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Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT

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.