Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes

TL;DR

This paper explores an odd analogue of Chern-Simons theory within the AKSZ-BV framework, constructing Lie algebra cocycles and linking perturbative partition functions to secondary characteristic classes, with explicit computations and reinterpretations of known models.

Contribution

It introduces an odd Chern-Simons analogue, explicitly constructs Lie algebra cocycles, and connects perturbative partition functions to secondary characteristic classes.

Findings

Perturbation theory matches that of classical Chern-Simons theory.

Explicit cocycle constructions for Lie algebra of Hamiltonian vector fields.

Reinterpretation of the Rozansky-Witten model in this framework.

Abstract

We investigate the generic 3D topological field theory within AKSZ-BV framework. We use the Batalin-Vilkovisky (BV) formalism to construct explicitly cocycles of the Lie algebra of formal Hamiltonian vector fields and we argue that the perturbative partition function gives rise to secondary characteristic classes. We investigate a toy model which is an odd analogue of Chern-Simons theory, and we give some explicit computation of two point functions and show that its perturbation theory is identical to the Chern-Simons theory. We give concrete example of the homomorphism taking Lie algebra cocycles to Q-characteristic classes, and we reinterpreted the Rozansky-Witten model in this light.