Strominger--Yau--Zaslow geometry, Affine Spheres and Painlevé III

Details Strominger--Yau--Zaslow geometry, Affine Spheres and Painlevé III Strominger--Yau--Zaslow geometry, Affine Spheres and Painlevé III

2008-09-17

arxiv.org/abs/0809.3015v4

Commun.Math.Phys.290:997-1024,2009

Mathematics

Differential Geometry

38 pages. Final version. To appear in Communications in Mathematical Physics

Scientific paper

10.1007/s00220-009-0861-x

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.

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