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

TL;DR

This paper derives bounds on parity-violating three-point functions in 3D conformal field theories, showing that certain parameters are confined within specific discs and that large N Chern-Simons theories lie on the boundary.

Contribution

It introduces conformal collider bounds for parity-violating correlators in 3D CFTs, establishing geometric constraints and linking them to large N Chern-Simons theories.

Findings

Parameters are bounded within discs in parameter space.

Large N Chern-Simons theories lie on the boundary circles.

The 't Hooft coupling determines the theories' positions on the boundary.

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 $\langle T T T \rangle$ 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.