2,445 research outputs found

    Frobenius manifolds from regular classical WW-algebras

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    We obtain polynomial Frobenius manifolds from classical WW-algebras associated to regular nilpotent elements in simple Lie algebras using the related opposite Cartan subalgebras

    Virasoro Symmetries of the Extended Toda Hierarchy

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    We prove that the extended Toda hierarchy of \cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators LmL_m, m‚Č•‚ąí1m\geq -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 CP1CP^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.Comment: A remark at the end of Section 5 is added; more detailed explanations in Appendix; references adde

    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

    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

    Frobenius Manifolds: Natural submanifolds and induced bi-Hamiltonian structures

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    Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being that these natural submanifolds carry a induced pencil of compatible metrics. It is then shown how one may constrain the bi-Hamiltonian hierarchies associated to a Frobenius manifold to live on these natural submanifolds whilst retaining their, now non-local, bi-Hamiltonian structure.Comment: 27 Pages, LaTeX, 1 figur

    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

    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
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