S^3/Z_n partition function and dualities

TL;DR

This paper studies the S^3/Z_n partition function of N=2 supersymmetric gauge theories, focusing on the role of holonomies and phases, and uses dualities to determine phase factors for different orbifold cases.

Contribution

It introduces a method to determine the relative phases in the holonomy sum of the partition function using dualities, especially for odd n cases.

Findings

Phase factors can be absorbed by modifying a single function for odd n.

Dualities help fix the phases in the holonomy sum.

The approach clarifies the structure of the partition function on orbifolds.

Abstract

We investigate S^3/Z_n partition function of N = 2 supersymmetric gauge theories. A gauge theory on the orbifold has degenerate vacua specified by the holonomy. The partition function is obtained by summing up the contributions of saddle points with different holonomies. An appropriate choice of the phase of each contribution is essential to obtain the partition function. We determine the relative phases in the holonomy sum in a few examples by using duality to non-gauge theories. In the case of odd n the phase factors can be absorbed by modifying a single function appearing in the partition function.