--- title: Homotopy Lattice Gauge Fields url: https://www.emergentmind.com/topics/homotopy-lattice-gauge-fields-hlgfs type: topic --- # Homotopy Lattice Gauge Fields Homotopy Lattice Gauge Fields (HLGFs) are homotopy-enriched discretizations of gauge fields in which ordinary lattice parallel transport is supplemented by data attached to homotopies of paths, higher cells, or relative homotopy classes, so that discretization retains information normally lost in standard lattice gauge theory, notably principal-bundle topology, higher holonomy, and topological-sector structure at finite cutoff. In the literature, the term encompasses several closely related constructions: extended lattice gauge fields built from path groupoids and cotriangulations, higher lattice gauge fields valued in crossed modules or strict 2-groups, strict higher-groupoid formulations based on cubical or globular homotopy groupoids, and lattice field spaces explicitly augmented by local homotopy variables [1701.00775] [1606.06639] [2311.02363] [2105.14571] [2603.19525]. ## 1. Historical emergence and conceptual scope A central motivation for HLGFs is the observation that the usual lattice truncation keeps parallel transport along discrete paths but generally loses the information needed to reconstruct the topology of the underlying principal bundle. One early resolution is the extended lattice gauge (ELG) field of Meneses and Zapata, where a lattice gauge field is enlarged by relative homotopy data of path families subordinate to a cotriangulation. In that framework, a standard lattice gauge field is only part of the data, while the additional local homotopy classes are sufficient to recover the bundle class and its Čech cocycle data [1701.00775]. A second lineage comes from higher gauge theory. Here the basic discretization does not stop at edge holonomies: faces carry 2-holonomy, and higher flatness constraints are imposed on 3-cells. Finite strict 2-groups, encoded as crossed modules, supply the algebraic target. This viewpoint underlies Yetter-type topological quantum field theories, higher Kitaev models, and Yang–Mills-like lattice 2-group models in which links and faces both carry dynamical degrees of freedom [1606.06639] [1702.00868] [2109.12097] [1802.01139]. A third development formulates HLGFs directly as strict morphisms from higher homotopy groupoids of a lattice into higher group objects. In “Higher homotopy and lattice gauge fields,” local lattice higher gauge fields are strict internal morphisms of groupoids in $\omega$-groupoids, and under the correct assumptions the resulting formalism subsumes both extended lattice gauge fields and other higher lattice gauge field constructions [2311.02363]. The 2026 paper “Homotopy lattice gauge fields 1: The fields and their properties” then adopts the name HLGF explicitly for higher parallel transport maps that retain how transport changes under homotopies of curves, with emphasis on bundle reconstruction and topological charge in dimensions two and three [2603.19525]. A related but distinct usage appears in the program of adding explicit homotopy variables to lattice fields. There the space of lattice fields becomes a covering space whose connected components reproduce the connected components of the continuum field space, and local integer-valued topological charges become exact lattice variables rather than continuum-limit artifacts [2105.14571]. ## 2. Foundational mathematical structures Two mathematical ideas recur throughout the subject. The first is that gauge fields may be described by holonomy data, not primarily by local connection 1-forms. The second is that the relevant transport object is not merely a path groupoid, but a higher path or homotopy groupoid. In the holonomy approach, gauge fields are encoded by homomorphisms from a loop or path groupoid modulo thin homotopy or intimacy, subject to smoothness conditions; this clarifies why homotopy relations finer than simple retracing are the correct invariances for holonomy-based gauge theory [1904.10822]. In the ELG framework, smooth parallel transport relative to a cotriangulation $\mathscr C$ is a groupoid homomorphism $$ PT_{\mathscr C}: P_{\mathscr C}\to G $$ defined on a cellular path groupoid, with a smoothness condition on finite-dimensional cellular path families. Two such maps are equivalent up to local gauge transformations constant on fibers above base points. Relative homotopy classes of these transports, fixed on a cellular network $P_\Gamma$, define ELG fields [1701.00775]. In the strict higher-groupoid formulation, the source is the homotopy $\omega$-groupoid of a filtered space. If $X$ is a simplicial set or cell complex, then $\Pi_\omega(X)$ is realized by the homotopy cubical $\omega$-groupoid of the skeletal filtration of $|X|$. A local lattice higher gauge field on a contractible patch is then a strict internal morphism $$ A: P^1\rho(|X|)\longrightarrow BK, $$ or equivalently, in globular form, $$ \hat A:\rho^\circ(|X|)\longrightarrow HK[-1], $$ with $K=\rho(G)$ for a filtered topological group $G$ [2311.02363]. The 2026 HLGF paper sharpens this viewpoint by defining a higher parallel transport map as a section $$ PT:\rho^{1,\bigcirc}X_\ast\to At^{1,\bigcirc}(X_\ast;\rho^\bigcirc G), $$ where $\rho^{1,\bigcirc}X_\ast$ is the globular $\omega$-groupoid of globes of paths and $At^{1,\bigcirc}$ is a higher Atiyah groupoid. In a trivialization over vertices, this is encoded by $$ PT_\phi:\rho^{1,\bigcirc}X_\ast\to \rho^{1,\bigcirc}BG. $$ This formulation makes explicit that HLGFs transport not only points in fibers but relative homotopies of initial conditions to relative homotopies of final conditions [2603.19525]. For higher gauge theory in the 2-group sense, the target is a crossed module $(E\xrightarrow{\partial}G,\triangleright)$ satisfying the Peiffer identities $$ \partial(g\triangleright e)=g\,\partial(e)\,g^{-1},\qquad \partial(e)\triangleright f=e\,f\,e^{-1}. $$ These relations are the algebraic expression of compatibility between 1- and 2-holonomy [1606.06639] [1702.00868] [2109.12097] [1802.01139]. ## 3. Extended lattice gauge fields and reconstruction of bundles The ELG construction gives one of the clearest answers to what extra data must be added to ordinary lattice gauge theory to recover principal bundles. Starting from a cotriangulated manifold $(M,\mathscr C)$, one defines gluing maps on flags of cells by $$ g_{\sigma\tau}(x)=PT_{\mathscr C}([\gamma^x_{\sigma\tau}]), $$ and clutching maps on overlaps of top-dimensional cells by $$ h_{vw}(x)=g_{\tau v}(x)^{-1}g_{\tau w}(x). $$ The usual lattice gauge variables are obtained by restricting these maps to the discrete $0$-cells, while the full ELG field remembers relative homotopy classes of extensions from cell boundaries to interiors [1701.00775]. This extra information is organized by the homotopy groups $\pi_k(G)$. For a $k$-cell, extension classes form torsors under $\pi_k(G)$, subject to boundary constraints and multiplicative compatibility relations. The clutching maps satisfy the standard cocycle condition on triple overlaps. Taken together, these local data determine a Čech $1$-cocycle class, hence a principal $G$-bundle class in $\check H^1(M,G)\cong [M,BG]$ [1701.00775]. The reconstruction theorem states that equivalence classes of smooth parallel transport maps are in bijection with principal $G$-bundles equipped with chosen fiber points over the base points of the cotriangulation and a gauge orbit of smooth connections modulo restricted gauge transformations. The ELG field therefore contains enough information to reconstruct both bundle topology and connection data up to the stated equivalence [1701.00775]. This framework also describes the space of fields. For fixed $(M,\mathscr C,\Gamma)$, the space $\mathcal M_{\mathscr C}$ of ELG fields fibers over the ordinary lattice gauge field space $\mathcal M_\Gamma$, and the forgetful map $$ pr_{\mathscr C}:\mathcal M_{\mathscr C}\to \mathcal M_\Gamma $$ is a covering map. Connected components of $\mathcal M_{\mathscr C}$ parametrize equivalence classes of principal $G$-bundles. The associated deck transformations are built from products of homotopy groups $\pi_k(G)$ with multiplicative and boundary relations [1701.00775]. A common misconception is that such bundle reconstruction is tied to a fixed triangulation. The ELG theory addresses this directly. Using dual Pachner moves and a notion of local relatives, it proves a criterion for when ELG fields over different cotriangulations define equivalent bundles. Bundle classes are thus invariant under refinement and Pachner moves up to local homotopy, even though the auxiliary cellular data themselves depend on the chosen cotriangulation [1701.00775]. ## 4. Higher gauge theory and crossed-module realizations In higher lattice gauge theory, HLGFs are naturally realized as discrete 2-connections. On an oriented lattice one assigns a group element $g_e\in G$ to each oriented edge and an element $e_f\in E$ or $h_f\in H$ to each oriented face. The primary face constraint is fake flatness, $$ \partial(h_f)=g_{\partial f}, $$ which ties 2-holonomy to the boundary 1-holonomy and is the discrete analogue of vanishing fake curvature [1606.06639] [1702.00868] [2109.12097] [1802.01139]. On 3-cells one defines transported products of face variables, yielding a 2-curvature or blob holonomy. In the Hamiltonian higher Kitaev-type models, triviality of this 2-holonomy is imposed by commuting projectors. The resulting ground-state sector is topological: for a closed manifold $M$, $$ \mathrm{GSD}(M;\mathbb G)=|[M,B\mathbb G]|. $$ Thus the ground-state degeneracy equals the number of homotopy classes of maps from $M$ to the classifying space of the underlying crossed module [1702.00868]. The same structure appears in the exactly solvable 3+1-dimensional model “Topological phases from higher gauge symmetry in 3+1D.” There the degrees of freedom are a $G$-valued 1-form on edges and an $E$-valued 2-form on faces, subject to fake-flatness and tetrahedral 2-flatness. The Hamiltonian is a sum of mutually commuting vertex, edge, and blob projectors, and the ground-state projector on $M^3$ equals the Yetter homotopy 2-type TQFT partition function on $M^3\times[0,1]$ [1606.06639]. A more dynamical version appears in the finite 2-group model of Bullivant, Martins, and Pfeiffer. On a 4D hypercubic lattice the partition function is $$ Z=\sum_{\{g_e\in G\},\{h_f\in H\}} \Big[\prod_f \delta_{\partial(h_f),g_{\partial f}}\Big]e^{-S[\{g_e\},\{h_f\}]}, $$ with $$ S[g,h]=J_1\sum_{p}f_1(g_p)+J_2\sum_c f_2(\Omega_c). $$ The limit $J_2\to\infty$ yields Yetter’s TQFT, while finite $J_1,J_2$ produce a Yang–Mills-like regime in which curvature excitations are allowed [2109.12097]. The surface-holonomy side of the theory was given a detailed lattice derivation in “Two-dimensional algebra in lattice gauge theory,” which uses string diagrams and crossed-module 2-groups to define nonabelian surface transport. A convergence theorem proves that the lattice surface-ordered product converges to continuum 2-holonomy and that the full two-dimensional ordering simplifies to a single vertical path ordering, recovering the Schreiber–Waldorf construction [1802.01139]. ## 5. Local-to-global principles and unification A decisive structural result is that HLGFs admit a genuine local-to-global theorem. In “Higher homotopy and lattice gauge fields,” the higher homotopy Seifert–van Kampen theorem is used in cubical and globular form to show that homotopy $\omega$-groupoids preserve suitable pushouts of good covers. Consequently, global HLGFs can be glued uniquely from compatible local HLGFs, and the set of global fields is the limit of the local Hom-sets [2311.02363]. This yields several dimension-dependent identifications. In dimension one, the formalism reduces to ordinary lattice gauge fields: the relevant Hom-set is equivalent to the Hom-set from the free groupoid on the edge set to the delooping groupoid $BG$. In dimension two, local lattice higher gauge fields are equivalent to strict 2-functors into a strict 2-groupoid, reproducing standard crossed-module higher gauge theory. Under the paper’s assumptions, global globular lattice higher gauge fields then restrict canonically to ELGFs, thereby unifying the Meneses–Zapata construction with the higher-groupoid formalism [2311.02363]. The 2026 paper sharpens this unification from the perspective of higher parallel transport. It treats simplices of paths as generators of the relevant globular $\omega$-groupoid and shows that evaluation on these generators produces an ELGF satisfying face-compatibility and interior-extendibility conditions. On bases of dimension two or three, these data determine a principal $G$-bundle over the base manifold. This depends crucially on the fact that for Lie groups $\pi_2(G)=0$, so the corresponding higher obstructions vanish in those dimensions [2603.19525]. A plausible implication is that the various HLGF formalisms differ less in substance than in the choice of source higher groupoid and the packaging of the target data. One formulation emphasizes cotriangulations and relative homotopy classes, another strict $\omega$-groupoid morphisms, and another crossed-module 2-connections, but all are designed to preserve the local-to-global homotopy data discarded by ordinary edge-only discretizations [1701.00775] [2311.02363] [2603.19525]. ## 6. Topological sectors, charges, and observables One of the main reasons to introduce HLGFs is to make topological sector structure exact at finite lattice spacing. In the extended lattice-field approach of “Homotopy data as part of the lattice field,” the field space is enlarged to a covering space $\mathcal M_{EL}\to\mathcal M_L$ whose connected components match the continuum topological sectors. In one-dimensional $U(1)$ sigma model, the additional link variables are real lifts $x_{ij}\in\mathbb R$ of the phase differences, and the exact lattice action and charge are $$ S_{EL}(q^{EL})=\sum_{ij}\frac12\left(\frac{x_{ij}}{\delta_t}\right)^2\delta_t,\qquad Q_{EL}(q^{EL})=\frac{1}{2\pi}\sum_{ij}\left(\frac{x_{ij}}{\delta_t}\right)\delta_t. $$ The integer winding is therefore an exact lattice variable, not a continuum reconstruction [2105.14571]. In two-dimensional $U(1)$ sigma model, the local homotopy variables satisfy a flatness constraint on each plaquette, $$ \sum_{(ij)<\partial P}x_{ij}=0, $$ and on a torus the noncontractible cycles carry exact integer charges $Q_I=\sum_{(ij)