WDVV Equations from Algebra of Forms

TL;DR

This paper constructs solutions to the WDVV equations using period matrices of hyperelliptic Riemann surfaces, linking algebraic structures to physical theories like Seiberg-Witten and quantum cohomology.

Contribution

It introduces a new algebraic framework for WDVV solutions based on hyperelliptic surfaces, extending previous polynomial-based methods and connecting to supersymmetric string models.

Findings

Solutions derived from hyperelliptic Riemann surfaces.

The algebra of differentials reflects associativity in the WDVV equations.

Potential extensions to quantum cohomology and string theory prepotentials.

Abstract

A class of solutions to the WDVV equations is provided by period matrices of hyperelliptic Riemann surfaces, with or without punctures. The equations themselves reflect associativity of explicitly described multiplicative algebra of (possibly meromorphic) 1-differentials, which holds at least in the hyperelliptic case. This construction is direct generalization of the old one, involving the ring of polynomials factorized over an ideal, and is inspired by the study of the Seiberg-Witten theory. It has potential to be further extended to reveal algebraic structures underlying the theory of quantum cohomologies and the prepotentials in string models with N=2 supersymmetry.