$q$-nonabelianization for line defects

We consider the $q$-nonabelianization map, which maps links $L$ in a 3-manifold $M$ to 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 $\mathfrak{g}\mathfrak{l}(N)$ on $M\times {\mathbb{R}}^{2,1}$, and we consider surface defects placed on $L \times {x^4 = x^5 = 0}$;intheIRwehavethe$(2,0)$theoryoftype$\mathfrak{gl}(1)$on$\widetilde{M} \times $\mathbb{R}$^{2,1}$, and put the defects on $\tilde{L}\times \{x^4=x^5=0\}$. 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}$$,whentheprojectionof$L$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 $\tilde{C}\to C$.