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

TL;DR

The paper derives universal bounds on parity-even and parity-odd data in three-dimensional conformal field theories via conformal collider bounds, showing the pairs $(a_2, α_J)$ and $(t_4, α_T)$ are confined to discs with radii 2 and 4, respectively. It then demonstrates that large-$N$ Chern-Simons theories with fundamental matter lie on the corresponding boundary circles, with the circle locations set by the ’t Hooft coupling $\theta$. This saturation is tied to the presence of an infinite tower of higher-spin currents, suggesting a deep link between parity violation, higher-spin symmetry, and holographic dual descriptions. Overall, the work maps out the allowed parity-violating CFT data in $d=3$ and shows precise circle-boundary structures in terms of the CS coupling, guiding future explorations of parity-violating holography and conformal blocks.

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 $α_T$ , where $α_T$ is the parity-violating contribution. Similarly let the parameters which determine $ \langle j j T \rangle$ correlation function be $a_2$, and $α_J$ , where $α_J$ is the parity-violating contribution. We show that the parameters $(t_4, α_T)$ and $(a_2, α_J)$ are bounded to lie inside a disc at the origin of the $t_4$ - $α_T$ plane and the $a_2$ - $α_J$ 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.

Figures (4)

• Figure 1: The space of conformal field theories in $d=3$ obeying the conformal collider bounds is shaded. The $y$-axis denotes the parity odd coefficient of either the 3-pt functions $\langle TTT\rangle$ or $\langle jjT\rangle$, the $x$-axis denotes the parity even coefficient. Large $N$ Chern-Simons theories lie on the boundary of the disc. The position on the disc corresponds to the 't Hooft coupling of these theories. If we choose the convention that $\lambda$ is positive, the top-half circle corresponds to fundamental fermions, and the bottom half corresponds to fundamental bosons.
• Figure 2: Diagram IA and IB
• Figure 3: Diagram IIA and IIB
• Figure 4: Diagram IIIA and IIIB