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225
Bihamiltonian Hierarchies in 2D Topological Field Theory At OneLoop Approximation
, 1997
"... We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical Walgebra; we compute the central charge of this algebra. We also express the generating function of ..."
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Cited by 74 (8 self)
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We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical Walgebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov Witten invariants via taufunction of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity.
Mirror symmetry for weighted projective planes and their noncommutative deformations
, 2004
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Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and GromovWitten invariants
, 2001
"... 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 ..."
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Cited by 49 (2 self)
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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.
Elliptic GromovWitten invariants and the generalized mirror conjecture
"... A conjecture expressing genus 1 GromovWitten invariants in mirrortheoretic terms of semisimple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torusequivariant Gromov Witten invariants of compact Kähler manifolds with isolated fixed ..."
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Cited by 46 (5 self)
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A conjecture expressing genus 1 GromovWitten invariants in mirrortheoretic terms of semisimple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torusequivariant Gromov Witten invariants of compact Kähler manifolds with isolated fixed points and for concave bundle spaces over such manifolds. Several results on genus 0 Gromov Witten theory include: a nonlinear Serre duality theorem, its application to the genus 0 mirror conjecture, a mirror theorem for concave bundle spaces over toric manifolds generalizing a recent result of B. Lian, K. Liu and S.T. Yau. We also establish a correspondence (see the extensive footnote in section 4) between their new proof of the genus 0 mirror conjecture for quintic 3folds and our proof of the same conjecture given two years ago. Research supported by NSF grants DMS9321915 and DMS9704774
Moduli spaces of higher spin curves and integrable hierarchies
 Compositio Math
"... Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable rspin curves assemble into a generating function which yields a solution of the s ..."
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Cited by 42 (8 self)
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Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable rspin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdVr equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank r −1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity Ar−1. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of GromovWitten invariants and quantum cohomology. The moduli space of stable curves of genus g with n marked points Mg,n is a fascinating object. Mumford [37] introduced tautological cohomology classes associated to the universal curve Cg,n
Another way to enumerate rational curves with torus actions
 Invent. Math
"... mirror symmetry to predict the numbers of rational curves of any degree on a quintic threefold [7]. This took geometers by surprise, as the best mathematical results at the time could only count the rational curves of degree three or less. Since then, the field of enumerative algebraic geometry has ..."
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Cited by 41 (5 self)
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mirror symmetry to predict the numbers of rational curves of any degree on a quintic threefold [7]. This took geometers by surprise, as the best mathematical results at the time could only count the rational curves of degree three or less. Since then, the field of enumerative algebraic geometry has
On Hamiltonian perturbations of hyperbolic systems of conservation laws
, 2004
"... We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially onedimensional systems of hyperbolic PDEs vt + [φ(v)]x = 0. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the pert ..."
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Cited by 40 (6 self)
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We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially onedimensional systems of hyperbolic PDEs vt + [φ(v)]x = 0. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the socalled quasiMiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following [35], the invariants of such bihamiltonian structures with respect to the group of Miuratype transformations depending
Flat pencils of metrics and Frobenius manifolds
 IN: PROCEEDINGS OF 1997 TANIGUCHI SYMPOSIUM ”INTEGRABLE SYSTEMS AND ALGEBRAIC GEOMETRY”, EDITORS M.H.SAITO, Y.SHIMIZU AND K.UENO
, 1998
"... 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, thes ..."
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Cited by 35 (6 self)
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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 M appear naturally in the classification of bihamiltonian structures of hydrodynamics type on the loop space L(M). This elucidates the relations between Frobenius manifolds and integrable hierarchies.