Abstract
We consider the q-nonabelianization map, which maps links L in a 3-manifold M to combinations of links \( \tilde{L} \) in a branched N -fold cover \( \tilde{M} \). 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 \( \mathfrak{gl} \)(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 gl (1) on \( \tilde{M} \) × ℝ2,1, and put the defects on \( \tilde{L} \) × {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 \( \mathcal{N} \) = 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 \( \tilde{C}\to C \).
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Neitzke, A., Yan, F. q-nonabelianization for line defects. J. High Energ. Phys. 2020, 153 (2020). https://doi.org/10.1007/JHEP09(2020)153
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DOI: https://doi.org/10.1007/JHEP09(2020)153