Related papers: A generalization of the Witten conjecture through …

According to a conjecture of E. Witten proved by M. Kontsevich, a certain generating function for intersection indices on the Deligne -- Mumford moduli spaces of Riemann surfaces coincides with a certain tau-function of the KdV hierarchy.…

By employing polynomial-reduced KP integrability, combined with the string equation, this work establishes explicit relationships between the generalized Kontsevich model, the topological recursion of the spectral curve, and the geometry of…

We consider the Hurwitz Dubrovin--Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also…

The Witten $r$-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin--Frobenius manifold…

We construct and study various properties of a negative spin version of the Witten $ r $-spin class. By taking the top Chern class of a certain vector bundle on the moduli space of twisted spin curves that parametrises $ r $-th roots of the…

We review progress on the generalized Witten conjecture and some of its major ingredients. This conjecture states that certain intersection numbers on the moduli space of higher spin curves assemble into the logarithm of the tau function of…

A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov -…

We define a theory of descendent integration on the moduli spaces of stable pointed disks. The descendent integrals are proved to be coefficients of the $\tau$-function of an open KdV heirarchy. A relation between the integrals and a…

We perform a key step towards the proof of Zvonkine's conjectural $r$-ELSV formula that relates Hurwitz numbers with completed $(r+1)$-cycles to the geometry of the moduli spaces of the $r$-spin structures on curves: we prove the…

For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to…

The main goal of this paper is to prove the following two conjectures for genus up to two: 1. Witten's conjecture on the relations between higher spin curves and Gelfand--Dickey hierarchy. 2. Virasoro conjecture for target manifolds with…

We study the intersection theory of the $\Theta^{r,s}$-classes, where $r \geq 2$ and $1 \le s \le r-1$, which are cohomological field theories obtained as the top degrees of Chiodo classes. We show that the recently introduced generalized…

In recent works, [20],[21], descendent integrals on the moduli space of Riemann surfaces with boundary were defined. It was conjectured in [20] that the generating function of these integrals satisfies the open KdV equations. In this paper…

We study the reduced descendent Gromov-Witten theory of K3 surfaces in primitive curve classes. We present a conjectural closed formula for the stationary theory, which generalizes the Bryan-Leung formula. We also prove a new recursion that…

This article treats various aspects of the geometry of the moduli of r-spin curves and its compactification. Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a…

Let S be a nonsingular projective K3 surface. Motivated by the study of the Gromov-Witten theory of the Hilbert scheme of points of S, we conjecture a formula for the Gromov-Witten theory (in all curve classes) of the Calabi-Yau 3-fold S x…

This is an expository paper based on the results in [12] and [16]. The main goal is to prove the following two conjectures for genus up to two. (1) Witten's conjecture on the relations between higher spin curves and Gelfand-Dickey…

We study spin Hurwitz numbers, which count ramified covers of the Riemann sphere with a sign coming from a theta characteristic. These numbers are known to be related to Gromov-Witten theory of K\"ahler surfaces and to representation theory…

The Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class - the so-called Legendre transformations - were introduced by Dubrovin.…

In this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the…