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

Abstract

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 $⟨TTT⟩$ and $⟨jjT⟩$ correlation functions which are unique to three space-time dimensions. Let the parameters which determine the $⟨TTT⟩$ correlation function be $t_4$ and ${\alpha }_T$ , where ${\alpha }_T$ is the parity-violating contribution. Similarly let the parameters which determine $⟨jjT⟩$ 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.

Minor typos corrected, Figures changed , References added