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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 project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov- Witten classes and their descendents.

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For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus ≤ 1 Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.-S.Xiong and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology. 1

We prove that the extended Toda hierarchy of [1] admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators Lm, m ≥ −1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the CP 1 Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy. 1

We present the Lax pair formalism for certain extension of the continuous limit of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the CP 1 topological sigma model and the extended Toda hierarchy. We also establish an equivalence of the latter with certain extension of the nonlinear Schrödinger hierarchy. 1

We show that all (n-component) KP tau-functions, which are related to the twisted loop group of GLn, give solutions of the Darboux-Egoroff system of PDE’s. Using the Geometry of the Grassmannian we construct from the corresponding wave function the deformed flat coordinates of the Egoroff metric and from this the corresponding solution of the Witten–Dijkgraaf–E. Verlinde–H. Verlinde equations 1

Given a space M, one may attempt to construct various natural cohomology algebras such as the ordinary (simplicial, singular, etc.) cohomology algebra H ∗ (M) and the quantum

Abstract. It is well-known that the partition function of the unitary ensembles of random matrices is given by a τ-function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are τ-functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles. We also find an explicit formula for the two point function Fnm which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the dispersionless Toda lattice hierarchy, and 1 Fnm are the Grunsky coefficients of the Faber polynomials. nm

We consider the deformations of the Whitham systems (Dubrovin problem) including the ”dispersion terms”. Under some non-degeneracy requirements we suggest an algorithm of the deformation of the Whitham system using the initial nonlinear system. The general form of the deformed Whitham system coincides with the form of the ”low-dispersion ” asymptotic expansions used by B.A. Dubrovin and Y. Zhang in the theory of deformations of Frobenius manifolds. 1 Introduction. The classical Whitham method ([1, 2, 3, 4]) is connected with the slow modulations of the exact periodic or quasiperiodic solutions of non-linear PDE’s F i (ϕ, ϕt, ϕx, ϕtt, ϕxt, ϕxx,...) = 0, i = 1,...,n (1.1)