Frobenius Manifolds as a Special Class of Submanifolds in Pseudo-Euclidean Spaces

摘要

We introduce a very natural class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space)and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanifold. We show that all potential submanifolds bear natural special structures of Frobenius algebras on their tangent spaces. These special 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 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 define locally the class of potential submanifolds. The problem of explicit realization of an arbitrary concrete Frobenius manifold as a potential submanifold in a pseudo-Euclidean space is reduced to solving a linear system of second-order partial differential equations. For concrete Frobenius manifolds, this realization problem can be solved explicitly in elementary and special functions. Moreover, 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. We prove that each flat torsionless submanifold in a pseudo-Euclidean space gives a nonlocal Hamiltonian operator of hydrodynamic type with flat metric, a special pencil of compatible Poisson structures, a recursion operator, infinite sets of integrals of hydrodynamic type in involution and a natural class of integrable hierarchies, which are all directly associated with this flat torsionless submanifold. In particular, using our construction of the reduction to the associativity equations, we obtain that each Frobenius manifold (in point of fact, each solution of the associativity equations) gives a natural nonlocal Hamiltonian operator of hydrodynamic type with flat metric, a natural pencil of compatible Poisson structures (local and nonlocal), anatural recursion operator, natural infinite sets of integrals of hydrodynamic type in involution and a natural class of integrable hierarchies, which are all directly associated with this Frobenius manifold.