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2014•Journal of Differential GeometryOpen access

Mirror symmetry for orbifold Hurwitz numbers

Vincent Bouchard, Daniel Hernández Serrano, Xiaojun Liu, Motohico Mulase

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Abstract

We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the $r$-Lambert curve. We argue that the $r$-Lambert curve also arises in the infinite framing limit of orbifold Gromov-Witten theory of $[\mathbb{C}^3 / (\mathbb{Z} / r\mathbb{Z})]$. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve.

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We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the $r$-Lambert curve. We argue that the $r$-Lambert curve also arises in the infinite framing limit of orbifold Gromov-Witten theory of $[\mathbb{C}^3 / (\mathbb{Z} / r\mathbb{Z})]$. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve.

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Available abstract

We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the $r$-Lambert curve. We argue that the $r$-Lambert curve also arises in the infinite framing limit of orbifold Gromov-Witten theory of $[\mathbb{C}^3 / (\mathbb{Z} / r\mathbb{Z})]$. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve.

Key concepts: Orbifold, Mathematics, Mirror symmetry, Laplace transform, Pure mathematics, Recursion (computer science), Mathematical physics, Mathematical analysis

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