IST Lecture Series in Algebraic Geometry & Physics

Stokes Phenomenon and Dynamics on Wild Character Varieties of Painlevé Equations (I)

Painlevé equations were discovered at the beginning of XX-th century by Paul Painlevé for purely mathematical reasons. Their solutions, the Painlevé transcendents, are, in general, "new transcendental functions" and, as the classical special functions, they appear in many problems of mathematics and physics. Applications "exploded" at the end of XX-th century: Einstein metrics, Frobenius manifolds, correlation function of the $2$-dimensional Ising model, reduction of integrable PDEs, reduction of self-dual Yang-Mills equations, random matrix theory, $2$-dimensional CFT (conformal blocks), non perturbative effects in strings theory ($2$d quantum gravity)...

First Lecture

Firstly, we will recall basics about Painlevé equations. Each Painlevé vector field initially defined on a trivial bundle of fibre $C^2$ (the naive phase space) can be extended to a fiber bundle (the Okamoto bundle) whose fiber is the Okamoto space of initial conditions: an affine rational surface endowed with a canonical symplectic structure. On this extension the Painlevé vector field is complete.

We will detail the "simplest case": Painlevé VI. The basis of the Okamoto bundle is $C\setminus \{0,1\}$ and we have a non-linear monodromy (with two generators) which induces a dynamics on each Okamoto space.

Painlevé VI can be interpreted as a traduction of the isomonodromic deformations of some linear second order equations (the linearized equations). The space of monodromy data of the linearized equations is an affine cubic surface: the character variety of Painlevé VI. It is endowed with a canonical symplectic structure.

For generic values of the parameters the character variety is non-singular and there exists an analytic (symplectic) diffeomorphism, the Riemann-Hilbert map (RH) between each Okamoto variety and the character variety. The dynamics on the Okamoto variety (non linear monodromy) is conjugated by RH to a dynamics on the character variety. An essential result is that this last dynamics is algebraic and can be explicitely calculated. As a consequence it is possible to prove that (generically) the dynamics of PVI is "rich" (chaotic...) and that the Galois-Malgrange differential groupoid is "as big as possible" (in particular PVI is not integrable!).

The main purpose of the minicourse is to describe a generalization of this picture to the others Painlevé equations (it is a work in progress...).

For PVI the linearized equation is Fuchsian. The character variety is the set of (classical) monodromy representations up to equivalence.

In the other cases the linearized equation has (as the Painlevé equation itself) irregular singularities.

There are Stokes phenomena and it is necessary to "add" in some sense these phenomena to the classical monodromy data. The corresponding character variety is the set of the extended (wild) monodromy data up to equivalence. It is also a cubic surface.

The non-linear monodromy remains but it is a "poor information". The induced dynamics on the Okamoto variety is "too small". Some years ago (2012), I proposed to define a "better dynamics", the wild dynamics. The idea is to generalize the wild dynamics I defined before in the linear case: this dynamics is generated by the classical dynamics, the Stokes phenomena and some continuous tori actions (exponential tori actions).

In the Painlevé case the definition of the non-linear Stokes phenomena and of the exponential tori actions are far to be trivial... In 2012 they were conjectural.The rigorous approach is due to A. Bittmann and quite recent: 2016.

The wild dynamics induces, via RH, a (symplectic) dynamics on the character variety. This dynamics is a priori local. I conjectured that it is in fact rational (2012).

In the second part of the first lecture I will detail the case of PII. I will give a simple and totally explicit description (partly conjectural...) of the wild dynamics on the character variety. An essential point is that it is rational.

The $9$ lines on the character variety of PII play an essential role in the wild dynamics and they are related to some special solutions (Boutroux truncated, tritrucated, bitruncated solutions). There are also related to the resurgence (in Ecalle spirit) of PII. An heuristic principle follows: the lines on the character varieties (affine cubic surfaces) play a central role in the theory of Painlevé equations.

Funded under FCT project UIDB/MAT/04459/2020.