Characteristic classes of Q-manifolds: classification and applications

TL;DR

This paper introduces characteristic classes for Q-manifolds, classifies their intrinsic types, and explores their applications in gauge theory anomalies and foliation theory.

Contribution

It defines and classifies the intrinsic characteristic classes of Q-manifolds, linking them to invariants in gauge theory and foliation theory.

Findings

Complete classification of intrinsic characteristic classes.

Representation of classes via universal tensor polynomials.

Application to anomalies in gauge theory quantization.

Abstract

A $Q$-manifold $M$ is a supermanifold endowed with an odd vector field $Q$ squaring to zero. The Lie derivative $L_Q$ along $Q$ makes the algebra of smooth tensor fields on $M$ into a differential algebra. In this paper, we define and study the invariants of $Q$-manifolds called characteristic classes. These take values in the cohomology of the operator $L_Q$ and, given an affine symmetric connection with curvature $R$, can be represented by universal tensor polynomials in the repeated covariant derivatives of $Q$ and $R$ up to some finite order. As usual, the characteristic classes are proved to be independent of the choice of the affine connection used to define them. The main result of the paper is a complete classification of the intrinsic characteristic classes, which, by definition, do not vanish identically on flat $Q$-manifolds. As an illustration of the general theory we interpret some of the intrinsic characteristic classes as anomalies in the BV and BFV-BRST quantization methods of gauge theories. An application to the theory of (singular) foliations is also discussed.