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The Exact WKB Method as Abelianisation of Connections
Fokko du Cloux (Institut Camille Jordan)
Fokko du Cloux
Institut Camille Jordan
Université Lyon 1,
21 av. Claude Bernard,
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.