Implications of ANEC for SCFTs in four dimensions

Abstract

Abstract We explore consequences of the Averaged Null Energy Condition (ANEC) for scaling dimensions ∆ of operators in four-dimensional N $$ \mathcal{N} $$ = 1 superconformal field theories. We show that in many cases the ANEC bounds are stronger than the corresponding unitarity bounds on ∆. We analyze in detail chiral operators in the 1 2 j 0 $$ \left(\frac{1}{2}j,0\right) $$ Lorentz representation and prove that the ANEC implies the lower bound Δ ≥ 3 2 j $$ \Delta \ge \frac{3}{2}j $$ , which is stronger than the corresponding unitarity bound for j > 1. We also derive ANEC bounds on 1 2 j 0 $$ \left(\frac{1}{2}j,0\right) $$ operators obeying other possible shortening conditions, as well as general 1 2 j 0 $$ \left(\frac{1}{2}j,0\right) $$ 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 N $$ \mathcal{N} $$ = 1 results for multiplets of N $$ \mathcal{N} $$ = 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.

Keywords

• Conformal Field Theory
• Superspaces