4-d semistrict higher Chern-Simons theory I

TL;DR

This paper develops a 4D higher gauge Chern-Simons theory using semistrict Lie 2-algebras, exploring its gauge invariance, quantizations, and potential for defining 2-knot invariants in four dimensions.

Contribution

It introduces a novel 4D higher gauge Chern-Simons framework based on semistrict Lie 2-algebras, including analysis of gauge invariance and two quantization approaches.

Findings

The theory is gauge invariant up to a higher winding number.

Two distinct canonical quantizations are identified.

Explicit higher WZW actions and Ward identities are derived.

Abstract

We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its symmetry is encoded in a semistrict Lie 2-algebra equipped with an invariant non singular bilinear form. We analyze the gauge invariance of the theory and show that action is invariant under a higher gauge transformation up to a higher winding number. We find that the theory admits two seemingly inequivalent canonical quantizations. The first is manifestly topological, it does not require a choice of any additional structure on the spacial 3-fold. The second, more akin to that of ordinary Chern-Simons theory, involves fixing a CR structure on the latter. Correspondingly, we obtain two sets of semistrict higher WZW Ward identities and we find the explicit expressions of two higher versions of the WZW action. We speculate that the model could be used to define 2-knot invariants of 4-folds.