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

Abstract

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 2πi κ 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 sin πλ πλ (where λ is the 't Hooft coupling), explaining 'anomalous' crossing properties observed in earlier direct large N computations.