# Supersymmetric localization of refined chiral multiplets on topologically twisted $ H^2 \times S^1$

## TL;DR

This paper computes the exact partition function of an $
=2$ chiral multiplet on topologically twisted $H^2 imes S^1$ using supersymmetric localization, revealing connections to holomorphic blocks and special functions.

## Contribution

It derives the partition function of a chiral multiplet on twisted $H^2 imes S^1$ with background fields, providing explicit formulas involving $q$-Pochhammer symbols and Zeta functions.

## Key findings

- Partition function expressed in terms of special functions.
- Reproduction of 3D holomorphic blocks from normalizable fields.
- Explicit dependence on $S^1$ radius and angular momentum fugacity.

## Abstract

We derive the partition function of an $\mathcal N=2$ chiral multiplet on topologically twisted $H^2\times S^1$. The chiral multiplet is coupled to a background vector multiplet encoding a real mass deformation. We consider an $ H^2\times S^1$ metric containing two parameters: one is the $S^1$ radius, while the other gives a fugacity $q$ for the angular momentum on $H^2$. The computation is carried out by means of supersymmetric localization, which provides a finite answer written in terms of $q$-Pochammer symbols and multiple Zeta functions. Especially, the partition function of normalizable fields reproduces three-dimensional holomorphic blocks.