Abstract
The Weak Gravity Conjecture has recently been re-formulated in terms of a particle with non-negative self-binding energy. Because of the dual conformal field theory (CFT) formulation in the anti-de Sitter space, the conformal dimension ∆(Q) of the lowest-dimension operator with charge Q under some global U(1) symmetry must be a convex function of Q. This property has been conjectured to hold for any (unitary) conformal field theory and generalized to larger global symmetry groups. Here we refine and further test the convex charge conjecture via semiclassical computations for fixed charge sectors of different theories in various dimensions. We analyze the convexity properties of the leading and next-to-leading order terms stemming from the semiclassical computation, de facto, extending previous tests beyond the leading perturbative contributions and to arbitrary charges. In particular, the leading contribution is sufficient to test convexity in the semiclassical computations. We also consider intriguing cases in which the models feature a transition from real to complex conformal dimensions either as a function of the charge or number of matter fields. As a relevant example of the first kind, we investigate the O(N) model in 4 + ϵ dimensions. As an example of the second type, we consider the U(N) × U(M) model in 4 − ϵ dimensions. Both models display a rich dynamics where, by changing the number of matter fields and/or charge, one can achieve dramatically different physical regimes. We discover that whenever a complex conformal dimension appears, the real part satisfies the convexity property.
Highlights
Conjecture is suggested to hold for general conformal field theory (CFT) and it has been extensively tested [5] via perturbation theory, large N and semiclassical expansions
We discover that whenever a complex conformal dimension arises, the real part satisfies the convexity property
The reason can be understood from our discussion above: at large charge one may rely on expectation from large charge effective field theory (EFT) to derive the convexity of the scaling dimension, while at small values of charge conventional perturbation theory can be used to reliably determine the convexity property
Summary
2.1 Mathematical preparation The convex charge conjecture proposed in ref. [5] involves functional inequalities of the form f (x + y) ≥ f (x) + f (y). If f (Q) is nonnegative in J [39] In using this criterion, one should note that f (Q) is required to be twice differentiable in the whole open interval J. One should note that f (Q) is required to be twice differentiable in the whole open interval J If this is not satisfied, for example due to some kink, this criterion is not applicable. When we examine the real part of the scaling dimension of fixed-charge operators in certain non-unitary theories we do expect such kinks at critical values of the charge. In such cases, one should resort to definitions at a more basic level to determine whether the functions involved are convex
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