From Diagrams to Geometry: An Operator-Theoretic Construction of Gauge-Invariant Quantum Sectors

! 1! From Diagrams to Geometry: An Operator-Theoretic Construction of GaugeInvariant Quantum Sectors Borros Arneth Philipps University Marburg and Justus Liebig University Giessen, Germany [email protected] Abstract We present a comprehensive operator-theoretic framework in which quantum states of gauge theories are represented by equivalence classes of combinatorial–topological diagrams. The construction begins with a diagrammatic Hilbert space whose basis elements encode connectivity patterns, representation labels, and topological invariants such as genus, linking, and center flux. We introduce a complete, rigorously defined operator algebra acting on this space, including local connectivity-changing vertex operators, topological shift operators, a positive self-adjoint complexity operator, and a triplet of commuting projection operators imposing gauge invariance, topological consistency, and scale-dependent entropic stationarity. A formally defined coarse-graining map assigns to each diagram class a scale-dependent multiplicity of microstates, leading to an entropy functional whose Hessian defines an emergent Riemannian metric on the manifold of coarse-grained sectors. We compute explicit examples illustrating how entropic curvature arises from topological fluctuations. The spectrum of the complexity operator yields a mass hierarchy without introducing additional degrees of freedom, while renormalization-group flow emerges naturally from the scale dependence of entropic curvatures. This formalism provides a unified, mathematically controlled setting in which nonperturbative structures, gauge constraints, and emergent geometric behavior can be studied simultaneously, offering a complementary approach to lattice gauge theory, tensor-network methods, and topological quantum field theory. 1. Introduction Quantum gauge theories possess a rich array of nonperturbative structures that fall outside the reach of naive perturbation theory. Seminal discoveries such as instantons [1– 3], monopoles [4,5], center vortices [6–9], and knotted or linked Wilson-loop sectors [10,11] demonstrate that topological features play a decisive role in phenomena including confinement, chiral symmetry breaking, and vacuum structure. These configurations are ! 2! inherently combinatorial and often encoded naturally in diagrammatic objects capturing connectivity, linking, color flow, and topological invariants. At the same time, major developments in quantum information and quantum gravity have revealed that geometric behavior can emerge from more microscopic structures. Entanglement and tensor-network methods [12–14], information geometry [15], and holographic insights relating geometry to coarse-grained information [16–18] all point toward deep connections between combinatorial data and geometric structure. Similar trends appear in spinfoams [19], causal dynamical triangulations [20], and random-tensor models [21], which show that geometric degrees of freedom can arise from algebraic or combinatorial building blocks. The goal of this work is to unify these perspectives by constructing a diagrammatic Hilbert space, a mathematically well-defined setting in which diagrams encode the combinatorial and topological content of quantum gauge sectors. Each diagram represents equivalence classes of local connectivity patterns, representation assignments, genus contributions, linking structures, and topological indices such as instanton number or center flux. This yields a countable, separable Hilbert space whose basis elements have explicit physical interpretation and whose inner product is regulated by a weight functional suppressing extreme topological growth. Upon this Hilbert space acts a rigorously defined operator algebra. Vertex operators model local interaction rules familiar from the Standard Model and non-Abelian gauge theory [22,23]. Topological shift operators implement changes in quantized invariants, echoing structures known from instanton calculus [1–3], center-vortex models [6–9], and lattice topology [24]. A positive self-adjoint complexity operator, inspired by ideas of computational and topological complexity in quantum systems [25–27], assigns a discrete eigenvalue to each diagram, allowing mass-like quantities to emerge from intrinsic combinatorial structure. A triplet of projection operators implements three fundamental constraints: gauge invariance; consistency of topological sectors, including anomaly constraints [28,29]; and scale-dependent entropy stationarity under coarse-graining. A central conceptual structure in this framework is the coarse-graining map, which assigns to each diagram class a scale-dependent number of compatible microdiagrams. The multiplicity defines an entropy functional whose Hessian generates a Riemannian metric on the manifold of coarse-grained sectors. This construction is mathematically analogous to the appearance of Fisher metrics in information geometry [15] and the entanglement-induced metrics in holography [16–18]. Explicit examples show how diagrammatic fluctuations generate curvature, providing a geometric representation of nonperturbative effects. Additionally, the framework yields physical consequences. The eigenvalue spectrum of the complexity operator gives rise to hierarchical masses without introducing additional quantum fields, resonating with ideas from spectral geometry and complexity-based dynamics [26,27]. The scale dependence of entropic curvature yields an RG-like flow whose functional form reproduces characteristic ! 3! features of non-Abelian running couplings [22,23]. The operator algebra embeds the Standard Model gauge factors at a structural level, while the projection operators isolate gauge-invariant, anomaly-free, topologically stable states. This framework is not intended as a replacement for continuum gauge theory, lattice QCD, or holography, but as a complementary operator-theoretic approach that brings together diagrams, topology, coarse-graining, and geometry in a unified and conceptually transparent way. The resulting structure is flexible enough to interface with lattice methods [24], categorical and topological approaches [10,11], and entanglement-driven geometric constructions [12–18], while maintaining a mathematically precise operatoralgebraic foundation suitable for Annals of Physics. 2. The Diagrammatic Hilbert Space Quantum states in this framework are represented by combinatorial–topological diagrams that encode connectivity, representation labels, and topological invariants. The resulting Hilbert space is countable, separable, and equipped with a rigorously controlled inner product. This section defines the diagrammatic objects, their equivalence classes, and the analytic structure of the Hilbert space. 2.1 Diagrammatic objects A diagram is defined as a finite combinatorial structure Γ=(𝑉,𝐸;𝒞,𝒯) where: • 𝑉 is a finite set of vertices, • 𝐸 is a finite set of oriented edges, • 𝒞:𝐸 →ℛ assigns representation labels drawn from a set ℛ (e.g. SU(3), SU(2), and U(1) representations [22,23]), • 𝒯 is a collection of topological invariants characterizing the global structure of Γ. The topological invariants include: • Genus 𝑔(Γ), • Instanton number 𝑄(Γ) [1–3], • Total linking number 𝐿(Γ) [10,11], • Center flux Φ(Γ)∈𝑍(𝑆𝑈(3))=ℤ! [6–9], • Braid index 𝑏(Γ), which encodes chiral orientation in diagrammatic strands, ! 4! • Local combinatorial complexity 𝐶loc(Γ), counting branching and valence patterns. The set of diagrams is denoted by 𝒟:={Γ finite} Each diagram should be understood as a discrete representation of a gauge-field configuration’s connectivity and topological sector, analogous to Wilson graph networks [10], spinfoam complexes [19], Berezin-type combinatorial structures [21], or latticegenerated topological clusters [24]. 2.2 Equivalence relations and topological classes Two diagrams Γ,Γ"∈𝒟 are considered equivalent if they share the same topological and representation data: Γ∼Γ"⟺𝒞(Γ)=𝒞(Γ") and 𝒯(Γ)=𝒯(Γ") and if one can be transformed into the other via a finite sequence of topology-preserving deformations, including: • graph isomorphisms, • smooth homotopies, • Reidemeister moves for braid segments [10], • gauge-equivalent rewirings of color flow in SU(N) [22,23]. This structure is analogous to the classification of configurations in topological quantum field theory [10,11], knot theory applied to gauge fields [11], and the equivalence classes appearing in lattice topological sectors [24]. The equivalence classes form the quotient: [Γ]∈𝒟/ ∼ The quotient set is countable, since every diagram is finite and finitely labeled. ! 5! 2.3 Hilbert-space completion and inner product Let ℬ ={ ∣ [Γ] ⟩:[Γ]∈𝒟/ ∼ } be a set of abstract orthonormal basis vectors. Define the pre-Hilbert space ℋ#={F𝑐$ % $&' ∣[Γ$]⟩ : 𝑁 <∞, KKK 𝑐$∈ℂ} To ensure finite norms and analytic control, we introduce a weight functional 𝑤([Γ])>0 chosen to suppress high topological and combinatorial complexity. A convenient and physically motivated choice is: 𝑤([Γ])=expK [−𝛼𝑔(Γ)−𝛽 ∣𝑄(Γ)∣−𝛾𝐿(Γ)−𝛿 ∣Φ(Γ)∣−𝜂𝐶loc(Γ)] where 𝛼,𝛽,𝛾,𝛿,𝜂 >0. Such exponential suppressions appear in several contexts: • large-topology suppression in random-tensor models [21], • genus-weighted configurations in matrix models [21], • exponential weighting of instantons and topological charges [1–3], • suppression of extended vortex configurations in SU(3) gauge theory [6–9]. The inner product on basis elements is: ⟨[Γ]∣[Γ"]⟩=𝑤([Γ]) 𝛿()],[)!- The Hilbert space is the norm completion: ℋ =ℋ# ‾∥⋅∥ Because 𝑤([Γ])>0 and 𝒟/ ∼ is countable, ℋ is separable. ! 6! This structure resembles weighted Hilbert spaces used in: • spin-network quantization [19], • diagrammatic expansions of strongly coupled gauge theories [24], • information-theoretic representations of quantum states [12–15]. 2.4 Operator domains and essential self-adjointness The operator algebra introduced later relies on the analytic properties of operators on ℋ. We begin by defining: 𝒟#:=ℋ# which is dense in ℋ. Every operator 𝑂 discussed in this work satisfies: (i) Locality in diagram space 𝑂 modifies only a finite substructure of a diagram: 𝑂 ∣[Γ]⟩=F𝑐0 1 0&' ∣[Γ0]⟩, 𝑀 <∞ This reflects the locality of gauge interactions and topological transitions [22–24]. (ii) Bounded shifts of topological invariants For all generated diagrams Γ0, ∣𝑔(Γ0)−𝑔(Γ)∣≤1, ∣𝑄(Γ0)−𝑄(Γ)∣≤1 and similarly for Φ and 𝐿. Such bounded transitions are characteristic of instanton steps [1–3], vortex branching [6– 9], and monopole moves [4,5]. (iii) Weight control There exists 𝐾2>0 such that: ! 7! 𝑤([Γ0])≤𝐾2 𝑤([Γ]) This ensures that 𝑂 maps Cauchy sequences to Cauchy sequences. Lemma 2.1 (Closability) Every operator satisfying (i)–(iii) is closable on 𝒟#. Lemma 2.2 (Essential self-adjointness) Operators that are diagonal in ∣[Γ]⟩ or generate only finitely many shifts are essentially self-adjoint. This includes: • the complexity operator, • projection operators, • topological shift operators, • the entropy-flow operator defined later. These analytic properties mirror those found in lattice Hamiltonians [24], TQFT operator representations [10,11], and operator-algebraic QFT constructions [22]. 3. Operator Algebra The dynamics and structural transformations of diagrams are encoded in a set of linear operators acting on the diagrammatic Hilbert space ℋ. These operators model local interaction processes, topological transitions, combinatorial complexity, and the physical constraints imposed by gauge invariance, topological consistency, and entropic stability. The operators introduced here form a closed algebra under adjoint, composition, and linear combination, and are defined on the dense subspace 𝒟#⊂ℋ. Their analytic properties follow from locality, bounded topological shifts, and weight control, discussed in Section 2.4. ! 8! 3.1 Vertex operators Vertex operators model elementary local modifications of diagrams corresponding to gauge-theoretic interactions. For each admissible local interaction pattern 𝑎, we define an operator 𝑉3:ℋ →ℋ which acts as 𝑉3∣[Γ]⟩= F 𝑐3,0 1(3,)) 0&' ∣[Γ3,0]⟩, 𝑀(𝑎,Γ)<∞ The transformations Γ→Γ3,0 include: • edge mergers and splits representing 3and 4-gluon vertices in SU(3) gauge theory [22], • color-line insertions representing quark–gluon couplings [22,23], • SU(2) weak-interaction vertices and U(1) hypercharge flows [23], • Yukawa-type diagrammatic contractions, • neutral diagrammatic transformations respecting representation flow. These operators mirror the role of elementary vertices in lattice and diagrammatic formulations of gauge theory [24], and they resemble the local moves in spinfoam amplitudes [19] and tensor-network evolution maps [12–14]. Locality ensures that vertex operators modify only a small, bounded region of the diagram, preserving weight-control and essential self-adjointness (Lemmas 2.1 and 2.2). 3.2 Topological shift operators Topological sectors play a central role in gauge dynamics. The operator algebra includes shift operators that change quantized invariants in a controlled, finite manner. Instanton-number shift operator 𝑇6∣[Γ]⟩=∣[Γ"]⟩,𝑄(Γ")=𝑄(Γ)+1 This models a unit change of instanton number, consistent with instanton-based transitions in SU(2) and SU(3) Yang–Mills theory [1–3]. ! 9! Center-flux shift operator 𝑇7∣[Γ]⟩=∣[Γ"]⟩,Φ(Γ")=Φ(Γ)+𝑧 with 𝑧∈𝑍(𝑆𝑈(3))=ℤ!. Such transitions reflect center-vortex dynamics known to underlie confinement in SU(3) [6–9]. Linking-number shift operator 𝑇8∣[Γ]⟩=∣[Γ"]⟩, 𝐿(Γ")=𝐿(Γ)+1 This corresponds to controlled changes in the linking structure of flux tubes or Wilson loops [10,11]. Genus shift operator 𝑇9∣[Γ]⟩=∣[Γ"]⟩, 𝑔(Γ")=𝑔(Γ)+1 This operator resembles expansion steps in random-tensor models [21] and topological expansions in string-inspired gauge formulations [26]. Each operator modifies only a finite region of the diagram, ensuring that ∣𝑇:[Γ]⟩ remains within the weight-controlled region of ℋ. 3.3 The complexity operator We define a positive, self-adjoint operator 𝐶 acting diagonally on the basis: 𝐶 ∣[Γ]⟩=𝐶) ∣[Γ]⟩,KKKK𝐶)∈[0,∞) The complexity value 𝐶) is constructed from a combination of topological and combinatorial data: 𝐶)=𝑎'𝑔(Γ)+𝑎;∣𝑄(Γ)∣+𝑎!𝐿(Γ)+𝑎<∣Φ(Γ)∣+𝑎=𝐶>?@(Γ),𝑎$>0 This operator quantifies the internal structural richness of a diagram and plays a role analogous to: ! 16! 4.5 Worked example: explicit metric and curvature To illustrate these constructions, consider a coarse-grained diagram class characterized by two parameters: 𝑥'=𝜆,KK𝑥;=𝜉 representing, respectively: • the coarse-graining scale, • a linking-density-like variable. Let the entropy functional take the form: 𝑆(𝜆,𝜉)=𝑎𝜆;+𝑏𝜉;+𝑐𝜆𝜉 with 𝑎,𝑏 > 0 and 𝑐 ∈ℝ. Such quadratic forms arise in Gaussian approximations of topological fluctuations [2,7,10]. Compute first derivatives: ∂D𝑆 =2𝑎𝜆+𝑐𝜉,∂Q𝑆=2𝑏𝜉+𝑐𝜆 The Hessian is: 𝑔3K =∂3∂K𝑆 =•2𝑎 𝑐 𝑐 2𝑏€ The determinant is: detK𝑔=4𝑎𝑏−𝑐; positive whenever 4𝑎𝑏 >𝑐;. Thus the entropy metric is positive-definite. If 𝑎,𝑏,𝑐 are constants, the metric is flat: ! 17! 𝑅 =0 To induce curvature, let the coefficient 𝑎 depend on 𝜆: 𝑎(𝜆)=𝑎#+𝑎'𝜆.(𝑎'≠0) Then: 𝑔DD =2𝑎#+4𝑎'𝜆,𝑔DQ =𝑐,𝑔QQ =2𝑏 A direct computation yields: 𝑅(𝜆)=− 𝑎' ; (4𝑎𝑏−𝑐;)(𝑎#+2𝑎'𝜆); This curvature arises solely from the scale dependence of the entropy functional, illustrating how diagrammatic fluctuations and coarse-graining induce geometric structure. The behavior is analogous to curvature generated in entanglement-driven geometry [16– 18] and in coarse-grained spinfoam dynamics [19]. 5. Complexity Spectrum and Physical Quantities The complexity operator 𝐶, defined in Section 3.3, provides a quantized measure of the structural richness of diagrams, combining topological, combinatorial, and representation-theoretic contributions. Its eigenvalues define a natural hierarchy of scales. Let 𝐶 ∣𝜓J⟩=𝑐J∣𝜓J⟩,𝑐#≤𝑐'≤𝑐;≤⋯ The eigenfunctions ∣𝜓J⟩ are linear combinations of diagrammatic classes with equal effective complexity. Physically, the spectrum can be interpreted in several ways: ! 18! 1. Mass-like quantities: In analogy with spectral geometry and operator-based mass generation [25–27], define 𝑚J=Λ𝑐J where Λ is a characteristic scale (e.g. confinement scale, electroweak scale, or a diagrammatic UV cutoff). The hierarchical structure of 𝑐J naturally generates hierarchical masses. 2. Complexity-driven excitation layers: Higher complexity diagrams correspond to states with more topological richness (e.g. additional loops, linking, instanton content), reminiscent of bound-state excitations in strongly coupled gauge theories [2,7,24]. 3. Topological stability: Projection by Πtop and Πent removes complexity eigenstates that are topologically anomalous or unstable under coarse-graining. 4. Spectrum finiteness: Because high-complexity diagrams are exponentially suppressed by the weight 𝑤(Γ), the effective spectrum contributing to ℋphys remains manageable even though 𝐶 has infinite discrete eigenvalues. 5.1 Two-sector example Consider two coarse-grained classes with complexity values: 𝐶)"=1,KKKK𝐶)#=4 Then: 𝑚R=Λ,KKKKK𝑚S=4Λ This simple example illustrates how structural complexity produces hierarchical scales without additional fields or couplings—consistent with operator-algebraic views of emergent mass generation [25–27]. 6. Scale Dependence and Entropic Renormalization Flow The renormalization-group (RG) flow emerges naturally from the entropy’s scale dependence. We use the entropic curvature: ! 19! 𝜅(𝜇)=∂(gauge) ;𝑆([Γ],𝜇) to encode how diagrammatic multiplicities respond to scale changes. We identify the inverse gauge coupling with the curvature: 𝑔I;(𝜇)=𝐴𝜅(𝜇) where 𝐴>0 is a normalization constant set by physical units. Differentiating, 𝜇𝑑 𝑑𝜇𝑔I;(𝜇)=𝐴 𝜇𝑑𝜅 𝑑𝜇 Using the RG identity, 𝜇𝑑 𝑑𝜇𝑔I; =−2𝑔I!𝛽(𝑔) we obtain: 𝛽(𝑔)=−𝐴 2𝑔!𝜇𝑑𝜅 𝑑𝜇 This demonstrates that the sign of the β-function is determined by whether entropic curvature increases or decreases with scale. 6.1 Example: logarithmic curvature Assume 𝜅(𝜇)=𝐵#+𝐵'lnK𝜇 𝜇#, 𝐵'>0 Then: 𝛽(𝑔)=−𝐴𝐵' 2𝑔! ! 20! This reproduces the qualitative behavior of asymptotic freedom in non-Abelian gauge theories [22,23], but derived here from coarse-graining of diagrammatic multiplicities. Such entropy-based flow equations echo developments in: • information-theoretic RG [15], • tensor network renormalization [12–14], • nonperturbative gauge-theory analyses [7,24]. 7. Discussion and Outlook We have developed a unified operator-theoretic framework in which diagrams serve as the fundamental microscopic objects representing quantum gauge configurations. The resulting Hilbert space is equipped with a rigorously defined operator algebra capturing: • local gauge interactions via vertex operators, • topological transitions through shift operators, • intrinsic structural richness via the complexity operator, • physical constraints via mutually commuting projection operators. Coarse-graining leads to a well-defined entropy functional whose Hessian induces an emergent Riemannian metric on the space of diagrammatic sectors. Explicit examples reveal how curvature is generated by topological fluctuations, drawing deep parallels with the emergence of geometry in entanglement-based and holographic frameworks [16–18]. The framework is broad enough to unify ideas from: • lattice gauge theory [24], • instanton and vortex physics [1–3,6–9], • knot-theoretic gauge formulations [10,11], • spinfoam and CDT approaches [19,20], • tensor-network and information-theoretic perspectives [12–15], • operator-algebraic quantum field theory [22,23], • complexity-based quantum dynamics [25–27]. Promising directions include: 1. Entropic gravity-like extensions: The entropic metric may couple to diagrammatic degrees of freedom, potentially leading to effective Einstein-like equations. ! 21! 2. Integration with tensor networks: Identifying diagram classes with tensor-network coarse-graining layers. 3. 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