Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry

TL;DR

The paper extends lattice gauge theory to higher gauge theory on 2-lattices by using crossed modules (2-groups) and fake-flat 2-gauge configurations to define discrete 2D surface holonomy. It constructs the higher Kitaev model as a solvable (3+1)-dimensional Hamiltonian with vertex, edge, and blob operators, and proves that the ground-state degeneracy is a topological invariant equal to the number of homotopy classes of maps M→B_G, connecting to Yetter-type TQFTs. It provides both algebraic-topological and combinatorial formulations of 2D holonomy, gauge transformations, and their invariance properties, and gives concrete S^3 examples to illustrate the framework. The work demonstrates how higher gauge symmetry yields a Hamiltonian realisation of a 3+1D topological phase and clarifies the relationship with Kitaev's familiar 2D model, enriching the landscape of topological quantum computation and 3+1D TQFTs.

Abstract

Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we will continue the study of Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. In particular, we show that a previously proposed construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in 3+1 dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly {\it combinatorialised} CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group. The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretised 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.

Figures (10)

• Figure 1: The action of an element $\gamma \in \pi_1(Y,C)$, with $\gamma(0)=d$ and $\gamma(1)=c$ on a $\Gamma \in \pi_2(X,Y,c)$.
• Figure 2: A 2-lattice decomposition $L$ of the 2-sphere $S^2$.
• Figure 3: The CW-decomposition ${Z_{P}}$ of the 1-sphere $S^1$.
• Figure 4: A 2-lattice decomposition $L$ of $D^2$, where ${Z_{P}}$ is the corresponding CW-decomposition of $S^1$: cf. Def, \ref{['2-lattice']}. The base-point $x_P$ of $P$ is $v_1$. We also show a fake-flat 2-gauge configuration in $(D^2,L)$.
• Figure 5: A 2-lattice decomposition of $(\Sigma,\mathrm{bd}(\Sigma))\cong(D^2,S^1)$. As shown, the attaching maps of the plaquettes $P_1$ and $P_3$ are oriented counterclockwise, whereas $P_2$ attaches clockwise. The base point of $P_i$ is $x_{P_i}.$ The quantised boundaries $\partial_L^Q(P_i)$ of the plaquettes $P_i,i=1,2,3$ are also shown Def. \ref{['qbp']}. The remaining information in the figure will be explained in Def. \ref{['pb']} and Ex. \ref{['patt']}.

Theorems & Definitions (136)

• Remark 1
• Definition 2: Crossed modules of groups; Peiffer relations
• Example 3
• Example 4: From groups to crossed modules I
• Example 5: From groups to crossed modules II
• Definition 6: Crossed module of groupoids
• Definition 7: Crossed module map
• Definition 8
• Remark 9
• Definition 10: The totally intransitive groupoid $\pi_2(X,Y,C)$