94 research outputs found

    On universality of critical behaviour in Hamiltonian PDEs

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    Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one spatial dimension. For the systems of order one or two we describe the local structure of singularities of a generic solution to the unperturbed system near the point of "gradient catastrophe" in terms of standard objects of the classical singularity theory; we argue that their perturbed companions must be given by certain special solutions of Painleve' equations and their generalizations.Comment: 59 pages, 2 figures. Amer. Math. Soc. Transl., to appea

    On almost duality for Frobenius manifolds

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    We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theory of Coxeter groups and Shephard groups, from the Seiberg - Witten duality.Comment: 62 pages, a reference adde

    Flat pencils of metrics and Frobenius manifolds

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    This paper is based on the author's talk at 1997 Taniguchi Symposium ``Integrable Systems and Algebraic Geometry''. We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumptions, these two objects are identical. The flat pencils of contravariant metrics on a manifold MM appear naturally in the classification of bihamiltonian structures of hydrodynamics type on the loop space L(M)L(M). This elucidates the relations between Frobenius manifolds and integrable hierarchies.Comment: 25 pages, no figures, plain Te

    Geometry and analytic theory of Frobenius manifolds

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    Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories.Comment: 11 pages, to appear in Proceedings ICM9

    Frobenius Manifolds And Virasoro Constraints

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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 ‚ȧ1\leq 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.Comment: Latex, 40 page

    Canonical structure and symmetries of the Schlesinger equations

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    The Schlesinger equations S(n,m)S_{(n,m)} describe monodromy preserving deformations of order mm Fuchsian systems with n+1n+1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of nn copies of m√ómm\times m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S(n,m)S_{(n,m)} for all nn, mm and we compute the action of the symmetries of the Schlesinger equations in these coordinates.Comment: 92 pages, no figures. Theorem 1.2 corrected, other misprints removed. To appear on Comm. Math. Phy

    Extended affine Weyl groups and Frobenius manifolds

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    We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F(t1,...,tn)F(t_1, ..., t_n) of WDVV equations of associativity polynomial in t1,...,tn‚ąí1,exp‚Ā°tnt_1, ..., t_{n-1}, \exp t_n.Comment: 69 pages, amslatex, some references added, position of Table 1 is corrected. Revised version for Compositio Mathematic

    Matrix resolvent and the discrete KdV hierarchy

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    Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are then obtained, and applications to enumeration of ribbon graphs with even valencies and to the special cubic Hodge integrals are considered.Comment: It was added a remark after Theorem 1; added Appendix A; added references; two statements moved to Introduction; corrected typos. 26 page
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