On Frobenius structures in symmetric cones

TL;DR

This paper demonstrates the existence of Frobenius structures satisfying the WDVV equation in symmetric cones across various algebraic and geometric settings, extending previous results and unifying algebraic, geometric, and analytic methods.

Contribution

It establishes the existence of Frobenius manifolds within symmetric cones over real division algebras, generalizing prior work and combining algebraic, geometric, and analytic approaches.

Findings

Existence of Frobenius structures in symmetric cones.

Extension to pseudo-Riemannian and Lorentzian geometries.

Unification of algebraic, geometric, and analytic methods.

Abstract

We prove that in any strictly convex symmetric cone $\Omega$ there exists a non empty locus where the WDVV equation is satisfied (i.e. there exists a hyperplane being a Frobenius manifold). This result holds over any real division algebra (with a restriction to the rank 3 case if we consider the field $\mathbb{O}$) but also on their linear combinations. This theorem holds as well in the case of pseudo-Riemannian geometry, in particular for a Lorentz symmetric cone of Anti-de-Sitter type. Our statement can be considered as a generalisation of a result by Ferapontov--Kruglikov--Novikov and Mokhov. Our construction is achieved by merging two different approaches: an algebraic/geometric one and the analytic approach given by Calabi in his investigations on the Monge--Amp\`ere equation for the case of affine hyperspheres.