On 2-form gauge models of topological phases

TL;DR

This paper investigates 2-form topological gauge theories in 3+1 dimensions, establishing their classification, continuous embeddings, lattice models, and connections to categorical structures and Walker-Wang models.

Contribution

It introduces a lattice Hamiltonian for 2-form gauge theories, explores their embedding into 2-group gauge theories, and links them to categorical and Walker-Wang models.

Findings

Constructed an exactly solvable lattice Hamiltonian model in (3+1)d.

Established the classification of 2-form gauge theories via cohomology of $B^2G$.

Connected 2-form cocycles to premodular categories and Walker-Wang models.

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.

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