Constraints on parity violating conformal field theories in $d=3$

We derive constraints on three-point functions involving the stress tensor, $T$, and a conserved $U(1)$ current, $j$, in 2+1 dimensional conformal field theories that violate parity, using conformal collider bounds introduced by Hofman and Maldacena. Conformal invariance allows parity-odd tensor-structures for the $langle T T T rangle$ and $ langle j j T rangle$ correlation functions which are unique to three space-time dimensions. Let the parameters which determine the $langle T T T rangle$ correlation function be $t_4$ and $alpha_T$ , where $alpha_T$ is the parity-violating contribution. Similarly let the parameters which determine $ langle j j T rangle$ correlation function be $a_2$, and $alpha_J$ , where $alpha_J$ is the parity-violating contribution. We show that the parameters $(t_4, alpha_T)$ and $(a_2, alpha_J)$ are bounded to lie inside a disc at the origin of the $t_4$ - $alpha_T$ plane and the $a_2$ - $alpha_J$ plane respectively. We then show that large $N$ Chern-Simons theories coupled to a fundamental fermion/boson lie on the circle which bounds these discs. The `t Hooft coupling determines the location of these theories on the boundary circles.