Modular factorization of superconformal indices

TL;DR

This paper explores the modular properties and factorization of superconformal indices in 4D $ abla=1$ gauge theories, proposing a new framework involving holomorphic blocks and modular transformations, with evidence from specific models.

Contribution

It introduces a novel modular factorization approach for superconformal indices, connecting them to $SL(3, ext{Z})$ and $SL(2, ext{Z})$ transformations, and conjectures a general formula for lens space indices.

Findings

Superconformal indices factorize into holomorphic blocks via modular transformations.

The modular properties of elliptic $ extGamma$ functions support the factorization.

A cohomology class related to $SL(3, ext{Z})$ is constructed from the indices.

Abstract

Superconformal indices of four-dimensional $\mathcal{N}=1$ gauge theories factorize into holomorphic blocks. We interpret this as a modular property resulting from the combined action of an $SL(3,\mathbb{Z})$ and $SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$ transformation. The former corresponds to a gluing transformation and the latter to an overall large diffeomorphism, both associated with a Heegaard splitting of the underlying geometry. The extension to more general transformations leads us to argue that a given index can be factorized in terms of a family of holomorphic blocks parametrized by modular (congruence sub)groups. We find precise agreement between this proposal and new modular properties of the elliptic $\Gamma$ function. This leads to our conjecture for the ``modular factorization'' of superconformal lens indices of general $\mathcal{N}=1$ gauge theories. We provide evidence for the conjecture in the context of the free chiral multiplet and SQED, and sketch the extension of our arguments to more general gauge theories. Based on this result, we systematically prove that a normalized part of superconformal lens indices defines a non-trivial first cohomology class associated with $SL(3,\mathbb{Z})$. Finally, we use this framework to propose a formula for the general lens space index.