Higher order BRST and anti-BRST operators and cohomology for compact Lie algebras

TL;DR

This paper introduces higher order BRST and anti-BRST operators for compact simple Lie algebras, analyzes their associated Laplacians and supersymmetry algebra, and provides explicit examples including the su(3) case.

Contribution

It defines and studies higher order BRST/anti-BRST operators and Laplacians, extending the standard framework and analyzing their algebraic properties and examples.

Findings

Higher order Laplacians are not solely determined by Casimir operators.

A higher order Hodge decomposition is established.

Explicit su(3) case analysis with Laplacian expression.

Abstract

After defining cohomologically higher order BRST and anti-BRST operators for a compact simple algebra {\cal G}, the associated higher order Laplacians are introduced and the corresponding supersymmetry algebra $\Sigma$ is analysed. These operators act on the states generated by a set of fermionic ghost fields transforming under the adjoint representation. In contrast with the standard case, for which the Laplacian is given by the quadratic Casimir, the higher order Laplacians $W$ are not in general given completely in terms of the Casimir-Racah operators, and may involve the ghost number operator. The higher order version of the Hodge decomposition is exhibited. The example of su(3) is worked out in detail, including the expression of its higher order Laplacian W.