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.

中文翻译：

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ANEC 对 SCFT 的四个维度的影响

我们探讨了平均零能量条件 (ANEC) 对四维 N $$ \mathcal{N} $$ = 1 超共形场理论中算子的缩放维数 Δ 的影响。我们表明，在许多情况下，ANEC 边界强于 ∆ 上相应的幺正性边界。我们详细分析了 1 2 j 0 $$ \left(\frac{1}{2}j,0\right) $$ Lorentz 表示中的手性算子，并证明了 ANEC 意味着下界 Δ ≥ 3 2 j $ $ \Delta \ge \frac{3}{2}j $$ ，它比 j > 1 对应的幺正界更强。我们还推导出了 1 2 j 0 $$ \left(\frac{1} {2}j,0\right) $$ 运算符遵守其他可能的缩短条件，以及一般 1 2 j 0 $$ \left(\frac{1}{2}j,0\right) $$ 运算符不遵守任何缩短条件。在这两种情况下，我们发现它们通常比相应的幺正界限更强。最后，我们阐明了运算符维度约束，这些约束来自 N $$ \mathcal{N} $$ = 1 结果的多个 N $$ \mathcal{N} $$ = 2 ，4 个四维超共形理论。通过将 ANEC 重新定义为凸优化问题并使用标准的半定规划方法，我们能够改进文献中有关非超对称情况的先前分析。