Special K\"ahler geometry of the Hitchin system and topological recursion

TL;DR

This paper explores the special K"ahler geometry of the Hitchin system's base using spectral curves and topological recursion, revealing new ways to compute geometric invariants and derivatives.

Contribution

It demonstrates how to compute the Taylor expansion of the K"ahler metric via topological recursion and relates key geometric invariants to spectral curve invariants.

Findings

Donagi-Markman cubic computed by $W^{(0)}_3$ invariant

Taylor expansion of the metric obtained from spectral curve invariants

Extended computation of derivatives of the period matrix

Abstract

We investigate the special K\"ahler geometry of the base of the Hitchin integrable system in terms of spectral curves and topological recursion. The Taylor expansion of the special K\"ahler metric about any point in the base may be computed by integrating the $g = 0$ Eynard-Orantin invariants of the corresponding spectral curve over cycles. In particular, we show that the Donagi-Markman cubic is computed by the invariant $W^{(0)}_3$. We use topological recursion to go one step beyond this and compute the symmetric quartic of second derivatives of the period matrix.