Elliptic quantum curves of class Sk

Abstract

Quantum curves arise from Seiberg-Witten curves associated to 4d 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 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 C-2/Z(k) singularity. In the limit where the compactified 2-torus has zero size, the corresponding 4d N = 2 theories are known as class Sk. 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.