Nous organisons un petit groupe de travail pour essayer de mieux comprendre les liens entre la methode de BKW complexe, la correspondance de Hodge nonabelienne sauvage et la recursion toplogique d'Eynard-Orantin (voir au dessous).

Les premiers orateurs seront:

Vendredi 21 mars, 14h10-16h,
P. Boalch, Introduction, connections on curves, nonabelian Hodge theory, Stokes phenomenon

Vendredi 28 mars, 14h10-16h
A. Getmanenko, Variations of the Stokes pattern for an order two equation with a small parameter, after Delabaere-Dillinger-Pham

Vendredi 4 avril, 10h30-12h30
M. Kontsevich, Geometry of integrable systems

Vendredi 4 avril, 14h10-16h
B. Eynard, Topological recursion and WKB formal asymptotic expansion. From ODE's to loop equations, and the notion of "quantum curve". (Abstract)

Vendredi 11 avril, 10h30-12h30
C. Simpson, Asymptotic behavior of monodromy


Amphithéâtre Léon Motchane, IHES


Si vous désirez participer merci de nous envoyer un petit message.

Bien cordialement,
Philip Boalch et Alexander Getmanenko



------------------------------------------------------------

Groupe de travail Non-Perturbatif

------------------------------------------------------------

The basic aim is to try to better understand the relation between exact WKB, wild nonabelian Hodge theory and the topological recursion of Eynard-Orantin, as well as links to the (nonlinear) Stokes phenomenon.

Some background:
Nonabelian Hodge theory gives a canonical, but non-explicit, correspondence between connections on vector bundles on Riemann surfaces, and simpler objects known as Higgs bundles. This theory has been extended to the case of meromorphic connections, and so makes contact with the classical theory of linear differential equations on the Riemann sphere. In turn there is an abelianization picture, which says that a meromorphic Higgs bundle may be described in terms of a line bundle on a spectral curve.

The WKB method studies differential equations with a small parameter, i.e. "close" to being a Higgs bundle, and yields asymptotic information in terms of the spectral curve of the limiting Higgs bundle. There is also a refined version, initiated by Voros, that yields exact information, using Borel-type summation.

The topological recursion associates a list of invariants to a spectral curve, and these can be assembled into a formal solution to a differential operator. Many generating functions of invariants fit into this framework, such as Gromov-Witten invariants, volumes of moduli spaces of curves, Hurwitz numbers etc. String theorists have adopted the term "quantum curve" for such differential operators attached to spectral curves.

Amongst the things we would like to better understand are the following:

1) Does the topological recursion give an explicit way to compute a formal expansion of the nonabelian Hodge correspondence in the neighbourhood of a Higgs bundle? Conversely does the existence of the nonabelian Hodge correspondence imply some sort of summability properties of the recursive generating functions?

2) Can the complex/exact WKB method be interpreted precisely in terms of the nonlinear Stokes phenomenon (on the moduli space of lambda connections)?


Some references:

Arinkin, Moduli of connections with a small parameter on a curve, 2005

Biquard-Boalch, Wild nonabelian Hodge theory on curves, 2003

Dumitrescu-Mulase, Quantum curves for Hitchin fibrations and the Eynard-Orantin theory, arXiv 2013

Eynard-Orantin, Invariants of algebraic curves and topological expansion, arXiv 2007

Gaiotto-Moore-Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, 2013

Getmanenko-Tamarkin, Microlocal properties of sheaves and complex WKB, 2013

Iwaki-Nakanishi, Exact WKB and cluster algebras, arXiv 2014

Kawai-Takei, Algebraic analysis of singular perturbation theory, 2005

Simpson, The Hodge filtration on nonabelian cohomology, arXiv 1996

Voros, Return of the quartic oscillator: the complex WKB method, 1983