Crossing symmetry in matter Chern–Simons theories at finite $N$ and $k$

We present a conjecture for the crossing symmetry rules for Chern-Simons gauge theories interacting with massive matter in $2+1$ dimensions. Our crossing rules are given in terms of the expectation values of particular tangles of Wilson lines, and reduce to the standard rules at large Chern-Simons level. We present completely explicit results for the special case of two fundamental and two antifundamental insertions in $SU(N)_k$ and $U(N)_k$ theories. These formulae are consistent with the conjectured level-rank, Bose-Fermi duality between these theories and take the form of a $q=e^{\frac{ 2 \pi i }{\kappa}}$ deformation of their large $k$ counterparts. In the 't Hooft large $N$ limit our results reduce to standard rules with one twist: the $S$-matrix in the singlet channel is reduced by the factor $\frac{\sin \pi \lambda}{\pi \lambda} $ (where $\lambda$ is the 't Hooft coupling), explaining `anomalous' crossing properties observed in earlier direct large $N$ computations.