The Painlevé III equation of type (0,0,4,-4), its associated vector bundles with isomonodromic connections, and the geometry of the movable poles

Abstract

The paper is about a Painlevé III equation of type P III (D 6 ) and its relation to isomonodromic families of vector bundles on P 1 with meromorphic connections.The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation of type P III (D 6 ), which we denote by P III (0, 0, 4, −4). This is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears very widely in geometry and physics.The length of the paper is due to the systematic development of the material in the language of vector bundles with meromorphic connections, together with their additional structures which take care of all relevant symmetries. Our motivation is to explain in a leisurely fashion the language of this general theory, by means of a concrete example.Complex multi-valued solutions on C * are the natural context for most of the paper, but in the last three chapters real solutions on R >0 (with or without singularities) are addressed. The vector bundles appearing there can be regarded as mixing holomorphic and antiholomorphic structures. They are generalizations of variations of Hodge structures, called TERP structures (for Twistor Extension Real Pairing) and are related to tt * geometry, and harmonic bundles. The paper gives a complete picture of semisimple rank 2 TERP structures. General results on TERP structures can be applied to study the asymptotics near 0 and ∞ of real solutions.Finally, results about the asymptotics of real solutions near 0 and near ∞ are combined with results on the global geometry of the moduli spaces of initial data and monodromy data. This leads to a new global picture of all zeros and poles of all real solutions of P III (0, 0, 4, −4) on R >0 . ContentsChapter 1. Introduction 1.1. A Painlevé equation and its space of initial data M ini 1.2. The space of monodromy data M mon 1.3. Real solutions of P III (0, 0, 4, −4) on R >0 1.4. User's guide to the results 1.5. Related work on Painlevé III and meromorphic connections 1.6. P 3D6 -TEJPA bundles 1.7. TERP(0) bundles, generalizations of Hodge structures 1.8. Open problems 1.9. Acknowledgements Chapter 2. The Riemann-Hilbert correspondence for P 3D6 bundles Chapter 3. (Ir)Reducibility Chapter 4. Isomonodromic families Chapter 5. Useful formulae: three 2 × 2 matrices Chapter 6. P 3D6 -TEP bundles Chapter 7. P 3D6 -TEJPA bundles and moduli spaces of their monodromy tuples Chapter 8. Normal forms of P 3D6 -TEJPA bundles and their moduli spaces Chapter 9. Generalities on the Painlevé equations Chapter 10. Solutions of the Painlevé equation P III (0, 0, 4, −4) Chapter 11. Comparison with the setting of Its, Novokshenov, and Niles Chapter 12. Asymptotics of all solutions near 0 Chapter 13. Rank 2 TEPA bundles with a logarithmic pole 4 CONTENTS Chapter 14. Symmetries of the universal family of solutions of P III (0, 0, 4, −4) Chapter 15. Three families of solutions on R >0 Chapter 16. TERP structures and P 3D6 -TEP bundles...