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66
GromovWitten invariants and quantization of quadratic Hamiltonians
, 2001
"... We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about GromovWitten invariants of compact symplectic manifolds and, more generally, Frobenius structures at ..."
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Cited by 87 (3 self)
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We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about GromovWitten invariants of compact symplectic manifolds and, more generally, Frobenius structures at higher genus. We state several results illustrating the formalism and its use. In particular, we establish Virasoro constraints for semisimple Frobenius structures and outline a proof of the Virasoro conjecture for Gromov – Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere.
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 44 (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.
Semisimple Frobenius structures at higher genus
, 2000
"... We describe genus g ≥ 2 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In GromovWitten theory, it becomes a conjecture expressing higher genus GWinvariants in terms of genus 0 GWinvariants of symplect ..."
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Cited by 41 (4 self)
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We describe genus g ≥ 2 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In GromovWitten theory, it becomes a conjecture expressing higher genus GWinvariants in terms of genus 0 GWinvariants of symplectic manifolds with generically semisimple quantum cupproduct. The conjecture is supported by the corresponding theorem about equivariant GWinvariants of tori actions with isolated fixed points. The parallel theory of gravitational descendents is also presented.
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 37 (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
Flat pencils of metrics and Frobenius manifolds., ArXiv: math.DG/9803106
 In: Proceedings of 1997 Taniguchi Symposium ”Integrable Systems and Algebraic Geometry”, editors M.H.Saito, Y.Shimizu and K.Ueno
, 1998
"... Abstracts 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 assumpt ..."
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Cited by 35 (6 self)
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Abstracts 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. 1
Frobenius manifolds and Virasoro constraints
 Selecta Math. (N.S
, 1999
"... 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 prove ..."
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Cited by 31 (4 self)
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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
The Virasoro conjecture for Gromov–Witten invariants
, 1998
"... The Virasoro conjecture is a conjectured sequence of relations among the descendent GromovWitten invariants of a smooth projective variety in all genera; the only varieties for which it is known to hold are a point (Kontsevich) and CalabiYau manifolds of dimension at least three. We review the s ..."
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Cited by 29 (2 self)
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The Virasoro conjecture is a conjectured sequence of relations among the descendent GromovWitten invariants of a smooth projective variety in all genera; the only varieties for which it is known to hold are a point (Kontsevich) and CalabiYau manifolds of dimension at least three. We review the statement of the conjecture and its proof in genus 0, following Eguchi, Hori and Xiong.
The GromovWitten potential of a point, Hurwitz numbers, and Hodge integrals
 Proc. London Math. Soc
, 1999
"... 1.1. Recursions and GromovWitten theory 2 ..."
The Toda equations and the GromovWitten theory of the Riemann sphere
 Lett. Math. Phys
"... 0.1. Toda equations. The GromovWitten theory of P1 has been intensively studied in a sequence of remarkable papers by Eguchi, Hori, Xiong, Yamada, and Yang [EHY], [EY], [EYY], [EHX]. A major step in this analysis was the discovery of a (conjectural) matrix model for ..."
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Cited by 28 (2 self)
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0.1. Toda equations. The GromovWitten theory of P1 has been intensively studied in a sequence of remarkable papers by Eguchi, Hori, Xiong, Yamada, and Yang [EHY], [EY], [EYY], [EHX]. A major step in this analysis was the discovery of a (conjectural) matrix model for