E-string Quantum Curve

TL;DR

This paper investigates the quantisation of the Seiberg-Witten curve for the E-string theory on a torus, revealing an elliptic quantum curve linked to integrable systems, defect operators, and symmetry enhancements.

Contribution

It introduces a novel elliptic quantum curve associated with the E-string theory, connecting it to elliptic integrable systems and symmetry enhancements from SO(16) to affine E8.

Findings

The quantum curve is a generalised van Diejen operator.

Eigenvalues correspond to co-dimension 4 Wilson surfaces.

IR symmetry enhances to affine E8 characters.

Abstract

In this work we study the quantisation of the Seiberg-Witten curve for the E-string theory compactified on a two-torus. We find that the resulting operator expression belongs to the class of elliptic quantum curves. It can be rephrased as an eigenvalue equation with eigenvectors corresponding to co-dimension 2 defect operators and eigenvalues to co-dimension 4 Wilson surfaces wrapping the elliptic curve, respectively. Moreover, the operator we find is a generalised version of the van Diejen operator arising in the study of elliptic integrable systems. Although the microscopic representation of the co-dimension 4 defect only furnishes an $\mathrm{SO}(16)$ flavour symmetry in the UV, we find an enhancement in the IR to representations in terms of affine $E_8$ characters. Finally, using the Nekrasov-Shatashvili limit of the E-string BPS partition function, we give a path integral derivation of the quantum curve.