On S-duality of the Superconformal Index on Lens Spaces and 2d TQFT

TL;DR

The paper addresses S-duality of the 4d ${ m N}=2$ superconformal index on ${S^1 imes L(r,1)}$ by isolating a one-parameter fugacity slice and recasting the index as a correlator in a putative 2d TFT. In the Macdonald-like limit ${p=0}$ with ${t=q^r}$ (and further ${t=q^r}$), the authors derive explicit expressions for three-point structure constants, introduce a basis of orthonormal functions, and organize the index into a 2d-TFT-like sphere with three punctures whose correlators depend on holonomies ${m}$ and fugacities. They show that crossing symmetry of these 2d correlators enforces the S-duality of the 4d index in the considered sector, encoded by a finite set of algebraic constraints among $q$-dependent coefficients ${h_ell^{(i)}(q)}$, all of which hold to high order in the expansion. The work provides a concrete framework connecting 4d ${ m N}=2$ indices on nontrivial 3-manifolds with 2d topological quantum field theories and suggests avenues for identifying the underlying TFT and extending the analysis beyond the specific fugacity slice.

Abstract

We consider the 4d superconformal index for ${\cal N}=2$ gauge theories on $S^1 \times L(r,1)$, where $L(r,1)$ is a Lens space. We focus on a one-parameter slice of the three-dimensional fugacity space and in that sector we show S-duality. We do so by rewriting the index in a way that resembles a correlation function of a 2d TFT, which however, we do not identify.

Figures (1)

• Figure 1: The four point correlation function should be crossing symmetric. On both sides one should sum over the holonomy $m$ of the internal state.