Related papers: The extended D-Toda hierarchy

Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d,g) of complex plane curves of degree d and genus g through 3d+g-1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has…

We investigate existence of solitonic solutions for higher-order partial differential equations with polynomial nonlinearities. Using the Hirota method we obtain classification for higher-order integrable systems of equations.

We discuss how the theory of quantum cohomology may be generalized to ``gravitational quantum cohomology'' by studying topological sigma models coupled to two-dimensional gravity. We first consider sigma models defined on a general Fano…

We give a simple proof that the Neumann coefficients of surface states in Witten's SFT satisfy the Hirota equations for dispersionless KP hierarchy. In a similar way we show that the Neumann coefficients for the three string vertex in the…

We introduce an extension of the \ell-reduced KP hierarchy, which we call the \ell-Bogoyavlensky hierarchy. Bogoyavlensky's 2+1-dimensional extension of the KdV equation is the lowest equation of the hierarchy in case of \ell=2. We present…

The first part of this work constructs positive-genus real Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the present part focuses on their properties that are essential for actually working…

We build the abstract theory of Gromov-Witten invariants of genus 0 for quantum minimal Fano varieties (a minimal natural (with respect to Gromov-Witten theory) class of varieties). In particular, we consider ``the minimal Gromov-Witten…

We study the enumerative geometry of stable maps to Calabi-Yau 5-folds $Z$ with a group action preserving the Calabi-Yau form. In the central case $Z=X \times \mathbb{C}^2$, where $X$ is a Calabi-Yau 3-fold with a group action scaling the…

We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology, and can…

By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL hierarchies as…

The present note overviews our recent construction of real Gromov-Witten theory in arbitrary genera for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold, its properties, and…

This paper addresses the issue of integrable structures in topological string theory on generalized conifolds. Open string amplitudes of this theory can be expressed as the matrix elements of an operator on the Fock space of 2D charged free…

We study elliptic families of solutions to the recently introduced constrained Toda hierarchy, i.e., solutions which are elliptic functions of some linear combination of the hierarchical times. Equations of motion for poles of such…

The hierarchy of the classical nonlinear integrable equations associated with relativistic Toda chain model is considered. It is formulated for the N-th powers of the quantum operators of the corresponding quantum integrable models.…

We extend the action for evolution equations of KdV and MKdV type which was derived in [Capel/Nijhoff] to the case of not periodic, but only equivariant phase space variables, introduced in [Faddeev/Volkov]. The difference of these…

The finite-genus solutions for the Hirota's bilinear difference equation are constructed using the Fay's identities for the theta-functions of compact Riemann surfaces.

The goal of this article is to motivate and describe how Gromov-Witten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from Gromov-Witten theory have led to both conjectures and…

For a complete toric variety, we obtain an explicit formula for the localized equivariant Todd class in terms of the combinatorial data -- the fan. This is based on the equivariant Riemann-Roch theorem and the computation of the equivariant…

In this paper we consider the orbifold curve, which is a quotient of an elliptic curve $\mathcal{E}$ by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov-Witten theory of the orbifold…

A method is proposed to construct a new extended KP hierarchy, which includes two types of KP equation with self-consistent sources and admits reductions to k-constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources. It…