Theory of Submanifolds, Associativity Equations in 2D Topological Quantum Field Theories, and Frobenius Manifolds

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 potential flat torsionless submanifolds. We show that all potential flat torsionless 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 in this paper that each N -dimensional Frobenius manifold can locally be represented as a potential flat torsionless submanifold in a 2N -dimensional pseudo-Euclidean space. By our construction this submanifold is uniquely determined up to motions. Moreover, in this paper we consider a nonlinear system, which 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. This research was supported by the Max-Planck-Institut für Mathematik (Bonn, Germany), the Russian Science Support Foundation, the Russian Foundation for Basic Research (Grant No. 05-0100170), and the Program of Support for Leading Scientific Schools (Grant No. NSh-4182.2006.1).