q-nonabelianization for line defects

Abstract: I will talk about joint work with Andrew Neitzke on the q-nonabelianization map, which maps links L in a 3-manifold M to links L' in a branched N-fold cover M'. From the quantum field theory point of view, qnonabelianization is an UV-IR map relating two different kinds of defects. In the UV, we consider the 6d (2,0) superconformal field theory of type gl(N) on M × R2,1, with surface defects inserted along L× Rt where Rt is the time direction. In the IR, we have the 6d (2,0) theory of type gl(1) on M' × R2,1 with surface defects inserted along L'× Rt . In two special cases, q-nonabelianization map computes familiar objects. When M=R3, the map gives Jones polynomial of the link L; when M=C×R where the projection of L on C is homologically nontrivial and doesn't contain crossings, the map computes protected spin character for framed BPS states in 4d theories of class-S. In this talk, I will give a concrete construction of the qnonabelianization map in the case of N=2 and M=C×R, using WKB foliation data associated with a holomorphic covering C'—>C.

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