## Journal of Differential Geometry

### Mirror symmetry for orbifold Hurwitz numbers

#### 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.

#### Article information

Source
J. Differential Geom., Volume 98, Number 3 (2014), 375-423.

Dates
First available in Project Euclid: 28 July 2014

https://projecteuclid.org/euclid.jdg/1406552276

Digital Object Identifier
doi:10.4310/jdg/1406552276

Mathematical Reviews number (MathSciNet)
MR3263522

Zentralblatt MATH identifier
1315.53100

#### Citation

Bouchard, Vincent; Serrano, Daniel Hernández; Liu, Xiaojun; Mulase, Motohico. Mirror symmetry for orbifold Hurwitz numbers. J. Differential Geom. 98 (2014), no. 3, 375--423. doi:10.4310/jdg/1406552276. https://projecteuclid.org/euclid.jdg/1406552276