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

TL;DR

This paper introduces potential submanifolds in pseudo-Euclidean spaces and demonstrates their equivalence to Frobenius manifolds, revealing new geometric structures and linking them to topological quantum field theories.

Contribution

It establishes that Frobenius manifolds can be represented as potential submanifolds in pseudo-Euclidean spaces, connecting algebraic structures with geometric realizations.

Findings

Frobenius manifolds correspond to potential submanifolds in pseudo-Euclidean spaces.

Potential submanifolds naturally carry Frobenius algebra structures.

The associativity equations are reductions of submanifold fundamental equations.

Abstract

We introduce a 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.