A Lie based 4-dimensional higher Chern-Simons theory

TL;DR

This paper introduces a 4-dimensional higher Chern-Simons theory based on Lie 2-algebras, exploring its gauge symmetries, partition function, and connections to topological and 3D gauge theories.

Contribution

It constructs a novel 4D higher Chern-Simons model using Lie 2-algebras, analyzing its symmetries, partition function, and relation to other topological theories.

Findings

Partition function relates to flat connections with a specific characteristic class.

The theory's gauge symmetry forms an infinite-dimensional strict Lie 2-group.

Connections to 3D gauge theory with a symplectic structure and Hamiltonian action.

Abstract

We present and study a model of 4-dimensional higher Chern-Simons theory, special Chern-Simons (SCS) theory, instances of which have appeared in the string literature, whose symmetry is encoded in a skeletal semistrict Lie 2-algebra constructed from a compact Lie group with non discrete center. The field content of SCS theory consists of a Lie valued 2-connection coupled to a background closed 3-form. SCS theory enjoys a large gauge and gauge for gauge symmetry organized in an infinite dimensional strict Lie 2-group. The partition function of SCS theory is simply related to that of a topological gauge theory localizing on flat connections with degree 3 second characteristic class determined by the background 3-form. Finally, SCS theory is related to a 3-dimensional special gauge theory whose 2-connection space has a natural symplectic structure with respect to which the 1-gauge transformation action is Hamiltonian, the 2-curvature map acting as moment map.