q-nonabelianization for line defects

We consider the q-nonabelianization map, which maps links L in a 3-manifold M to combinations of links $$ \tilde{L} $$ L ˜ in a branched N -fold cover $$ \tilde{M} $$ 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 $$ \mathfrak{gl} $$ 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 $$ \tilde{M} $$ M ˜ × ℝ2,1, and put the defects on $$ \tilde{L} $$ 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 $$ \mathcal{N} $$ 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 $$ \tilde{C}\to C $$ C ˜ → C .