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 〈T T T〉 and 〈jjT〉 correlation functions which are unique to three space-time dimensions. Let the parameters which determine the 〈T T T〉 correlation function be t4 and αT , where αT is the parity-violating contribution. Similarly let the parameters which determine 〈jjT〉 correlation function be a2, and αJ , where αJ is the parity-violating contribution. We show that the parameters (t4, αT ) and (a2, αJ ) are bounded to lie inside a disc at the origin of the t4 - αT plane and the a2 - αJ 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.