Abelianization of flat connections, and its q-deformation

Abstract: Abelianization of flat connections is a construction motivated by supersymmetric quantum field theory, which has turned out to be connected to various bits of geometry -- in particular, to Donaldson-Thomas theory, cluster algebra, the exact WKB method for analysis of ODEs, and hyperkahler geometry. In some of these subjects it is known that there exists a natural q-deformation which takes us from the commutative to the noncommutative world. This suggests that there ought to exist a q-deformation of abelianization as well. I will explain joint work in progress with Fei Yan on constructing this q-deformation in a geometric way using spectral networks. This construction is inspired by related work by various authors, especially Bonahon-Wong, Gabella, Gaiotto-Witten. One byproduct is a new scheme for computing known polynomial invariants of links in R^3, which generalizes the usual "vertex models".

algebraic geometrysymplectic geometry

Audience: researchers in the topic

M-seminar