Results on the Wess-Zumino consistency condition for arbitrary Lie algebras

TL;DR

This paper extends the covariant Poincare lemma to arbitrary Lie algebras, enabling the computation of solutions to the Wess-Zumino consistency condition for various algebraic structures, with applications to Chern-Simons theory.

Contribution

It generalizes the covariant Poincare lemma to non-reductive Lie algebras and provides a method to solve the Wess-Zumino consistency condition in these cases.

Findings

Solution of the Wess-Zumino condition for arbitrary Lie algebras.

Complete solution for abelian ideals.

Application to 2+1 dimensional Chern-Simons theory.

Abstract

The so-called covariant Poincare lemma on the induced cohomology of the spacetime exterior derivative in the cohomology of the gauge part of the BRST differential is extended to cover the case of arbitrary, non reductive Lie algebras. As a consequence, the general solution of the Wess-Zumino consistency condition with a non trivial descent can, for arbitrary (super) Lie algebras, be computed in the small algebra of the 1 form potentials, the ghosts and their exterior derivatives. For particular Lie algebras that are the semidirect sum of a semisimple Lie subalgebra with an ideal, a theorem by Hochschild and Serre is used to characterize more precisely the cohomology of the gauge part of the BRST differential in the small algebra. In the case of an abelian ideal, this leads to a complete solution of the Wess-Zumino consistency condition in this space. As an application, the consistent deformations of 2+1 dimensional Chern-Simons theory based on iso(2,1) are rediscussed.