##
**Mirror symmetry for orbifold Hurwitz numbers.**
*(English)*
Zbl 1315.53100

The principle of mirror symmetry has become a dominant guideline in mathematical physics. There exist various versions of mirror symmetry, such as homological mirror symmetry of Kontsevich, mirror symmetry of singularities developed by Fan-Jarvis-Ruan recently and so on. An interesting phenomenon is that the mirror symmetries appear in many counting problems involving the moduli space of Riemannian surfaces \(\overline{\mathcal{M}}_{g,n}\). To make this more precisely, for counting problems mentioned above there exists a mirror symmetric counterpart determined by a universal integral recursion formula. A typical example of such counting problem are the Gromov-Witten invariants of symplectic manifolds or symplectic orbifolds. Consider the 1-dimensional complex projective space \(\mathbb{P}^{1}\). A simple counting problem for \(\mathbb{P}^{1}\) (resp. \(\mathbb{P}^{1}[r]\)) is to count the automorphism weighted number of the topological types of a given simple Hurwitz cover. This leads to the definition of simple Hurwitz number (resp. orbifold Hurwitz number). The main purpose of the paper under review is to study the orbifold Hurwitz number of the orbifold \(\mathbb{P}^{1}[r]\) with one stack point \([0/(\mathbb{Z}/r\mathbb{Z})]\) from the mirror symmetry point of view.

The main results of this paper are the following:

(1) In Section 3, the authors realize the generating functions of the orbifold Hurwitz numbers for \(\mathbb{P}^{1}[r]\); and furthermore, they show that such generating functions can be obtained in the infinite framing limit of the orbifold topological vertex generating functions for the open orbifold Gromov-Witten invariants of \([\mathbb{C}^{3}/(\mathbb{Z}/\mathbb{Z})]\) in Theorem 3.2.

(2) In Section 4, the authors construct the spectral curve for orbifold Hurwitz numbers via Laplace transform, which is called the r-Lambert curve. Furthermore, they prove that the generating functions of the orbifold Hurwitz numbers are determined by a series of partial differential recursion equations in Theorem 1.1.

(3) In Section 6, the authors give the explicit partition functions for orbifold Hurwitz numbers. In particular, this implies the existence of a quantum curve of orbifold Hurwitz numbers {Theorem 1.3}.

(4) In Section 7, the authors prove that the generating functions of orbifold Hurwitz numbers satisfy the integral recursion of Eynard-Orantin with the spectral curve constructed in Section 4, {Theorem 1.7}.

This is a very interesting and meaningful work. It gives a new perspective to understand the orbifold Hurwitz numbers from the viewpoint of mirror symmetry.

The main results of this paper are the following:

(1) In Section 3, the authors realize the generating functions of the orbifold Hurwitz numbers for \(\mathbb{P}^{1}[r]\); and furthermore, they show that such generating functions can be obtained in the infinite framing limit of the orbifold topological vertex generating functions for the open orbifold Gromov-Witten invariants of \([\mathbb{C}^{3}/(\mathbb{Z}/\mathbb{Z})]\) in Theorem 3.2.

(2) In Section 4, the authors construct the spectral curve for orbifold Hurwitz numbers via Laplace transform, which is called the r-Lambert curve. Furthermore, they prove that the generating functions of the orbifold Hurwitz numbers are determined by a series of partial differential recursion equations in Theorem 1.1.

(3) In Section 6, the authors give the explicit partition functions for orbifold Hurwitz numbers. In particular, this implies the existence of a quantum curve of orbifold Hurwitz numbers {Theorem 1.3}.

(4) In Section 7, the authors prove that the generating functions of orbifold Hurwitz numbers satisfy the integral recursion of Eynard-Orantin with the spectral curve constructed in Section 4, {Theorem 1.7}.

This is a very interesting and meaningful work. It gives a new perspective to understand the orbifold Hurwitz numbers from the viewpoint of mirror symmetry.

Reviewer: Xiaojun Chen (Chengdu)

### MSC:

53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |

14J33 | Mirror symmetry (algebro-geometric aspects) |