On the WDVV Equation and M-Theory

TL;DR

This paper demonstrates that certain Seiberg-Witten models derived from M-theory, described by non-hyperelliptic Riemann surfaces, possess an associative algebra of holomorphic differentials, advancing understanding of their relation to the WDVV equation.

Contribution

It extends known results about hyperelliptic Seiberg-Witten models to non-hyperelliptic cases constructed via M-theory, establishing an associative algebra structure.

Findings

Existence of an associative algebra of holomorphic differentials in these models

Connection between algebraic structures and M-theory brane configurations

Progress towards proving these models satisfy the WDVV equation

Abstract

A wide class of Seiberg-Witten models constructed by M-theory techniques and described by non-hyperelliptic Riemann surfaces are shown to possess an associative algebra of holomorphic differentials. This is a first step towards proving that also these models satisfy the Witten-Dijkgraaf-Verlinde-Verlinde equation. In this way, similar results known for simpler Seiberg-Witten models (described by hyperelliptic Riemann surfaces and constructed without recourse to M-theory) are extended to certain non-hyperelliptic cases constructed in M-theory. Our analysis reveals a connection between the algebra of holomorphic differentials on the Riemann surface and the configuration of M-theory branes of the corresponding Seiberg-Witten model.