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

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 ( 2 n + 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.