Séminaire Physique mathématique ICJ

The Exact WKB Method as Abelianisation of Connections

by Prof. Nikita Nikolaev

Fokko du Cloux (Institut Camille Jordan)

Fokko du Cloux

Institut Camille Jordan

Université Lyon 1, Bât. Braconnier, 21 av. Claude Bernard, 69100 Villeurbanne
I will propose a geometric formalism for the WKB method in terms of searching for an invariant splitting of a given oper (i.e., a meromorphic connection on a filtered vector bundle). I will explain why this seems to be the correct formalism by considering the special case of an oper which arises  from a Schrödinger equation. An advantage of this point of view on the WKB method is being able to relate this operation to a much more general operation on meromorphic connections, called abelianisation. Abelianisation has a nontrivial history, but it came into spotlight in the last decade in the context of supersymmetric gauge theories through the work of Gaiotto-Moore-Neitzke on spectral networks. Recently, I have begun developing the mathematical theory of abelianisation and spectral networks. I will explain how abelianisation provides a vast generalisation of the exact WKB method. This talk is based on [arXiv:1902.03384], [arXiv:1909.04011], and a few more papers in preparation, including one joint with M. Gualtieri, K. Iwaki, A. Neitzke.
Organized by

Nguyen-Viet Dang