Elliptic quantum curves of class $$ {\mathcal{S}}_k $$

Abstract

Abstract Quantum curves arise from Seiberg-Witten curves associated to 4d $$ \mathcal{N} $$ N = 2 gauge theories by promoting coordinates to non-commutative operators. In this way the algebraic equation of the curve is interpreted as an operator equation where a Hamiltonian acts on a wave-function with zero eigenvalue. We find that this structure generalises when one considers torus-compactified 6d $$ \mathcal{N} $$ N = (1, 0) SCFTs. The corresponding quantum curves are elliptic in nature and hence the associated eigenvectors/eigenvalues can be expressed in terms of Jacobi forms. In this paper we focus on the class of 6d SCFTs arising from M5 branes transverse to a ℂ2/ℤk singularity. In the limit where the compactified 2-torus has zero size, the corresponding 4d $$ \mathcal{N} $$ N = 2 theories are known as class $$ {\mathcal{S}}_k $$ S k . We explicitly show that the eigenvectors associated to the quantum curve are expectation values of codimension 2 surface operators, while the corresponding eigenvalues are codimension 4 Wilson surface expectation values.