5D SYM on 3D Sphere and 2D YM

TL;DR

The paper shows that localization of five-dimensional ${\cal N}=1$ SYM on ${\bf R}^2\times S^3$ reduces the computation of correlation functions in the BRST-invariant sector to those of a two-dimensional bosonic Yang-Mills theory on ${\bf R}^2$. By constructing a nilpotent BRST charge and a suitable deformation, the authors identify fixed points with Cartan-subalgebra backgrounds and demonstrate cancellations between bosonic and fermionic fluctuations, up to a residual contribution that reproduces the 2D YM action. This establishes an exact 5D→2D correspondence for observables in the localized sector and suggests natural extensions, such as incorporating hypermultiplets and exploring connections to related dualities. The result deepens our understanding of localization in higher-dimensional gauge theories and provides a bridge to lower-dimensional effective descriptions with potential implications for exact results and dualities in supersymmetric field theories.

Abstract

It is shown by using localization that in five-dimensional N=1 supersymmetric Yang-Mills theory on a three-dimensional sphere, correlation functions in a sector are identical to correlation functions in two-dimensional bosonic Yang-Mills theory.