E-string Quantum Curve

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.

arXiv commentv2 typos corrected, added explanations on brane constructions, added references

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Cited by in corpus (7)

• Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model
• Elliptic Quantum Curves of 6d SO(N) theories
• Topological strings and Wilson loops
• Quantum Representation of Affine Weyl Groups and Associated Quantum Curves
• $\mathcal{N}=1$ Curves on Generalized Coulomb Branches
• Weyl invariant $E_8$ Jacobi forms and $E$-strings
• Aspects of 4d supersymmetric dynamics and geometry