Elliptic Quantum Curves of Class $\mathcal{S}_k$

TL;DR

This paper extends the concept of quantum curves from 4d $ abla$2 gauge theories to elliptic quantum curves associated with 6d SCFTs, revealing their eigenvectors and eigenvalues as surface operator expectations.

Contribution

It introduces elliptic quantum curves for 6d SCFTs from M5 branes and relates their eigenvectors and eigenvalues to surface operators and Wilson surfaces.

Findings

Quantum curves generalize from 4d to elliptic 6d theories.

Eigenvectors correspond to surface operator expectation values.

Eigenvalues relate to Wilson surface expectation values.

Abstract

Quantum curves arise from Seiberg-Witten curves associated to 4d $\mathcal{N}=2$ 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 $\mathcal{N}=(1,0)$ 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 $\mathbb{C}^2/\mathbb{Z}_k$ singularity. In the limit where the compactified 2-torus has zero size, the corresponding 4d $\mathcal{N}=2$ theories are known as class $\mathcal{S}_k$. 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.