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arXiv · 2008.05155

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

Jin Chen·Babak Haghighat·Hee-Cheol Kim·Marcus Sperling
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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.

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Jin Chen, Babak Haghighat, Hee-Cheol Kim, Marcus Sperling. 2020-08-12. Elliptic Quantum Curves of Class $\mathcal{S}_k$. https://doi.org/10.1007/jhep03(2021)028

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