Graded geometry in gauge theories and beyond

TL;DR

This paper introduces equivariant Q-cohomology, a graded geometric framework that generalizes gauging in gauge theories, enabling analysis of symmetries and construction of invariant functionals like sigma models.

Contribution

It generalizes equivariant cohomology to Q-manifolds, providing new tools for analyzing gauge symmetries and constructing invariant functionals in gauge theories.

Findings

Developed the concept of equivariant Q-cohomology.

Applied to twisted Poisson sigma models with infinite-dimensional gauge groups.

Potential applications in supersymmetric gauge theories and higher structures.

Abstract

We study some graded geometric constructions appearing naturally in the context of gauge theories. Inspired by a known relation of gauging with equivariant cohomology we generalize the latter notion to the case of arbitrary Q-manifolds introducing thus the concept of equivariant Q-cohomology. Using this concept we describe a procedure for analysis of gauge symmetries of given functionals as well as for constructing functionals (sigma models) invariant under an action of some gauge group. As the main example of application of these constructions we consider the twisted Poisson sigma model. We obtain it by a gauging-type procedure of the action of an essentially infinite dimensional group and describe its symmetries in terms of classical differential geometry. We comment on other possible applications of the described concept including the analysis of supersymmetric gauge theories and higher structures.