2020/02/19 by Andrew Neitzke, Fei Yan · 1 citation
Physics and Astronomy · Mathematics · #hep-th #math.GT #math.QA #math.RT
paper · pdf · doi:10.1007/jhep09(2020)153
71 pages, 77 figures
arxiv created 2020/02/19 · arxiv updated 2020/10/28
We consider the q-nonabelianization map, which maps links L in a 3-manifold M to links \widetildeL in a branched N-fold cover \widetildeM. In quantum field theory terms, q-nonabelianization is the UV-IR map relating two different sorts of defect: in the UV we have the six-dimensional (2,0) superconformal field theory of type \mathfrakgl(N) on M × ℝ2,1, and we consider surface defects placed on L × \x4 = x5 = 0\; in the IR we have the (2,0) theory of type \mathfrakgl(1) on \widetildeM × ℝ2,1, and put the defects on \widetildeL × \x4 = x5 = 0\. In the case M = ℝ3, q-nonabelianization computes the Jones polynomial of a link, or its analogue associated to the group U(N). In the case M = C × ℝ, when the projection of L to C is a simple non-contractible loop, q-nonabelianization computes the protected spin character for framed BPS states in 4d N=2 theories of class S. In the case N=2 and M = C × ℝ, we give a concrete construction of the q-nonabelianization map. The construction uses the data of the WKB foliations associated to a holomorphic covering \widetildeC → C.