Holonomy Saddles and Supersymmetry

TL;DR

This paper investigates how supersymmetric gauge theories on a circle decompose into multiple lower-dimensional theories at different holonomies, revealing new links between 4d and 3d partition functions and reinterpreting anomaly-related phenomena.

Contribution

It introduces the concept of $H$-saddles, showing that supersymmetric theories reduce to a sum over distinct holonomies, and explores their implications for 4d-3d relations and anomaly studies.

Findings

Identification of $H$-saddles in supersymmetric gauge theories.

New relationships between 4d and 3d partition functions.

Re-examination of Cardy exponents, Casimir energies, and anomalies.

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 $d=4$ $\cal N=1$ theories on $T^2$ fibred over $\Sigma_g$, in the limits of elongated $T^2$. This naturally generates novel relationships between 4d and 3d partition functions, including ones between 4d and 3d Witten indices, and also leads us to re-examine recent studies of the Cardy exponents and the Casimir energies and of their purported connections to the 4d anomalies.