Characteristic classes of gauge systems

TL;DR

This paper introduces universal characteristic classes for gauge systems modeled as (anti-)Poisson supermanifolds with homological vector fields, providing new invariants and algebraic structures relevant to gauge theories and related geometric frameworks.

Contribution

It defines and computes universal characteristic classes for gauge systems using homological vector fields, expanding the understanding of invariants in gauge theory and related structures.

Findings

Characteristic classes constructed from first derivatives of the homological vector field.

The space of characteristic classes admits a Lie bracket structure different from the Poisson bracket.

Examples relate characteristic classes to anomalies and foliation theory.

Abstract

We define and study invariants which can be uniformly constructed for any gauge system. By a gauge system we understand an (anti-)Poisson supermanifold provided with an odd Hamiltonian self-commuting vector field called a homological vector field. This definition encompasses all the cases usually included into the notion of a gauge theory in physics as well as some other similar (but different) structures like Lie or Courant algebroids. For Lagrangian gauge theories or Hamiltonian first class constrained systems, the homological vector field is identified with the classical BRST transformation operator. We define characteristic classes of a gauge system as universal cohomology classes of the homological vector field, which are uniformly constructed in terms of this vector field itself. Not striving to exhaustively classify all the characteristic classes in this work, we compute those invariants which are built up in terms of the first derivatives of the homological vector field. We also consider the cohomological operations in the space of all the characteristic classes. In particular, we show that the (anti-)Poisson bracket becomes trivial when applied to the space of all the characteristic classes, instead the latter space can be endowed with another Lie bracket operation. Making use of this Lie bracket one can generate new characteristic classes involving higher derivatives of the homological vector field. The simplest characteristic classes are illustrated by the examples relating them to anomalies in the traditional BV or BFV-BRST theory and to characteristic classes of (singular) foliations.