Characteristic classes associated to Q-bundles

TL;DR

This paper introduces a new framework for characteristic classes using Q-bundles in graded manifolds, generalizing Chern-Weil theory and connecting to topological sigma models, Poisson fibrations, and Lie algebra cohomology.

Contribution

It develops the concept of Q-bundles in graded manifolds, extending characteristic class theory and providing new examples linked to topological field theories.

Findings

Generalizes Chern-Weil classes via Q-bundles

Constructs cohomology classes from gauge fields and fiber forms

Relates to topological sigma models and Poisson fibrations

Abstract

A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. Any principal bundle yielding canonically a Q-bundle, this construction generalizes Chern-Weil classes. Novel examples include cohomology classes that are locally the de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate to equivariant cohomology and Lecomte's characteristic classes of exact sequences of Lie algebras.