Appears in collection : Resurgence in Mathematics and Physics

The talk is based upon a joint work with Y. OHYAMA and J. SAULOY. Classically the space of Monodromy data (or character variety) of PV I (the sixth Painlevé differential equation) is the space of linear representations of the fundamental group of a 4-punctured sphere up to equivalence of representations. If one fixes the local representation data it “is” a cubic surface. We will describe a q-analog: the space of q-Monodromy data of the q-Painlevé V I equation. For the q-analogs of the Painlevé equations (which are non-linear q-difference equations), according to H. SAKAI work, “everything” is well known on the “left side” of the (q-analog of the) Riemann-Hilbert map (the varieties of “initial conditions”), but the “right side” (the q-analogs of the spaces of Monodromy data or character varieties) remained quite mysterious.We will present a complete description of the space of Monodromy data of q−PV I (some local data being fixed). It is a “modification” of an elliptic surface and we will explicit some “natural” parametrizations. This surface is analytically, but not algebraically isomorphic to the Sakai surface of ”initial conditions”. Our description uses a new tool, the Mano decompositions, which are a q-analog of the classical pants decompositions of surfaces. We conjecture that our constructions can be extended to the others q-Painlevé equations. This involves q-Stokes phenomena.