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

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.

arXiv comment38 pages. Final version. To appear in Communications in Mathematical Physics

References in corpus (4)

• Mirror Symmetry is T-Duality
• Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability
• Hyper-complex four-manifolds from the Tzitzéica equation
• Tzitzéica transformation is a dressing action

Cited by in corpus (5)

• Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painleve' Equation: II
• Abelian vortices from Sinh--Gordon and Tzitzeica equations
• Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results
• Long-time asymptotics of the Tzitzéica equation on the line
• Scalar--flat Kähler metrics with conformal Bianchi V symmetry