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 TMF, 2007, Volume 152, Number 2, Pages 368–376 (Mi tmf6093)

Theory of submanifolds, associativity equations in 2D topological quantum field theories, and Frobenius manifolds

O. I. Mokhovab

a M. V. Lomonosov Moscow State University
b Landau Institute for Theoretical Physics, Centre for Non-linear Studies

Abstract: We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of flat torsionless potential submanifolds. We show that all flat torsionless potential submanifolds in pseudo-Euclidean spaces bear natural structures of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (in fact, the structural constants are given by the set of the Weingarten operators of the submanifolds). We prove that each $N$-dimensional Frobenius manifold can be locally represented as a flat torsionless potential submanifold in a $2N$-dimensional pseudo-Euclidean space. By our construction, this submanifold is uniquely determined up to motions. Moreover, we consider a nonlinear system that is a natural generalization of the associativity equations, namely, the system describing all flat torsionless submanifolds in pseudo-Euclidean spaces, and prove that this system is integrable by the inverse scattering method.

Keywords: Frobenius manifold, submanifold in a pseudo-Euclidean space, flat submanifold, submanifold with flat normal bundle, flat submanifold with zero torsion, associativity equation in two-dimensional topological quantum field theory, integrable system

DOI: https://doi.org/10.4213/tmf6093

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English version:
Theoretical and Mathematical Physics, 2007, 152:2, 1183–1190

Bibliographic databases:

Citation: O. I. Mokhov, “Theory of submanifolds, associativity equations in 2D topological quantum field theories, and Frobenius manifolds”, TMF, 152:2 (2007), 368–376; Theoret. and Math. Phys., 152:2 (2007), 1183–1190

Citation in format AMSBIB
\Bibitem{Mok07} \by O.~I.~Mokhov \paper Theory of submanifolds, associativity equations in 2D topological quantum field theories, and Frobenius manifolds \jour TMF \yr 2007 \vol 152 \issue 2 \pages 368--376 \mathnet{http://mi.mathnet.ru/tmf6093} \crossref{https://doi.org/10.4213/tmf6093} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2429286} \zmath{https://zbmath.org/?q=an:1134.81405} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2007TMP...152.1183M} \elib{http://elibrary.ru/item.asp?id=9541941} \transl \jour Theoret. and Math. Phys. \yr 2007 \vol 152 \issue 2 \pages 1183--1190 \crossref{https://doi.org/10.1007/s11232-007-0101-5} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000249211500013} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34548457326} 

• http://mi.mathnet.ru/eng/tmf6093
• https://doi.org/10.4213/tmf6093
• http://mi.mathnet.ru/eng/tmf/v152/i2/p368

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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, “Duality in a special class of submanifolds and Frobenius manifolds”, Russian Math. Surveys, 63:2 (2008), 378–380
2. Konopelchenko B.G., “Quantum deformations of associative algebras and integrable systems”, J. Phys. A, 42:9 (2009), 095201, 18 pp.
3. O. I. Mokhov, “Realization of Frobenius Manifolds as Submanifolds in Pseudo-Euclidean Spaces”, Proc. Steklov Inst. Math., 267 (2009), 217–234
4. 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.
5. 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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