Strominger--Yau--Zaslow geometry, Affine Spheres and Painlevé III
arXiv:0809.3015 · doi:10.1007/s00220-009-0861-x
Abstract
We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitzéica 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 Painlevé transcendents.
38 pages. Final version. To appear in Communications in Mathematical Physics