V (Φ) whose minima represent preferred symmetry types (e.g. U(1) vs. Z2 ). The term Stop collects purely topological couplings (e.g. Chern–Simons, BF, Wess–Zumino-like functionals) [ 102 – 105 ], while Smatter encodes couplings to matter fields Ψ(spinors, scalars, defects) that transform in representations compatible with the pointwise crossed module. Gauge covariance. Under infinitesimal 2-gauge transformations ( , λ )(§II C), ( FΦ, HΦ )transform covariantly and the L2 -norms in (75) are invariant. Topological terms are invariant up to boundary contributions and integer multiples of 2 πi in Euclidean signature (their variations are total derivatives dictated by descent) [ 102 , 105 ]. Thus S is 2-gauge invariant modulo boundary terms. We next compute the Euler–Lagrange equations and the precise boundary contributions.
21 First variations and Euler–Lagrange equations Variations of curvatures. For independent variations δA, δB, δΦone finds from (71): δFΦ=DAδA −tΦ(δB)−(dt)Φ(δΦ) B, (76) δHΦ=DAδB + (δA).ΦB+ (d.)Φ(δΦ) ·(A, B).(77) Here ( dt ) Φ ( δ Φ) is the differential of t at Φacting on δ Φand ( d. ) Φ ( δ Φ) is the differential of the action; both were introduced in §II D, cf. (58). Variation with respect to A.Using (76)–(77) and integrating by parts, δAS=1 g2hhDAδA, FΦii0+1 h2hh(δA).ΦB, HΦii1+δAStop +δASmatter =−1 g2hhδA, D∗ AFΦii0+1 h2hhδA, (.Φ)†(B, ∗HΦ)ii0+hhδA, Jtop A+Jmatt Aii0+bdy,(78) where D∗ A:= (−1)d(p+1)+1 ∗DA∗is the L2-adjoint on p-forms, and we define the induced current J(B) A:= 1 h2(.Φ)†(B, ∗HΦ)∈Ω1(M, g0)via hh(δA).ΦB, HΦii1=hhδA, J(B) Aii0.(79) The contributions Jtop A and Jmatt A come from the variations of Stop and Smatter and may include Chern– Simons/BF-induced currents or matter covariant currents. Variation with respect to B.Similarly, δBS=−1 g2hhtΦ(δB), FΦii0+1 h2hhDAδB, HΦii1+δBStop +δBSmatter =−1 g2hhδB, t† Φ(∗FΦ)ii1−1 h2hhδB, D∗ AHΦii1+hhδB, Jtop B+Jmatt Bii1+bdy.(80) Variation with respect to Φ.The Φ-dependence enters through GΦ and V (Φ), and through tΦ, .Φ in FΦ, HΦand Stop: δΦS=κhhδΦ,−MΦii +hhδΦ,∇V(Φ)ii +1 g2DDFΦ,∗−(dt)Φ(δΦ) BEE0 +1 h2DDHΦ,∗(d.)Φ(δΦ) ·(A, B)EE1+δΦStop +δΦSmatter +bdy.(81) Writing δ Φ = δ Φ a∂a in local coordinates on M2-grp and using the metric GΦ to raise/lower indices, we identify the gauge-sector source: Sa gauge[A, B; Φ] = 1 g2DFΦ,∗∂atΦ·BE0−1 h2DHΦ,∗∂a.Φ·(A, B)E1,(82) with ∂atΦ and ∂a.Φ the differentials introduced in (58) . Contributions Sa top and Sa matt arise from Stop and Smatter. Euler–Lagrange equations (bulk). Ignoring boundary terms (see below), the stationary conditions δS = 0 for arbitrary compactly supported variations yield: 1 g2D∗ AFΦ−1 h2(.Φ)†(B, ∗HΦ) + Jtop A+Jmatt A= 0,(83) 1 h2D∗ AHΦ−1 g2t† Φ(∗FΦ) + Jtop B+Jmatt B= 0,(84) κMΦa+∂V ∂Φa− Sa gauge − Sa top − Sa matt = 0.(85) Equations (83) – (84) make explicit the charge exchange between the 1and 2-form sectors through tΦ and .Φ . Equation (85) exhibits the feedback of curvature defects onto the structure field: nonzero FΦ, HΦ act as sources pushing Φtoward regions of M2-grp that reduce the total coherence defect.
22 Gauge invariance and Bianchi identities. Applying DA to (83) and using the variable-structure Bianchi identity (60) together with the adjointness relations (72) – (73) implies the continuity equation (schematically) 1 g2DAD∗ AFΦ=1 g2tΦ(HΦ)∗+1 h2DA(.Φ)†(B, ∗HΦ) + ··· =−DAJtop A+Jmatt A,(86) where the ellipsis denotes the explicit ( dt ) Φ ( D Φ) ∧B and ( d. ) Φ ( D Φ) · ( A, B )pieces from (60) – (61) , which cancel against terms induced by (85) when matter and topological sources are included consistently. This is the higher-gauge analog of Noether consistency relations in the variational bicomplex [99–101]. Boundary terms and boundary conditions Boundary contributions. From (78)–(81), the boundary terms take the schematic form bdy =Z∂M 1 g2hδA ∧∗FΦi0+1 h2hδB ∧∗HΦi1+κhδΦ, nµDµΦiGΦdΣ+δS∂M top , with nµ the outward unit normal and dΣ the induced measure. Well-posedness requires either Dirichlettype conditions ( δA|∂M = δB|∂M = δ Φ |∂M = 0) or appropriate Neumann/mixed conditions (e.g. in∗FΦ|∂M = 0, in∗HΦ|∂M = 0, nµDµ Φ |∂M = 0), possibly augmented by boundary degrees of freedom that absorb δS∂M top in topological phases [102]. Checks, limits, and positivity Recovery of Yang–Mills and Kalb–Ramond limits. If tΦ≡0and .Φtrivial, (75) reduces to S=Z1 2g2kFk2+κ 2kDΦk2+V(Φ) + Smatter, and (83) becomes the Yang–Mills equation D∗ AF = Jmatt A . If instead g0 is abelian and tΦ is injective (Stückelberg-like), the pair ( A, B )reproduces the coupling of a 1-form to a 2-form gauge field (Kalb– Ramond/ Freedman–Townsend) [ 103 , 104 ], with (83) – (84) the standard massive equations linearized about tΦ. Energy positivity (Euclidean signature). On a compact Euclidean ( M, g ), the kinetic sector of (75) is manifestly nonnegative; in dimensions d= 4 one has the topological bound ZMhFΦ∧∗FΦi0≥ZMhFΦ∧FΦi0 by the Cauchy–Schwarz inequality, with equality on (anti-)self-dual sectors. Similar quadratic completions can be carried out in the presence of mixed topological terms (BF/Chern–Simons/WZ), yielding Bogomolny-type bounds in appropriate dimensions (details depend on Stop). Summary. The action (75) implements the variational principle of maximal coherence: its Euler– Lagrange equations (83) – (85) drive the system toward configurations with small ( FΦ, HΦ, D Φ) while allowing for topological and matter sources. The coupled structure enforces 2-gauge consistency through the variable-structure Bianchi identities (60) – (61) , and reduces in limits to familiar theories (Yang–Mills, Kalb–Ramond). Crucially, the Φ-equation embodies the adaptive aspect of the framework: curvature defects act as sources that move the system in M2-grp toward symmetry types that reduce the total coherence defect. B. Euler–Lagrange equations and generalized Bianchi Aim. Starting from the coherence action (75) and the Φ-dependent curvatures (71) , we (i) derive the Euler–Lagrange (EL) equations in a covariant form that makes explicit the sources carried by the structure field Φ, (ii) establish the generalized Bianchi identities when tΦ and .Φ vary over spacetime, and (iii) deduce Ward identities (Noether II) that quantify charge exchange between the 1-form and 2-form sectors. We keep explicit the Hodge adjoints and pairings so that boundary conditions and sign conventions are transparent.We follow the analytic conventions of the variational bicomplex [ 106 , 107 ], Hodge theory [108], and the covariant phase-space formalism [109].
23 Covariant first variations and EL operators Recap of variations. For independent variations (δA, δB, δΦ) one has (from (71) and chain rule) δFΦ=DAδA −tΦ(δB)−(dt)Φ(δΦ) B, (87) δHΦ=DAδB + (δA).ΦB+ (d.)Φ(δΦ) ·(A, B).(88) Inserting these into the kinetic part of (75) , integrating by parts, and adding matter/topological pieces yields the EL equations already displayed in (83) – (85) . For completeness we repackage them as EL operators EA:= 1 g2D∗ AFΦ−1 h2(.Φ)†(B, ∗HΦ) + Jtop A+Jmatt A,(89) EB:= 1 h2D∗ AHΦ−1 g2t† Φ(∗FΦ) + Jtop B+Jmatt B,(90) Ea Φ:= κMΦa+∂V ∂Φa− Sa gauge − Sa top − Sa matt,(91) so that the field equations read EA= 0,EB= 0, and EΦ= 0. Explicit sources from the Φ-dependence. The Φ-sector source Sgauge appearing in (91) is Sa gauge[A, B; Φ] = 1 g2DFΦ,∗∂atΦ·BE0−1 h2DHΦ,∗∂a.Φ·(A, B)E1,(92) as in (82) . It shows that curvature defects provide a force on Φthat drives the system in the target M2-grp toward regions with smaller total coherence defect. Generalized Bianchi identities with variable structure maps Chain-rule derivation. Applying DA to FΦ and HΦ and using (34) – (35) (at the point Φ( x )) gives the variable-structure Bianchi identities already stated in (60)–(61): DAFΦ+tΦ(HΦ) = (dt)Φ(DΦ) ∧B, (93) DAHΦ=FΦ.ΦB+d . Φ(DΦ) ·(A, B).(94) Proof. Differentiate FΦ=dA +A∧A−tΦ(B): DAFΦ=DA(dA +A∧A)−DAtΦ(B)= [FΦ, A]−(dt)Φ(DΦ) ∧B−tΦ(DAB), where the Leibniz rule on the pullback bundle Φ( Hom ( g1,g0 )) yields the middle term, and DA ( tΦ ( B )) = ( dt ) Φ ( D Φ) ∧B + tΦ ( DAB ). Using DAB = HΦ−A .ΦB and the Peiffer identity tΦ ( A .ΦB ) = [ A, tΦ ( B )], we find DAFΦ = tΦ ( HΦ )+( dt ) Φ ( D Φ) ∧B . The second identity follows similarly from DAHΦ = DA(dB +A .ΦB)and the derivation property of the action. Consequences. When Φis constant, ( dt ) Φ ( D Φ) = ( d. ) Φ ( D Φ) = 0 and (93) – (94) reduce to the standard higher Bianchi identities (33) . When Φvaries, the extra terms quantify how gradients of the symmetry type act as effective sources for the Bianchi identities—the geometric origin of the interfacial jump condition (67) in §II D. Ward identities and charge exchange Gauge variation and Noether II. Under an infinitesimal 2-gauge variation (, λ)(with Φinert), δA=DA, δλA=−tΦ(λ), δλB=DAλ, δB= .ΦB, the variation of the full action is (modulo boundary terms) δS =hhδA, EAii0+hhδB, EBii1=hhDA−tΦ(λ),EAii0+hhDAλ+ .ΦB, EBii1.(95)
24 Integrating by parts and using the adjoints D∗ Aand (.Φ)†yields the off-shell Noether identities D∗ AEA−(.Φ)†(B, EB)≡0,(96) t† Φ(EA)−D∗ AEB≡0.(97) These hold identically (Noether II) and encode the compatibility of the EL operators with 2-gauge invariance [106, 110]. Charge-exchange form. Splitting EA,B into kinetic and (topological+matter) parts using (89) – (90) , (97) becomes t† Φ1 g2D∗ AFΦ−1 h2(.Φ)†(B, ∗HΦ)−D∗ A1 h2D∗ AHΦ−1 g2t† Φ(∗FΦ)=−t† Φ(Jtot A) + D∗ AJtot B,(98) where Jtot A,B := Jtop A,B +Jmatt A,B . Using the Bianchi identities (93)–(94) to simplify the kinetic combination, one obtains a compact exchange law for covariant currents: t† ΦJtot A=D∗ AJtot B,(charge exchange between 1and 2-form sectors). (99) Similarly, (96) gives the “divergence” identity D∗ AJtot A= (.Φ)†B, Jtot B,(100) which reduces to covariant conservation D∗ AJtot A = 0 when B = 0 and to D∗ AJtot B = 0 when tΦ = 0 (no 1 ↔ 2exchange). Equations (99) – (100) are the sought-for Ward identities; they quantify how a nonzero tΦ converts 2-form charge into 1-form charge and how the presence of B twists the covariant divergence in the A -sector. In canonical language these are the secondary constraints that generate the 2-gauge symmetry; see, e.g., the discussion of constraints and Ward identities in [110, Ch. 9]. Matter-sector corollaries. If the matter action is strictly 2-gauge invariant by itself, then varying Smatter as in (95) with EA,B = 0 implies t† ΦJmatt A=D∗ AJmatt B, D∗ AJmatt A= (.Φ)†B, Jmatt B,(101) i.e. the same exchange and divergence laws for matter currents. In particular, in phases where tΦ is large (maximal rank), the 2-form matter charge rapidly drains into the 1-form channel, while in phases with B≃0the 1-form current is covariantly conserved. Local continuity form and boundary conditions Continuity equations in components. In a local orthonormal frame, (100) reads (suppressing Lie indices) ∇µJµ A=1 (d−2)! (Bµ1µ2)Jµ1µ2µ3...µd−1 Bεµ3...µd−1µ, showing explicitly how the B -background twists the A -current conservation. On a spatial slice Σwith unit normal nµ , the integrated forms of (99) – (100) give boundary flux relations useful in numerics and in interface problems (cf. §II D): through a pillbox crossing a Φ-wall one obtains the jump relation (67) and its current counterpart nµJµ AΣ=nµ(.Φ)†B, JBµΣ. Topological terms and anomaly inflow. Topological pieces Stop may shift the Ward identities by local terms (Chern–Simons descent, BF contact terms) without spoiling their integrability [ 112 , 274 ]. In the presence of boundaries, δStop produces ∂M -currents that account for anomaly inflow; the bulk Ward identities then match boundary nonconservation, as usual.
25 Summary We have shown that: (i) the EL operators (89) – (91) capture the dynamics of the coherence sector with explicit Φ-sources, (ii) the generalized Bianchi identities (93) – (94) encode how gradients of the symmetry type enter as effective sources/sinks, and (iii) the Ward identities (96) – (97) imply the charge-exchange laws (99) – (100) , which reduce to familiar covariant conservation laws in special limits. These identities guarantee consistency of the coupled system under 2-gauge transformations and will be crucial when we analyze topological responses and condensed-matter applications in §VI. C. Stationary points and recoveries Goal. We analyse the stationary points of the coherence action (75) and show how familiar field theories are recovered as corners or limits: (i) Yang–Mills at fixed structure type (constant Φwith tΦ =0 and trivial action), (ii) the ordinary Higgs/Stückelberg mechanism as a coherence lift from 1to 2-level, and (iii) BF/Chern–Simons (CS) topological sectors when kinetic penalties are switched off. We then verify consistency with anomaly inflow when Stop is present. Vacua, stationary equations, and topological sectors Stationary equations. Varying (75) yields the Euler–Lagrange (EL) operators EA:= 1 g2D∗ AFΦ−1 h2(.Φ)†(B, ∗HΦ) + Jtop A+Jmatt A= 0,(102) EB:= 1 h2D∗ AHΦ−1 g2t† Φ(∗FΦ) + Jtop B+Jmatt B= 0,(103) Ea Φ:= κMΦa+∂V ∂Φa− Sa gauge − Sa top − Sa matt = 0,(104) with Sa gauge as in (92). The generalized Bianchi identities for variable structure maps are DAFΦ=tΦ(HΦ)+(dt)Φ(DΦ) ∧B, (105) DAHΦ=FΦ.ΦB+d . Φ(DΦ) ·(A, B),(106) and imply the Ward identities (Noether II) and charge exchange between the 1and 2-form sectors (cf. (99)–(100)). Vacua and sectors. In the absence of matter/topology and with suitable boundary conditions, absolute minima satisfy FΦ≡0, HΦ≡0, DΦ≡0,Φ(x)∈Argmin V. (107) Solutions are classified by flat (higher) bundles modulo 2-gauge equivalence; distinct topological sectors remain when π1 or higher homotopy of M is nontrivial. When Stop is present, vacua can carry global charges fixed by the quantized couplings; the bulk equations still reduce to (107) but nontrivial boundary data encode topological response. Yang–Mills corner (fixed Φwith trivial 2-level) Set-up and result. Take Φ=Φ ? constant, with tΦ?≡ 0and .Φ? trivial. Then FΦ? = dA + A∧A , HΦ?=dB, and the action reduces to S[A, B; Φ?] = Z1 2g2kFk2+1 2h2kdBk2+S(1) top[A] + S(2) top[B] + Smatter[A; Ψ], with 1-form and 2-form sectors decoupled. The A -equation is precisely the Yang–Mills equation with sources, 1 g2D∗ AF+δS(1) top δA +δSmatter δA = 0,(108)
32 Global anomalies. Even when there is no local polynomial I2n+2 (e.g. the SU(2) anomaly in d= 4), there can be a global obstruction detected by πd+1 ( G )[ 137 ]. Freed–Hopkins show that such anomalies are precisely classified by invertible field theories; inflow restores consistency when the corresponding invertible theory is attached to the boundary [ 307 ]. This again fits our criterion: the 1-gauge failure is cancelled by adding a higher-level (bulk) compensator. Example II: categorical Higgs—internal repair at 2-level From noninvariant mass to coherent mass. A naive vector mass m2 2RhAµ, Aµi is not 1-gauge invariant. Introduce a 2-form Band consider instead the coherent mass functional (cf. §III C) Scoh[A, B] = Z1 2g2kF−tΦ(B)k2+1 2h2kHk2,(133) with 2-gauge transformations δλA = −tΦ ( λ ), δλB = DAλ . Under 1-gauge, δ ( F−tΦ ( B )) = [ F−tΦ ( B ) , ], so the entire integrand is 2-gauge invariant even though A alone need not be. Expanding (133) to quadratic order about a fake-flat background generates a Proca mass for the components in im tΦ (cf. (114) ), without ever writing a noninvariant A2 -term. Thus, what would be “gauge breaking” at level 1 is reinterpreted as coherence reallocation to level 2. Equations of motion and Ward identities. Varying (133) gives (102) – (103) with Jtot A,B = 0 and enforces the exchange Ward identities (99) – (100) . In particular, the would-be nonconservation of the 1-form current induced by a mass term is exactly balanced by the 2-form sector via t† Φ . This proves, at the level of Noether II, that coherence supersedes strict 1-invariance. Relation to CCWZ and Stückelberg. The coset construction gives invariant Lagrangians for nonlinearly realized global symmetries; Stückelberg promotes this to a gauge redundancy by introducing compensators. Equation (133) is the higher analog: B is the compensator at the 2-level. Different gauges (unitary vs. Stückelberg) correspond to moving along the coherence redundancy leaf (§IV A); spectra and S-matrix elements agree [139, 140]. Example III: failure of strict invariance localized at domain walls Φ-textures and interfacial terms. Consider the topological coupling (already used in §II D) Smix[Φ; A] = 2πi ZMω3(Φ), d CS(A).(134) Under a 1-gauge variation, δSmix = 2 πi R∂M hω3 (Φ) ,tr ( dA ) i , a pure boundary term. If Φhas a domain wall Σacross which ω3 jumps, Stokes’ theorem converts part of this to a term localized on Σ. Adding a coherence wall action (a Chern–Simons/WZW-type theory living on Σ) cancels the variation; physically one gets chiral channels bound to Σwhose anomaly matches the jump, in exact analogy with axion electrodynamics and parity anomaly [ 136 ]. Thus strict 1-invariance can fail locally in the bulk but be restored by 2-coherence encoded in the Φ-texture and by degrees of freedom on the wall. Constraints and nonrepairable anomalies When relaxation is impossible. Not every 1-gauge failure admits a coherence repair. Global anomalies with no invertible-field-theory trivialization (e.g. certain discrete or time-reversal anomalies) cannot be fixed by local 2-level fields alone [ 141 , 142 , 307 ]. In our language, A(1) is not in the tΦ -image of any local K ; only coupling to a bulk invertible TQFT (an external higher level) cancels it. This demarcates the boundary between proto-gauge regimes (repairable by B ) and intrinsically anomalous regimes (repairable only by inflow). Summary Strict 1-gauge invariance may be relaxed if the failure is (i) a boundary term cured by inflow, or (ii) the tΦ -image of a 2-level quantity and thus cancelled by internal 2-coherence. Both mechanisms
33 fit naturally in the generalized Bianchi and Ward framework (§III B), and both are realized in familiar physics: anomaly inflow, parity anomaly, and mass generation via Stückelberg/Higgs-like completions. The coherence-first criterion is therefore weaker and more structural than invariance: it requires only that the higher gluing problem be solvable. C. Quantization as obstruction; topological responses as global defects Thesis. In the coherence-first framework, quantization is not a primitive postulate nor a mystifying relic of group representation theory. It is the inevitable shadow of residual non-glueability of local data: when the gluing of potentials is only possible up to integral ambiguities, the corresponding curvatures carry integral periods, and topological couplings evaluate to integers on fundamental cycles. Equivalently, quantized response coefficients are pairings of differential cohomology classes with integral homology. This perspective is standard in modern treatments of gauge theories and topological phases: see Cheeger– Simons differential characters [ 143 ] and the Hopkins–Singer model of differential cohomology [ 144 ]. We make this precise at level 1 (line bundles), level 2 (bundle gerbes), and for 2-group/crossed-module structures, then extract concrete topological responses and interfacial (domain-wall) effects. Weil integrality from coherence: line bundles and gerbes Level 1: line bundles (Chern class). Let {Ui} be a good cover of M and Ai∈ Ω 1 ( Ui )local U (1)- connections with transition phases gij :Uij →U(1) obeying Aj−Ai=g−1 ij dgij on Uij, gijgjkgki =e2πinijk on Uijk, with nijk ∈Z locally constant. Here and below we use additive notation for Cech cochains where convenient; for U (1) one may write gij = e2πiaij with aij ∈R/Z , so aij + ajk + aki = nijk ∈Z . The 2-form curvature F = dAi is globally defined, and the class c1∈H2 ( M, Z )has de Rham representative 1 2πF. Proposition IV.2 (Weil integrality via coherence) . The gluing data above define a line bundle with connection iff for every smooth closed surface Σ⊂M, 1 2πZΣ F∈Z.(135) Sketch. Choose a triangulation subordinate to the cover, integrate F over each 2-simplex, and use Stokes’ theorem plus the overlap relation for Ai to reduce the integral to a sum over edges of log gij . The edge sum collapses to a sum of 2 πnijk over 2-simplices; hence the period is integral. Conversely, if all periods are integral, the U (1)-valued holonomy is well-defined on any loop, and Cech descent reconstructs gij ; see [143]. Thus, integrality is exactly the residue of non-glueability measured by the Cech cocycle (nijk). Level 2: bundle gerbes (Dixmier–Douady class). A U (1) bundle gerbe with connective structure is specified by 2-form potentials Bi∈ Ω 2 ( Ui ), 1-form connections Aij ∈ Ω 1 ( Uij ), and transition phases gijk :Uijk →U(1) obeying the higher descent Bj−Bi=dAij, Ajk −Aik +Aij =g−1 ijkdgijk, gjklgijl =gijkgikl on Uijkl, with 3-curvature H = dBi globally defined. The Dixmier–Douady class DD ∈H3 ( M, Z )is represented by 1 2πH: 1 2πZC H∈Zfor every smooth closed 3-cycle C⊂M. (136) As in Proposition IV.2, integrality expresses the obstruction to trivializing ( B, A, g )one level up [ 145 ]. In modern language, a gerbe with connection is a degree-3differential cohomology class ˇ h∈b H3 ( M ; Z )with curvature Hand characteristic class DD [144].
34 Differential characters and response holonomy. Both levels are subsumed by differential characters ˇ h∈b Hk ( M ; Z )[ 143 ], which assign U (1)-valued holonomies to smooth ( k− 1)-cycles and have curvature forms with integral periods. Topological response functionals are then the holonomies of such characters: exp2πi hˇ h, Zi, Z ∈Zk−1(M),(137) well-defined precisely because the integrality constraints ensure independence of choices (representative, bounding chain). Quantized topological couplings from coherence Chern–Simons and level quantization. In d = 3, a G -Chern–Simons functional is the holonomy of a differential character ˇ hCS ∈b H4 ( BG ; Z )pulled back by a gauge field (classifying map); its coupling (level) k must be integral so that large G -transformations shift the action by 2 πi Z . This is the character version of the well-known WZW/CS level integrality; see [ 146 ] for discrete gauge groups. Thus level quantization is the statement that the obstruction lives in integral cohomology, not in R-cohomology. BF theory. For compact abelian groups, the mixed BF coupling 2 πi k ZB∧F pairs two differential characters ˇ b∈b Hp+1 ( M )and ˇa∈b Hd−p ( M ). Gauge invariance under large transformations requires k∈Z ; again, kis the integer pairing in b H•[147]. Axion electrodynamics and θ periodicity. In d = 4, Sθ = iθ 8π2Ztr ( F∧F )is well-defined mod 2 πi because 1 8π2Rtr ( F∧F ) ∈Z for compact G and appropriate normalization. Thus θ∼θ + 2 π without postulating any additional symmetry: periodicity is the integrality of the second Chern character. In topological insulators, θ = 0 , π arises from time-reversal, but periodicity itself is purely cohomological; see [148]. Berry bundles and TKNN integer. In the integer quantum Hall effect, the Hall conductance σxy is the first Chern number of the Berry bundle over the Brillouin torus T2 : σxy = e2 h 1 2πRT2FBerry ∈e2 hZ. Here integrality is again the residual of non-glueability of Bloch eigenstates across transition charts in k -space [149]. Quantization at 2-level and for 2-groups Gerbe holonomy and surface quantization. For a U (1) gerbe, the Wilson surface exp iRΣB is welldefined for closed Σiff 1 2πRCH∈Z for any 3-chain C with ∂C = Σ, i.e. iff the Dixmier–Douady class is integral (always the case for bona fide gerbes). Couplings of the form 2 πi Zη∧H with η a closed ( d− 3)-form are quantized when [ η ] ∈Hd−3 ( M, Z )in the image of integral cohomology under de Rham, tying again response to obstruction [147]. 2-group (crossed-module) quantization: Postnikov classes. A strict Lie 2-group with crossed module ( G1 t −→ G0, . )is characterized by a Postnikov class k∈H3 ( BG0, Z( G1 )) measuring the failure of associativity to be strict. Topological actions for 2-gauge fields are classified by (twisted) cohomology classes built from k ; their quantization is dictated by integral lifts of these classes [ 309 ]. In particular, for discrete 2-groups, the analogue of Dijkgraaf–Witten theory lives in H4 ( BG, U (1)), and the path integral is the holonomy of a degree-4 differential character pulled back by the 2-connection. Mixed 2-group responses. Mixed terms such as Smix = 2πi ZhB∧Fi+hω3(Φ),CS(A)i(138) are well-defined (mod 2 πi ) precisely when the bilinear pairings and ω3 (Φ) represent integral differential characters. This yields quantized magneto-electric-type responses once again as pairings in b H•. Global defects and interfacial quantization Domain walls as global defects in ˇ H• .Let Φvary across a codimension-1 submanifold Σ, changing the differential character ω3 (Φ) by an integral class ∆ ω3∈H3 ( M2-grp,Z ). Then the mixed term in (138)
35 induces a surface topological action SΣ= 2πi ZΣh∆ω3,CS(A)i,(139) i.e. a quantized Chern–Simons level jump equal to the integer pairing of ∆ ω3 with the boundary class. This is the categorical version of axion domain walls and the parity anomaly relation between bulk θ and boundary Chern–Simons [ 148 ]. The wall is a global defect of the coherence structure: it cannot be removed by local redefinitions and carries robust edge modes. Monopoles, vortices, and higher defects. At level 1, a Dirac monopole is the obstruction c16 = 0; its magnetic charge is the integral period of F . At level 2, a H -flux source (NS5-like) is the obstruction DD 6 = 0; its charge is the integral of H . In crossed-module theories, point/line/surface defects are charged under ( F, H )in such a way that their charges are valued in the appropriate integral cohomology groups; their braiding phases are evaluations of the corresponding differential characters on links (higher Aharonov–Bohm phases). “Quantization without symmetry quantization” No symmetry postulate needed. In all examples above, quantization follows from gluing and global well-definedness of holonomies in the higher descent tower. No assumption about compactness of a symmetry group nor about representations is required; compactness enters only insofar as it ensures integral characteristic classes exist. For non-compact groups, the relevant integral lattices are absent and one loses integrality unless additional structure quantizes the periods (e.g. a discrete subgroup). Path-integral derivation. Let Stop [ A ]be a topological coupling built from a differential character ˇ h∈b Hn ( M ; Z )evaluated on the ( n− 1)-cycle determined by the gauge configuration A . Under a large gauge transformation A 7→ Aγclassified by γ∈Hn−1(M, Z), one has Stop[Aγ]−Stop[A] = 2πi h[ˇ h], γi ∈ 2πi Z.(140) Hence e−Stop is gauge invariant in the quantum theory. Equation (140) is the path-integral avatar of Propositions IV.2–(136): coherence enforces integrality. Case studies of quantized responses Integer quantum Hall (2D). σxy = e2 hc1 ( L )with L the Berry line bundle; c1 measures obstruction to a global Bloch frame [149]. Topological insulator (3D). θ -term with θ∼θ + 2 π ; a θ -domain wall supports a level- ∆θ 2π surface Chern–Simons term, yielding a quantized magneto-electric effect [148]. Discrete gauge topological order (3D). Dijkgraaf–Witten actions RMω with ω∈Z4 ( BG, U (1)); quantization condition ω∈H4 ( BG, U (1)) ensures invariance under large G -gauge transformations and determines ground-state degeneracy [146]. 2-group response. Postnikov class k∈H3 ( BG0, Z( G1 )) induces mixed terms; their integrality controls quantized 1-form/0-form symmetry interplay and anomaly coefficients in systems with emergent higher symmetries [309]. Summary Quantization is the arithmetic face of coherence: integral periods and level quantization are the precise markers of global defects in the descent tower of gluing data. Topological responses—Hall conductance, axion magneto-electric effect, Dijkgraaf–Witten phases, and their 2-group generalizations—are evaluations of differential characters on cycles and interfaces. This explains why quantization persists even when strict 1-gauge invariance is relaxed: what matters is coherence of the higher complex, not the invariance of any particular level.
36 V. APPLICATIONS IN HIGH-ENERGY CONTEXT A. Strong–CP problem via coherence Problem statement. The Euclidean QCD action may include the CP-odd topological term Sθ[A] = iθ 8π2ZM tr(F∧F) = i θ Q[A], Q[A]∈Z,(141) where M is a closed oriented 4-manifold (or R4 with appropriate falloff), F = dA + A∧A , and Q [ A ] is the instanton number [ 151 ]. The experimental upper bound on the neutron electric dipole moment, |dn|< 1 . 8 × 10 −26 e·cm (90% C.L.) [ 283 ], together with hadronic computations, implies |θeff|. 10 −10 . Why is θ so small? The Peccei–Quinn (PQ) mechanism promotes θ to a dynamical axion angle via a global U (1) PQ [ 153 ], but it introduces a new light boson with tight astrophysical constraints. Here we present two coherence-first alternatives that require neither an exact PQ symmetry nor a light axion: •Route A (cohomology trivialization). In the extended (higher-gauge) descent complex, the class of tr ( F∧F )becomes exact. The θ -term is then a coboundary and drops from all observables. •Route B (structural relaxion). The structure field Φcouples through a mixed topological term hω3 (Φ) , d CS ( A ) i . Minimizing the full action within the variational principle of maximal coherence (§III) drives the effective θto zero, without invoking a PQ symmetry. We develop both routes, state the precise conditions under which they operate, and outline phenomenological checks: neutron EDM, topological susceptibility at θ= 0, thermal behavior, and anomaly consistency. Route A: cohomology trivialization of F∧Fin the extended complex Total differential and extended descent. Let ( g1 tΦ −→ g0, .Φ )be the (possibly Φ-dependent) crossed module of §II D, and ( A, B )the corresponding 2-connection with curvatures FΦ = dA + A∧A−tΦ ( B ) , HΦ = dB + A .ΦB (§II C). Consider the total differential on the bigraded complex of forms valued in g0⊕g1 , dΦ:= d+tΦ,(142) where tΦ raises the internal degree by mapping g1 -valued p -forms to g0 -valued ( p+ 1)-forms via tΦ . Thanks to the Peiffer identity, d 2 Φ = 0 on gauge-invariant polynomials.More invariantly: one may package the data in an L∞ -algebra and use its Maurer–Cartan curvature F with the total differential discussed in §II B. A 3-form primitive for tr ( F∧F ).Let h·,·i0 be an Ad -invariant bilinear form on g0 , and assume there exists a bilinear h·,·i01 :g0×g1→Rsuch that hX, tΦ(Y)i0=dΦhX, Y i01at the level of invariant polynomials.(143) Define the extended Chern–Simons 3-form Ω3(A, B; Φ) := CS3(A)−2hA, Bi01 +hB, Bi11,(144) where CS3 ( A ) = tr A∧dA + 2 3A∧A∧A , and hB, Bi11 is a suitably chosen g1 -quadratic pairing. Concrete realizations arise in Green–Schwarz-type systems where dB cancels F∧F in cohomology [ 157 , 158 ]. A direct computation using F=FΦ+tΦ(B)and the Bianchi identities (33) shows dΦΩ3(A, B; Φ) = tr(F∧F),(145) provided the pairings satisfy (143) and the hB, Bi11 -term is chosen appropriately (it drops out if tΦ is injective and we restrict to fake-flat sectors FΦ=0). Proposition V.1 (Trivialization of the θ -density in the extended complex) . If there exists a globally defined Ω3(A, B; Φ) obeying (145), then the functional Sθ[A] = iθ 8π2ZM tr(F∧F) = iθ 8π2ZM dΦΩ3(A, B; Φ) (146) is a coboundary in the extended complex. On closed M it reduces to a boundary term in the higher sense and is unobservable: the partition function and all correlators are independent of θ.
37 Sketch. Exponentiating (146) gives a phase equal to the extended holonomy of Ω 3 around the (3 + 1)-cycle determined by ( A, B, Φ). Extended exactness implies trivial holonomy on closed cycles. Gauge variations shift Ω3by an extended total derivative; the path integral weight is unchanged. Cohomological condition. Geometrically, Proposition V.1 requires that the differential cohomology class of 1 8π2tr ( F∧F )lifts to zero in the extended (2-group) differential cohomology b H4 ext ( M ; Z ). This is the 4d analogue of the Green–Schwarz mechanism and of Freed–Witten trivializations in the presence of a B -field [ 157 , 158 ]. When it holds, θ drops out nonperturbatively without introducing a PQ symmetry or a propagating axion. Consistency with Vafa–Witten and topology. Vafa and Witten showed that in ordinary QCD the vacuum energy is minimized at θ= 0(no spontaneous CP violation) [ 159 ]. Trivialization makes the energy flat in θ , hence consistent with—and stronger than—their result. On nontrivial bundles where Rtr ( F∧F ) ∈Z is quantized, exactness in the extended complex still removes physical θ -dependence because the phase is the extended holonomy of a coboundary. Route B: structural relaxion from the mixed topological coupling Coupling and effective angle. Augment the coherence action (75) by the mixed term Smix[Φ; A] = 2πi κ∗ZMω3(Φ), d CS(A)=i κ∗ZM f(Φ) tr(F∧F),(147) where ω3 (Φ) is the closed 3-form on M2-grp introduced in §II D and f (Φ) the induced scalar density on M. The QCD functional then depends on the effective angle θeff(Φ) := θ+κ∗f(Φ).(148) Integrating out Yang–Mills (near θ= 0). At energies below the confinement scale, the Yang–Mills contribution to the vacuum energy is (for small θ) [154–156] EYM(θ) = 1 2χtθ2+O(θ4), χt:= Zd4xh0|1 32π2tr(F˜ F)(x)1 32π2tr(F˜ F)(0) |0i,(149) the topological susceptibility. In our case, replacing θ7→ θeff(Φ) gives the effective potential Veff(Φ) = V(Φ) + 1 2χtθ+κ∗f(Φ)2+··· ,(150) where dots denote higher harmonics and small corrections from the (A, B)-sector. Proposition V.2 (Dynamical relaxation of θ ) . If V (Φ) is sufficiently shallow compared to χt in the direction of f(Φ), then the minimum of (150) satisfies θeff(Φ?) = θ+κ∗f(Φ?) = 0 + O(θ3),(151) and small fluctuations ϕabout Φ?have mass m2 Φ=V00(Φ?) + κ2 ∗χtf0(Φ?)2,(152) with a CP-even kinetic mixing to the topological density suppressed by f0(Φ?). Proof. Stationarity gives V0 (Φ ? ) + κ∗χtf0 (Φ ? ) θ + κ∗f (Φ ? ) = 0. If V0 (Φ) is small in the relevant direction or vanishes by symmetry, then (151) follows. Expanding to quadratic order yields (152). Comparison to PQ axion. Unlike the PQ axion, no global shift symmetry is postulated: f (Φ) need not be periodic, and Φlives on the moduli M2-grp of symmetry types. Nevertheless, the low-energy physics near the minimum resembles axion physics with decay constant F−2 a∼κ2 ∗f0 (Φ ? ) 2 and mass m2 Φ∼χt/F2 a [ 154 , 155 ]. If V00 (Φ ? ) χt/F2 a ,Φcan be heavy and evade stellar bounds that apply to light axions.
38 Phenomenological checks Neutron EDM. In chiral perturbation theory, dn≃cnθeff efm with cn = O (10 −2− 10 −1 )depending on low-energy constants. Route A predicts θeff ≡ 0; Route B gives θeff minimized as in (151) , naturally consistent with |dn| bounds [ 283 ]. A small residual θeff may remain if V0 (Φ ? ) 6 = 0, which can be tuned below 10−10 without delicate cancellations provided Vis smooth. Topological susceptibility at θ= 0.Both routes preserve the value of χt at θ= 0as an intrinsic property of the Yang–Mills vacuum (or QCD with quarks) [ 156 , 163 ]. In Route B, Φreceives the additive mass term (152) from χt ; in Route A , χt controls the response to small deformations that break the trivialization (e.g. away from the exact pairing condition). Thermal behavior. At high temperature, the dilute instanton gas predicts χt ( T ) ∝T−b with b computable semiclassically [ 160 ], and lattice data support a rapid fall-off above the crossover [ 162 , 285 ]. Route B then implies m2 Φ ( T ) ∝χt ( T )decreases with T ; θeff tracks the minimum of (150) and remains small provided V does not induce large drifts of Φ. Route A is temperature-blind: trivialization holds at all T. Anomalies and global issues. Both routes respect the axial U (1) A anomaly structure: the tr ( F∧F ) density appears only in the invariant combination (Route B) or as an extended coboundary (Route A). No new ’t Hooft anomaly is introduced: the 2-level fields transform to cancel any induced 1-level variation in the sense of §IV B. Domain walls and cosmology. If f (Φ) is periodic or V (Φ) has multiple minima, Route B may produce domain walls with tension ∼√χtFa ; choosing V with a unique minimum avoids cosmological issues that plague axion models. Route A produces no CP-violating walls (the θ -sector is trivial), though Φ-textures may still exist as discussed in §II D. Summary of predictions •Route A: Strong CP is absent because tr ( F∧F )is exact in the extended complex. No axion-like particle is required; θ-periodicity is subsumed by extended exactness. •Route B: Strong CP is dynamically relaxed by the structure field Φthrough (147) . The Φ-mass receives a calculable contribution ∝χtbut Φneed not be ultralight. •Common checks: (i) |dn| bound satisfied by design; (ii) χt ( θ= 0) unchanged and consistent with lattice; (iii) thermal scaling tied to χt(T); (iv) anomaly consistency ensured by higher coherence. B. Higgs mechanism as lifting coherence (1→2) Idea in one line. In the coherence-first framework, gauge-boson masses measure the energetic cost of maintaining 2-coherence after relaxing strict 1-coherence. Concretely, when the structure map tΦ : g1→g0 selected by Φis nonzero, the fake-curvature penalty in (75) , 1 2g2kFΦk2 = 1 2g2kdA + A∧A−tΦ ( B ) k2, forces the 1-form connection A and the 2-form compensator B to align. Integrating out the 2-level fluctuations (or completing the square) yields a mass operator for the 1-level gauge field proportional to tΦt† Φ . The Higgs mass, in turn, is the stiffness (curvature of V (Φ) and of GΦ ) along directions of Φthat change the norm and rank of tΦ . Unlike the symmetry-breaking axiomatics of [ 164 – 167 ], the mechanism here is a coherence lift: vector masses arise by reallocating coherence from level 1 to level 2, while higher-gauge invariance remains intact. Quadratic analysis: mass from 2-coherence Linearization and mass operator. Expand (75) to quadratic order about a background with constant Φand fake-flat curvatures FΦ =0, HΦ =0. Write A7→ A + a , B7→ B + b , with background set to zero for clarity.Backgrounds with nonzero flat holonomy only conjugate the final mass matrix. To quadratic order, L(2) =1 2g2kda −tΦ(b)k2+1 2h2kdbk2+··· ,(153)
39 where dots indicate gauge-fixing and 2-level interaction terms that do not affect the mass at zero momentum. In momentum space and in transverse gauges p·a = 0, p·b = 0, the b -equation is algebraic at p2h2/`2(with `a UV scale), giving b=g2 h2 1 p2+···t† Φ(da).(154) Plugging (154) back yields the effective quadratic form for a, L(2) eff [a] = 1 2g2kdak2+1 2ha, M2 Aai+··· ,M2 A=1 αtΦt† Φ+O(p2), α := g2 h2.(155) Thus the mass matrix is the positive operator 1 αtΦt† Φ acting on g0 -valued 1-forms. Its kernel is ker ( t† Φ ), i.e. precisely the unbroken directions. No explicit symmetry-breaking source has been added; masses arise from the energetic cost of enforcing 2-coherence. Proposition V.3 (Gauge-boson masses from coherence lift) . Let Φbe constant and FΦ = HΦ =0. Then the physical vector-boson masses are the positive eigenvalues of 1 αtΦt† Φ . Massless vectors span ker ( t† Φ ) (stability subgroup). Proof. Equations (153) – (155) show that, to leading order in p2 , the quadratic form for a is 1 2ha, ( + M2 A ) ai . Diagonalizing tΦt† Φ gives massive Klein–Gordon equations for eigenvectors with nonzero eigenvalues and massless Maxwell equations for the kernel. Standard Model case: SU(2)×U(1) →U(1)em Data and normalization. Take g0 = su (2) ⊕u (1) Y with inner product hTa, Tbi0 = δab , hY, Y i0 = 1, and hTa, Y i0 = 0. Choose g1 = R3 with orthonormal basis {e1, e2, eZ} . Define tΦ at its vacuum value by tΦ(e1) = v 2g T1, tΦ(e2) = v 2g T2, tΦ(eZ) = v 2(g T3−g0Y),(156) with g, g0the weak and hypercharge couplings and va positive scale. Theorem V.4 (SM mass spectrum from tΦ ) . For tΦ as in (156) , the mass matrix 1 αtΦt† Φ has eigenvectors W±=1 √2(T1∓iT2),Z0= cos θWT3−sin θWY, and A= sin θWT3+ cos θWY, with M2 W=g2v2 4α, M2 Z=(g2+g02)v2 4α, M2 γ= 0,cos θW=g pg2+g02.(157) Hence ρ≡M2 W/(M2 Zcos2θW)=1at tree level (custodial relation). Proof. Compute tΦt† Φ on the {T1, T2, T 3, Y } basis: tΦt† Φ ( T1 ) = v2g2 4T1, tΦt† Φ ( T2 ) = v2g2 4T2, tΦt† ΦT3 Y = v2 4g2−gg0 −gg0g02T3 Y. Diagonalizing the neutral block gives eigenvectors Z0 and A with eigenvalues v2 4 ( g2 + g02 )and 0, respectively; the charged block is already diagonal. Divide by α to obtain (157) . Interpretation. The vector-boson masses are not put in by a symmetry-breaking scalar doublet; they are the cost of 2-coherence fixed by tΦ . The massless photon spans ker ( t† Φ ), i.e. the 1-coherent directions that require no 2-level repair. This reproduces the textbook spectrum [ 164 – 167 ] with the identification v2 SM = v2/α. In gauges where B is eliminated (unitary-like), one recovers the usual massive Yang–Mills Lagrangian; in Stückelberg-like gauges, B carries the would-be Nambu–Goldstone modes as 2-morphisms. Higgs (scalar) mass as stiffness of coherence Structure-field radial mode. Let s be a radial coordinate on M2-grp controlling the overall scale of tΦ (at fixed direction and rank), so that tΦ(s) = eσ(s)btwith btconstant and σ0(s)>0. Then M2 V(s) = e2σ(s) αλV(bt), λV(bt)an eigenvalue of btbt†.(158)
40 The canonically normalized fluctuation ϕ along s has kinetic term κ 2Gss ( ∂ϕ ) 2 and potential Veff ( s )from (75). Expanding at the vacuum s?gives the Higgs mass m2 H=κ Gss(s?)∂2 sVeffs?.(159) Using (158), ∂ln M2 V ∂s s? = 2 σ0(s?),⇒m2 H=κ Gss 4∂ln M2 V ∂s −2 s?∂2 sVeffs?.(160) Equation (160) makes explicit that the Higgs mass is the stiffness of 2-coherence: it is the curvature of Veff along the direction that scales tΦ , weighted by the metric stiffness κGss and the sensitivity of vector masses to coherence (∂sln M2 V). Radiative stability (soft running). Radiative corrections shift Gss and Veff ; however, the higher-gauge Ward identities (99) – (100) constrain the combined renormalization so that at fixed MW, MZ the physical m2 H inherits at most logarithmic sensitivity to heavy thresholds under broad conditions (decoupling) [ 172 ]. In the abelian 1 → 2 toy model, an explicit one-loop computation shows that power divergences in the scalar stiffness can be reabsorbed into ( κ, Gss )while M2 A is renormalized multiplicatively, leading to a soft-naturalness pattern (cf. Coleman–Weinberg-type logarithms) [ 170 ]. The upshot is conceptual: the would-be quadratic sensitivity of a symmetry-breaking mass parameter is traded for the renormalization of a coherence stiffness, whose running is milder by Ward identities tied to 2-coherence, echoing the naturalness heuristic of [171] from a different angle. Observable differences from the fine-tuning perspective 1) Sum rule linking mH to MW, MZ .From (160) , small departures of m2 H from the SM expectation at fixed {MW, MZ}measure the curvature of Veff and the metric GΦon M2-grp: m2 H=∂ln M2 Z ∂s −2 κ Gss ∂2 sVeffs? =m2 H,SM | {z } tree-level + ∆coh ,(161) with a calculable ∆ coh once Gss and Veff are specified by the microphysics of Φ. In particular, if Gss has approximate shift isometry near s?,∆coh is loop-suppressed. 2) Universality of Higgs-coupling shifts. Fluctuations ϕ rescale tΦ and hence all vector masses according to (158) . Therefore deviations of Higgs couplings to W, Z obey a universal relation δghV V /ghV V = 1 2 ( ∂sln M2 V ) δs up to small custodial breaking. This predicts a correlated pattern in h→WW∗, ZZ∗ rates, distinct from generic SMEFT coefficient scans. 3) Custodial protection and ρ -parameter. Because tΦ in (156) singles out an SO (4) ≃SU (2) L×SU (2) R - covariant direction, ρ = 1 at tree level. Small deviations arise only from departures of tΦ from the custodial orbit or from curvature of GΦthat breaks SO(4); both are constrained by precision data. 4) Absence of additional light scalars. Unlike PQ-like solutions, no unavoidable ultralight excitations appear: the radial ϕ can be heavy if Veff is stiff in the s -direction (cf. (159) ), while angular modes along the Aut0-orbit of Φare redundancies (coherence gauge) rather than physical fields (cf. §IV A). Remarks on renormalization and unitarity Equivalence to the textbook formulation. Choosing a gauge where B is eliminated and parameterizing Φso that tΦ∝vbt reproduces the standard massive SU (2) ×U (1) theory with a scalar order parameter. Renormalizability and unitarity then follow from the classic analyses [ 168 , 169 ]. In Stückelberg-like gauges, the coherence Ward identities implement the cancellations that in the textbook picture are encoded by the scalar Goldstones. Radiative electroweak symmetry preservation. Coleman–Weinberg radiative breaking [170] is reinterpreted here as a radiative lift of tΦ from zero to nonzero; the structure-field potential selects a nontrivial Φ ? and hence a nontrivial tΦ? . Conversely, symmetry restoration at high T corresponds to thermal flattening of Veff in the s-direction.
41 Summary Vector-boson masses are the spectral values of 1 αtΦt† Φ : the energy required to maintain 2-coherence after relaxing strict 1-coherence. The Higgs mass is the stiffness of 2-coherence along the direction that scales tΦ. This “coherence lift” reproduces the Standard Model spectrum and predicts a pattern of deviations that is distinct from generic fine-tuning narratives: at fixed MW, MZ , the scalar mass and couplings probe the geometry GΦ and potential Veff on the moduli of symmetry types, with softened UV sensitivity controlled by higher-gauge Ward identities. C. Functorial link between quantization and Higgs Executive summary. Prequantization turns integral closed forms into geometric objects with connection (line bundles for 2-forms, gerbes for 3-forms); integrality reflects residual non-glueability (§IV C). Higgs as coherence lift (§V B) turns a 1-connection into a 2-connection by introducing a compensator B through the structure map tΦ , thereby reallocating coherence from level 1 to level 2. Here we make precise the analogy: Quantization: Ω2 cl,Z(M)PreQ1 −−−−−→ Line bundles with connection, Ω3 cl,Z(M)PreQ2 −−−−−→ Gerbes with connection; Higgs lift: Conn1(G0)HIGtΦ −−−−−→ Conn2(GΦ), with GΦ the crossed-module 2-group selected by Φ. Structurally, PreQ is a left adjoint to curvature, and HIGtΦ is a left adjoint to fake curvature FΦ = dA + A∧A−tΦ ( B ). This identifies the Higgs mechanism as an internal quantization functor for gauge degrees of freedom. Discrete defects (vortices/monopoles) appear as quantized obstructions to the existence of a global lift; functoriality fixes their charges and protects universal features. Prequantization functors and their universal property Classical story (review). Let Ω k cl,Z ( M )denote closed k -forms with integral periods. The prequantization theorems of Kostant–Souriau and their higher analogues state that there are functors PreQ1: Ω2 cl,Z(M)→LineConn(M),PreQ2: Ω3 cl,Z(M)→GerbeConn(M),(162) assigning to ω2 a line bundle ( L, ∇ )with curvature F∇ = ω2 and to H3 a bundle gerbe ( G, B )with dB =H3[173, 174, 176, 316]. These are left adjoints to the curvature functors curv1,curv2: HomLineConnPreQ1(ω2),(L, ∇)∼ =HomΩ2 cl,Zω2,curv1(L, ∇),(163) and similarly in degree 3 [ 176 , 181 ]. Equivalently: PreQ chooses, functorially and universally, a geometric object whose curvature is the prescribed integral form. Higgs lift as an internal prequantization (left adjoint to fake curvature) Categories of connections. Let Conn1 ( G0 )be the groupoid of G0 -connections A on principal bundles over M . Let Conn2 ( GΦ )be the groupoid of 2-connections ( A, B )for the crossed module GΦ = ( G1 tΦ −→ G0, .Φ)(§II C, §II D). Define the fake curvature functor FakeΦ:Conn2(GΦ)−→ Ω2(M, g0),(A, B)7−→ FΦ:= dA +A∧A−tΦ(B).(164) Definition (Higgs lift functor). Fix couplings ( g, h )and the inner products as in §III A. For A∈ Conn1 ( G0 ), define HIGtΦ ( A ) = ( A, B∗ [ A ]), where B∗ [ A ]is the unique (up to coherence redundancy, cf. §IV A) solution to the internal prequantization problem B∗[A] = arg min B∈Ω2(M,g1)EΦ(A, B)with EΦ(A, B) := 1 2g2kFΦk2+1 2h2kDABk2.(165)
48 Currents from inhomogeneous θeff and axion electrodynamics Bulk response to θ-textures. Varying (178) yields the axion-electrodynamics current jµ θ=e2 2πh µνρσ (∂νθeff)Fρσ/2,i.e. jθ=e2 2πh (∇θeff )×E+˙ θeff B.(186) For a step θeff ( z ) = θ1 + ∆ θ Θ( z )this reproduces (183) . For smooth θ -textures (e.g. strain gradients), (186) predicts bulk gyrotropic/optical activity; the response is topological to the extent that θeff is pinned to quantised values except near walls. Gauge completion and higher Ward identities. Including the B -sector ensures exact gauge invariance even when θeff varies (§IV B): the 1-level nonconservation of the surface current is precisely compensated by a 2-level inflow, and the Ward identities (99) – (100) control current exchange between the surface and bulk 2-form sectors. From band invariants to measurable coefficients Ab-initio computation. In practice, θME and its dependence on Φcan be computed from (177) using hybrid Wannier methods and nonabelian Wilson loops [ 202 ]. A crystalline perturbation (strain, magnetisation) is encoded in Φ; the change ∆ θeff across a heterostructure or domain wall follows from the change ∆ P3 . Equation (183) then gives the predicted σH ; (185) fixes the thermal counterpart when a gravitational response is present. Experimental signatures. (i) Quantised Faraday/Kerr rotation at terahertz frequencies in thin films with opposite surface gaps [ 204 , 205 ]. (ii) Line-localised chiral channels at Φ-wall intersections on the surface (e.g. magnetic domain walls), with conductance quanta e2/h per channel and thermal conductance quanta ( π2k2 B/ 3 h ) T set by ∆ c . (iii) Strain-tunable magnetoelectric effect: controlled variation of Φby piezoelectric substrates produces stepwise changes in Faraday rotation as strain crosses a critical value (band inversion). Summary In the extended higher-gauge complex, the electromagnetic density F∧F is exact, so a uniform axion term is bulk-inert and only interfaces contribute. Surfaces carry Chern–Simons terms quantised by ∆ θ/ 2 π , yielding half-quantised Hall responses in strong TIs. When the structure field Φvaries, domain walls support quantised electrical, spin (when applicable), and thermal Hall steps set by ∆ ω3 (Φ). These results refine axion electrodynamics by tying all responses to coherence and its defects, providing robust, programmable pathways to topological transport in solids. B. U(1)↔Z2spin-liquid crossover Synopsis. We formulate a controlled crossover between a gapless U (1) spin liquid and a gapped Z2 spin liquid as a coherence lift (1 → 2) driven by the structure field Φ. In the U (1) regime the Maxwell photon is gapless in 2 + 1dimensions (dual to a compact scalar), while in the Z2 regime the gauge sector is gapped and described at long distances by a level-2BF theory (visons and spinons with mutual semionic statistics). The crossover is encoded by an abelian crossed module with tΦ : u (1) →u (1), and the photon mass mγ (Φ) arises from the internal mass generation mechanism of §V B. Spatial textures of Φgenerate domain walls supporting bound modes: a gauge bound mode described by a Pöschl–Teller problem and a naturon (coherence mode) bound to the same wall. We derive neutron-scattering reweighting across the crossover and predict quantized edge steps (electric/spin/thermal) from naturality inflow even when the bulk topological order does not change.
49 Microscopic setting and effective fields Microscopic model. Consider a frustrated Mott insulator on a kagome or triangular lattice at (near) half filling described by H=JX hiji Si·Sj+J2X hhijii Si·Sj+KX hex Si·SjSk·Sl+··· ,(187) with additional ring-exchange or Dzyaloshinskii–Moriya terms as appropriate. A fermionic (or bosonic) parton representation ciα with Si = 1 2c† iασαβciβ introduces an emergent U (1) gauge redundancy; meanfield ansätze and gauge fluctuations produce U (1) or Z2 spin liquids depending on whether a charge-2 Higgs bilinear condenses [209–213]. Higher-gauge completion. At long distances (continuum), we place the emergent abelian gauge sector in the crossed-module framework: g0=u(1),g1=u(1), . ≡0, tΦ=m(Φ) id. Let Abe the emergent U(1) 1-form connection, Ban abelian 2-form. The curvatures FΦ=dA −m(Φ) B, HΦ=dB (188) enter the coherence action of the gauge sector Sgauge[A, B; Φ] = ZM2+1 1 4g2(FΦ)µν(FΦ)µν +1 12h2HµνρHµνρ,(189) supplemented by matter spinons minimally coupled to A and by a slow structure field Φcontrolling m (Φ) via a double-well potential V(Φ) (cf. §II D). Phases. •U (1) regime ( m (Φ) = 0): FΦ = dA , B decouples; Maxwell theory in 2 + 1d has one gapless mode (photon) with C ( T ) ∝T2 and algebraic correlations (Coulomb phase or algebraic spin liquid) [211, 213]. •Z2regime ( m (Φ) 6 = 0): internal mass generation (Abelian version of Proposition V.3) gives a photon gap m2 γ(Φ) = m(Φ)2h2 g2,(190) and integrating out the massive modes leaves at Emγ a level-2BF topological field theory (vison/spinon mutual semions), i.e. Z2topological order [214]. Crossover field theory and spectrum Quadratic spectrum and vison gap. In momentum space (Lorenz gauge pµAµ = 0, pµBµν = 0) the coupled equations from (189) give the dispersion ω2 ±(p) = p2+1 2m2 γ+αp2±q(m2 γ+αp2)2−4αp2m2 γ, α := g2 h2.(191) For mγ→ 0, ω−→ |p| is the photon; for p→ 0with mγ6 = 0, ω−→mγ , identifying the photon gap. The orthogonal combination ω+ is heavier and can be integrated out at low energies. The vison gap ∆ v arises as the vortex of B(or π-flux of A) and scales as ∆v∼λvmγ,(192) with a dimensionless constant λv determined by short-distance physics (core energy); in the BF limit λv→ O(1).
50 Thermodynamics. The free energy density of the gauge sector interpolates as fgauge(T; Φ) = π 6T3/v2 γ(mγ(Φ) = 0, U(1)), −T mγ(Φ) 2πK1mγ(Φ) T+··· (mγ(Φ) T, Z2), (193) with K1 a modified Bessel function and vγ the photon velocity. Thus the heat capacity exhibits a crossover C(T; Φ) = π 2 T2 v2 γ (U(1)), ∼mγ 2πT 1/2m2 γe−mγ/T (Z2). (194) Domain wall physics and bound modes Kink profile for m (Φ).Let x be the coordinate normal to a domain wall (DW) and assume a smooth kink profile m(x) = m0tanhx ξ,(195) with width ξ controlled by V (Φ) and the gradient stiffness of Φ. Linearizing the gauge equations for a mode propagating along the wall with momentum ky and temporal frequency ω , one obtains a 1D Schrödinger equation for the transverse profile a(x)of the relevant component of A: −d2 dx2+U(x)a(x) = κ2a(x), U(x) = m2 0 αtanh2x ξ−m0 αξ sech2x ξ,(196) with κ2 := ω2−k2 y . The potential U ( x )is of Pöschl–Teller type and supports a finite number of bound states [215]. Proposition VI.1 (Gauge bound mode on a Φ-wall) . For ν := m0ξ √α> 1, (196) admits a single normalizable bound state with dispersion ω2 DW(ky) = k2 y+m2 γ(∞)1−1 ν2, mγ(∞) = m0h g.(197) For ν≤1the mode merges with the continuum. The spatial profile is a(x)∝sechν(x/ξ). Naturon bound mode. Small fluctuations ϕof Φabout the kink satisfy −∂2 t+∂2 y+∂2 x−V00Φkink(x)ϕ(x, y, t) = S[A, B],(198) with a source S quadratic in gauge fields (backreaction). For a φ4 -type V , V00 is again Pöschl–Teller, yielding a zero mode ϕ0 ( x ) ∝∂x Φ kink (translation) and possibly one massive bound mode—the naturon— with gap mnat ∼ξ−1 . Gauge–naturon hybridization is suppressed by O ( α )but leads to an avoided crossing in the 1D spectrum along the wall. Neutron-scattering signatures: reweighting across the crossover Dynamical structure factor. The inelastic neutron intensity is S ( q, ω ) = Pab ( δab −ˆqaˆqb ) Im χab ( q, ω ), where χis the spin susceptibility. In the parton description, χ(q, ω)≃χspinon(q, ω) + χgauge(q, ω; Φ),(199) with χgauge controlled by the photon (or gapped mode) propagator. U(1) regime. For an algebraic spin liquid with Dirac spinons, one finds [212, 213] S(q, ω)U(1) ∼ωFω vsp|q|+γ |q|(for small ω, magnonless continuum),(200) the second term being the gauge contribution with coefficient γ∝g2.
51 Z2regime. When mγ(Φ) 6= 0, the low-ωweight is suppressed as S(q, ω)Z2∼Θ(ω−2∆sp)pω−2∆sp + Θ(ω−mγ)qω2−m2 γ ω,(201) where ∆ sp is the spinon gap (if any). Thus, at fixed q , the spectral weight reweights from low ω (algebraic tail) toward the threshold(s) mγ and 2∆ sp as Φmoves into the Z2 stratum. In materials like herbertsmithite, this manifests as a redistribution of continuum weight detectable by polarized inelastic neutron scattering [216, 217]. Wall-bound channels. For a Φ-wall, (197) yields a 1D channel along the wall. Its dynamic structure factor has a sharp ridge at ω = ωDW ( ky )superimposed on the 2D bulk continuum. Scanning the sample to cross a domain wall produces a line-localized enhancement in S ( q, ω )with dispersion set by m0, ξ —a direct signature of the coherence mechanism. Naturality inflow and quantized edge steps BF limit and mixed coupling. At Emγthe gauge sector reduces to SBF =i k 2πZB∧dA, k = 2,(202) and couples to external probes (electric or spin U (1)) via a mixed topological term analogous to (184) . Across a Φ-wall that changes ω3 (Φ) by an integral ∆ ω3∈H3 ( M2-grp,Z ), the effective action on the wall contains Sedge Σ= 2πi ZΣ∆σs e2/h CSA(s)+ ∆κCSgrav(Γ),(203) with quantized steps ∆σs=νs e2 h,∆κxy T=νT π2k2 B 3h, νs,T ∈1 2Z,(204) even when the bulk remains in the same Z2 topological order on both sides. This is a pure naturality inflow: the wall compensates the change in 2-coherence specified by ∆ ω3 (Φ) without a change in bulk anyon content. Controlled predictions and numbers Crossover scales. Let m(Φ) interpolate between 0and m0with width ξ. Then mγ(Φ) = h gm(Φ),∆v≃λv h gm(Φ), ωDW(0) = mγ(∞)p1−1/ν2(ν > 1).(205) The specific-heat crossover occurs near T≃ 0 . 4 mγ (from (194) ). The neutron reweighting sets in when the instrumental window straddles mγ: a shift of spectral weight δW ∼ O(mγ)over a bandwidth W. Protocol. Prepare a sample with tunable pressure/strain or weak out-of-plane field that couples to Φ. (i) Measure C ( T ): fit T2 at small strain and activated form at larger strain to extract mγ (Φ). (ii) Map S ( q, ω )via polarized neutrons across a DW engineered by a strain gradient; look for the 1D ridge ωDW ( ky ). (iii) In materials with approximate U (1) Sz (e.g. easy-plane anisotropy), detect spin Hall steps along DWs via nonlocal spin transport; in chiral contexts, test thermal Hall steps. Relation to prior approaches The crossover is usually described by a charge-2Higgs field condensing to gap the U (1) photon and leave Z2 order [ 210 , 211 ]. Here the same physics is realized as an internal quantization functor HIGtΦ (Sec. V C), with B the 2-level compensator. This yields (i) a transparent formula for the photon gap (190) ; (ii) a geometric interpretation of wall modes and quantized edge steps via ∆ ω3 (Φ); (iii) Ward identities that control corrections in the proto-gauge regime.
52 Summary A single structure field Φcontrolling the map tΦ suffices to organize the U (1) ↔Z2 spin-liquid crossover. The photon mass mγ (Φ), vison gap ∆ v , and wall-bound dispersions are calculable from (190) – (197) . Neutron spectra and thermodynamics undergo characteristic reweightings, and quantized edge steps occur at Φ-walls as a manifestation of naturality inflow, independent of changes in bulk topological order. C. Fracton elasticity duality Aim. We recast planar elasticity and its topological defects into the coherence-first language by identifying the categorical 2-form with (a potential for) stress, and its curvature with disclination density: B←→ (stress potential), H =dB ←→ (disclination density 2-form).(206) The coherence penalty kHΦk2 then becomes the elastic energy, while sources for H encode disclinations (fracton charges) and dislocations (fracton dipoles). Varying the structure field Φunfreezes 2-coherence, driving a transition between plastic (mobile dipoles, proliferated curvature) and fractonic (suppressed curvature, restricted mobility) regimes. We derive the duality, establish the defect–fracton correspondence, and give quantitative predictions for domain patterns and mobility crossovers. Elasticity: fields, energy, and defects Kinematics and energy. In a 2D crystal, the displacement field ui ( x, t )( i = 1 , 2) defines the linear strain uij =1 2(∂iuj+∂jui).The elastic energy (Hooke law) is Hel =1 2Zd2x Cijkl uijukl,(207) with Cijkl the elastic tensor (for isotropy, Cijkl = λ δijδkl + µ ( δikδjl + δilδjk ), Lamé moduli λ, µ ). The stress is σij =∂Hel/∂uij =Cijklukl, and static equilibrium obeys force balance ∂iσij = 0. Defects. Adisclination is a point defect characterized by Frank angle Ω; its density 2-form is s(x)d2x=ik∂iωkd2x, ωk:= 1 2k` ∂`umδmm(208) (the local rotation 1-form ω ). A dislocation with Burgers vector bi has density 1-form current jdisloc i given by jdisloc i µ := µνρ ∂ν∂ρui, µ, ν, ρ ∈ {0,1,2},(209) a conserved 3D current in ( t, x1, x2 ). Indices are raised/lowered with δij . ij is the 2D Levi–Civita symbol with 12 = +1. Classical compatibility relates strain, disclinations, and dislocations [218–220]. Dual formulation: from stress to a 2-form gauge potential Hubbard–Stratonovich and force balance. Introduce a symmetric auxiliary stress σij and write (imaginary time) L=1 2C−1 ijkl σijσkl +i σij uij +ρ 2(∂tui)2,(210) with ρ the mass density. Integrating out ui (up to boundary terms and defect sources) enforces force balance ∂iσij = 0 in the static sector. By the Airy representation there exist (non-unique) stress potentials bj(x, t)such that σij =ik ∂kbj+jk ∂kbi.(211) Equation (211) makes symmetry and force balance manifest: σij =σji and ∂iσij = 0 identically.
53 Categorical 2-form and curvature. Pack the stress potential into an abelian 2-form on spacetime: B:= bidt ∧dxi,so that (dB)0ij =∂ibj−∂jbi.(212) Define the curvature 3-form H=dB and its Hodge dual (a scalar in 2+1D) ˜ H:= ∗H=∂t(ij∂ibj)−ij∂i˙ bj,(213) which, in the static sector, reduces to ˜ H = ij∂ibj . Comparing with the compatibility relations, one identifies H←→ s(x)dt ∧d2x, (214) i.e. the curvature His the disclination density 2-form. Elastic energy from coherence. Using (211) , the quadratic form C−1 ijklσijσkl can be written in terms of bderivatives. For isotropic elasticity one finds (after integrations by parts) 1 2Zd3x C−1 ijkl σijσkl =K 2Zd3x(ij∂ibj)2+µ0 2Zd3x(∂ibi)2,(215) with K := λ+2µ µ(λ+µ) and µ0 := 1 µ for concreteness. In terms of the 2-form, the curvature term ( ij∂ibj ) 2 is simply ∝ kHk2, i.e. Hel ≡κFrank 2kHk2+κdiv 2(∂·b)2, κFrank ∝K, κdiv ∝µ0,(216) making explicit that the coherence penalty kHk2 reproduces the Frank energy of curvature (disclinations), while the (∂·b)2term enforces symmetric stress (shear). Coupling to defects. Disclinations and dislocations couple as sources for B: Sdef =iZd3x˜ Bµjdisclin µ+iZd3x bijdisloc i0,˜ Bµ:= 1 2µνρBνρ,(217) so that varying B reproduces (208) – (209) as source terms in d∗H and ∂·b equations. The first term charges disclinations under the 1-form symmetry generated by ˜ B ; the second attaches Burgers charge to b. Fracton dictionary and mobility constraints Rank-2 gauge dual. The stress conservation ∂iσij = 0 allows a dual description by a symmetric tensor gauge field Aij with electric field Eij [221, 222]: Eij ←→ σij, ∂i∂jEij ←→ s(x) (disclinations), ∂jEij ←→ bi(x) (dislocations).(218) The fracton Gauss laws ∂i∂jEij =ρfr, ∂jEij =ρdip i(219) express conservation of charge and dipole moment, implying the fracton immobility of disclinations and the glide-only mobility of dislocations (dipoles), in direct correspondence with elasticity [ 222 , 223 ]. In our 2-form picture, these laws are the Bianchi identities and source equations for Hand B. Energy and dynamics. The quadratic rank-2 gauge action ( E2 + B2 )is dual to the elastic energy and phonon dynamics. Equivalently, in the 2-form formulation the action S[B; Φ] = ZκFrank(Φ) 2kHk2+κdiv(Φ) 2(∂·b)2+i˜ B·jdisclin +i bijdisloc i0(220) reproduces the phonon spectrum in the defect-free sector and the fracton/dipole energetics when sources are present, with moduli promoted to structure-field–dependent couplings κ(Φ).
54 Plastic ↔fracton transition as (un)freezing of coherence Control parameter. Let κFrank(Φ) = κ0+α δΦ + ··· (221) govern the cost of curvature (disclinations). Large κFrank freezes curvature (disclinations are heavy) and one is in an elastic / fracton regime: only dislocation dipoles move (glide), with restricted mobility and subdiffusive dynamics [ 221 ]. As κFrank is lowered (by tuning Φ), curvature unfreezes: disclinations proliferate at a critical scale κFrank,c, screening dipole conservation and producing a plastic fluid. Defect fugacities and RG. Assign fugacities ydisclin, ydisloc to defect worldlines. Coarse-graining the Coulomb gas obtained from (220) yields RG equations (schematic) d dl 1 κFrank =a1y2 disclin +a2y2 disloc +··· ,(222) dydisclin dl = (2 −πκFrank)ydisclin +··· ,dydisloc dl = (2 −πκdiv)ydisloc +··· , with nonuniversal a1,2> 0. When πκFrank drops below 2, disclination fugacity is relevant and the system crosses into the plastic phase; above that threshold, the disclination gas is bound and one has a fracton-elastic phase (dislocations may still proliferate, realizing a “hexatic” analogue with mobile dipoles but immobile charges). Transport: mobility crossover. In the fracton-elastic regime, a single dislocation of Burgers vector b has anisotropic mobility tensor M ij ∝bibj , i.e. glide only. Approaching the plastic side, the effective κFrank (Φ) softens, defect–defect screening weakens dipole conservation, and a crossover to isotropic diffusion occurs: Dk(Φ) ∼D0, D⊥(Φ) ∼D0e−c κFrank(Φ),(223) with Dk,⊥ mobilities parallel/perpendicular to b and c a constant set by core energies. Equation (223) predicts an exponential enhancement of transverse mobility as curvature unfreezes. Domains, patterns, and line modes Φ-textures. Spatial variations of Φmodulate κFrank (Φ), creating domain patterns with alternating fracton-elastic and plastic regions. At the interface where κFrank varies on a length ξΦ , the b -field equations reduce to a waveguide problem supporting line modes (along the interface) with dispersion ω2(kk)≃v2 Tk2 k+ ∆2(ξΦ),∆(ξΦ)∝ξ−1 Φ,(224) where vT is the transverse phonon velocity on the elastic side. These modes are the elastic analogues of the gauge-bound modes of §VI B; they localize energy and channel momentum and are testable by Brillouin light scattering or surface acoustic wave probes. Naturality inflow in elasticity. Across a Φ-wall, the change in the differential character ω3 (Φ) induces aquantized shift of lineal responses (spin/thermal analogues discussed in §VI A). Here, the counterpart is a topological momentum inflow along the interface: the integrated Peach–Koehler force on a dislocation bunch equals the jump in the coherence charge RH across the wall, a robust, geometry-controlled effect independent of microscopic pinning. Experimental predictions Ultrasound and rheology. As Φtunes through the unfreezing point, ultrasonic attenuation crosses over from Rayleigh-like ( ∝ω4 ) to defect-dominated (approximately ∝ω ), while the storage/loss moduli (G0, G00)show a kink where κFrank(Φ) softens. Fitting to (216) extracts κFrank(Φ) directly. Imaging defects and mobility. Direct imaging (TEM or confocal in colloidal crystals) should reveal anisotropic dislocation trajectories at large κFrank , crossing over to isotropic random walks as predicted by (223). The crossover line in the (T, Φ) plane provides a quantitative fracton–plastic phase diagram. Acoustic waveguides. Engineered Φ-patterns (via strain or substrate modulation) act as phononic guides supporting line modes (224) , with frequencies set by ξΦ . These can route energy elastically around obstacles, an application of coherence engineering in metamaterials.
55 Relation to prior work Our 2-form formulation is equivalent to the rank-2 U(1) tensor gauge description of crystalline elasticity and its defects [ 221 , 222 ], itself rooted in the gauge theory of defects [ 218 – 220 ]. The new ingredient is the structure field Φthat controls the coherence penalty κFrank (Φ) and thereby dynamically tunes between fractonic (curvature-frozen) and plastic (curvature-unfrozen) regimes, with quantized interfacial responses inherited from the general coherence framework. D. Hydrodynamic electron flow Two lenses and a coherence map. Electronic transport may be viewed through two complementary “lenses”: a kinetic lens F(Boltzmann/quantum kinetic) that resolves the single-particle distribution f ( x,p, t ), and a hydrodynamic lens Gthat resolves only a few conserved or slowly relaxing fields (charge density n , momentum density π = ρu with ρ the mass density, and entropy/temperature s, T ). When electron–electron (ee) scattering dominates over momentum-relaxing processes (impurities, phonons), the kinetic description glues to the hydrodynamic one: f is rapidly driven toward a drifting local equilibrium fle [ µ, T, u ], and the long-time dynamics is governed by Navier–Stokes-type equations [ 224 – 226 , 230 , 231 ]. In our coherence-first language, this gluing is mediated by a structure field Φ(encoding, e.g., disorder strength, phonon bath occupation, band geometry) that controls transport maps from the kinetic to hydrodynamic variables and, in particular, the viscosity tensor η (Φ). We will (i) write the hydrodynamic equations with explicit Φ-dependence, (ii) reinterpret viscous stresses as a categorical 2-form B (as in §VI C), (iii) identify a curvature K=Dη +··· (225) whose nonvanishing quantifies the failure of perfect gluing between Fand Gwhen η varies in space, time, or across the moduli M of structure types, and (iv) extract predictions: a gapped naturon collective mode (coherence fluctuation), nonlocal resistance “hotspots” characteristic of viscous flow, and quantized pumping under cyclic drives of Φ. From kinetic to hydrodynamic: regimes and equations Kinetic prerequisites. Let τee be the ee-scattering time and τmr the momentum-relaxing time (impurities/phonons). Hydrodynamics applies in the window [230, 231] `ee w`mr, `ee =vFτee, `mr =vFτmr,(226) for devices of width w (Poiseuille regime). In linear response about a static background with density n0(Φ) and temperature T0(Φ), the hydrodynamic fields (δn, u, δT)satisfy ∂tδn +n0∇·u= 0,(227) ρ0∂tu=−∇δp +η(Φ) ∇2u+ζ(Φ) + η(Φ) 3∇(∇·u) −ρ0 τmr(Φ) u+n0eE+u×B+··· ,(228) cn(Φ) ∂tδT =κ(Φ) ∇2δT +··· ,(229) where ρ0 is the effective inertia (areal mass density in 2D or enthalpy density in relativistic Dirac fluids), δp = ( ∂p/∂n ) Tδn + ( ∂p/∂T ) nδT , η and ζ are the shear and bulk viscosity, and κ is the thermal conductivity. Gurzhi length and viscous resistivity. In the stationary (Stokes) limit of (228) with incompressibility (∇·u=0) one obtains η(Φ) ∇2u−ρ0 τmr(Φ) u+n0eE= 0.(230) The Gurzhi length emerges: `G(Φ) = sη(Φ) τmr(Φ) ρ0 ,(231)
56 as the crossover scale between viscous and ohmic momentum relaxation. In a strip of width w with no-slip boundaries, (230) yields a Poiseuille profile and a viscous contribution to the longitudinal resistance Rvisc =12 η(Φ) L (n0e)2w3(2D strip, length L).(232) The hallmark Gurzhi effect is a decrease of R upon heating (since η drops with increasing T in many materials) before ohmic phonon scattering takes over [224, 225]. Categorical stress and coherence curvature B as stress potential; H = dB as vorticity/disclination in flow. Following §VI C, introduce a vector potential bifor the symmetric stress tensor σij and define a 2-form on spacetime, B:= bidt ∧dxi, H := dB, (233) so that in the stationary, incompressible regime the scalar ∗H = ij∂ibj coincides (up to a modulus) with the vorticity ωz = ( ∇×u ) z , while kHk2 measures the curvature of the flow. As in (216) , a quadratic functional F[B; Φ] = κη(Φ) 2kHk2+κdiv(Φ) 2(∂·b)2(234) reconstructs the viscous sector of (228) upon minimization with the constraint that b encodes the Newtonian constitutive relation. In the language of §V B, the Newtonian law σij = 2 η (Φ) usym ij plays the role of cancelling the fake curvature between the 1-level velocity gradient and the 2-level stress via a map tΦproportional to η(Φ). Curvature on the transport bundle. Let η = η (Φ ,x, t )vary because Φvaries in space and time (e.g. inhomogeneous disorder, strain, or controlled gates). Define a connection A on the fiber bundle of transport coefficients over M×M and its curvature K: Dη := dη +Aη, K:= Dη +. . . , (235) where the dots include Hall-viscous and thermo-electric coefficient covariant derivatives. Then K 6 = 0 quantifies the failure of strict commutativity between spatial/temporal changes and the constitutive identification between u and B . Physically, nonzero K sources vorticity and nonlocal voltages even for uniform drives; below we compute its signatures. Naturon: a coherence collective mode Coupled u –Φequations and dispersion. Augment (228) with a dynamics for Φ(structure fluctuations) and a bilinear coherence coupling to vorticity: χΦ∂2 tΦ + γΦ∂tΦ−ξ2 Φ∇2Φ + m2 ΦΦ = λ ωz, ωz= (∇×u)z,(236) with susceptibility χΦ , damping γΦ , coherence length ξΦ , and mass mΦ set by V (Φ). Linearizing (228) for transverse (divergence-free) flows and combining with (236) we obtain, in Fourier space ( ω, k ), the coupled system −iωρ0+ηk2+ρ0/τmr −iλk +iλk −χΦω2−iγΦω+ξ2 Φk2+m2 Φu⊥ Φ= 0.(237) The eigenfrequencies solve −iωρ0+ηk2+ρ0/τmr−χΦω2−iγΦω+ξ2 Φk2+m2 Φ+λ2k2= 0.(238) When λ6 = 0 and damping is modest, one root becomes an under-damped naturon with smallk dispersion ωnat(k)≃qω2 0+v2 natk2−i 2Γnat(k), ω2 0=λ2 ρ0χΦ , v2 nat =ξ2 Φ+η ρ0 λ2 ρ0m2 Φ ,(239) interpolating between a reactive (propagating) coherence wave and diffusive shear, depending on η/ρ0 and mΦ . The linewidth Γ nat is linear in γΦ and in ηk2/ρ0 (viscous damping). Observation: ω0 scales with the coherence coupling λ and is tunable by Φ(gate/strain), offering a new spectroscopic knob distinct from cyclotron resonance.
57 Nonlocal resistance and “hotspots” from coherence curvature Viscous negative nonlocal response. In a two-terminal 2D strip, current injected at a point produces a potential profile with negative nonlocal resistance RNL ( x )near the injector due to viscous backflow and current vortices [ 226 , 227 , 232 , 233 ]. Solving (230) in a half-plane with a point source and no-slip boundary, one finds [226] RNL(x)≈ − ρ2 π η 1 1 + 4(x/w)2(near-contact, x.w),(240) with ρ the bulk resistivity and w the effective width (the precise prefactor depends on boundary slip). Equation (240) captures the sign change of RNL and the “hotspot” of negative voltage near the injector observed experimentally [227, 232, 233]. Effect of inhomogeneous η (Φ).When η varies slowly, the extended curvature K in (235) acts as a source term in the Stokes equation, adding a piece η∇2u+ (∇η)·∇u−ρ0 τmr u+n0eE= 0,(241) which reweights the negative-lobe pattern and moves the hotspot position x∗ (Φ). For a monotone gradient ∂xη > 0, the minimum of RNL shifts by δx∗≃w2 8 ∂xη η,(242) offering a direct meter of Dη and thus of the coherence curvature K. Corner flows and vorticity imaging. In Hall-bar geometries, K 6 = 0 focuses vorticity at corners where ∇η changes direction, amplifying ωz hot-spots that are resolvable by scanning magnetometry or NV centers [ 233 ]. The scaling ωmax z∝ |∇η| and the proportional drift of hotspot positions with gate voltage or strain are robust signatures of the coherence-first structure. Quantized pumping by cyclic coherence drives Cyclic drives in Φ-space. Consider a closed spatio-temporal cycle C of the structure field Φ( x, t ) defined on a strip Ωthat returns to its initial configuration after a period T but sweeps a nontrivial 3-cycle in (x, t, M). The mixed topological term (the hydrodynamic analogue of (179)) Smix[Φ; Aem] = 2πi ZM2+1 ω3(Φ), d CS(Aem)(243) implies a pumped edge charge along the boundary ∂Ωper cycle Qpump =e2 h[ω3],[C]∈e2 hZ,(244) provided the cycle winds an integral class of ω3 . Using the convention that Rd CS ( Aem ) = h e , the prefactor e2 h may equivalently appear as a conductance unit. We adopt the electromagnetic normalization in which the exterior derivative of the Chern–Simons three–form, dCS ( Aem ) , has an integral equal to a single flux quantum, RdCS ( Aem ) = h/e. With this convention, the prefactor e2/h in Eq. (244) carries units of conductance and converts the dimensionless topological pairing h [ ω3 ] , [ C ] i ∈ Z into a physical charge in coulombs. Thus the pumped charge per cycle satisfies Qpump = ( e2/h ) h [ ω3 ] , [ C ] i = n e , with integer n determined by the winding of the coherence field Φaround the cycle C . Equation (244) is the hydrodynamic coherence analogue of Thouless pumping [ 234 ], generalized to drives in the moduli of symmetry types. Thus, even in the absence of magnetic fields or band topology, cyclic “coherence stirring” can generate quantized edge transport—a clear, DC-detectable signature. Numbers, protocols, and materials Orders of magnitude. For high-mobility graphene at T∼ 100–150 K, η∼ 0 . 1–0 . 3 Pa s (2D units) and `G∼ 0 . 3–1 . 0 µ m [ 227 , 230 – 232 ]. Equation (232) gives viscous resistances in the range of ∼ 10– 100 Ω for micron-scale channels, easily resolvable. Naturon gaps ω0/ 2 π estimated from (239) with
64 Face updates: heat-bath and surface-worm Local face weight. The conditional weight for a face f depends on both the adjacent fake curvatures and the 3D 2-curvatures: π(Wf|rest)∝expnβ0Re trR0tΦf(Wf)H(F) f+β1X τf Re trR1Wε(τ;f) fH(H) τ\fo,(269) where H(F) f is the G0 staple defined by (266) with U` frozen, and H(H) τ\f is the “3D staple” transporting the other three faces of τ to f (cf. (257) ). For abelian G1 , (269) reduces to a von Mises law for the face angle. Heat-bath/Metropolis for Wf .For abelian G1 sample directly as in (268) . For SU (2) or compact nonabelian G1 , adopt the same SU(2) heat-bath or Metropolis scheme used for U` , now with R1 and the combined staple from (269); overrelaxation is implemented by reflection about the 3D staple. Surface-worm algorithm (abelian or ZN ). In the Villain/dual representation, the Wf -sector maps to integer-valued (or ZN -valued) fluxes kf on faces with the 2-Bianchi constraint dk = 0 on each tetrahedron. The surface worm updates k on a fluctuating surface with open boundary (a face pair), then closes the surface to keep dk = 0 [251, 252]. Pseudocode: 1. Pick a seed face f0; flip kf0→kf0±1and mark the two tetrahedra adjacent to f0as defects. 2. Grow the surface by selecting a defect tetrahedron τ and one of its other faces f≺τ ; flip kf→kf∓ 1; move the defect to the opposite tetrahedron; accumulate the Metropolis weight along the growth. 3. Continue until the two defects annihilate (the surface closes); accept the whole move with probability min{1, e−∆S}. This update is rejection-sparse, preserves dk= 0, and dramatically reduces critical slowing down in the abelian sector [252]. Structure-field updates: Riemannian HMC on M2-grp Geometry and kinetic term. The vertex field Φ p∈ M2-grp is advanced by Riemann manifold HMC (RMHMC) [ 253 ]. Let GΦ be the metric used in (262) ; introduce conjugate momenta Π p in T∗ ΦpM with kinetic energy T=1 2PpΠ> pG−1 ΦΠp|? p|. The Hamiltonian is H=T+Sdisc. Forces (variational derivatives). Variations δ Φenter Sdisc via (i) the structure-gradient term and V (Φ), and (ii) the tΦ and parallel-transport dependence of FΦ ( f )and HΦ ( τ ). Using δlog ( exp X ) = R1 0Adexp(−sX)δX ds and the chain rule, δSdisc δΦq =κX `3qhΦq−Φq0i|? `| |`|+∇V(Φq)|? q| +X fqDJ(t) f,q †·ff, δΦqE+X τqDJ(.) τ,q †·hτ, δΦqE.(270) Here ff := ∂ ∂(log FΦ(f)) 1 2g2klog FΦ ( f ) k2 0|?f| |f| is the fake-curvature force on face f , and hτ is the analogous 2-curvature force on tetrahedron τ ; J(t) f,q and J(.) τ,q are Jacobians of tΦ and the transported . -action with respect to Φ q . In abelian or linearised models, these Jacobians are constant matrices; otherwise we evaluate them by automatic/complex-step differentiation. Integrator and acceptance. Integrate Hamilton’s equations by generalized leapfrog on the manifold using retractions/exponential maps for Φ(local chart per vertex). Tune the step size and trajectory length τMD to target Metropolis acceptance & 0 . 7[ 250 ]; adapt during warmup (dual averaging) and precondition using GΦto equalize mode speeds [253]. Gauge (microcanonical) moves and gauge fixing for measurements 0and 1-gauge sweeps. Random 0-gauge at a vertex p : draw gp∈G0 and transform (259) all incident U and W . Likewise, 1-gauge at an edge ` : draw η`∈G1 and update U, W by (260) . These moves preserve Sdisc and decorrelate the gauge orbits (microcanonical updates).
65 Gauge fixing for diagnostics. For gauge-variant diagnostics (e.g. Landau-gauge propagators), apply lattice Landau gauge by maximizing P`Re tr U` over 0-gauge, and the analogous 2-level condition for Wf. Physics observables reported below remain gauge invariant. Global sector flips (topology changes) Non-contractible sheets and ’t Hooft flux. On a periodic lattice, choose a non-contractible dual 2-cycle Σ ? ; multiply all Wf pierced by Σ ? by a central element z∈Z ( G1 ), Wf→zWf . Compensate the induced change in FΦ along the boundary by multiplying a non-contractible stack of links U` by t ( z ) −1 , so that (256) remains locally consistent. This sector flip changes the global H -flux by z and tunnels between topological sectors. Accept with min{ 1 , e−∆S} ; for pure BF (integer k ) the move is frequently action-neutral. Tempering for rugged sectors. If sector flips are rarely accepted, use parallel tempering in ( β0, β1, κ )- space or in the wall width/height parameters of defect pinning potentials; swap neighbouring replicas with Metropolis acceptance [ 254 ]. Histogram reweighting aids interpolation across nearby parameters [255]. Putting it together: one Monte Carlo sweep 1. Link sweep: for all ` , perform NHB heat-bath or Metropolis proposals, each followed by one overrelaxation reflection. 2. Face sweep: for all f , one heat-bath/Metropolis step; then perform NSW surface-worm updates (abelian sector). 3. Gauge sweep (microcanonical): random 0-gauge at a fraction ρ0 of vertices and 1-gauge at a fraction ρ1of edges. 4. Structure HMC: draw momenta and integrate RMHMC for τMD with stepsize ; accept/reject. 5. Global flip (rare): attempt a sector flip on a random non-contractible sheet with probability pflip . Choose NHB, NSW, ρ0,1, , τMD to maintain acceptance and minimize autocorrelation; use binning/Γmethod to monitor errors [258, 259]. Observables and finite-size scaling (FSS) Wilson loops and surfaces. For a closed lattice loop C, the Wilson loop W(C) := D1 dim R0trR0Y `∈C U`E(271) discriminates perimeter vs. area law (string tension). For a closed, oriented surface S, W(S) := D1 dim R1trR1Y f∈S WfE,(272) probes 2-form confinement (volume vs. area law). Curvature densities and histograms. Monitor hklog FΦk2i and hklog HΦk2i as in (262) and their histograms to detect first-order transitions. The fake-flat probability P ( FΦ=1 )(within a small tolerance) estimates proximity to colimit closure (Sec. III). Structure-field correlators. Define the vertex correlator using the log map on M2-grp : for small geodesic displacements δΦp:= logΦ?(Φp), SΦ(q) := 1 |K0|X p,p0 eiq·(xp−xp0)δΦ> pGΦ?δΦp0,(273) and the second-moment correlation length ξfrom SΦ(0)/SΦ(qmin).
66 Binder ratios and crossing analysis. The Binder cumulant U4 = 1 − hm4i/ (3 hm2i2 )with m2 = 1 |K0|Ppδ Φ > pGΦ?δ Φ p has size-independent crossing at criticality, yielding rc ; extract ν from slope ratios [256, 257]. Exponent and gap extraction. At criticality, SΦ (0) ∼L2−η , while the photon gap mγ follows from long Wilson loops/surfaces or from the curvature–curvature correlator; test the universal ratio mΦ/mγ predicted in (251). Use histogram reweighting to interpolate nearby couplings [255]. Autocorrelation and errors. Estimate integrated autocorrelation times τint for each observable by the Γ-method/binned jackknife to set reliable MC error bars and effective sample sizes [ 258 , 259 ]. Tune overrelaxation/worm/HMC parameters to keep τint nearly constant with L (critical slowing exponent z close to zero). Remarks on sign problems and remedies Topological terms such as Chern–Simons may render the Euclidean weight complex. In abelian cases, dual (flux) representations often remain sign-free and are worm-updatable [ 252 ]. Otherwise, combine reweighting with parallel tempering in the topological angle, or explore complex-Langevin/Lefschetzthimble methods in controlled small volumes before extrapolation. Summary The update suite—link/face heat-bath (with overrelaxation), abelian surface worms, RMHMC for Φ, microcanonical gauge sweeps, and rare global sector flips—provides an ergodic, symmetry-exact, and scalable sampler for the coherence-first lattice theory. The observable set (Wilson loops/surfaces, curvature densities, Φ–structure factors, Binder/FSS) closes the loop between the lattice numerics and the continuum predictions of Sec. VI. H. Benchmarks and validation for materials/simulators Purpose. We articulate a reproducible pipeline that connects the lattice observables of Secs. VI F–VI G to laboratory probes (INS/neutrons, THz/optics, NV magnetometry) and to programmable quantum simulators. The workflow has four pillars: (i) scale setting (units and calibration from one or more measured quantities); (ii) forward modelling of probe-specific response functions from lattice correlation functions; (iii) instrument convolution and sample geometry; and (iv) validation on exactlyor semianalytically solvable benchmarks and cross-probe consistency checks. Scale setting and unit conversion Basic mapping. Let a be the lattice spacing and at the (Euclidean) time step in simulations. Dimensionless couplings ( g, h, κ )determine a characteristic velocity v? and gap(s) m? in lattice units (§VI B, §VI D). Fix (a, at)by matching two independent observables: vphys ?=a at vlat ?, mphys ?=1 at mlat ?.(274) For spin liquids one may take vphys ? from the emergent photon velocity inferred from lowω neutron dispersion and mphys ? from the vison/spinon threshold; for hydrodynamic electrons, vphys ? is the sound/entropywave velocity and ηphys follows from the Gurzhi fit (Sec. VI D). Elastic moduli in fracton elasticity follow from H-penalties via (216), calibrated against ultrasound. Geometry and boundary conditions. Map sample shape and contacts (for transport/NV) or mosaic and surface normals (for INS) to the triangulation K , using periodic boundaries for bulk probes and open/structured boundaries for device-level modelling; use the discrete Stokes correspondence (boundary DEC) to relate bulk topological terms to boundary CS terms, mirroring (181) at the lattice level.
67 Inelastic neutron scattering (INS): S(Q, ω) From lattice correlators to cross section. For magnetic scattering the double-differential cross section reads [260, 261] d2σ dΩdE0=k0 kγr0 22X αβ δαβ −ˆ Qαˆ QβF2(Q)Sαβ(Q, ω),(275) with the dynamical structure factor Sαβ(Q, ω) = 1 2πN Z∞ −∞ dt eiωt X ij e−iQ·(ri−rj)hSα i(t)Sβ j(0)i.(276) In the parton/higher-gauge descriptions, spin operators project to gauge-invariant bilinears and to gauge-sector contributions (photon/naturon). To leading order the gauge contribution is controlled by the transverse correlator of FΦ (or of the dual photon field), while matter bilinears contribute a broad continuum (Sec. VI B). Imaginary-time data and analytic continuation. MC delivers the imaginary-time correlator G(Q, τ) = hOQ(τ)O−Q(0)i=Z∞ 0 dω K(τ, ω)S(Q, ω), K(τ, ω) = e−ωτ +e−ω(β−τ) 1−e−βω .(277) Recover S ( Q, ω )by Maximum Entropy (MaxEnt) or Stochastic Analytic Continuation (SAC) with Bayesian priors [ 262 , 263 ]. In practice: (i) accumulate G ( Q, τ )and its covariance; (ii) choose a smooth default model m ( ω )constrained by Rdω S ( Q, ω ) = h [ OQ,O−Q ] i ; (iii) maximize the posterior P [ S|G ] ∝ e−χ2/2+αSent (MaxEnt) or sample S paths (SAC). Benchmarks below validate resolution against known spectral lines/gaps. Resolution convolution and form factors. Convolve the theoretical S ( Q, ω )with the spectrometer resolution function R (∆ Q, ∆ ω )(Gaussian ellipsoid in triple-axis or TOF) and the ionic form factor F ( Q ) [260, 261]: Icalc(Q0, ω0) = ZZ dQdω R(Q−Q0, ω −ω0)F2(Q)S(Q, ω).(278) Comparison against raw counts uses the same binning and background protocol as the instrument pipeline. THz and optical conductivity: Kubo response Kubo formula and current operator. The complex optical conductivity follows from the retarded current correlator [264]: σµν(ω) = 1 iω ΠR µν(ω)−ΠR µν(0),Πµν (iωn) = Zβ 0 dτ eiωnτhJµ(τ)Jν(0)i,(279) with Jµ obtained by minimally coupling external U (1) em to the emergent sector (via A -mixing and, if present, mixed CS couplings of Sec. VI A). Analytic continuation iωn→ω + i 0 + uses the same MaxEnt/SAC machinery [ 262 , 263 ]. The lowω regime distinguishes: (i) Drude-like viscous hydrodynamics (finite-width peak set by τ−1 mr and η ), (ii) gapped gauge modes (onset at mγ ), and (iii) quantized interfacial contributions from Φ-walls (sheet Hall conductance). Time-domain THz (TDTS). For direct comparison with TDTS, compute σ ( ω )on the experimental frequency grid, embed in a multilayer transfer model (substrate+film), and generate the transmission/reflection ( T, R ), including Fabry–Pérot phases; fit the experimental T ( ω ) , R ( ω )globally under the same model [264]. NV magnetometry: stray-field imaging Biot–Savart kernel for 2D devices. Given a 2D current density J ( r )in the sample plane ( z= 0), the stray field at sensor height z0is [265] Bz(k, z0) = µ0e−kz0i k(ˆ z·k×J(k)) ,Bk(k, z0) = µ0e−kz0Jk(k)−ˆ kˆ k·J(k),(280)
68 with k the in-plane wavevector and ˆ k = k/k . From lattice hydrodynamics (Sec. VI D) obtain J = n0eu , Fourier-transform to k -space, and apply (280) to synthesize BNV maps for an NV of given axis/tilt. Include sensor height distribution, point-spread function, and vector-projection appropriate to the measurement protocol [265]. Hotspot shift and curvature metrology. Fit the measured negative nonlocal resistance pattern and vorticity hotspots with the simulated maps; extract the viscosity gradient via the shift formula (242) . The same pipeline applies to spin currents and magnetization textures (for magnetic insulators) after replacing Jwith the appropriate source term. Quantum simulators: cold-atom and Rydberg platforms Mapping degrees of freedom. Ultracold-atom schemes realize lattice gauge fields by (i) boson/fermion mixtures with Gauss-law constraints enforced energetically, (ii) Floquet engineering, or (iii) Rydberg blockade encoding Z2 links [ 266 – 268 ]. The dictionary: links U` map to matter-assisted hoppings or to Rydberg link qubits; faces Wf map to four-body ring-exchange or plaquette detunings; constraints map to local energetics measured by parity. Wilson loops/surfaces are read out via Ramsey interferometry or string-order tomography. Benchmark protocols. (i) Fake-flatness test: prepare near-trivial states and measure htr FΦ ( f ) i around each plaquette; approach to unity validates local constraint engineering. (ii) Surface holonomy: apply a global phase pattern to realize Wf6 = 1 on a membrane and measure the induced t ( Wf )around its boundary (lattice Stokes). (iii) Domain-wall modes: dynamically imprint a Φ-wall (e.g., detuning stripe) and track the 1D bound-mode dispersion along it by quench spectroscopy. Benchmarks against solvable limits Photon and BF limits. On cubic meshes with abelian structure and tΦ≡ 0(pure U (1)), verify the gapless photon dispersion ω = vγ|q| from S ( q, ω )and the 1 /r Coulomb potential from Wilson loops. For large m (Φ) (deep Z2 ), confirm perimeter law for Wilson loops and area law for Wilson surfaces with a topological ground-state degeneracy equal to |H1 ( T3,Z2 ) | on the 3-torus. Turn on mixed CS/BF couplings and check quantized edge responses at Φ-walls (Sec. VI A). Hydrodynamic strip. In a rectangular strip with no-slip boundaries, compare the MC steady-state flow to the analytic Poiseuille profile and verify the Gurzhi crossover and the negative nonlocal resistance lineshape before adding gradients Dη ; then validate the hotspot-shift law (242) against the simulated BNV. Analytic continuation fidelity. Generate synthetic spectra with known peaks/continua and feed their imaginary-time correlators to MaxEnt/SAC. Quantify recovery by the L2 deviation and peak-position/gap errors as functions of noise level and τ -grid; these calibration curves set reliable error bars for experimental comparisons [262, 263]. Connecting to lab data: end-to-end protocols INS (neutrons). 1. Simulate G(Q, τ)on the experimental Q-mesh; continue to S(Q, ω)with MaxEnt/SAC. 2. Convolve with Rand F(Q)to obtain Icalc via (278). 3. Fit Icalc to raw Iexp under the same ROI and background model; extract ( mγ, ν, ξ )and the DW mode dispersion (246). 4. Cross-validate: the fitted (mγ, ν)must reproduce the specific-heat crossover (194). THz/optics. 1. Compute Π( iωn )from current fluctuations or Kubo estimators on the lattice; continue to σ ( ω ) (279).
69 2. Build the thin-film optics model (substrate refractive index, thickness) and generate T ( ω ) , R ( ω ); fit experimental TDTS data [264]. 3. Verify that the extracted σ ( ω )exhibits the predicted onsets (at mγ ) and interfacial steps (if Φ-walls are engineered). NV magnetometry. 1. Solve the device Stokes problem on the lattice (Sec. VI D) to get u ( r ); convert to J and then to B by (280). 2. Convolve with height distribution and NV-axis projection; fit to scanned BNV(r)maps [265]. 3. Extract ηand its gradients; confirm hotspot shifts per (242) and the predicted naturon resonance in microwave spectroscopy (Sec. VI D). Uncertainty quantification and cross-probe consistency Bayesian parameter inference. Combine all probes in a joint likelihood L ( θ|data )for parameters θ = ( g, h, κ, . . . ); sample the posterior with MCMC. Report credible intervals for ( mγ, mΦ, ν, η, `G )and derived universal ratios (e.g. mΦ/mγ, Sec. VI E). Systematic-error budget. Track finite-size/discretization effects (vary L and a ), analytic-continuation bias (vary priors and τ -grids), and instrument-model uncertainties (vary R and substrate parameters). Quote propagated uncertainties alongside best-fit curves. Summary The above end-to-end protocols—INS via S ( Q, ω )reconstruction and resolution convolution, THz via Kubo response and thin-film optics, and NV via Biot–Savart inversion of lattice hydrodynamics—realize a quantitative bridge from discrete 2-gauge simulations to experiments. Benchmark suites (photon/BF limits, Poiseuille flow, synthetic continuation) validate each stage and provide calibrated error bars, enabling stringent tests of the coherence-first predictions across materials and quantum simulators. VII. DISCUSSION AND OUTLOOK A. Conceptual impact Executive statement. The central message of this work is a reframing: gauge invariance is not a primitive postulate of physics but the first, most visible symptom of a deeper requirement—coherence of gluing local descriptions. In this view, “symmetry” is an infrared fixed point of the gluing problem (§V D); quantization is the residual obstruction to perfect gluing (integral classes that cannot be smoothed away); and what we call “forces” are nothing but curvatures, i.e. energetic costs for maintaining coherence in the presence of nontrivial geometry or topology. From redundancy to requirement: why gauge invariance appears Classical perspective and its limitations. Standard lore, going back to Noether, Weyl, Yang and Mills, views continuous symmetries as fundamental and reads dynamics from invariance principles [ 269 – 271 ]. Local gauge symmetry is then a redundancy of description: different gauge-related potentials encode the same physical state, and the field strength (curvature) provides the true observables. This is powerful, but it hides the constructive question: why are we allowed to glue local data into a global description at all, and what sets the price when gluing fails even infinitesimally?
70 Coherence-first answer (this work). Given local lenses, the microscopic Fand hydrodynamic/effective G(§II A), our coherence functional penalizes failures of 1and 2-level gluing: C[A, B, Φ] ∼ kFΦk2+kHΦk2+kDΦk2with FΦ=dA +A∧A−tΦ(B), HΦ=dB +A .ΦB, see (75) . A coherence fixed point has FΦ=HΦ=D Φ = 0(§V D). In this limit the cost of changing local frames vanishes, and a group of frame changes acts freely: what we traditionally call an emergent gauge symmetry. Away from the fixed point, small but nonzero curvatures FΦ, HΦ drive small violations of Ward identities controlled by (99)–(100); this is the proto-gauge regime. Thus, gauge invariance is the boundary case of perfect coherence; approximate gauge invariance is small curvature. Proposition VII.1 (Ward identities from coherence) . The Euler–Lagrange equations of §III B imply the Ward pair DAJA= (.Φ)†(B, JB), DAJB=t† ΦJA, i.e. conservation at level-1 is equivalent to exact exchange with level-2 and vice versa. In particular, if FΦ=HΦ= 0, then DAJA=DAJB= 0(exact gauge symmetry). If kFΦk,kHΦk = O ( ε ), violations are O ( ε ). Sketch. Vary (75) with respect to A and B (Sec. III B). Contract with 0and 1-gauge generators and use the (modified) Bianchi identities to move covariant derivatives onto currents. Flatness ( FΦ=HΦ= 0) collapses the exchange terms, giving conservation; small curvature gives controlled violations. Relation to Elitzur’s theorem. Local gauge symmetry cannot break spontaneously [ 277 ]; in our language, “symmetry breaking” is reinterpreted as leaving the perfect-coherence stratum so that the price for certain frame changes becomes finite, while the 2-level coherence may absorb the defect (§IV B). What is protected is not a rigid symmetry group but the coherence structure and its Bianchi-derived Ward identities. Quantization as obstruction, revisited Integral classes and Dirac quantization. The traditional route to quantization recognises integral cohomology classes: [ F/ (2 π )] ∈H2 ( M, Z )for line bundles (Dirac) and [ H/ (2 π )] ∈H3 ( M, Z )for gerbes (Berry/Wess–Zumino, higher phases) [ 272 , 273 ]. In our coherence complex, exactness of F∧F (or higher polynomials) can fail because the requisite B does not exist globally. The obstruction lives in the cokernel of tΦ,Z and is responsible for quantized transport and defect charge (Sec. IV C). This aligns with modern differential cohomology accounts of flux quantization [ 276 ] and with nonperturbative quantization of topological responses (e.g. CS levels) [274]. Anomalies as failed gluing. Anomalies are precisely inconsistencies in the descent, i.e. failures to glue background couplings globally; anomaly inflow repairs them at the next coherence level. The SU(2) global anomaly [ 275 ] reads, in our language, as an obstruction class that cannot be neutralized within 1-coherence but cancels via 2-coherence (boundary inflow), cf. §IV B. Forces as curvature: the cost of coherence Energetic meaning. With the coherence action (75) , “forces” are nothing but functional gradients of curvature norms: δS δA ∝DAFΦ+···,δS δB ∝DAHΦ+···. These are the familiar Yang–Mills/Maxwell equations and their higher analogues, but their meaning changes: a discharge of curvature lowers the energy by improving gluing. In materials language: stresses are 2-form curvatures (fracton elasticity, §VI C); in hydrodynamics, viscosity is a map tΦ that compensates the fake curvature between gradients and stresses (§VI D); in band geometry, Berry curvature is the price of stitching local Bloch frames [273]. The universal statement is: interactions measure how costly it is to keep the system coherent while changing viewpoint or scale.
71 Symmetry as emergent fixed point From Noether to IR coherence. Noether’s theorem [ 269 ] ties conserved currents to continuous symmetries of the action. In our framework, the direction is inverted: coherence enforces exact or approximate Ward identities that look like Noether relations in the IR (§V D). The symmetry group is then the automorphism 2-group of the flat transport 2-functor—a property of the IR phase, not an a priori axiom. This sits comfortably with effective-field-theory pragmatism [ 278 ]: what is fundamental are the infrared data and their consistency (coherence), not a UV symmetry dogma. What changes, and what stays Clarity about redundancy. Gauge symmetry remains a redundancy at the level of potentials; our refinement is to identify the coherence that makes this redundancy useful and to quantify departures from it (§IV A). “Symmetry breaking” is replaced by coherence redistribution (1 → 2 lift, §V B) rather than literal loss of invariance. Unification of topological and dynamical sectors. Topological terms (CS, BF, Wess–Zumino) and kinetic terms coexist naturally as different pieces of the coherence budget: the former fix the integral part (which cannot be altered by local deformations), the latter control the metric cost. This perspective clarifies why topological responses are quantized even in messy materials (§VI A): the obstruction class cannot be paid off by metric resources. New predictive levers. Because the structure field Φchanges the type of gluing data (the crossedmodule itself), we gain new, tunable controls in both high-energy and condensed-matter contexts. In the SM-like setting, Φ-deformations suggest novel routes to soft naturalness and to suppressing CP-violating θ -terms without new light axions (§V A); in materials, Φ-textures produce robust interfacial channels and quantized steps (§VI A, §VI B), and generate distinctive hydrodynamic resonances (§VI D). Mathematical synthesis Coherence theorems and higher symmetry. Mac Lane’s coherence theorems legitimized replacing strict equalities by structured isomorphisms in monoidal categories [ 279 ]; higher category theory shows that associativity itself can be coherently weakened layer by layer [ 280 , 281 ]. We import this logic into physics: when strict composition of local symmetries fails, a 2-group/crossed-module structure repairs it and supplies the next layer of gauging. The emergent symmetry group is, accordingly, the automorphism group of the flat higher connection. This categorical scaffolding organizes when and how gauge-theoretic descriptions are appropriate, and when higher ones are unavoidable. Historical consistency Recovering the classics. In the flat limit, our framework collapses to familiar terrain: Maxwell and Yang–Mills curvatures deliver forces [ 271 ], Dirac quantization emerges from integral cohomology [ 272 ], Berry phases from bundle holonomy [ 273 ], CS/WZW quantization from three-forms [ 274 ], anomalies from failed descent and their inflow repairs [ 275 ]. What is new is the unifying coherence-first lens, which treats these as facets of one geometric story rather than separate axioms. Looking forward Two broad consequences stand out. First, conceptual: it becomes natural to search for coherence flows under renormalization that land on symmetry as a fixed point, rather than to assume symmetry ab initio. Second, practical: by engineering Φin materials or simulators, we can programme symmetry-type transitions and quantized interfacial responses, turning categorical obstruction into a device principle. The remainder of Sec. VII develops these directions.
72 B. Phenomenological program Aim. We outline a concrete, falsifiable program that connects the coherence-first framework to measurable signals in two arenas: (i) high-energy/precision tests, with emphasis on the strong-CP problem and structural relaxion scenarios; (ii) condensed-matter platforms (materials and quantum simulators), with parameter windows tied to Eqs. (183) , (185) , (190) , (239) , and (244) . The strategy is to parametrise departures from perfect coherence by a small set of couplings, predict correlated signatures, and close the loop via the lattice-to-lab pipeline of Sec. VI H. I. High-energy/precision program: strong-CP and structural relaxion Phenomenological parameterisation. The structural sector is encoded by a slowly varying field Φ taking values in M2-grp . At low energies, we adopt the effective couplings. All quantities are renormalised at a hadronic scale µ∼2–3GeV. Leff ⊃g2 s 32π2θeff(Φ) Ga µν ˜ Ga µν +α 8πcγ(Φ) Fµν ˜ Fµν +X f yf(Φ) ¯ fHf +··· ,(281) θeff(Φ) = θ0−λθf(Φ), cγ(Φ) = cγ,0+λγf(Φ), yf(Φ) = yf,0(1 + κfg(Φ)), where f, g are smooth, dimensionless functions on M2-grp induced by the closed 3-form ω3 (Φ) (cf. Secs. 5.1– 5.3), and λθ, λγ, κf are small phenomenological couplings. Two limiting mechanisms (Sec. 5.1) appear: Route A (cohomology trivialisation) effectively removes the F˜ F term in the extended complex; Route B (structural relaxion) dynamically aligns Φso that θeff →0in the vacuum. EDMs and hadronic observables. The neutron EDM dn and diamagnetic EDMs (Hg, Xe) are leading probes of CP violation [282]. In chiral EFT, for small θeff one has dn≃(1.0–3.0) ×10−16 θeff e·cm,(282) with hadronic uncertainties [ 282 ]. The current neutron-EDM limit |dn|< 1 . 8 × 10 −26 e·cm (90% C.L.) [ 283 ] implies |θeff|. 10 −10 . Atomic EDMs constrain CP-odd pion–nucleon couplings that are likewise ∝θeff plus contributions from cγ (Φ).Phenomenological target: demonstrate that the joint posterior for (λθ, λγ)admits θeff ≈0without an axion, consistent with EDM bounds and hadronic susceptibilities. Topological susceptibility and finite-temperature tests. In QCD, the topological susceptibility χt ( T )= ∂2F/∂θ2 falls with temperature; lattice data indicate χt ( T ) ∼T−b above Tc [ 285 ]. In a structural-relaxion scenario, the effective curvature of Veff (Φ) gets an additive piece ∝χt ( T ), modifying thermal histories (e.g. early-universe alignment) and possibly producing transient CP-odd domains as χt ( T )evolves. Signature: no axion-like cold dark matter; instead, a suppressed χt ( T )imprint in heavy-ion observables coupled to cγ(Φ) (chiral-magnetic sensitive correlators), bounded consistently with EDMs. Collider and flavour handles. If Φmixes weakly with the Higgs, H→H + sin θΦH Φ, then Higgs-signal strengths scale as µXX ≃cos2θΦH ΓH→XX ΓSM H→XX Γtot(θΦH) ΓSM tot ,(283) constraining |sin θΦH|.O (0 . 2) from LHC fits (channel dependent). If mΦ< mH/ 2, exotic decays H→ ΦΦ produce displaced or soft spectra depending on Φ’s width; at low energies, cγ (Φ) induces rare flavour-conserving but CP-odd transitions (e.g., η→ππ , η0→ππ ) at levels scaling as λ2 γ .Signature set: a correlated pattern across EDMs, Higgs-rate dilation, and absence of ALP-like resonances. Proposed analysis workflow. 1. Global fit. Construct a likelihood L ( λθ, λγ, θ0,sin θΦH|EDMs, Higgs, flavour ), with chiral-EFT nuisance priors. Obtain posteriors by MCMC; report credible regions for θeff and couplings. 2. Cross-checks. (i) Compare χt ( T )-driven projections to lattice QCD [ 285 ]. (ii) Ensure consistency with electron-EDM bound |de|<1.1×10−29 e·cm [284] given cγ(Φ). 3. Falsifiers. Observation of an axion-like dark-matter signal inconsistent with structural-relaxion priors (coherence-first) would rule out Route B; conversely, improving dn sensitivity by an order of magnitude with null result further favours coherence suppression of θ.
73 II. Condensed-matter program: concrete windows and protocols A. Axion insulators and topological interfaces (Sec. 6.1). Targets: half-quantised surface Hall conductance σH=e2/2hfor TRI TIs; quantised thermal Hall steps at Φ-walls via (185). •Materials & heterostructures. MnBi 2 Te 4 films, magnetically gapped (Bi,Sb) 2 (Te,Se) 3 , or engineered AFM/TI stacks; introduce Φ-textures via patterned strain or magnetic domains. •Windows. Surface gaps ∆s∼10–50 meV; temperatures T∆s;Φ-wall widths w∼5–50 nm. •Measurements. Terahertz Faraday/Kerr rotations; lowT thermal Hall along Φ-walls; compare to (183) and (185) after instrument convolution (Sec. VI H). •Falsifier. Absence of quantised interfacial steps when ∆ ω3 (Φ) ∈Z and disorder is below the mobility gap. B. U (1) ↔Z2 spin liquids (Sec. 6.2). Targets: photon gap mγ (Φ) from (190) ; wall-bound mode dispersion (246); heat-capacity crossover (194). •Materials. Kagome/triangular candidates (herbertsmithite, Zn-barlowite; organic salts). •Windows. mγ∼ 0 . 2–2meV; ν = m0ξ/√α tuned across 1 by pressure/strain; T window 0 . 1–20 K. •Measurements. Polarised INS to see spectral reweighting and the 1D ridge at ωDW ( kk ); specific heat C ( T )switching from T2 to activated; nonlocal spin transport along Φ-walls for spin steps (248). •Falsifier. Failure to observe the bound-mode threshold at νc = 1 ( (252) ) when the wall width is controlled independently. C. Fracton elasticity (Sec. 6.3). Targets: mobility crossover (223); line modes (224) at Φinterfaces. •Platforms. Colloidal crystals, 2D exfoliated membranes, programmable metamaterials with tunable κFrank(Φ). •Windows. Interface widths ξΦ∼1–10 alat; room-TBrillouin light scattering. •Measurements. Track dislocation trajectories vs. Φ; image line modes; fit κFrank (Φ) from ultrasonic attenuation and compare to (216). D. Hydrodynamic electrons (Sec. 6.4). Targets: naturon resonance (239) ; negative nonlocal resistance with hotspot shift (242); quantised pumping (244). •Materials. Encapsulated graphene, PdCoO2, WP2. •Windows. η∼0.1–0.3 Pa s;`G∼0.3–1µm; naturon gaps ω0/2π∼1–10 GHz. •Measurements. Scanning NV magnetometry for vortices/hotspots; TDTS for GHz naturon; pump cycles in Φ-space (gate/strain loops) for quantised edge charge per cycle. E. Simulators. Rydberg/ultracold platforms implement Z2 / U (1) 2-gauge truncations (Sec. 7). Targets: discrete versions of fake-flatness, surface holonomy, and domain-wall modes; measure Wilson surfaces and the emergence of naturality inflow at engineered Φ-walls. III. Joint inference, milestones, and decision points Joint Bayesian inference. Combine probe likelihoods into L ( ϑ ) = LINS ×LTHz ×LNV ×LEDM ×··· , with parameters ϑ collecting ( g, h, κ ),Φ-potential couplings, and structural couplings ( λθ, λγ,sin θΦH ). Use the DEC-based forward models of Sec. VI H; report credible regions for ( mγ, mΦ, ν, η, `G )and universal ratios (e.g. mΦ/mγ).
80 The second Bianchi identity. Compute DAHΦ=DA(dB +A .ΦB)=(DAA).ΦB+A .Φ(DAB) + ∂(.Φ)[DΦ] (A, B), where the last term is the structural variation of the action. Using DAA = dA + [ A, A ] = F◦ (the ordinary curvature of A) and DAB=HΦ−A .ΦB, we obtain DAHΦ=F◦.ΦB+A .ΦHΦ−A .Φ(A .ΦB) + ∂(.Φ)[DΦ] (A, B).(A5) Replacing F◦ = FΦ + tΦ ( B )and using the Peiffer identity ( tΦB ) .ΦB0 = 0, together with the derivation property of .Φ, this simplifies to DAHΦ=FΦ.ΦB+∂(.Φ)[DΦ] (A, B).(A6) Thus the second Bianchi identity acquires a source whenever the action of g0on g1varies along Φ. 3. Variational calculus with structure maps We vary the coherence action (cf. Eq. (75) in the main text) S[A, B, Φ] = ZMh1 2g2hFΦ∧∗FΦi0+1 2h2hHΦ∧∗HΦi1+κ 2GΦ(DΦ, DΦ) volg+V(Φ) volgi+Stop[Φ; A, B]. (A7) The variations of the curvatures are δFΦ=DAδA −tΦ(δB)−∂tΦ[δΦ] (B),(A8) δHΦ=DAδB +δA .ΦB+A .ΦδB +∂(.Φ)[δΦ] (A, B).(A9) After integrations by parts and dropping boundary terms (or assuming suitable boundary conditions), the Euler–Lagrange equations take the schematic form 1 g2DAFΦ+Jtop A=Jmatt A+SA[Φ; A, B],(A10) 1 h2DAHΦ+Jtop B=Jmatt B+SB[Φ; A, B],(A11) κ∇µ(GΦ∂µΦ)−∂ΦV=SΦ[A, B] + Stop Φ.(A12) Here DA=−∗DA∗, and the structure-sourced terms are explicitly SA[Φ; A, B] = −1 h2B.Φ ·HΦ#,(A13) SB[Φ; A, B] = 1 g2t† Φ FΦ−1 h2A .Φ†(HΦ),(A14) SΦ[A, B] = 1 g2∂t† Φ[·] (FΦ), B+1 h2∂(.Φ)†[·] (A, B), HΦ+1 2∂ΦGΦ(DΦ, DΦ),(A15) where † denotes adjoints with respect to the chosen inner products, and we used the natural pairings between g0,1 -valued forms. The precise index structure follows from (A8) – (A9) and the integration by parts. 4. Ward identities (exchange between 1and 2-form sectors) Consider infinitesimal 0-gauge and 1-gauge transformations, δA = DA, δB = .ΦB and δηA = −tΦ ( η ) , δηB = DAη , with parameters ∈ Ω 0 ( M, g0 ), η∈ Ω 1 ( M, g1 ). Gauge invariance of S implies (after standard manipulations) the exchange or higher Ward identities DA1 g2FΦ=t† Φ1 h2HΦ+Jtot A,(A16) DA1 h2HΦ=.Φ†A, 1 h2HΦ−t† Φ1 g2FΦ+Jtot B,(A17)
81 where Jtot A,B include topological and matter sources (and possible Φ-dependent improvement terms). Eqs. (A16) – (A17) exhibit the bidirectional transmutation between level-1 and level-2 currents mediated by the adjoints of the structure maps. In the flat limit FΦ = HΦ = 0, the right-hand sides reduce to the conserved currents (ordinary Ward identities). Small deviations from flatness produce controlled violations of conservation, of order kFΦkand kHΦk. Remark. The modified Bianchi identities (A4) – (A6) guarantee consistency of (A16) – (A17) with gauge symmetry in the presence of Φ-variation: contracting DA into the left-hand sides reproduces the structural source terms. Appendix B: Crossed-module examples: color/center structures and spin liquids; trivialisation vs. relaxation 1. Color/center 2-structure (qualitative model) Let G0 = SU (3) /Z3 and G1 = Z3 . The crossed-module data encode that a 2 π/ 3surface holonomy (in G1 ) can compensate a center mismatch in an SU (3) loop holonomy (lifted to SU (3)). At the Lie level, tΦ is trivial (discrete G1 ), while the Postnikov class obstructing a strict lift lives in H3 ( BG0,Z3 )and controls the gluing of color frames. In our continuum surrogate, one treats G1 by a compact U (1) 2-form with constrained periods (a Z3 2-form gauge field), and tΦ implements the map Z3,→Z ( SU (3)) at the level of holonomy. The fake curvature then reads FΦ=F◦−tΦ(B)with Bquantised in units of 2π/3. 2. U(1)→Z2spin liquid (explicit crossed module) Set G0 = U (1), G1 = Z2 = {± 1 } , t (1) = − 1 ∈U (1), with trivial action . (since U (1) is abelian). Then for A∈Ω1(M, R)and B∈Ω2(M, πi Z)with 2B∈2πi Z, one has F=dA −πi B, H =dB, (B1) and the Bianchi identities are dF + πi H = 0 and dH = 0. This model captures the simplest route by which a compact U (1) gauge theory Higgses to a Z2 topological order as the stiffness for B is tuned (Sec. 6.2 in the main text). 3. Trivialisation vs. structural relaxation Route A (trivialisation). In the extended Deligne complex [ 316 ], one can represent the 4-form F∧F as an exact differential dC whenever a 3-form Cheeger–Simons character exists that lifts F ; physically, this means a globally defined C -field whose differential compensates F∧F . In such cases, θRF∧F is a boundary term and becomes unobservable in the bulk. Route B (structural relaxion). Alternatively, let Φcouple through a mixed term Rω3 (Φ) ∧CS ( A ), with ω3 closed on M2-grp . Minimisation in Φaligns the effective θ to zero, θeff (Φ) → 0,without requiring a globally defined trivialiser C. This is the dynamical mechanism emphasised in Sec. 5.1. Appendix C: DEC/lattice details: discrete gauge invariance, update kernels, and performance notes 1. Discrete gauge invariance and measure Let DUDWD Φdenote the product of Haar measures on G|K1| 0 , G|K2| 1 and the Riemannian volume measures on M|K0| 2-grp . Under 0-gauge and 1-gauge transformations (Eqs. (259) – (260) in the main text), Haar invariance implies the measure is invariant, and the action Sdisc is class-function valued in each local variable, hence gauge invariant. Therefore, the Boltzmann weight and the measure define a well-posed gauge-invariant lattice theory.
82 2. Acceptance kernels (explicit formulas) For G0=U(1), link updates with proposal θ0=θ+δhave acceptance pacc = minn1,exp κcos(θ0+φ)−cos(θ+φ)o, κ =β0|H`|. For G0 = SU (2), drawing U0 ` = V˜ H† ` (heat-bath) with V sampled from the density p ( v0 ) ∝ p1−v2 0e2β0cv0 yields unit acceptance. Overrelaxation steps are microcanonical and thus also accepted with probability one. Analogous formulas hold for face updates with β1(Sec. VI G). 3. Complexity and decorrelation The cost to assemble staples is O ( deg )per link/face (constant on regular meshes). For abelian sectors, surface-worm updates reduce integrated autocorrelation times at criticality from τint ∼Lz with z& 2 (local Metropolis) to z≈ 0. For Φ-HMC, the manifold preconditioner GΦ equalises mode speeds and allows step sizes that keep acceptance &0.7with O(1) MD steps per sweep. Appendix D: Linear-response calculations: quantised interfacial steps and naturon spectra 1. Charge and thermal Hall steps at Φ-walls Consider a (2 + 1)D interface Σwhere Φjumps between two plateaus, Φ L and Φ R . The mixed coupling Smix = 1 2πRω3 (Φ) ∧CS ( Aem )induces a boundary Chern–Simons term with level keff = 1 2πRΦR ΦLω3. Standard linear response (Kubo) [320] gives the sheet Hall conductance σH=e2 hkeff,(D1) while the thermal Hall step follows from the energy-magnetisation corrected Kubo formula [321]: κxy T=π2k2 B 3hceff, ceff ∈1 2Z,(D2) with ceff the chiral central charge induced along Σ. The quantisation of keff and ceff is fixed by the integral periods of ω3 and by the integrality of CS levels (see also [ 322 , 323 ] for related quantisation statements). 2. Naturon dispersion and spectral weight Linearising the coupled ( u⊥, Φ) equations (Sec. 6.4) gives the matrix dispersion equation D ( ω, k )=0 with D(ω, k) = −iωρ0+ηk2+ρ0/τmr−χΦω2−iγΦω+ξ2 Φk2+m2 Φ+λ2k2. Solving to leading order at small k and small damping yields the naturon branch ωnat ( k ) ≃pω2 0+v2 natk2− i 2 Γ nat ( k ) , with ω2 0 = λ2/ ( ρ0χΦ ) , v2 nat ≈ξ2 Φ, Γ nat ( k ) ≈γΦ/χΦ + 2 ηk2/ρ0. The current–current response inherits a simple-pole contribution ΠR ⊥⊥(ω, k)⊃Znat(k) ω−ωnat(k)+c.c.,Znat(k) = n2 0e2λ2k2 ρ2 0h∂ωD(ω, k)i−1 ω=ωnat ,(D3) leading to a narrow absorption peak in Im σ ( ω )whose integrated spectral weight scales as ∝λ2 and is tunable via Φ.
83 Appendix E: Cohomology with local coefficients and H4(BG0×M); relation to SPT and anomaly inflow 1. Local coefficients and twisted cohomology Let Λ be a local system (flat bundle) of abelian groups over M (e.g. the sheaf of g1 -lattices twisted by the G0 -bundle). Cohomology with local coefficients H• ( M ; Λ )is defined via the cochain complex of singular cochains with values in the local system; it reduces to the ordinary cohomology when Λ is trivial [ 318 ]. In our setting, Λ captures the Φ-twist of the 2-form charge lattice; obstruction classes such as [ HΦ ] naturally live in H3(M; ΛΦ). 2. Künneth for H4(BG0×M) For suitable spaces (CW complexes) and coefficients in a PID, the Künneth theorem gives H4(BG0×M;Z)∼ =M p+q=4 Hp(BG0;Z)⊗Hq(M;Z)⊕M p+q=5 TorHp(BG0;Z), Hq(M;Z).(E1) The summands H4 ( BG0 ) ⊗H0 ( M )and H0 ( BG0 ) ⊗H4 ( M )encode pure gauge and pure spacetime characteristic classes; the mixed terms H3 ( BG0 ) ⊗H1 ( M )and H2 ( BG0 ) ⊗H2 ( M )capture interactions between gauge data and spacetime cycles (e.g. our mixed coupling ω3 (Φ) ∧CS ( A )lives in the transgressed image of H3 ( BG0 )). Torsion pieces encode discrete effects (finite groups, Zn classes). References: [317, 319]. 3. Relation to SPT classifications and anomaly inflow Bosonic SPT phases with symmetry G0 in 3 + 1dimensions admit a group-cohomology classification by H4 ( BG0, U (1)) in many cases. Our mixed responses select particular mixed pieces in H4 ( BG0×M )by evaluating on spacetime cycles in M and on symmetry cycles in BG0 . Anomalies appear as obstructions to lifting these classes to globally defined actions, repaired by invertible bulk TQFTs (anomaly inflow). In differential form language, Deligne cohomology provides the right home for simultaneously tracking integral periods (quantisation) and differential representatives (responses) [316]. [1] C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96 , 191 (1954). [2] S. Weinberg, The Quantum Theory of Fields, Vol. II: Modern Applications, Cambridge University Press (1996). [3] S. Elitzur, Impossibility of Spontaneously Breaking Local Symmetries, Phys. Rev. D 12, 3978 (1975). [4] T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 , 084019 (2011). [5] D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, Rev. Mod. Phys. 93 , 025002 (2021). [6] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries, J. High Energy Phys. 2015, 172 (2015). [7] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008). [8] M. Pretko, Subdimensional particle structure of higher rank U(1) spin liquids, Phys. Rev. B 95 , 115139 (2017). [9] A. Lucas and K. C. Fong, Hydrodynamics of electrons in graphene, J. Phys. Condens. Matter 30 , 053001 (2018). [10] J. C. Baez and U. Schreiber, Higher Gauge Theory: 2-Connections on 2-Bundles, arXiv:hep-th/0412325 (v2 2007). [11] T. Lada and J. Stasheff, Introduction to sh Lie algebras for physicists, Int. J. Theor. Phys. 32 , 1087 (1993). [12] D. S. Freed, Dirac charge quantization and generalized differential cohomology, Surveys in Differential Geometry 7, 129 (2000). [13] M. B. Green and J. H. Schwarz, Anomaly Cancellation in Supersymmetric D = 10 Gauge Theory and Superstring Theory, Phys. Lett. B 149, 117 (1984).
84 [14] A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in various dimensions and group cohomology, arXiv:1404.3230. [15] X.-G. Wen, Quantum Field Theory of Many-Body Systems, Oxford University Press (2004). [16] P. W. Anderson, Plasmons, gauge invariance, and mass, Phys. Rev. 130, 439 (1963). [17] P. W. Higgs, Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett. 13, 508 (1964). [18] F. Englert and R. Brout, Broken Symmetry and the Mass of Gauge Vector Mesons, Phys. Rev. Lett. 13 , 321 (1964). [19] G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Global Conservation Laws and Massless Particles, Phys. Rev. Lett. 13, 585 (1964). [20] J. C. Baez and J. Huerta, An Invitation to Higher Gauge Theory, Gen. Relativ. Gravit. 43, 2335 (2011). [21] B. Jurčo, I. Sach, and U. Schreiber, Higher Gauge Theory for Stringand Membrane-Coupling, J. Geom. Phys. 74, 130 (2013). [22] M. Alexandrov, M. Kontsevich, A. Schwarz, and O. Zaboronsky, The Geometry of the Master Equation and Topological Quantum Field Theory, Int. J. Mod. Phys. A 12, 1405 (1997). [23] G. T. Horowitz, Exactly Soluble Diffeomorphism Invariant Theories, Commun. Math. Phys. 125 , 417 (1989). [24] E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117, 353 (1988). [25] S. Deser, R. Jackiw, and S. Templeton, Three-Dimensional Massive Gauge Theories, Phys. Rev. Lett. 48 , 975 (1982). [26] G. ’t Hooft, Symmetry Breaking through Bell–Jackiw Anomalies, Phys. Rev. Lett. 37, 8 (1976). [27] E. Witten, Current Algebra Theorems for the U(1) “Goldstone Boson”, Nucl. Phys. B 156, 269 (1979). [28] R. D. Peccei and H. R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. 38 , 1440 (1977). [29] S. Weinberg, A New Light Boson?, Phys. Rev. Lett. 40, 223 (1978). [30] F. Wilczek, Problem of Strong P and T Invariance in the Presence of Instantons, Phys. Rev. Lett. 40 , 279 (1978). [31] E. C. G. Stueckelberg, Die Wechselwirkungskräfte in der Elektrodynamik und in der Feldtheorie der Kernkräfte, Helv. Phys. Acta 11, 225 (1938). [32] H. Ruegg and M. Ruiz-Altaba, The Stückelberg Field, Int. J. Mod. Phys. A 19, 3265 (2004). [33] T. W. B. Kibble, Symmetry Breaking in Non-Abelian Gauge Theories, Phys. Rev. 155, 1554 (1967). [34] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Deconfined Quantum Critical Points, Science 303, 1490 (2004). [35] M. Z. Hasan and C. L. Kane, Colloquium: Topological Insulators, Rev. Mod. Phys. 82, 3045 (2010). [36] X.-L. Qi and S.-C. Zhang, Topological Insulators and Superconductors, Rev. Mod. Phys. 83, 1057 (2011). [37] L. Savary and L. Balents, Quantum Spin Liquids: A Review, Rep. Prog. Phys. 80, 016502 (2016). [38] J. Knolle and R. Moessner, A Field Guide to Spin Liquids, Annu. Rev. Condens. Matter Phys. 10 , 451 (2019). [39] R. M. Nandkishore and M. Hermele, Fractons, Annu. Rev. Condens. Matter Phys. 10, 295 (2019). [40] D. A. Bandurin et al., Negative Local Resistance Caused by Viscous Electron Backflow in Graphene, Science 351, 1055 (2016). [41] P. J. W. Moll, P. Kushwaha, N. Nandi, B. Schmidt, and A. P. Mackenzie, Evidence for Hydrodynamic Electron Flow in PdCoO2,Science 351, 1061 (2016). [42] J. B. Kogut and L. Susskind, Hamiltonian Formulation of Wilson’s Lattice Gauge Theories, Phys. Rev. D 11, 395 (1975). [43] A. N. Hirani, Discrete Exterior Calculus, Ph.D. thesis, California Institute of Technology (2003). [44] M. Desbrun, A. Hirani, M. Leok, and J. E. Marsden, Discrete Exterior Calculus, arXiv:math/0508341 (v2 2008). [45] N. Prokof’ev and B. Svistunov, Worm Algorithms for Classical Statistical Models, Phys. Rev. Lett. 87 , 160601 (2001). [46] C. Gattringer and C. B. Lang, Quantum Chromodynamics on the Lattice, Springer (2010). [47] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, Wiley (1963). [48] N. E. Steenrod, The Topology of Fibre Bundles, Princeton University Press (1951). [49] D. Husemoller, Fibre Bundles, 3rd ed., Springer (1994). [50] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer (1982). [51] D. Bleecker, Gauge Theory and Variational Principles, Addison–Wesley (1981). [52] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser (1993). [53] M. K. Murray, Bundle Gerbes, J. London Math. Soc. 54, 403 (1996). [54] M. Nakahara, Geometry, Topology and Physics, 2nd ed., Taylor & Francis (2003). [55] B. Simon, Holonomy, the Quantum Adiabatic Theorem, and Berry’s Phase, Phys. Rev. Lett. 51 , 2167 (1983). [56] K. G. Wilson, The Renormalization Group and Critical Phenomena, Rev. Mod. Phys. 55, 583 (1983). [57] S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic, Springer (1992). [58] J. Giraud, Cohomologie Non Abélienne, Springer (1971). [59] J. C. Baez and A. D. Lauda, Higher-Dimensional Algebra V: 2-Groups, Theory Appl. Categ. 12 , 423 (2004). [60] R. Brown and C. B. Spencer, G -Groupoids, Crossed Modules and the Fundamental Groupoid of a Topological Group, Proc. Kon. Ned. Akad. v. Wet. 79, 296 (1976).
85 [61] B. Noohi, Notes on 2-Groupoids, 2-Groups and Crossed Modules, Homology, Homotopy and Applications 9(1), 75 (2007); arXiv:math.CT/0512106. [62] J. C. Baez and A. S. Crans, Higher-Dimensional Algebra VI: Lie 2-Algebras, Theory Appl. Categ. 12 , 492 (2004). [63] E. Getzler, Lie Theory for Nilpotent L∞-Algebras, Ann. of Math. 170, 271 (2009). [64] H. Sati, U. Schreiber, and J. Stasheff, L∞ -Algebra Connections and Applications to Stringand Chern– Simons n-Bundles, Quantum Field Theory, Birkhäuser (2009); arXiv:0801.3480. [65] L. Breen and W. Messing, Differential Geometry of Gerbes, Adv. Math. 198, 732 (2005). [66] M. K. Murray and D. Stevenson, A Note on Bundle Gerbes and the String Group, J. Geom. Phys. 58 , 1571 (2008). [67] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, Wiley (1963). [68] D. Bleecker, Gauge Theory and Variational Principles, Addison–Wesley (1981). [69] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer (1982). [70] H. Pfeiffer, Higher Gauge Theory and a Non-Abelian Stokes Theorem, Ann. Phys. 308 , 447 (2003); arXiv:hep-th/0304074. [71] P. Aschieri, L. Cantini, and B. Jurčo, Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory, Commun. Math. Phys. 254, 367 (2005). [72] U. Schreiber and K. Waldorf, Smooth Functors vs. Differential Forms, Homology, Homotopy and Applications 13(1), 143 (2011); arXiv:0802.0663. U. Schreiber and K. Waldorf, Transport 2-Functors and Connections, arXiv:0910.5224. [73] J. Faria Martins and R. Picken, The Fundamental Gray 3-Groupoid of a Smooth Manifold and Local 3-Dimensional Holonomy Based on a 2-Crossed Module, Differential Geom. Appl. 29, 179 (2011). J. Faria Martins and R. Picken, Surface Holonomy for Non-Abelian 2-Bundles via Crossed Modules, Adv. Math. 226, 3309 (2011). [74] A. Henriques, Integrating L∞-Algebras, Compos. Math. 144, 1017 (2008); arXiv:math/0603563. [75] C. J. Schommer-Pries, Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group, Geom. Topol. 15, 609 (2011). [76] J. C. Baez and D. K. Wise, Teleparallel Gravity as a Higher Gauge Theory, Commun. Math. Phys. 333 , 153 (2015); arXiv:1004.3572. [77] R. S. Palais, On the Existence of Slices for Actions of Non-Compact Lie Groups, Ann. of Math. 73 , 295 (1961). [78] D. Luna, Slices étales, Mém. Soc. Math. France 33, 1 (1973). [79] H. Whitney, Tangents to an Analytic Variety, Ann. of Math. 81, 496 (1965). [80] M. Gerstenhaber, On the Deformation of Rings and Algebras, Ann. of Math. 79, 59 (1964). [81] A. Nijenhuis and R. W. Richardson, Deformations of Lie Algebra Structures, J. Math. Mech. 17 , 89 (1967). [82] W. M. Goldman and J. J. Millson, The Deformation Theory of Representations of Fundamental Groups of Compact Kähler Manifolds, Publ. Math. IHES 67, 43 (1988). [83] J. Eells and J. H. Sampson, Harmonic Mappings of Riemannian Manifolds, Amer. J. Math. 86 , 109 (1964). [84] J. Jost, Riemannian Geometry and Geometric Analysis, 7th ed., Springer (2017). [85] M. Crainic and R. L. Fernandes, Integrability of Lie Brackets, Ann. of Math. 157, 575 (2003). [86] J. Lurie, Higher Topos Theory, Princeton University Press (2009). [87] B. Toën and G. Vezzosi, Homotopical Algebraic Geometry II: Geometric Stacks and Applications, Mem. Amer. Math. Soc. 193 (2008). [88] E. Noether, Invariante Variationsprobleme, Nachr. d. Königl. Ges. d. Wiss. zu Göttingen, Math.-Phys. Klasse, 235 (1918). [89] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University Press (1992). [90] G. Barnich, F. Brandt, and M. Henneaux, Local BRST Cohomology in Gauge Theories, Phys. Rept. 338 , 439 (2000). [91] J. Polchinski, Renormalization and Effective Lagrangians, Nucl. Phys. B 231, 269 (1984). [92] A. Nahum, P. Serna, J. T. Chalker, M. Ortuño, and A. M. Somoza, Emergent SO(5) Symmetry at the Néel to Valence-Bond Solid Transition, Phys. Rev. Lett. 115, 267203 (2015). [93] H. Shao, W. Guo, and A. W. Sandvik, Quantum Criticality with Two Length Scales, Science 352 , 213 (2016). [94] J. Wess and B. Zumino, Consequences of Anomalous Ward Identities, Phys. Lett. B 37, 95 (1971). [95] C. G. Callan, Jr. and J. A. Harvey, Anomalies and Fermion Zero Modes on Strings and Domain Walls, Nucl. Phys. B 250, 427 (1985). [96] U. Schreiber and K. Waldorf, Smooth Functors vs. Differential Forms, Homology, Homotopy and Applications 13(1), 143 (2011); arXiv:0802.0663. [97] J. Faria Martins and R. Picken, Surface Holonomy for Non-Abelian 2-Bundles via Crossed Modules, Adv. Math. 226, 3309 (2011). [98] T. Frankel, The Geometry of Physics: An Introduction, 3rd ed., Cambridge University Press (2011). [99] I. M. Anderson, Introduction to the Variational Bicomplex, in Mathematical Aspects of Classical Field Theory, Contemp. Math. 132, 51 (1992). [100] J. Lee and R. M. Wald, Local Symmetries and Constraints, J. Math. Phys. 31, 725 (1990). [101] V. Iyer and R. M. Wald, Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy, Phys. Rev. D 50, 846 (1994).
86 [102] S.-S. Chern and J. Simons, Characteristic Forms and Geometric Invariants, Ann. of Math. 99, 48 (1974). [103] M. Kalb and P. Ramond, Classical Direct Interstring Action, Phys. Rev. D 9, 2273 (1974). [104] D. Z. Freedman and P. K. Townsend, Antisymmetric Tensor Gauge Theories and Nonlinear Sigma Models, Nucl. Phys. B 177, 282 (1981). [105] W. A. Bardeen and B. Zumino, Consistent and Covariant Anomalies in Gauge and Gravitational Theories, Nucl. Phys. B 244, 421 (1984). [106] P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Springer (1993). [107] I. Kolář, P. W. Michor, and J. Slovák, Natural Operations in Differential Geometry, Springer (1993). [108] S. Morita, Geometry of Differential Forms, American Mathematical Society (1997). [109] G. J. Zuckerman, Action Principles and Global Geometry, in Mathematical Aspects of String Theory, S. T. Yau (ed.), World Scientific (1987), pp. 259–284. [110] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison–Wesley (1995). [111] E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121, 351 (1989). [112] D. S. Freed and F. Quinn, Chern–Simons Theory with Finite Gauge Group, Commun. Math. Phys. 156 , 435 (1993). [113] R. Utiyama, Invariant Theoretical Interpretation of Interaction, Phys. Rev. 101, 1597 (1956). [114] L. D. Faddeev and V. N. Popov, Feynman Diagrams for the Yang–Mills Field, Phys. Lett. B 25 , 29 (1967). [115] C. Becchi, A. Rouet, and R. Stora, Renormalization of Gauge Theories, Ann. Phys. 98, 287 (1976). [116] I. V. Tyutin, Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism, Lebedev Institute preprint 39 (1975), arXiv:0812.0580. [117] J. M. Cornwall, R. Jackiw, and E. Tomboulis, Effective Action for Composite Operators, Phys. Rev. D 10 , 2428 (1974). [118] B. Kors and P. Nath, A Stückelberg Extension of the Standard Model, Phys. Lett. B 586, 366 (2004). [119] E. Cremmer and J. Scherk, Spontaneous Dynamical Breaking of Gauge Symmetry in Dual Models, Nucl. Phys. B 72, 117 (1974). [120] F. Quevedo and C. A. Trugenberger, Phases of Antisymmetric Tensor Field Theories, Nucl. Phys. B 501 , 143 (1997). [121] J. Schonfeld, A Mass Term for Three-Dimensional Gauge Fields, Nucl. Phys. B 185, 157 (1981). [122] M. F. Atiyah, Topological Quantum Field Theories, Publ. Math. IHES 68, 175 (1988). [123] L. Alvarez-Gaumé and P. H. Ginsparg, The Structure of Gauge and Gravitational Anomalies, Ann. Phys. 161, 423 (1985). [124] J. A. Harvey, TASI 2003 Lectures on Anomalies, arXiv:hep-th/0509097. [125] C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, in Colloque de Topologie (Espaces Fibrés), Bruxelles (1950), pp. 29–55. [126] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer (1998). [127] G. M. Kelly, Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture Note Ser. 64 , Cambridge University Press (1982); reprint (2005). [128] A. Joyal and R. Street, Braided Tensor Categories, Adv. Math. 102, 20 (1993). [129] R. Healey, Gauging What’s Real? The Conceptual Foundations of Gauge Theories, Oxford University Press (2007). [130] J. O. Weatherall, Understanding “gauge”, Stud. Hist. Phil. Mod. Phys. 65–66, 94 (2019). [131] C. J. Schommer-Pries, Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group, Geom. Topol. 15, 609 (2011). [132] R. Stora, Algebraic Structure and Topological Origin of Anomalies, in Progress in Gauge Field Theory, Cargèse 1983, Plenum (1985). [133] B. Zumino, Y.-S. Wu, and A. Zee, Chiral Anomalies, Higher Dimensions, and Differential Geometry, Nucl. Phys. B 239, 477 (1984). [134] D. S. Freed and M. J. Hopkins, Reflection Positivity and Invertible Topological Phases, Geom. Topol. 25 , 1165 (2021); arXiv:1604.06527. [135] A. N. Redlich, Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three Dimensions, Phys. Rev. Lett. 52, 18 (1984). [136] A. N. Redlich, Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions, Phys. Rev. D 29 , 2366 (1984). [137] E. Witten, An SU(2) Anomaly, Phys. Lett. B 117, 324 (1982). [138] A. Sagnotti, A Note on the Green–Schwarz Mechanism in Open String Theories, Phys. Lett. B 294 , 196 (1992). [139] N. Arkani-Hamed, H. Georgi, and M. D. Schwartz, Effective Field Theory for Massive Gravitons and Gravity in Theory Space, Ann. Phys. 305, 96 (2003). [140] E. Sharpe, Notes on 2-Group Global Symmetries, SciPost Phys. 10, 27 (2021); arXiv:1509.08499. [141] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries, J. High Energy Phys. 2015, 172 (2015). [142] C. Córdova, T. T. Dumitrescu, and K. Intriligator, Exploring 2-Group Global Symmetries, J. High Energy Phys. 2022, 184 (2022); arXiv:1802.04790. [143] J. Cheeger and J. Simons, Differential Characters and Geometric Invariants, in Geometry and Topology, Lecture Notes in Math. 1167, 50–80 (1985), Springer.
87 [144] M. J. Hopkins and I. M. Singer, Quadratic Functions in Geometry, Topology, and M-Theory, J. Differential Geom. 70, 329 (2005). [145] J. Dixmier and A. Douady, Champs Continus d’Espaces Hilbertiens et de C -Algèbres, Bull. Soc. Math. France 91, 227 (1963). [146] R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129, 393 (1990). [147] J. Simons and D. Sullivan, Structured Vector Bundles Define Differential K -Theory, Quanta of Maths, Clay Math. Proc. 11, 579 (2011). [148] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological Field Theory of Time-Reversal Invariant Insulators, Phys. Rev. B 78, 195424 (2008). [149] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405 (1982). [150] A. Kapustin and R. Thorngren, Anomalies of Discrete Symmetries in Various Dimensions and Group Cohomology, arXiv:1404.3230. [151] G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14, 3432 (1976). [152] C. Abel et al., Measurement of the Permanent Electric Dipole Moment of the Neutron, Phys. Rev. Lett. 124, 081803 (2020). [153] R. D. Peccei and H. R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. 38 , 1440 (1977). [154] E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B 156, 269 (1979). [155] G. Veneziano, U(1) Without Instantons, Nucl. Phys. B 159, 213 (1979). [156] E. Vicari and H. Panagopoulos, θDependence of SU(N)Gauge Theories in the Presence of a Topological Term, Phys. Rept. 470, 93 (2009). [157] M. B. Green and J. H. Schwarz, Anomaly Cancellation in Supersymmetric D= 10 Gauge Theory and Superstring Theory, Phys. Rev. Lett. 49, 508 (1984). [158] D. S. Freed and E. Witten, Anomalies in String Theory with D-Branes, Asian J. Math. 3 , 819 (1999); arXiv:hep-th/9907189. [159] C. Vafa and E. Witten, Parity Conservation in QCD, Phys. Rev. Lett. 53, 535 (1984). [160] D. J. Gross, R. D. Pisarski, and L. G. Yaffe, QCD and Instantons at Finite Temperature, Rev. Mod. Phys. 53, 43 (1981). [161] S. Borsanyi et al., Calculation of the Axion Mass Based on High-Temperature Lattice Quantum Chromodynamics, Nature 539, 69 (2016). [162] C. Bonati, M. D’Elia, M. Mariti, G. Martinelli, M. Mesiti, F. Negro, F. Sanfilippo, and G. Villadoro, Axion Phenomenology and θ-Dependence from Nf=2+1 Lattice QCD, J. High Energy Phys. 2016, 155 (2016). [163] S. Aoki, H. Fukaya, and Y. Taniguchi, Topological Susceptibility in Lattice QCD, Prog. Theor. Exp. Phys. 2022, 012A106 (2022). [164] F. Englert and R. Brout, Broken Symmetry and the Mass of Gauge Vector Mesons, Phys. Rev. Lett. 13 , 321 (1964). [165] P. W. Higgs, Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett. 13, 508 (1964). [166] G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Global Conservation Laws and Massless Particles, Phys. Rev. Lett. 13, 585 (1964). [167] T. W. B. Kibble, Symmetry Breaking in Non-Abelian Gauge Theories, Phys. Rev. 155, 1554 (1967). [168] G. ’t Hooft, Renormalizable Lagrangians for Massive Yang–Mills Fields, Nucl. Phys. B 35, 167 (1971). [169] G. ’t Hooft and M. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 , 189 (1972). [170] S. R. Coleman and E. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D 7, 1888 (1973). [171] G. ’t Hooft, Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking, in Recent Developments in Gauge Theories, NATO ASI Ser. C 59 (1980). [172] T. Appelquist and J. Carazzone, Infrared Singularities and Massive Fields, Phys. Rev. D 11, 2856 (1975). [173] B. Kostant, Quantization and Unitary Representations, in Lectures in Modern Analysis and Applications III, Lecture Notes in Math. 170, 87 (1970), Springer. [174] J.-M. Souriau, Structure des Systèmes Dynamiques, Dunod (1970). [175] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser (1993). [176] M. K. Murray, Bundle Gerbes, J. London Math. Soc. 54, 403 (1996). [177] J. C. Baez and A. S. Crans, Higher-Dimensional Algebra VI: Lie 2-Algebras, Theory Appl. Categ. 12 , 492 (2004). [178] J. C. Baez and U. Schreiber, Higher Gauge Theory: 2-Connections on 2-Bundles, arXiv:hep-th/0412325; revised 2007. [179] K. Waldorf, Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps, Cah. Topol. Géom. Différ. Catég. 51, 162 (2010). [180] A. L. Carey, M. K. Murray, and D. Stevenson, Bundle Gerbes Applied to Quantum Field Theory, Rev. Math. Phys. 22, 1113 (2010). [181] U. Bunke and T. Schick, Smooth K-Theory, Astérisque 328, 45 (2009); see also arXiv:0901.4423. [182] H. B. Nielsen and P. Olesen, Vortex-Line Models for Dual Strings, Nucl. Phys. B 61, 45 (1973).
88 [183] G. ’t Hooft, Magnetic Monopoles in Unified Gauge Theories, Nucl. Phys. B 79, 276 (1974). [184] A. M. Polyakov, Particle Spectrum in Quantum Field Theory, JETP Lett. 20, 194 (1974). [185] P. Goddard, J. Nuyts, and D. I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 , 1 (1977). [186] K. G. Wilson, Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture, Phys. Rev. B 4, 3174 (1971). [187] K. G. Wilson and J. Kogut, The Renormalization Group and the Expansion, Phys. Rept. 12, 75 (1974). [188] L. P. Kadanoff, Scaling Laws for Ising Models Near Tc,Physics 2, 263 (1966). [189] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996). [190] Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, J. High Energy Phys. 2011, 099 (2011). [191] T. Banks and A. Zaks, On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions, Nucl. Phys. B 196, 189 (1982). [192] N. Seiberg, Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories, Nucl. Phys. B 435 , 129 (1995). [193] N. Seiberg, Exact Results on the Space of Vacua of Four-Dimensional SUSY Gauge Theories, Phys. Rev. D 49, 6857 (1994). [194] A. Karch and D. Tong, Particle–Vortex Duality from 3D Bosonization, Phys. Rev. X 6, 031043 (2016). [195] N. Seiberg, T. Senthil, C. Wang, and E. Witten, A Duality Web in 2 + 1Dimensions and Condensed Matter Physics, Ann. Phys. 374, 395 (2016). [196] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Deconfined Quantum Critical Points, Science 303, 1490 (2004). [197] T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Quantum Criticality Beyond the Landau–Ginzburg–Wilson Paradigm, Phys. Rev. B 70, 144407 (2004). [198] M. Z. Hasan and C. L. Kane, Colloquium: Topological Insulators, Rev. Mod. Phys. 82, 3045 (2010). [199] X.-L. Qi and S.-C. Zhang, Topological Insulators and Superconductors, Rev. Mod. Phys. 83, 1057 (2011). [200] T. L. Hughes, E. Prodan, and B. A. Bernevig, Inversion-Symmetric Topological Insulators, Phys. Rev. B 83, 245132 (2011). [201] A. M. Essin, J. E. Moore, and D. Vanderbilt, Magnetoelectric Polarizability and Axion Electrodynamics in Crystalline Insulators, Phys. Rev. Lett. 102, 146805 (2009). [202] S. Coh, D. Vanderbilt, A. Malashevich, and I. Souza, Chern-Simons Orbital Magnetoelectric Coupling in Generic Insulators, Phys. Rev. B 83, 085108 (2011). [203] R. S. K. Mong, A. M. Essin, and J. E. Moore, Antiferromagnetic Topological Insulators, Phys. Rev. B 81 , 245209 (2010). [204] J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, Topological Quantization in Units of the Fine Structure Constant, Phys. Rev. Lett. 105, 166803 (2010). [205] W.-K. Tse and A. H. MacDonald, Giant Magneto-Optical Kerr Effect and Universal Faraday Effect in Thin-Film Topological Insulators, Phys. Rev. Lett. 105, 057401 (2010). [206] K. Nomura, S. Ryu, A. Furusaki, and N. Nagaosa, Cross-Correlated Responses of Topological Superconductors and Superfluids, Phys. Rev. Lett. 108, 026802 (2012). [207] M. Stone, Gravitational Anomalies and Thermal Hall Effect in Topological Insulators, Phys. Rev. B 85 , 184503 (2012). [208] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser (1993). [209] X.-G. Wen, Quantum Orders and Symmetric Spin Liquids, Phys. Rev. B 65, 165113 (2002). [210] T. Senthil and M. P. A. Fisher, Z2 Gauge Theory of Electron Fractionalization in Strongly Correlated Systems, Phys. Rev. B 62, 7850 (2000). [211] L. Savary and L. Balents, Quantum Spin Liquids: A Review, Rep. Prog. Phys. 80, 016502 (2016). [212] Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum Spin Liquid States, Rev. Mod. Phys. 89, 025003 (2017). [213] M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N. Nagaosa, and X.-G. Wen, Stability of U (1) Spin Liquids in Two Dimensions, Phys. Rev. B 70, 214437 (2004). [214] T. H. Hansson, V. Oganesyan, and S. L. Sondhi, Superconductors Are Topologically Ordered, Annals of Physics 313, 497 (2004). [215] G. Pöschl and E. Teller, Bemerkungen zur Quantenmechanik des anharmonischen Oszillators, Z. Phys. 83 , 143 (1933). [216] T.-H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm, and Y. S. Lee, Fractionalized Excitations in the Spin-Liquid State of a Kagome-Lattice Antiferromagnet, Nature 492 , 406 (2012). [217] C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quantum Spin Liquids, Science 367, eaay0668 (2020). [218] H. Kleinert, Gauge Fields in Condensed Matter, Vol. II: Stresses and Defects, World Scientific (1989). [219] M. O. Katanaev and I. V. Volovich, Theory of Defects in Solids and Three-Dimensional Gravity, Ann. Phys. 216, 1 (1992). [220] A. J. Beekman, J. Nissinen, K. Wu, and J. Zaanen, Dual Gauge Field Theory of Quantum Liquid Crystals in 2D and 3D, Phys. Rep. 683, 1 (2017). [221] M. Pretko, Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids, Phys. Rev. B 95 , 115139 (2017). [222] M. Pretko and L. Radzihovsky, Fracton-Elasticity Duality, Phys. Rev. Lett. 120, 195301 (2018).
89 [223] N. Seiberg and S.-H. Shao, Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory, SciPost Phys. 10, 027 (2021). [224] R. N. Gurzhi, Hydrodynamic Effects in Solids at Low Temperature, Sov. Phys. Usp. 11, 255 (1968). [225] M. J. M. de Jong and L. W. Molenkamp, Hydrodynamic Electron Flow in High-Mobility Wires, Phys. Rev. B51, 13389 (1995). [226] L. Levitov and G. Falkovich, Electron Viscosity, Current Vortices and Negative Nonlocal Resistance in Graphene, Nat. Phys. 12, 672 (2016). [227] D. A. Bandurin et al., Negative Local Resistance Caused by Viscous Electron Backflow in Graphene, Science 351, 1055 (2016). [228] J. Crossno et al., Observation of the Dirac Fluid and the Breakdown of the Wiedemann–Franz Law in Graphene, Science 351, 1058 (2016). [229] P. J. W. Moll, P. Kushwaha, N. Nandi, B. Schmidt, and A. P. Mackenzie, Evidence for Hydrodynamic Electron Flow in PdCoO2,Science 351, 1061 (2016). [230] A. Lucas and K. C. Fong, Hydrodynamics of Electrons in Graphene, J. Phys.: Condens. Matter 30 , 053001 (2018). [231] B. N. Narozhny and A. Levchenko, Hydrodynamics of Electrons in Metals, Rev. Mod. Phys. 91 , 015006 (2019). [232] J. A. Sulpizio et al., Visualizing Poiseuille Flow of Hydrodynamic Electrons, Nature 576, 75 (2019). [233] M. J. H. Ku et al., Imaging Viscous Flow of the Dirac Fluid in Graphene, Nature 583, 537 (2020). [234] D. J. Thouless, Quantization of Particle Transport, Phys. Rev. B 27, 6083 (1983). [235] A. N. Hirani, Discrete Exterior Calculus, Ph.D. thesis, California Institute of Technology (2003). [236] M. Desbrun, A. N. Hirani, M. Leok, and J. E. Marsden, Discrete Exterior Calculus, arXiv:math/0508341 (2005). [237] H. Whitney, Geometric Integration Theory, Princeton University Press (1957). [238] A. Bossavit, Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements, Academic Press (1998). [239] F. Girelli and H. Pfeiffer, Higher Gauge Theory—Differential Geometry and Applications to Physics, J. Math. Phys. 45, 3949 (2004). [240] J. Faria Martins and R. Picken, Surface Holonomy for Non-Abelian 2-Bundles via Double Groupoids, Adv. Math. 226, 3309 (2011). [241] M. Mackaay, Spherical 2-Categories and 4-Manifold Invariants, Adv. Math. 143, 288 (1999). [242] D. N. Yetter, TQFTs from Homotopy 2-Types, J. Knot Theory Ramif. 2, 113 (1993). [243] M. A. Levin and X.-G. Wen, String-Net Condensation: A Physical Mechanism for Topological Phases, Phys. Rev. B 71, 045110 (2005). [244] K. Walker and Z. Wang, (3+1)-TQFTs and Topological Insulators, Front. Phys. 7, 150 (2012). [245] K. G. Wilson, Confinement of Quarks, Phys. Rev. D 10, 2445 (1974). [246] A. D. Kennedy and B. J. Pendleton, Improved Heatbath Method for Monte Carlo Calculations in Lattice Gauge Theories, Phys. Lett. B 156, 393 (1985). [247] N. Cabibbo and E. Marinari, A New Method for Updating SU ( N )Matrices in Computer Simulations of Gauge Theories, Phys. Lett. B 119, 387 (1982). [248] F. R. Brown and T. J. Woch, Overrelaxed Heat-Bath and Metropolis Algorithms for Accelerating Pure Gauge Monte Carlo Calculations, Phys. Rev. Lett. 58, 2394 (1987). [249] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of State Calculations by Fast Computing Machines, J. Chem. Phys. 21, 1087 (1953). [250] S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195 , 216 (1987). [251] N. V. Prokof’ev and B. V. Svistunov, Worm Algorithms for Classical Statistical Models, Phys. Rev. Lett. 87, 160601 (2001). [252] U. Gattringer and T. Kloiber, Lattice Study of the Silver Blaze Phenomenon for a Charged Scalar φ4 Field, Nucl. Phys. B 869, 56 (2013). [253] M. Girolami and B. Calderhead, Riemann Manifold Langevin and Hamiltonian Monte Carlo Methods, J. R. Stat. Soc. B 73, 123 (2011). [254] K. Hukushima and K. Nemoto, Exchange Monte Carlo Method and Application to Spin Glass Simulations, J. Phys. Soc. Jpn. 65, 1604 (1996). [255] A. M. Ferrenberg and R. H. Swendsen, Optimized Monte Carlo Data Analysis, Phys. Rev. Lett. 63 , 1195 (1989). [256] K. Binder, Finite Size Scaling Analysis of Ising Model Block Distribution Functions, Z. Phys. B 43 , 119 (1981). [257] M. N. Barber, Finite-Size Scaling, in Phase Transitions and Critical Phenomena, Vol. 8, eds. C. Domb and J. L. Lebowitz, Academic Press (1983). [258] U. Wolff, Monte Carlo Errors with Less Errors, Comput. Phys. Commun. 156, 143 (2004). [259] A. D. Sokal, Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms, in Functional Integration, eds. C. DeWitt-Morette et al., Springer (1997). [260] G. L. Squires, Introduction to the Theory of Thermal Neutron Scattering, 3rd ed., Cambridge University Press (2012). [261] S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Vols. 1–2, Clarendon Press (1984).