On 2-form gauge models of topological phases

TL;DR

This work develops a comprehensive framework for (3+1)d 2-form topological gauge theories rooted in the classifying space $B^2G$ of a finite abelian group $G$. It bridges continuum and lattice perspectives by proving that discrete 2-form theories embed into strict 2-group gauge theories via Deligne–Beilinson cohomology, and by constructing an exactly solvable lattice Hamiltonian using a 2-form cohomology class in $H^4(G_{[2]},\mathbb{U}(1))$. The paper shows that a 2-form 4-cocycle encodes an abelian braided monoidal category data $(\alpha,R)$, thereby linking the bulk 2-form theory to the Walker–Wang model for $G$-graded vector spaces, with the 2-form cocycle giving the amplitude data for Pachner moves and braidings. This establishes a precise correspondence between higher-form gauge theories and modular/premodular categories, clarifying the role of the Pontrjagin square and quadratic forms in classifying bulk topological order and surface anyon structure. The results pave the way for exploring weak 2-groups, Postnikov-tower generalizations, and higher-form braiding phenomena in both continuum and lattice settings, with potential applications to symmetry-protected and symmetry-enriched topological phases.

Abstract

We explore various aspects of 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space $B^2G$ of the symmetry group $G$, and they are classified by cohomology classes of $B^2G$. Discrete topological gauge theories can typically be embedded into continuous quantum field theories. In the 2-form case, the continuous theory is shown to be a strict 2-group gauge theory. This embedding is studied by carefully constructing the space of $q$-form connections using the technology of Deligne-Beilinson cohomology. The same techniques can then be used to study more general models built from Postnikov towers. For finite symmetry groups, 2-form topological theories have a natural lattice interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d that is exactly solvable. This construction relies on the introduction of a cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified with the simplicial cocycles of $B^2G$ as provided by the so-called $W$-construction of Eilenberg-MacLane spaces. We show algebraically and geometrically how a 2-form 4-cocycle reduces to the associator and the braiding isomorphisms of a premodular category of $G$-graded vector spaces. This is used to show the correspondence between our 2-form gauge model and the Walker-Wang model.

Figures (5)

• Figure 1: Polyhedral decomposition of a 1-cycle $\mathfrak L^{(1)}=\Sigma_{i=1}^{3}\mathfrak l^{(1)}_i$ subordinate to a choice of open cover $\mathcal{U}=\bigcup_{i=1,2,3}U_{i}$. The 1-chains $\mathfrak l^{(1)}_i\in U_{i}$ and 0-chains $\mathfrak l^{(0)}_{ij}\in U_{ij}$.
• Figure 2: Polyhedral decomposition of a 2-cycle $\mathfrak L^{(2)}=\Sigma_{i=1}^{4}\mathfrak l^{(1)}_i$ subordinate to a choice of open cover $\mathcal{U}=\bigcup_{i=1,2,3,4}U_{i}$. The open cover has not been illustrated in the figure above to avoid clutter but it is such that the 2-chains $\mathfrak l^{(1)}_{i}\in U_{i}$, the 1-chains $\mathfrak l^{(2)}_{ij}\in U_{ij}$ and the 0-chains $\mathfrak l^{(0)}_{ijk}\in U_{ijk}$.
• Figure 3: Consistency condition of the 2--3 Pachner move whose amplitude is given by the 2-form 4-cocycle $\omega \in Z^4(G_{[2]},{\rm U}(1))$. Starting from the union of three 3-simplices, there exist two different successions of 2--3 Pachner moves which lead to the same union of six 3-simplices. On each arrow we indicate the 4-simplex bounded by the five 3-simplices involved in the Pachner move, on which the 2-form cocycle is evaluated.
• Figure 4: Graphical depiction of the 2-form cocycle condition $d^{(4)}\omega(0,0,c,d|e,0,0|0,0|0) = 1$ and $d^{(4)}\omega(0,0,c,0|0,0,g|h,0|0) = 1$. The dashed line are labeled by the identity group element $\text{\small $0$} \in G$. Each arrow of the diagram is labeled by a 4-simplex $(abcde)$ such that $\langle \omega, (abcde)\rangle$ is the evaluation of the 2-form 4-cocycle $\omega$ that is the amplitude of the corresponding 2-3 Pachner move, as well as a symbol $\alpha$, $R$ or ${\rm id}$ depending on whether the 2-3 Pachner move effectively reduces to a 2-2 Pachner move, a braiding move or a trivial move, respectively. Together, these two consistency conditions effectively reduce to a so-called hexagon relation.
• Figure 5: Graphical depiction of the 2-form cocycle condition $d^{(4)}\omega(a,\text{\small $0$},c,d|\text{\small $0$},\text{\small $0$},\text{\small $0$}|h,\text{\small $0$}|\text{\small $0$}) \equiv \langle d^{(4)}\omega ,(012345)\rangle = 1$. This illustrates how the term $\omega(a,c,d|h,\text{\small $0$}|\text{\small $0$}) = \langle \omega ,(01345)\rangle$ encodes all the defining steps of the plaquette operator in the Walker-Wang model: the splitting of the four-valent node into three valent ones, the combination of $\mathcal{P}_{2 \mapsto 2}$ moves and braiding moves as well as the recombination of three-valent nodes into a single four-valent one.

Theorems & Definitions (27)

• Example 2.1: $X$ is the $q$-th classifying space $B^qG$ of a finite abelian group $G$
• Example 2.2: $X$ is a two-stage Postnikov tower
• Example 2.3: $q$-form lattice gauge theories
• Definition 2.1: Quadratic form
• Definition 3.1: Oriented and ordered open cover
• Definition 3.2: Polyhedral decomposition
• Example 3.1
• Example 3.2
• Definition 3.3: Deligne-Beilinson cohomology
• Example 3.3: 1-form ${\rm U}(1)$ connections