Inventiones mathematicae

, Volume 141, Issue 1, pp 55–147 | Cite as

Monodromy of certain Painlevé–VI transcendents and reflection groups

  • B. Dubrovin
  • M. Mazzocco
Article

Abstract.

We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β=γ=0, δ=1/2 and 2α=(2μ-1)2 with arbitrary μ, 2μ≠∈ℤ. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • B. Dubrovin
    • 1
  • M. Mazzocco
    • 1
  1. 1.International School of Advanced Studies, via Beirut 2-4, 34014 Trieste, Italy¶(e-mail: dubrovin@sissa.it, mazzocco@sissa.it)Italy

Personalised recommendations