Feynman diagrams and the KdV hierarchy

Quadratic double ramification integrals and the noncommutative KdV hierarchy

In this paper we compute the intersection number of two double ramification cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic double ramification integrals are the main ingredient for the computation of the double ramification hierarchy associated to the infinite dimensional partial cohomological field theory given by $\exp(μ^2 Θ)$ where $μ$ is a parameter and $Θ$ is Hain's theta class, appearing in Hain's formula for the double ramification cycle on the moduli space of curves of compact …

Extended $r$-spin theory in all genera and the discrete KdV hierarchy

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's $r$-spin classes. They are parameterized by a phase space which has one extra dimension and in genus $0$ they correspond to the extended $r$-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the $r$-spin class. They do not form a cohomological field theory, but a more general object which we call F-Coh…

Space Curves and Solitons of the KP Hierarchy. I. The $l$-th Generalized KdV Hierarchy

It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connections with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the $l$-th generalized KdV hierarchy with $l\ge 2$ is related to the rational space curves associated with the nu…

Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy

We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro constraint condition on a generating function for these volumes. We also show that the generating function for psi and kappa_1 intersections on the moduli space of stable algebraic curves is a 1-parameter solution to the KdV hierarchy. It recovers the Witten-Kontsevich generating function when the parameter …

量子 KdV 层级与拟模形式

Quantum KdV hierarchy and quasimodular forms

本文将无色散KdV层级量子化谱与拟模形式的关联扩展至完全量子KdV层级及量子Intermediate Long Wave层级，并提出拟模性的一般有效判据作为主要贡献。

The Toda hierarchy and the KdV hierarchy

The Toda hierarchy of size $N$ is well known to be analogous to the KdV hierarchy at $N$ goes to infinity. This paper shows that given $f$ a periodic function, there is a canonical way of defining the initial data for the Toda lattice equations so that the evolution of this data under the Toda lattice hierarchy looks asymptotically like the evolution of $f$ under the KdV hierarchy. Further, the conserved quantities of $f$ and those of the Toda hierarchy match.

GUE相关子项在KdV可积性中的可积性

KdV integrability in GUE correlators

本文基于GUE相关子项与Witten交集数的关系，通过Toda晶格的tau函数证明了Witten-Kontsevich定理，揭示了KdV可积结构与GUE相关子项的联系。

Toda and KdV

The main object of this paper is to produce a deformation of the KdV hierarchy of partial differential equations. We construct this deformation by taking a certain limit of the Toda hierarchy. This construction also provides a deformation of the Virasoro algebra.

A_{n-1} singularities and nKdV hierarchies

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. The generating function is naturally generalized under the name the {\em total descendent potential} in the theory of Gromov -- Witten invariants of symplectic manifolds. The papers arXiv: math.AG/0108100 and arXive: math.DG/0108160 contain two equivalent constructions, motivated by some results in Gromov -- Witten theory, which associate a total descendent p…

Hodge-GUE correspondence and the discrete KdV equation

We prove the conjectural relationship recently proposed in [9] between certain special cubic Hodge integrals of the Gopakumar--Mariño--Vafa type [17, 28] and GUE correlators, and the conjecture proposed in [7] that the partition function of these Hodge integrals is a tau function of the discrete KdV hierarchy.

Semi-infinite variations of Hodge structures and integrable hierarchies of KdV type

We introduce integrable KdV type hierarchy associated naturally with arbitrary semi-simple Frobenius manifold. We present hierarchy in a Lax form and show that it admits bihamiltonian description.

Construction of Two Parametric Deformation of KdV-Hierarchy and Solution in Terms of Meromorphic Functions on the Sigma Divisor of a Hyperelliptic Curve of Genus 3

Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we constructed the field of meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3 and solutions of the systems for $g=3$ by these functions. In this paper, as an application of our previous results, we construct two parametric deformation of the KdV-hierarchy. This new system is integrated in the meromorphic functions on the sigma divisor of hyp…