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 ℂ2/ℤk 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.
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Chen, J., Haghighat, B., Kim, HC. et al. Elliptic quantum curves of class \( {\mathcal{S}}_k \). J. High Energ. Phys. 2021, 28 (2021). https://doi.org/10.1007/JHEP03(2021)028
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DOI: https://doi.org/10.1007/JHEP03(2021)028