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Strominger–Yau–Zaslow Geometry, Affine Spheres and Painlevé III

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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 \({SL(3, \mathbb{C})}\) 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.

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Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Maciej Dunajski & Prim Plansangkate

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  1. Maciej Dunajski
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  2. Prim Plansangkate
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Correspondence to Maciej Dunajski.

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Communicated by G. W. Gibbons

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Dunajski, M., Plansangkate, P. Strominger–Yau–Zaslow Geometry, Affine Spheres and Painlevé III. Commun. Math. Phys. 290, 997–1024 (2009). https://doi.org/10.1007/s00220-009-0861-x

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  • DOI: https://doi.org/10.1007/s00220-009-0861-x

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