Strominger--Yau--Zaslow geometry, Affine Spheres and Painlev\'e III

TL;DR

This paper characterizes affine sphere equations and their relation to Painlevé III through gauge invariance, linking geometric structures to integrable systems and explicit Calabi-Yau metrics.

Contribution

It provides a gauge invariant characterization of affine sphere equations as reductions of anti-self-dual Yang--Mills equations and connects these to Painlevé transcendents via explicit metric constructions.

Findings

Explicit expression for semi-flat Calabi--Yau metrics in terms of affine sphere solutions

Identification of affine sphere equations as reductions of Hitchin equations

Connection between Painlevé III transcendents and special Calabi--Yau metrics

Abstract

We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitz\'eica equation as reductions of real forms of $SL(3, \C)$ anti--self--dual Yang--Mills equations by two translations, or equivalently as a special case of the Hitchin equation. We use the Loftin--Yau--Zaslow construction to give an explicit expression for a six--real dimensional semi--flat Calabi--Yau metric in terms of a solution to the affine-sphere equation and show how a subclass of such metrics arises from 3rd Painlev\'e transcendents.