Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement

TL;DR

The paper develops a 4d confining TQFT framework based on a discrete 2-form valued in the magnetic gauge group $\Pi_2$ and shows that discrete theta-angles are classified by quadratic functions on $\Pi_2$, realized in both continuum and lattice formulations. It proves a duality to a topological gauge theory when the discrete theta-angles vanish, while allowing nontrivial duality-breaking phases tied to the Pontryagin square and manifold signature. A concrete lattice Yang-Mills coupling is given that preserves a well-defined '$t\;\text{Hooft flux}$' in $\Pi_2$ and incorporates the discrete theta-angles; explicit expressions for $\Pi_2=\mathbb{Z}_r$ on $T^4$ illustrate the interplay between topology and lattice observables. Collectively, the work connects the AST discrete theta-angle classification to a lattice-implementable confining phase, clarifying how topological data govern confinement and duality in 4d gauge theories.

Abstract

We study a topological field theory describing confining phases of gauge theories in four dimensions. It can be formulated on a lattice using a discrete 2-form field talking values in a finite abelian group (the magnetic gauge group). We show that possible theta-angles in such a theory are quantized and labeled by quadratic functions on the magnetic gauge group. When the theta-angles vanish, the theory is dual to an ordinary topological gauge theory, but in general it is not isomorphic to it. We also explain how to couple a lattice Yang-Mills theory to a TQFT of this kind so that the 't Hooft flux is well-defined, and quantized values of the theta-angles are allowed. The quantized theta-angles include the discrete theta-angles recently identified by Aharony, Seiberg and Tachikawa.