Exact non-Lagrangian Schur index in closed form

The Schur index is a powerful tool to probe the spectrum and dualities of 4d $\mathcal{N}=2$ superconformal field theories (SCFTs), deeply related to 2d 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 $D_p^b(\mathfrak{sl}_N,[Y])$ realized as the $SU(2)$ gauging of two AD matter theories, where we identify several infinite families with interesting central charge relations analogous to the $a_\text{4d} = c_\text{4d}$ of $\mathcal{N} = 4$ theories. We focus on $D_{N-4}(\mathfrak{sl}(N),[N-4,4])$ and $D_{N-2}(\mathfrak{sl}(N),[N-3,3])$, 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 $SU(2)$ SQCD flavored index, also analogous to the relation among the $a_\text{4d} = c_\text{4d}$ 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 non-vacuum modules of the corresponding VOAs.