More on the weak gravity conjecture via convexity of charged operators

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.

Convexity, large charge and the large-N phase diagram of the φ4 theory

“…More in general, having a non-perturbative description of the effective potential, we can relate the stability of the model to the convexity of its free energy and -at criticality, via the state-operator correspondence -to the convexity of the conformal dimension ∆(Q) of the lowest operator of given charge. This provides further evidence in favor of the recent proposal by Aharony and Palti related to the weak gravity conjecture [47], confirming the recent large-charge analysis in [48].…”

Repulsive black holes and higher-derivatives

“…Naively, any conjecture about repulsive forces appears to require modification or clarification in this setting -due to the confining potential of AdS space, no two massive objects can be truly self-repulsive at large distances. A very interesting recent idea in this direction is that theories of quantum gravity in AdS should have states with a non-negative binding energy [46,47]. Previous discussions of the WGC in AdS include [13,[48][49][50][51][52][53][54].…”

A Scaling Limit for Line and Surface Defects

“…It would be very interesting to study the convexity properties of the DCFT under the light of [29] (see also [30] for a discussion in a context related to ours). The defect interaction contributes oppositely to the bulk piece, but is suppressed so as to be able to compensate for it.…”

“…One could also wonder wether a similar limit can be defined in gauge theories for Wilson lines in higher representations (for instance, the n-fold symmetric of the fundamental) by taking large n at fixed g 2 Y M n -somewhat similarly to [28]-, and wether such limit would be related in any way to [8,27]. Another interesting aspect to study is the potential relation to the convexity conjecture in [29] (see also [30] for a discussion in a context related to ours) when applied to defects. We leave these aspects for future studies.…”