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 Funktsional. Anal. i Prilozhen., 2006, Volume 40, Issue 1, Pages 14–29 (Mi faa15)

Nonlocal Hamiltonian Operators of Hydrodynamic Type with Flat Metrics, Integrable Hierarchies, and the Associativity Equations

O. I. Mokhov

Landau Institute for Theoretical Physics, Centre for Non-linear Studies

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.

DOI: https://doi.org/10.4213/faa15

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English version:
Functional Analysis and Its Applications, 2006, 40:1, 11–23

Bibliographic databases:

UDC: 517.9

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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa15
• https://doi.org/10.4213/faa15
• http://mi.mathnet.ru/eng/faa/v40/i1/p14

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. I. Mokhov, “Systems of integrals in involution and associativity equations”, Russian Math. Surveys, 61:3 (2006), 568–570
2. 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
3. O. I. Mokhov, “Duality in a special class of submanifolds and Frobenius manifolds”, Russian Math. Surveys, 63:2 (2008), 378–380
4. O. I. Mokhov, “Realization of Frobenius Manifolds as Submanifolds in Pseudo-Euclidean Spaces”, Proc. Steklov Inst. Math., 267 (2009), 217–234
5. 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.
6. 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
7. Kath I., Nagy P.-A., “A Splitting Theorem for Higher Order Parallel Immersions”, Proc. Amer. Math. Soc., 140:8 (2012), 2873–2882
8. Mikhail B. Sheftel, Devrim Yazici, “Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański”, SIGMA, 12 (2016), 091, 17 pp.
9. 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
10. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937
11. 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
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