Andy Neitzke (Yale University) “Abelianization of flat connections, and its q-deformation”

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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”.