Seiberg-Witten Theory and Topological Recursion

TL;DR

This paper explores the connection between Seiberg-Witten curves and topological recursion, extending formulas for prepotentials from Hitchin systems to more general families of curves within foliated symplectic surfaces.

Contribution

It generalizes the formula relating Seiberg-Witten prepotentials to topological recursion from Hitchin systems to broader classes of curves.

Findings

Derived a new formula linking Seiberg-Witten prepotential to genus zero topological recursion.

Extended existing formulas to include curves embedded in foliated symplectic surfaces.

Provided geometric insights into the relationship between moduli spaces and topological recursion.

Abstract

Kontsevich-Soibelman (2017) reformulated Eynard-Orantin topological recursion (2007) in terms of Airy structure which provides some geometrical insights into the relationship between the moduli space of curves and topological recursion. In this work, we investigate the analytical approach to this relationship using the Seiberg-Witten family of curves as the main example. In particular, we are going to show that the formula computing the Hitchin systems' Special Kahler's prepotential from the genus zero part of topological recursion as obtained by Baraglia-Huang (2017) can be generalized for a more general family of curves embedded inside a foliated symplectic surface, including the Seiberg-Witten family. Consequently, we obtain a similar formula relating the Seiberg-Witten prepotential to the genus zero part of topological recursion on a Seiberg-Witten curve.