Adjusting Higher Chern-Simons Theory

TL;DR

This paper introduces half-adjusted higher Chern-Simons theories to address the construction of consistent higher gauge theories, overcoming obstructions related to inner products and fake-flatness constraints, with detailed frameworks in various dimensions.

Contribution

It proposes half-adjusted higher Chern-Simons theories that resolve key obstructions, providing a comprehensive construction and differential cohomology framework for higher gauge theories.

Findings

Successful formulation of half-adjusted higher Chern-Simons theories

Explicit construction in four dimensions

Complete differential cohomological framework developed

Abstract

A fundamental problem in formulating higher Chern-Simons theories is the construction of a consistent higher gauge theory that circumvents the fake-flatness constraint. Here, we propose a solution to this problem using adjusted higher connections. In particular, we shall demonstrate that there is an obstruction to constructing such action functionals since, generically, adjusted higher gauge algebras do not admit an inner product. To overcome this obstruction, we introduce half-adjusted higher Chern-Simons theories. These theories have both well-defined underlying kinematic data as well as the expected properties of a higher generalisation of Chern-Simons theory. We develop the general construction of these theories in arbitrary dimensions and provide explicit details for the four-dimensional case. We also present the complete differential cohomological framework for principal 2-bundles with half-adjusted connections. Finally, we discuss an alternative approach introducing additional trivial symmetries.