Holonomy saddles and supersymmetry

Abstract

In gauge theories on a spacetime equipped with a circle, the holonomy variables, living in the Cartan torus, play special roles. With their periodic nature properly taken into account, we find that a supersymmetric gauge theory in d dimensions tends to reduce in the small radius limit to a disjoint sum of multiple (d-1) dimensional theories at distinct holonomies, called H-saddles. The phenomenon occurs regardless of the spacetime dimensions, and here we explore such H-saddles for d=4 N=1 theories on T2 fibered over Σg, in the limits of elongated T2. This naturally generates novel relationships between 4d and 3d partition functions, including ones between 4d and 3d Witten indices, and also leads us to reexamine recent studies of the Cardy exponents and the Casimir energies and of their purported connections to the 4d anomalies.