相关论文: Simple singularities and integrable 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.…

It was proved in 2010 that the principal Kac--Wakimoto hierarchy of type $D$ is a reduction of the 2-component BKP hierarchy. On the other hand, it is known that the total descendant potential of a singularity of type $D$ is a tau-function…

For the Kac-Wakimoto hierarchy constructed from the principal vertex operator realization of the basic representation of the affine Lie algebra $D_n^{(1)}$, we compute the coefficients of the corresponding Hirota bilinear equations, and…

Semisimple Dubrovin-Frobenius manifolds can be used to construct integrable hierarchies, following the work of Dubrovin-Zhang and Buryak. Examples of such hierarchies include the Kac-Wakimoto hierarchies, the KP hierarchy, among others. In…

We propose a new construction of an integrable hierarchy associated to any infinite series of Frobenius manifolds satisfying a certain stabilization condition. We study these hierarchies for Frobenius manifolds associated to $A_N$, $D_N$…

A well known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper we present a generalization of this construction for the…

We construct a Hermitian matrix model for the total descendant potential of a simple singularity of type D similar to the Kontsevich matrix model for the generating function of intersection numbers on the Deligne--Mumford moduli spaces…

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finite-dimensional Lie algebra, and W its Weyl group. The set of g-invariants in the basic representation of the affine Kac-Moody…

Fano orbifold lines are classified by the Dynkin diagrams of type $A,D,$ and $E$. It is known that the corresponding total descendant potential is a tau-function of an appropriate Kac--Wakimoto hierarchy. It is also known that in the A-case…

Given a semisimple Frobenius manifold, we construct a class of integrable deformations of its hierarchy of topological type. We show that these integrable deformations have polynomial tau-structures, and conjecture that for the…

The total descendent potential of a simple singularity satisfies the Kac-Wakimoto principal hierarchy. Bakalov and Milanov showed recently that it is also a highest weight vector for the corresponding W-algebra. This was used by Liu, Yang…

We prove the equivalence of two hierarchies of soliton equations associated to a simply-laced finite Dynkin diagram. The first was defined by Kac and Wakimoto using the principal realization of the basic representations of the corresponding…

K. Saito's theory of primitive forms gives a natural semi-simple Frobenius manifold structure on the space of miniversal deformations of an isolated singularity. On the other hand, Givental introduced the notion of a total ancestor…

In our earlier papers we proposed a new approach to integrable hierarchies of soliton equations and their quantum deformations. We have applied this approach to the Toda field theories and the generalized KdV and modified KdV (mKdV)…

We prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D_4 with symmetry group <J> and D_4^T with symmetry group G_{max}, respectively, are both tau-functions of the D_4…

We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov - Witten invariants of all genera into the theory of integrable systems. The…

We describe genus g>1 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In Gromov-Witten theory, it becomes a conjecture expressing higher genus…

We construct an integrable hierarchy in terms of vertex operators and Hirota Quadratic Equations (HQE shortly) and we show that the equivariant total descendant potential of $\C P^1$ satisfies the HQE. Our prove is based on the quantization…

We study the local bihamiltonian structures of the asymmetric rational reductions of the 2D-Toda hierarchy (RR2T) of types $(2,1)$ and $(1,2)$ at the full-dispersive level, and construct a three-dimensional generalized Frobenius manifold…

We propose a generalization of the Witten conjecture, which connects a descendent enumerative theory with a specific reduction of KP integrable hierarchy. Our conjecture is realized by two parts: Part I (Geometry) establishes a…