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# Advances in Theoretical and Mathematical Physics

## Volume 27 (2023)

### Number 1

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

Pages: 193 – 310

DOI: https://dx.doi.org/10.4310/ATMP.2023.v27.n1.a5

#### Authors

#### 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 $q = e^{\frac{2 \pi i}{\kappa}}$ deformation of their large $k$ counterparts. In the ’t Hooft large $N$ limit our results reduce to standard rules with one twist: the Smatrix in the singlet channel is reduced by the factor $\frac{\operatorname{sin} \pi \lambda}{\pi \lambda}$ (where $\lambda$ is the ’t Hooft coupling), explaining ‘anomalous’ crossing properties observed in earlier direct large $N$ computations.

Published 13 July 2023