Lie algebra cohomology and group structure of gauge theories

TL;DR

This paper explores the Lie algebra cohomology of gauge groups, establishing Hodge decomposition and Poincaré duality, and linking these mathematical structures to physical concepts like BRST symmetry and gauge anomalies.

Contribution

It explicitly constructs the adjoint of the coboundary operator, linking it to the BRST adjoint generator, and reveals duality relations relevant to gauge anomalies and topological terms.

Findings

Established Hodge decomposition and Poincaré duality in gauge Lie algebra cohomology.

Identified the BRST adjoint generator with the adjoint of the coboundary operator.

Linked duality relations to gauge anomalies and Wess-Zumino-Witten terms.

Abstract

We explicitly construct the adjoint operator of coboundary operator and obtain the Hodge decomposition theorem and the Poincar\'e duality for the Lie algebra cohomology of the infinite-dimensional gauge transformation group. We show that the adjoint of the coboundary operator can be identified with the BRST adjoint generator $Q^{\dagger}$ for the Lie algebra cohomology induced by BRST generator $Q$. We also point out an interesting duality relation - Poincar\'e duality - with respect to gauge anomalies and Wess-Zumino-Witten topological terms. We consider the consistent embedding of the BRST adjoint generator $Q^{\dagger}$ into the relativistic phase space and identify the noncovariant symmetry recently discovered in QED with the BRST adjoint N\"other charge $Q^{\dagger}$.