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We present a conjecture for the crossing symmetry rules for Chern-Simonsgauge theories interacting with massive matter in $2+1$ dimensions. Ourcrossing rules are given in terms of the expectation values of particulartangles of Wilson lines, and reduce to the standard rules at large Chern-Simonslevel. We present completely explicit results for the special case of twofundamental 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 Hooftlarge $N$ 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.

Crossing Symmetry in Matter Chern-Simons Theories at finite $N$ and $k$