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22
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.
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
On structure constants of sl(2) theories
 Nucl. Phys. B
, 1995
"... Structure constants of minimal conformal theories are reconsidered. It is shown that ratios of structure constants of spin zero fields of a nondiagonal theory over the same evaluated in the diagonal theory are given by a simple expression in terms of the components of the eigenvectors of the adjace ..."
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Cited by 7 (1 self)
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Structure constants of minimal conformal theories are reconsidered. It is shown that ratios of structure constants of spin zero fields of a nondiagonal theory over the same evaluated in the diagonal theory are given by a simple expression in terms of the components of the eigenvectors of the adjacency matrix of the corresponding Dynkin diagram. This is proved by inspection, which leads us to carefully determine the signs of the structure constants that had not all appeared in the former works on the subject. We also present a proof relying on the consideration of lattice correlation functions and speculate on the extension of these identities to more complicated theories. 10/94 to be submitted to Nuclear Physics B
Compatible Dubrovin–Novikov Hamiltonian operators, Lie derivative and integrable systems of hydrodynamic type
"... In the present paper, we prove that a local Hamiltonian operator of hydrodynamic type K ij 1 (Dubrovin–Novikov Hamiltonian operator [1]) is compatible with a nondegenerate local ..."
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Cited by 5 (1 self)
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In the present paper, we prove that a local Hamiltonian operator of hydrodynamic type K ij 1 (Dubrovin–Novikov Hamiltonian operator [1]) is compatible with a nondegenerate local
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.
On quantum symmetries of the non–ADE graph F4
, 2004
"... We describe quantum symmetries associated with the F4 Dynkin diagram. Our study stems from an analysis of the (Ocneanu) modular splitting equation applied to a partition function which is invariant under a particular congruence subgroup of the modular group. CPT2004/P.045 1 ..."
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Cited by 2 (1 self)
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We describe quantum symmetries associated with the F4 Dynkin diagram. Our study stems from an analysis of the (Ocneanu) modular splitting equation applied to a partition function which is invariant under a particular congruence subgroup of the modular group. CPT2004/P.045 1