BRST invariance and de Rham-type cohomology of 't Hooft-Polyakov monopole

TL;DR

This paper explores the BRST invariance and de Rham-type cohomology structure of the 't Hooft-Polyakov monopole system using quantum field operator algebra and topological charge analysis.

Contribution

It introduces a framework for analyzing monopole systems through BRST invariance and cohomology, linking gauge fixing, ghost terms, and topological bounds.

Findings

BRST invariant Hamiltonian constructed for monopole system

De Rham-type cohomology group structure identified

Topological charge relates to Bogomol'nyi bound

Abstract

We exploit the 't Hooft-Polyakov monopole to define closed algebra of the quantum field operators and the BRST charge $Q_{BRST}$. In the first-class configuration of the Dirac quantization, by including the $Q_{BRST}$-exact gauge fixing term and the Faddeev-Popov ghost term, we find the BRST invariant Hamiltonian to investigate the de Rham-type cohomology group structure for the monopole system. The Bogomol'nyi bound is also discussed in terms of the first-class topological charge defined on the extended internal 2-sphere.