This article is cited in 5 scientific papers (total in 5 papers)
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
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.
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
PDF file (393 kB)
Theoretical and Mathematical Physics, 2007, 152:2, 1183–1190
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
\paper Theory of submanifolds, associativity equations in 2D topological quantum field theories, and Frobenius manifolds
\jour Theoret. and Math. Phys.
Citing articles on Google Scholar:
Related articles on Google Scholar:
This publication is cited in the following articles:
O. I. Mokhov, “Duality in a special class of submanifolds and Frobenius manifolds”, Russian Math. Surveys, 63:2 (2008), 378–380
Konopelchenko B.G., “Quantum deformations of associative algebras and integrable systems”, J. Phys. A, 42:9 (2009), 095201, 18 pp.
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.
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
|Number of views:|