All Stable Characteristic Classes of Homological Vector Fields

TL;DR

This paper classifies all stable characteristic classes of homological vector fields on supermanifolds, revealing their structure as invariants derived from $Q$-invariant tensors and cohomology.

Contribution

It provides a complete classification of characteristic classes of homological vector fields, expanding understanding of their invariants in supergeometry.

Findings

Classification of characteristic classes in terms of $Q$-invariant tensors

Representation of classes via cohomology of $L_Q$

Identification of stable invariants in supermanifold geometry

Abstract

An odd vector field $Q$ on a supermanifold $M$ is called homological, if $Q^2=0$. The operator of Lie derivative $L_Q$ makes the algebra of smooth tensor fields on $M$ into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator $L_Q$ and are represented by $Q$-invariant tensors made up of the homological vector field and a symmetric connection on $M$ by means of tensor operations.