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30
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 86 (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.
Coxeter arrangements
 Proceedings of Symposia in Pure Mathematics 40
, 1983
"... Let V be an ℓdimensional Euclidean space. Let G ⊂ O(V) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the ..."
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Cited by 60 (6 self)
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Let V be an ℓdimensional Euclidean space. Let G ⊂ O(V) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the derivation module D (m) (A) = {θ ∈ DerS  θ(αH) ∈ Sα m H}. The module is known to be a free Smodule of rank ℓ by K. Saito (1975) for m = 1 and L. SolomonH. Terao (1998) for m = 2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D (m) (A). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m − 1)h/2) + mi(1 ≤ i ≤ ℓ) (when m is odd). Here m1 ≤ · · · ≤ mℓ are the exponents of G and h = mℓ + 1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G.) Some new results concerning the primitive derivation D are obtained in the course of proof of the main result.
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.
Geometry and analytic theory of Frobenius manifolds
, 1998
"... 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 manifol ..."
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Cited by 36 (3 self)
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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.
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.
The primitive derivation and freeness of multiCoxeter arrangements
, 2002
"... We will prove the freeness of multiCoxeter arrangements by constructing a basis of the module of vector fields which contact to each reflecting hyperplanes with some multiplicities using K. Saito’s theory of primitive derivation. ..."
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Cited by 9 (3 self)
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We will prove the freeness of multiCoxeter arrangements by constructing a basis of the module of vector fields which contact to each reflecting hyperplanes with some multiplicities using K. Saito’s theory of primitive derivation.
An−1 singularities and nKdV hierarchies
"... Abstract. According to a conjecture of E. Witten [18] proved by M. Kontsevich [11], a certain generating function for intersection indices on the Deligne – Mumford moduli spaces of Riemann surfaces coincides with a certain taufunction of the KdV hierarchy. The generating function is naturally genera ..."
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Cited by 7 (0 self)
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Abstract. According to a conjecture of E. Witten [18] proved by M. Kontsevich [11], a certain generating function for intersection indices on the Deligne – Mumford moduli spaces of Riemann surfaces coincides with a certain taufunction of the KdV hierarchy. The generating function is naturally generalized under the name the total descendent potential in the theory of Gromov – Witten invariants of symplectic manifolds. The papers [5, 4] contain two equivalent constructions, motivated by some results in Gromov – Witten theory, which associate a total descendent potential to any semisimple Frobenius structure. In this paper, we prove that in the case of K.Saito’s Frobenius structure [14] on the miniversal deformation of the An−1singularity, the total descendent potential is a taufunction of the nKdV hierarchy. We derive this result from a more general construction for solutions of the nKdV hierarchy from n − 1 solutions of the KdV hierarchy. 1. Introduction: Singularities and Frobenius
Coxeter multiarrangements with quasiconstant multiplicities
, 2008
"... We study structures of derivation modules of Coxeter multiarrangements with quasiconstant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasiconstant multiplicity is combinatorially computable. ..."
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Cited by 5 (5 self)
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We study structures of derivation modules of Coxeter multiarrangements with quasiconstant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasiconstant multiplicity is combinatorially computable.
On almost duality for Frobenius manifolds
"... on the occasion of his 65th birthday. Abstract. 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 ..."
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Cited by 5 (1 self)
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on the occasion of his 65th birthday. Abstract. 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. 1.