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72
GromovWitten classes, quantum cohomology, and enumerative geometry
 Commun. Math. Phys
, 1994
"... The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological ..."
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Cited by 364 (3 self)
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The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given. Let V be a projective algebraic manifold. Methods of quantum field theory recently led to a prediction of some numerical characteristics of the space of algebraic curves in V, especially of genus zero, eventually endowed with a parametrization and marked points. It turned out that
Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 343 (2 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
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 73 (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.
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.
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 30 (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
On organizing principles of discrete differential geometry. Geometry of spheres
 RUSSIAN MATH. SURVEYS 62:1 1–43
, 2007
"... ..."
Symplectic and Poisson geometry on loop spaces of manifolds and nonlinear equations
 Advances in Soviet Mathematics. S.P.Novikov
, 1995
"... We consider some differential geometric classes of local and nonlocal Poisson and symplectic structures on loop spaces of smooth manifolds which give natural Hamiltonian or multihamiltonian representations for some important nonlinear equations of mathematical physics and field theory such as nonlin ..."
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Cited by 20 (4 self)
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We consider some differential geometric classes of local and nonlocal Poisson and symplectic structures on loop spaces of smooth manifolds which give natural Hamiltonian or multihamiltonian representations for some important nonlinear equations of mathematical physics and field theory such as nonlinear sigma models with torsion, degenerate Lagrangian systems of field theory, systems of hydrodynamic type, Ncomponent systems of Heisenberg magnet type, MongeAmpère equations, the KricheverNovikov equation and others. In particular, complete classification of all nondegenerate Poisson bivectors ω ij (x, u, ux, uxx,...) depending on derivatives of the field variables u i (x) and the independent space variable x is obtained (u i, i = 1,..., N, are local coordinates on smooth manifold M). In other words, all Poisson brackets of the following form {u i (x), u j (y)} = ω ij (x, u, ux, uxx,...)δ(x − y), det(ω ij) ̸ = 0, are explicitly described. In addition, we shall prove integrability of some class of nonhomogeneous systems of hydrodynamic type and give a description of nonlinear partial differential equations of associativity in 2D topological field theories (for some special type solutions of the WittenDijkgraafE.VerlindeH.Verlinde (WDVV) system) as integrable
D.H.: Symplectic forms in the theory of solitons
 Surveys in Differential Geometry IV
, 1998
"... We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form ω = 1 2 Res ∞ < Ψ ∗ 0 ..."
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Cited by 13 (3 self)
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We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form ω = 1 2 Res ∞ < Ψ ∗ 0 δL ∧ δΨ0> dk. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finitegap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in nonlinear WKB theory, topological field theory, and SeibergWitten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role.