The tensor product in the theory of Frobenius manifolds

TL;DR

This paper introduces and constructs the tensor product operation for Frobenius manifolds, extending formal results to analytic cases and establishing existence, uniqueness, and initial conditions for tensor products.

Contribution

It extends the tensor product concept to analytic Frobenius manifolds, including structures with Euler fields and flat identities, and proves existence and uniqueness of tensor product diagrams.

Findings

Tensor product of pointed germs of Frobenius manifolds exists.

Existence of tensor product diagrams with factorizable flat identities is proven.

Initial conditions for tensor products of semi-simple Frobenius manifolds are characterized.

Abstract

We introduce the operation of forming the tensor product in the theory of analytic Frobenius manifolds. Building on the results for formal Frobenius manifolds which we extend to the additional structures of Euler fields and flat identities, we prove that the tensor product of pointed germs of Frobenius manifolds exists. Furthermore, we define the notion of a tensor product diagram of Frobenius manifolds with factorizable flat identity and prove the existence such a diagram and hence a tensor product Frobenius manifold. These diagrams and manifolds are unique up to equivalence. Finally, we derive the special initial conditions for a tensor product of semi--simple Frobenius manifolds in terms of the special initial conditions of the factors.