Affine geometry and Frobenius algebra

TL;DR

This paper establishes a geometric interpretation of the WDVV equation by linking it to the vanishing of sectional K-curvature in Frobenius manifolds, connecting algebraic and curvature properties.

Contribution

It shows that the associativity condition of Frobenius manifolds is equivalent to zero sectional K-curvature, providing a new curvature-based perspective on the WDVV equation.

Findings

Associativity of Frobenius manifold multiplication is equivalent to vanishing sectional K-curvature.

The WDVV equation corresponds to the condition of zero sectional K-curvature.

Provides a curvature interpretation for the algebraic WDVV equation.

Abstract

The associativity of the multiplication on a Frobenius manifold is equivalent to the WDVV equation of a symmetric cubic form in flat coordinates. Frobenius manifold could be regarded a very special type of statistical manifold. There is a natural commutative product on each tangent space of a statistical manifold. We show that it is associative, hence making it into a manifold with Frobenius algebra structure, if and only if the sectional $K$-curvature vanishes. In other words, WDVV equation is equivalent to zero sectional $K$-curvature. This gives a curvature interpretation for WDVV equation.