Hermiticity and the Cohomology Condition in Topological Yang-Mills Theory

TL;DR

This paper investigates the symmetries of topological Yang-Mills theory in the Hamiltonian formalism, focusing on the algebraic structure, hermiticity properties, and the derivation of the BRST cohomology condition for physical states.

Contribution

It explicitly constructs the twisted N=2 superPoincaré algebra generators and clarifies the relation between Lorentz and internal symmetries, including hermiticity considerations.

Findings

Twisted Lorentz generators do not generate Lorentz symmetry.

Boost generators are non-hermitian.

BRST cohomology condition is derived from representation theory.

Abstract

The symmetries of the topological Yang-Mills theory are studied in the Hamiltonian formalism and the generators of the twisted N=2 superPoincar\'e algebra are explicitly constructed. Noting that the twisted Lorentz generators do not generate the Lorentz symmetry of the theory, we relate the two by extracting from the latter the twisted version of the internal SU(2) generator. The hermiticity properties of the various generators are also considered throughout, and the boost generators are found to be non-hermitian. We then recover the BRST cohomology condition on physical states from representation theory arguments.