Exact Schur index in closed form for non-Lagrangian theories

The Schur index is a powerful tool to probe the spectrum and dualities of four-dimensional N = 2 superconformal field theories, deeply related to two-dimensional vertex operator algebras (VOAs). In this paper, we compute the Schur index in closed form for two series of non-Lagrangian theories. We explore and classify the Argyres-Douglas (AD) theories D p b ( sl N , [ Y ] ) realized as the S U ( 2 ) gauging of two AD matter theories, where we identify several infinite families with interesting central charge relations analogous to the a 4 d = c 4 d of N = 4 theories. We focus on D N − 4 ( sl ( N ) , [ N − 4 , 4 ] ) and D N − 2 ( sl ( N ) , [ N − 3 , 3 ] ) and compute their flavored and unflavored Schur and Wilson line indices in compact form. We also explore their large- N behavior and show that they arise as special limits of the S U ( 2 ) SQCD flavored index, also analogous to the relation among the a 4 d = c 4 d theories. We also generalize the elliptic function integration formula in the presence of higher-order poles to compute in closed form the partially flavored indices of the Minahan-Nemeschansky E 6 and E 7 theories. Our results point to a universal structure underlying the residues of elliptic integrands, Wilson loop indices, and nonvacuum modules of the corresponding VOAs.