The other topological twisting of N=4 Yang-Mills

TL;DR

This paper proposes a novel topological twist of N=4 Yang-Mills in four dimensions in which the partition function localizes on flat connections of the complexified gauge group rather than instantons. It constructs a two-BRST-operator theory (Q and Q~) with a spectrum reorganized into a complexified gauge field and auxiliary fields, and demonstrates that the action is Q-invariant and metric-independent. The ground-state structure on manifolds with nonpositive Euler characteristic is governed by the moduli space of flat complexified connections, leading to a single invariant Omega—analogous to the Casson invariant in three dimensions—that depends only on the gauge group and topology. The work discusses reducible and noncompact moduli space issues, and suggests connections to broader mathematical structures and potential links to string theory methods.

Abstract

We present the alternative topological twisting of N=4 Yang-Mills, in which the path integral is dominated not by instantons, but by flat connections of the COMPLEXIFIED gauge group. The theory is nontrivial on compact orientable four-manifolds with nonpositive Euler number, which are necessarily not simply connected. On such manifolds, one finds a single topological invariant, analogous to the Casson invariant of three-manifolds.