The weak gravity conjecture is typically stated as a bound on the mass-to-charge ratio of a particle in the theory. Alternatively, it has been proposed that its natural formulation is in terms of the existence of a particle which is self-repulsive under all long-range forces. We propose a closely related, but distinct, formulation, which is that it should correspond to a particle with non-negative self-binding energy. This formulation is particularly interesting in anti–de Sitter space, because it has a simple conformal field theory (CFT) dual formulation: let $\mathrm{\Delta }(q)$ be the dimension of the lowest-dimension operator with charge $q$ under some global $U(1)$ symmetry, then $\mathrm{\Delta }(q)$ must be a convex function of $q$. This formulation avoids any reference to holographic dual forces or even to locality in spacetime, and so we make a wild leap, and conjecture that such convexity of the spectrum of charges holds for any (unitary) conformal field theory, not just those that have weakly coupled and weakly curved duals. This charge convexity conjecture, and its natural generalization to larger global symmetry groups, can be tested in various examples where anomalous dimensions can be computed, by perturbation theory, $1/N$ expansions and semiclassical methods. In all examples that we tested we find that the conjecture holds. We do not yet understand from the CFT point of view why this is true.

中文翻译：

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CFTs 中带电算子的凸性和弱引力猜想

弱引力猜想通常被表述为理论中粒子质荷比的界限。或者，有人提出它的自然公式是根据一个粒子的存在，该粒子在所有长程力下都具有自斥性。我们提出了一个密切相关但又截然不同的公式，即它应该对应于具有非负自结合能的粒子。这个公式在反德西特空间中特别有趣，因为它有一个简单的共形场论 (CFT) 对偶公式：让$\mathrm{\Delta }(q)$ 是带电荷的最低维算子的维数 $q$ 在一些全球 $你(1)$ 对称性，那么 $\mathrm{\Delta }(q)$ 必须是一个凸函数 $q$. 这个公式避免了对全息对偶力甚至时空局部性的任何提及，因此我们进行了一次疯狂的飞跃，并推测电荷谱的这种凸性适用于任何（酉）共形场理论，而不仅仅是那些具有弱耦合和弱弯曲的对偶。这种电荷凸性猜想及其对更大全局对称群的自然推广，可以在各种例子中进行测试，其中可以通过微扰理论计算异常维度，$1/N$扩展和半经典方法。在我们测试的所有例子中，我们发现这个猜想成立。从 CFT 的角度来看，我们还不明白为什么这是真的。