Higher (gauged) Wess--Zumino--Witten terms based on Lie crossed modules

TL;DR

This paper develops higher Wess--Zumino--Witten and gauged WZW terms within strict higher Chern--Simons gauge theory using Lie crossed modules, revealing gauge invariance properties and boundary effects.

Contribution

It introduces a systematic derivation of higher WZW and gWZW terms from higher Chern--Simons forms based on Lie crossed modules, extending gauge theory concepts.

Findings

Higher WZW term vanishes for symmetric invariant polynomials.

Higher gWZW term is exact, leading to gauge invariance on closed manifolds.

Boundary gauge dependence is captured by boundary terms.

Abstract

We derive higher Wess--Zumino--Witten (WZW) and gauged WZW (gWZW) terms within strict higher Chern--Simons (CS) gauge theory. Starting from the Cartan homotopy formula, we obtain the $(2n+2)$-dimensional higher CS forms and transgression forms for strict Lie 2-groups presented by Lie crossed modules. Given two 2-connections related by a higher gauge transformation, higher transgression forms yield canonical higher WZW and gWZW terms. We prove that, for the symmetric invariant polynomial associated with differential crossed modules, the pure-gauge higher WZW term vanishes identically, whereas the higher gWZW term is exact. Consequently, the higher CS action is higher-gauge invariant on closed manifolds, and on manifolds with boundary all gauge dependence is encoded in boundary terms.