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24
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 45 (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
 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.
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 6 (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
On almost duality for Frobenius manifolds
, 2004
"... 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 theo ..."
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Cited by 5 (1 self)
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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.
Solutions to WDVV from generalized DrinfeldSokolov hierarchies, www.arxiv.org mathph/0003020
, 2003
"... The dispersionless limit of generalized DrinfeldSokolov hierarchies associated to primitive regular conjugacy class of Weyl groupW (g) is discussed. The map from these generalized Drinfeld Sokolov hierarchies to algebraic solutions to WDVV equations has been constructed. Example of g = D4 and [w] ..."
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Cited by 4 (0 self)
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The dispersionless limit of generalized DrinfeldSokolov hierarchies associated to primitive regular conjugacy class of Weyl groupW (g) is discussed. The map from these generalized Drinfeld Sokolov hierarchies to algebraic solutions to WDVV equations has been constructed. Example of g = D4 and [w] = D4(a1) is considered in details and corresponding Frobenius structure is found. 1 Introduction and
COMPATIBLE FLAT METRICS
, 2002
"... We solve the problem of description for nonsingular pairs of compatible flat metrics in the general Ncomponent case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are foun ..."
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Cited by 4 (1 self)
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We solve the problem of description for nonsingular pairs of compatible flat metrics in the general Ncomponent case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. The integrating of these equations is based on reducing to a special nonlinear differential reduction of the Lamé equations and using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).