Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT
Jaewon Song
TL;DR
The work develops a unified TQFT-based framework for computing superconformal indices of 4d N=2 class S theories with irregular punctures I_{k,N}, covering Schur, Hall-Littlewood, and Macdonald limits. It introduces and tests wave functions for irregular punctures, with coprime (k,N) cases yielding clean Schur expressions and specific N cases yielding HL/Macdonald analogs, including detailed AD theory realizations. The results connect to known chiral algebras, Virasoro/W-algebra minimal models, and affine algebras, providing new checks against S-duality and 3d mirrors, and enabling mixed Schur calculations for N=1 class S. These findings enhance our ability to extract protected operator data and OPE structure from non-Lagrangian theories, with potential implications for dualities and holographic interpretations.
Abstract
We study superconformal indices of 4d N=2 class S theories with certain irregular punctures called type $I_{k, N}$. This class of theories include generalized Argyres-Douglas theories of type $(A_{k-1}, A_{N-1})$ and more. We conjecture the superconformal indices in certain simplified limits based on the TQFT structure of the class S theories by writing an expression for the wave function corresponding to the puncture $I_{k, N}$. We write the Schur limit of the wave function when $k$ and $N$ are coprime. When $k=2$, we also conjecture a closed-form expression for the Hall-Littlewood index and the Macdonald index for odd $N$. From the index, we argue that certain short-multiplet which can appear in the OPE of the stress-energy tensor is absent in the $(A_1, A_{2n})$ theory. We also discuss the mixed Schur indices for the N=1 class S theories with irregular punctures.