Abstract:
We solve the problem of describing all nonlocal Hamiltonian operators of
hydrodynamic type with flat metrics. This problem is equivalent to
describing all flat submanifolds with flat normal bundle in a
pseudo-Euclidean space. We prove that every such Hamiltonian operator (or
the corresponding submanifold) specifies a pencil of compatible Poisson
brackets, generates bihamiltonian integrable hierarchies of hydrodynamic
type, and also defines a family of integrals in involution. We prove that
there is a natural special class of such Hamiltonian operators
(submanifolds) exactly described by the associativity equations of
two-dimensional topological quantum field theory (the
Witten–Dijkgraaf–Verlinde–Verlinde and Dubrovin equations). We show that
each N-dimensional Frobenius manifold can locally be represented by a
special flat N-dimensional submanifold with flat normal bundle in a
2N-dimensional pseudo-Euclidean space. This submanifold is uniquely
determined up to motions.
Citation:
O. I. Mokhov, “Nonlocal Hamiltonian Operators of Hydrodynamic Type with Flat Metrics, Integrable Hierarchies, and the Associativity Equations”, Funktsional. Anal. i Prilozhen., 40:1 (2006), 14–29; Funct. Anal. Appl., 40:1 (2006), 11–23
\Bibitem{Mok06}
\by O.~I.~Mokhov
\paper Nonlocal Hamiltonian Operators of Hydrodynamic Type with Flat Metrics, Integrable Hierarchies, and the Associativity Equations
\jour Funktsional. Anal. i Prilozhen.
\yr 2006
\vol 40
\issue 1
\pages 14--29
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\jour Funct. Anal. Appl.
\yr 2006
\vol 40
\issue 1
\pages 11--23
\crossref{https://doi.org/10.1007/s10688-006-0002-7}
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Linking options:
https://www.mathnet.ru/eng/faa15
https://doi.org/10.4213/faa15
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This publication is cited in the following 14 articles:
Alexander A. Balinsky, Victor A. Bovdi, Anatolij K. Prykarpatski, “On the Quantum Deformations of Associative Sato Grassmannian Algebras and the Related Matrix Problems”, Symmetry, 16:1 (2023), 54
Prykarpatski A.K., Balinsky A.A., “On Symmetry Properties of Frobenius Manifolds and Related Lie-Algebraic Structures”, Symmetry-Basel, 13:6 (2021), 979
Casati M. Lorenzoni P. Vitolo R., “Three Computational Approaches to Weakly Nonlocal Poisson Brackets”, Stud. Appl. Math., 144:4 (2020), 412–448
Prykarpatski A.K., “On the Solutions to the Witten-Dijkgraaf-Verlinde-Verlinde Associativity Equations and Their Algebraic Properties”, J. Geom. Phys., 134 (2018), 77–83
Sheftel M.B., Yazici D., Malykh A.A., “Recursion operators and bi-Hamiltonian structure of the general heavenly equation”, J. Geom. Phys., 116 (2017), 124–139
O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937
Mikhail B. Sheftel, Devrim Yazici, “Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański”, SIGMA, 12 (2016), 091, 17 pp.
Kath I., Nagy P.-A., “A Splitting Theorem for Higher Order Parallel Immersions”, Proc. Amer. Math. Soc., 140:8 (2012), 2873–2882
O. I. Mokhov, “Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type”, Theoret. and Math. Phys., 167:1 (2011), 403–420
O. I. Mokhov, “Realization of Frobenius Manifolds as Submanifolds in Pseudo-Euclidean Spaces”, Proc. Steklov Inst. Math., 267 (2009), 217–234
Sergyeyev A., “Infinite hierarchies of nonlocal symmetries of the Chen-Kontsevich-Schwarz type for the oriented associativity equations”, J. Phys. A, 42:40 (2009), 404017, 15 pp.
O. I. Mokhov, “Duality in a special class of submanifolds and Frobenius manifolds”, Russian Math. Surveys, 63:2 (2008), 378–380
O. I. Mokhov, “Theory of submanifolds, associativity equations in 2D topological quantum field theories, and Frobenius manifolds”, Theoret. and Math. Phys., 152:2 (2007), 1183–1190
O. I. Mokhov, “Systems of integrals in involution and associativity equations”, Russian Math. Surveys, 61:3 (2006), 568–570
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