Implications of ANEC for SCFTs in four dimensions

TL;DR

This paper investigates how the Averaged Null Energy Condition (ANEC) imposes stronger bounds on operator dimensions in 4D superconformal field theories than traditional unitarity bounds, using convex optimization techniques.

Contribution

It derives new, stronger ANEC bounds on operator dimensions in 4D $ ext{SCFTs}$ and extends these results to theories with extended supersymmetry, improving previous bounds via semidefinite programming.

Findings

ANEC bounds are often stronger than unitarity bounds for operator dimensions.

Derived explicit lower bounds on $ ext{SCFT}$ operator dimensions using ANEC.

Improved analysis of the nonsupersymmetric case with convex optimization methods.

Abstract

We explore consequences of the Averaged Null Energy Condition (ANEC) for scaling dimensions $\Delta$ of operators in four-dimensional $\mathcal{N}=1$ superconformal field theories. We show that in many cases the ANEC bounds are stronger than the corresponding unitarity bounds on $\Delta$. We analyze in detail chiral operators in the $(\frac12 j,0)$ Lorentz representation and prove that the ANEC implies the lower bound $\Delta\ge\frac32j$, which is stronger than the corresponding unitarity bound for $j>1$. We also derive ANEC bounds on $(\frac12 j,0)$ operators obeying other possible shortening conditions, as well as general $(\frac12 j,0)$ operators not obeying any shortening condition. In both cases we find that they are typically stronger than the corresponding unitarity bounds. Finally, we elucidate operator-dimension constraints that follow from our $\mathcal{N}=1$ results for multiplets of $\mathcal{N}=2,4$ superconformal theories in four dimensions. By recasting the ANEC as a convex optimization problem and using standard semidefinite programming methods we are able to improve on previous analyses in the literature pertaining to the nonsupersymmetric case.