Abstract
We explore consequences of the Averaged Null Energy Condition (ANEC) for scaling dimensions ∆ 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 ∆. We analyze in detail chiral operators in the left(frac{1}{2}j,0right) Lorentz representation and prove that the ANEC implies the lower bound Delta ge frac{3}{2}j , which is stronger than the corresponding unitarity bound for j > 1. We also derive ANEC bounds on left(frac{1}{2}j,0right) operators obeying other possible shortening conditions, as well as general left(frac{1}{2}j,0right) 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.
Highlights
Introduction and summary of resultsIn recent years attention has been brought to the utility of expectation values of integrated projections of the stress-energy tensor along null lines in conformal field theories (CFTs)
In order to apply the various constraints originating from the Averaged Null Energy Condition (ANEC) to our three-point function in superspace we need to express its components in a basis of nonsupersymmetric three-point functions
In this paper we studied effects of the ANEC on the operator spectrum of CFTs
Summary
In recent years attention has been brought to the utility of expectation values of integrated projections of the stress-energy tensor along null lines in conformal field theories (CFTs). There, it was shown that an energy-positivity condition implies constraints on the coefficients in the three-point function of the stress-energy tensor. Given a state |ψ of a local CFT with stress-energy tensor Tμν and a null geodesic parametrized by λ with tangent vector uμ, the following inequality, called the Averaged Null Energy Condition (ANEC), holds:. It is known that in CFTs scaling dimensions of operators are bounded from below as a consequence of unitarity [11, 12] This is true independently of locality properties of the CFT, i.e. it does not rely on the presence of a stress-energy tensor in the CFT spectrum. The rest of the paper carefully goes through the details of our calculations
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