General Yang-Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical framework OR From Bianchi identities to twisted Courant algebroids

TL;DR

This paper develops a unified mathematical framework for higher gauge theories using Q-structures and Lie infinity algebroids, connecting physical gauge identities with advanced geometric structures like twisted Courant algebroids.

Contribution

It introduces a comprehensive approach to higher gauge theories via Q-manifolds and Lie infinity algebroids, bridging physics and mathematics.

Findings

Structural identities form a Q-structure or Lie infinity algebroid.

Gauge theories are represented as bundles in the category of Q-manifolds.

For p=2, the gauge system corresponds to a Lie 2-algebroid and a twisted Courant algebroid.

Abstract

Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into a so-called Q-structure or, equivalently, a Lie infinity algebroid. This has many technical and conceptual advantages: Complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator Q squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines only and is given by the so-called derived bracket construction. This article aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are valid for arbitrary highest form-degree p, we pay particular attention to p=2, i.e. 1- and 2-form gauge fields coupled non-linearly to scalar fields (0-form fields). The structural identities of the coupled system correspond to a Lie 2-algebroid in this case and we provide different axiomatic descriptions of those, inspired by the application, including e.g. one as a particular kind of a vector-bundle twisted Courant algebroid.