Painlevé III A Case Study in the Geometry of Meromorphic Connections

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Synopsis

The purpose of this monograph is two-fold: it introduces a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation of type PIII (D6), called PIII (0, 0, 4, −4), describing its relation to isomonodromic families of vector bundles on P1 with meromorphic connections. This equation is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears widely in geometry and physics. It is used here as a very concrete and classical illustration of the modern theory of vector bundles with meromorphic connections. Complex multi-valued solutions on C* are the natural context for most of the monograph, but in the last four chapters real solutions on R>0 (with or without singularities) are addressed. These provide examples of variations of TERP structures, which are related to tt∗ geometry and harmonic bundles. As an application, a new global picture o0 is given.

Book details

Series:
Lecture Notes in Mathematics (Book 2198)
Author:
Martin A. Guest, Claus Hertling
ISBN:
9783319665269
Related ISBNs:
9783319665252
Publisher:
Springer International Publishing
Pages:
N/A
Reading age:
Not specified
Includes images:
Yes
Date of addition:
2018-10-12
Usage restrictions:
Copyright
Copyright date:
2017
Copyright by:
Springer International Publishing, Cham 
Adult content:
No
Language:
English
Categories:
Mathematics and Statistics, Nonfiction