Topology of $SU(N)$ lattice gauge theories coupled with $\mathbb{Z}_N$ $2$-form gauge fields

TL;DR

This paper extends L"uscher's lattice topological charge to 4D $SU(N)$ gauge theories coupled with $bZ_N$ 2-form gauge fields, preserving key symmetries and deriving the mixed 't Hooft anomaly on the lattice.

Contribution

It introduces a lattice definition of topological charge for $SU(N)$ gauge theories with $bZ_N$ 2-form fields, maintaining gauge invariance and elucidating the mixed anomaly.

Findings

Maintains locality, gauge invariance, and $bZ_N$ symmetry in the lattice formulation.

Provides a lattice derivation of the mixed 't Hooft anomaly.

Highlights the role of 1-form gauge invariance in the construction.

Abstract

We extend the definition of L\"uscher's lattice topological charge to the case of $4$d $SU(N)$ gauge fields coupled with $\mathbb{Z}_N$ $2$-form gauge fields. This result is achieved while maintaining the locality, the $SU(N)$ gauge invariance, and $\mathbb{Z}_N$ $1$-form gauge invariance, and we find that the manifest $1$-form gauge invariance plays the central role in our construction. This result gives the lattice regularized derivation of the mixed 't Hooft anomaly in pure $SU(N)$ Yang-Mills theory between its $\mathbb{Z}_N$ $1$-form symmetry and the $\theta$ periodicity.