$q$-nonabelianization for line defects

Abstract

We consider the $q$-nonabelianization map, which maps links $L$ in a 3-manifold $M$ to links $\widetilde{L}$ in a branched $N$-fold cover $\widetilde{M}$. 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 $M \times \mathbb{R}^{2,1}$, and we consider surface defects placed on $L \times \{x^4 = x^5 = 0\}$; in the IR we have the $(2,0)$ theory of type $\mathfrak{gl}(1)$ on $\widetilde{M} \times \mathbb{R}^{2,1}$, and put the defects on $\widetilde{L} \times \{x^4 = x^5 = 0\}$. In the case $M = \mathbb{R}^3$, $q$-nonabelianization computes the Jones polynomial of a link, or its analogue associated to the group $U(N)$. In the case $M = C \times \mathbb{R}$, 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 $M = C \times \mathbb{R}$, we give a concrete construction of the $q$-nonabelianization map. The construction uses the data of the WKB foliations associated to a holomorphic covering $\widetilde{C} \to C$.