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The 4d Superconformal Index from q-deformed 2d Yang-Mills

Abhijit Gadde, Leonardo Rastelli, Shlomo S. Razamat, Wenbin Yan

TL;DR

The paper identifies a 2d q-deformed Yang-Mills model as the underlying framework computing the reduced 4d N=2 superconformal index for a broad class of SU(N) generalized quivers, including T_N theories and nontrivial SCFTs like E6. By expressing indices through q-dimensions and SU(N)_q structure constants and exploiting S-duality, it enables explicit computations for strongly coupled theories and clarifies how punctures and gluing correspond to 2d topological data. The results provide a concrete 4d–2d duality that yields insights into protected operator spectra and Higgs branch structure, with promising avenues toward the full index and potential connections to elliptic deformations and the 6d (2,0) origin on Riemann surfaces.

Abstract

We identify the 2d topological theory underlying the N=2 4d superconformal index with an explicit model: q-deformed 2d Yang-Mills. By this route we are able to evaluate the index of some strongly-coupled 4d SCFTs, such as Gaiotto's T_N theories.

The 4d Superconformal Index from q-deformed 2d Yang-Mills

TL;DR

The paper identifies a 2d q-deformed Yang-Mills model as the underlying framework computing the reduced 4d N=2 superconformal index for a broad class of SU(N) generalized quivers, including T_N theories and nontrivial SCFTs like E6. By expressing indices through q-dimensions and SU(N)_q structure constants and exploiting S-duality, it enables explicit computations for strongly coupled theories and clarifies how punctures and gluing correspond to 2d topological data. The results provide a concrete 4d–2d duality that yields insights into protected operator spectra and Higgs branch structure, with promising avenues toward the full index and potential connections to elliptic deformations and the 6d (2,0) origin on Riemann surfaces.

Abstract

We identify the 2d topological theory underlying the N=2 4d superconformal index with an explicit model: q-deformed 2d Yang-Mills. By this route we are able to evaluate the index of some strongly-coupled 4d SCFTs, such as Gaiotto's T_N theories.

Paper Structure

This paper contains 7 sections, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $SU(2)$ generalized quivers
  3. $SU(3)$ generalized quivers
  4. Three Maximal Punctures
  5. Two Maximal and One $U(1)$ Puncture
  6. General statement
  7. Discussion

Figures (1)

  • Figure 1: An example of the rule to associate flavor fugacities for a non-maximal puncture. Illustrated here is a puncture for $N=26$ with flavor symmetry $S(U(3)U(2)^{2}U(1))$. The $S(\dots)$ constraint imposes $(ab)^{5}(cde)^{4}f^{2}gh=1$.