On Convexity of Charged Operators in CFTs and the Weak Gravity Conjecture

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 $\Delta(q)$ be the dimension of the lowest-dimension operator with charge $q$ under some global $U(1)$ symmetry, then $\Delta(q)$ 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 semi-classical 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.


Introduction
In this note we consider properties of local operators in unitary Conformal Field Theories (CFTs) with continuous global symmetries (in d > 2 space-time dimensions). We propose that such operators should satisfy a certain convexity-like property: Abelian Convex Charge Conjecture: Consider any CFT with a U (1) global symmetry. Denote by ∆ (q) the dimension of the lowest dimension operator of charge q. Then this must satisfy a convex-like constraint ∆ (n 1 q 0 + n 2 q 0 ) ≥ ∆ (n 1 q 0 ) + ∆ (n 2 q 0 ) , (1.1) for any positive integers n 1 , n 2 , for some q 0 of order one.
Equivalently, the conjecture states that the operator product expansion (OPE) of the lowest-dimension operators with positive charges has no singular terms, whenever the charges are integer multiples of q 0 . The U (1) charge can always be normalized to be an integer (with the minimal charge equal to one). As we will discuss below, the non-trivial statement here is that q 0 is not parameterically large in any parameter of the CFT. We propose that this is an exact property of any CFT whose continuous global symmetry is precisely U (1).
There is a natural generalization of the proposal also to theories with larger symmetry groups : Convex Charge Conjecture: Consider any CFT with a continuous global symmetry group G, and consider a simple factor G 0 ⊂ G. Denote by ∆ (r) the dimension of the lowest dimension operator in the representation r of G. Then, there is always some representation r 0 , which is non-trivial in G 0 and has weights of order one, such that the dimensions∆(q) ≡ ∆(Sym q (r 0 )) satisfy a convex-like constraint ∆ (n 1 + n 2 ) ≥∆ (n 1 ) +∆ (n 2 ) , (1.2) for any positive integers n 1 , n 2 . 1 Our proposal is motivated by, but is much more general than, the Weak Gravity Conjecture [1] (WGC) in Anti-de Sitter space. Indeed, we propose that the WGC should be formulated (also in flat space) as a statement about the self-binding energy of a particle: Positive Binding Conjecture: For a (weakly coupled) gravitational theory with a U (1) gauge field, there should exist at least one charged particle in the theory, with charge of order one, which has a non-negative self-binding energy.
By a self-binding energy here we mean the difference of energies between the lowest twoparticle state and twice the energy of the one-particle state. This formulation is closely related to previous formulations, most closely to self-repulsive statements [2,3], but has some important differences which become more pronounced in anti-de Sitter (AdS) space.
Specifically, in AdS space the binding energy receives important contributions from contact terms, and not only from long-range forces.
The organization of this paper is as follows. In section 2 we discuss the formulation of the WGC in terms of binding energy, in flat space and in anti-de-Sitter space. In section 3 we introduce the CFT dual statements, in terms of convexity of the spectrum, and discuss some related details. In section 4 we discuss how the conjecture fits in with what is known generally about CFTs. Our discussion in these sections is mostly for the U (1) case, but it can be generalized in a straightforward manner also to the non-Abelian cases mentioned above (at least in dimensions where the gauge theory in the bulk is IR-free). In section 5 we perform preliminary tests of the conjecture in many different simple examples of CFTs for which the anomalous dimensions are computable in some expansion, and we find that it is always satisfied. We also find that it is not satisfied in theories like the O(N ) model in 4 < d < 6, which are believed not to be unitary. We end with a summary in section 6.

The Weak Gravity Conjecture and Binding Energy
The Swampland program aims to understand constraints on effective theories coming from the requirement of an ultraviolet completion into quantum gravity [4] (see [5,6] for reviews).
One such conjectured constraint is the Weak Gravity Conjecture [1], which places a bound on the mass of a charged particle under a U (1) gauge field -there must be a particle obeying Here m is the mass of the particle, g is the gauge coupling, q its quantised charge, and In [2] (see also [3]) it was argued that the more precise formulation of the WGC should be in terms of the existence of a self-repulsive charged particle. This formulation differs from (2.1) in the presence of massless scalar fields, since they mediate an additional self-attractive long-range force. Specifically, the generalization to this case reads where µ is the coupling of the particle to the scalar fields. This has been tested in string theory, for example see [3,7,8].
One motivation for the self-repulsive statement is that it ensures the absence of a large number n ∼ g −1 of stable bound states formed from copies of the particle with the largest charge-to-mass ratio in the theory [1]. It is as yet unclear why such states would be inconsistent. But we can assume such a requirement and study its consequences.
The reason that a self-attractive charged particle would form a stable bound state with itself (or n copies of itself) is because the charge of the bound state would be twice that of the particle, while the energy of the bound state would be less than twice the mass of the particle due to the binding energy. If that particle is the one with the largest charge-to-mass ratio in the theory, then the bound state cannot decay to it or to any other state in the theory.
If one is motivated by the absence of such stable bound states, then actually the most direct and natural formulation of the WGC is as the Positive Binding Conjecture discussed in the introduction.
In flat space we can apply this statement to a particle placed an infinite distance from its copy. In that case, the binding energy positivity is mapped directly to the particle being self-repulsive under the long-range Coulomb forces 2 , yielding the formulation (2.2). However, global AdS space behaves effectively like a box, so there is a limit as to how far the particle can be displaced from its copy. In particular there is no obvious relation between the longrange force and the binding energy. This makes the binding energy formulation of the WGC somewhat different from both the statements about unstable Reissner-Nordstrom black holes (2.1) and about repulsive long-range forces (2.2). In AdS space there is a contribution from contact interactions to the binding energy, which cannot be dismissed.
Let us make this more precise for the case of 5-dimensional AdS space, following the calculation in [9]. We consider a 5-dimensional gravitational theory with a U (1) gauge field and a scalar field ϕ charged under it with charge q, that has an AdS 5 solution: Here κ is the 5-dimensional Planck scale. We measure every dimensionful quantity in units of the AdS radius R AdS , and set this radius to unity R AdS = 1. The coefficients a and b are arbitrary constants. The action (2.3) captures all the relevant contributions to the self-binding energy of ϕ (up to two-derivative order). Let us denote the self-binding energy of ϕ as γ ϕ 2 . This was calculated at leading order in perturbation theory in [9], which found Here we have introduced ∆ (the dimension of the CFT operator dual to ϕ) and N ∆ as The Positive Binding Conjecture implies that in all UV-complete theories γ ϕ 2 ≥ 0, and there are no counter-examples to this as far as we know. In the case where the action (2.3) is completed into a supersymmetric one, and ϕ is taken as a BPS state, we have the relations This leads to the exact relation γ ϕ 2 = 0. It also manifests explicitly that the contact contribution to the binding energy becomes negligible relative to the other contributions in the ∆ → ∞ limit, where the particle is very heavy compared to the AdS scale.
Note that the action (2.3) is not the most general low-energy effective action, in that it does not include the contribution from additional massless scalar fields. But the main point is that in AdS the contact interactions are important, and so the binding energy formulation of the WGC is significantly different from the one based on black holes or long-range forces.
It is also the one which is precisely saturated with supersymmetry, and so it seems like the more natural one.

A Convex Dimension Conjecture
The AdS/CFT correspondence [10][11][12]  to a statement about CFTs, which should hold at least for CFTs with weakly coupled and weakly curved gravitational duals, and, as we propose, also more generally.
The holographic dual of the WGC in AdS was first explored in [13], where it was formulated as a statement on the ratio of the dimension to the charge of an operator in the CFT (for 4-dimensional CFTs) ∆ 2 q 2 ≤ 9 40 Here ∆ is the dimension of the operator, q its charge, and C T and C V are the coefficients of the two-point functions of the energy-momentum tensor and the global symmetry current, respectively (schematically, these measure the number of degrees of freedom in the CFT, with C T counting all of them and C V only the ones charged under the global U (1)). Subsequently, the holographic dual of the Weak Gravity Conjecture was further developed in [14][15][16][17][18][19][20][21][22].
In [23,24] the holographic dual of the distance conjecture was developed.
In this paper we discussed a different formulation of the WGC in terms of binding energy, and this has a simpler formulation in the dual CFT, as the Convex Charge Conjecture (1.1) that we presented in the introduction. 4 While the motivation for the conjecture comes from CFTs with weakly coupled and weakly curved gravitational duals, we will boldly conjecture that it holds for all CFTs, and perform various tests of this conjecture below 5 .
In the case of weakly-coupled theories, we can formulate the conjecture in terms of specific operators related to the fields appearing in the action. For example, consider a primary scalar operator φ charged under a global symmetry, with charge of order one. Denote by φ n the first (lowest dimension) primary operator appearing in the (symmetrised) OPE of n φ's. Then, the dimension of φ n should satisfy (1.1) (or its generalization (1.2)). We can introduce a more condensed notation as and then write (1.1) or (1.2) as The last statement depends on the boundary conditions, but it is always true for d ≥ 3. In any case the inverse statement is always true; a continuous global symmetry in the CFT is related to a gauge field in the bulk. 4 Note that (1.1) is not quite equivalent to convexity of the function ∆ (q). More precisely, (1.1) follows if ∆ (q) is convex as a function for all real q ≥ 0. But since in the CFT ∆(q) is defined only over the integers (and more precisely, in the conjecture it appears just for integer multiples of q 0 ), it is possible to have ∆ (q) which is not convex as a real function, and still satisfy (1.1). 5 Note that our conjecture is not directly related to previous conjectures on the convexity of the black We will use this notation in the weakly-coupled tests that we will perform below.
There are various extensions of the Weak Gravity Conjecture that have been studied in the literature, see the review [5], which will also play a role here. Specifically, it is possible that when the global symmetry group G contains multiple U (1)'s that can mix together, one has to define a more subtle notion of convexity in the multi-dimensional space of U (1) charges, perhaps along the lines of the analysis of [3,16,26,27]. A possible simple restriction which avoids needing to consider this higher dimensional space is that if there are n U (1)'s, then the conjecture should hold when G 0 is chosen to be one of n specific linearly independent combinations of the U (1)'s. However, we expect that stronger statements are also likely to hold.
We note that all the examples we will discuss below are consistent with a stronger version of our conjecture, which is that the charge q 0 (or the representation r 0 which appears) is the charge of the charged operator with the lowest dimension in the CFT. This is related to stronger versions of the WGC, stating that it should hold for the charged particle with the smallest mass. Such strong versions of the Weak Gravity Conjecture have been studied on the gravity side, starting already from the original paper [1], and we refer to the review [5] for more details. Note that in the case of multiple U (1)'s there is a candidate counterexample to such a statement proposed in [26]. More specifically, it is proposed as a possible Another possible stronger version of the conjecture, which is consistent with all of our examples, is that the spectrum of all charged scalar operators is convex. In particular, q 0 can always be chosen to be the lowest charge of a charged scalar operator.

Application to General CFTs
Before discussing the conjecture for general CFTs, let us begin by reviewing the behavior of the spectrum of operators ∆(q) in general CFTs (see [28] for a recent review), at very large charge q for a U (1) global symmetry (where large here means larger than any other large parameter that the CFT may have). Recall that ∆(q) is the same as the energy of the hole spectrum in flat space made by N. Arkani-Hamed [25], where it was conjectured that m/q is a convex function of the charge q in flat space. 6 We note that another possible strong version of the conjecture would be that q 0 should be associated to the smallest possible charge, but the example in section 5.4 contradicts this, as does the example in [26].
lightest state of charge q in the CFT on S d−1 . It is believed (though not proven) that in all CFTs this behavior falls into one of three classes: • In generic CFTs, one expects the large charge behavior on a sphere to be similar locally to the behavior at large charge density in flat space. In this regime, the low-energy theory includes a Nambu-Goldstone boson related to the spontaneous breaking of the global symmetry. This leads to a universal behavior [29][30][31] of the energy at large q, where the constant A depends on the specific theory (and a specific subleading term in ∆(q) is universal and can be computed explicitly [29,32]).
• In free scalar theories where the scalar φ is charged under the U (1), the operators φ q give ∆(q) = q∆(1). A similar behavior arises also in supersymmetric theories with a moduli space, whenever there is some BPS operator O(x) carrying the global charge which can obtain an expectation value on the moduli space, since this implies ∆(O n ) = n∆(O).
The operators O n are BPS operators in this case, for all n (namely, they do not obey any relations in the chiral ring). Note that generally theories with moduli spaces have a continuous R-symmetry group, and then the charged operators discussed here would always carry some R-symmetry charge (and perhaps also additional charges).
• In free fermionic theories, where the fermion carries U (1) charge, the lightest charged operators are not just products of fermions, but rather because of Pauli's principle they involve products of fermions and their derivatives. This leads to states filling a Fermi sea (similar to the situation in flat space). In this case one also finds ∆(q) = Aq d/(d−1) , where the constant A depends on the type of spinor and on d, but the low-lying excitations over the Fermi sea look different than in the generic case mentioned above.
It is interesting that in this case the function ∆(q) is not analytic even in the large q limit beyond the leading order [33], but this does not affect our analysis.
Note that in all these cases, the large charge spectrum is convex (marginally so in the free scalar / SUSY case), so that for a large enough q 0 (where the large charge expansion in valid) our conjecture is always satisfied. In the second and third cases, we can compute the spectrum exactly and the conjecture is satisfied (for q 0 that is equal to the charge of the free scalar/fermion, or to the lowest charge carried by a BPS operator O that is non-zero on the moduli space). In the fermion case, if we take q 0 equal to the number of components of the fermion field, the spectrum is not even marginally convex [33]. This means that convexity (for that value of q 0 ) is maintained under any small-parameter perturbation of the free fermion theory. Therefore, non-trivial tests of the conjecture in fermionic theories require a large number of fermion fields, for example as studied below in section 5.9.
Thus, the non-trivial question is whether the conjecture is satisfied in the first case, for some q 0 of order one. This conjecture is only non-trivial if the CFT has some large parameter (which can be the inverse of some small parameter), since in this case the large charge behavior described above only sets in when q scales as some power of this large parameter. Our conjecture is that for smaller values of q, even though the large charge expansion is not valid, convexity of the spectrum still holds, with q 0 of order one (namely, not scaling with the large parameter).
The original motivation for the conjecture, discussed above, is relevant for CFTs with a weakly-curved and weakly-coupled gravitational dual (in such theories the large charge expansion mentioned above is related to large black holes in AdS space). But since there is no sharp distinction in the space of CFTs between such CFTs and general CFTs, it is believed that any CFT has in some sense a quantum gravity AdS dual, and it seems reasonable to assume that any conjecture applying to "holographic CFTs" could apply to all CFTs. It is therefore natural to formulate the general version of the conjecture (1.1) (or (1.2)), which would apply to any CFT.
We do not have any direct arguments for this conjecture, but we can check it in various examples (see below), corresponding to various types of expansions where the dimensions ∆(q) are computable, and it seems to always hold. In generic perturbative expansions in some parameter (which can be for instance a small coupling, the space-time dimension, or the inverse number of fields), the leading order correction to ∆(q) − q∆(1) will go as a P (q) for some positive a and for some polynomial P (q) (often proportional to q(q − 1)), whose leading term can be positive or negative. In the first case, convexity holds at the leading order in perturbation theory. Higher orders in perturbation theory cannot change this within the regime of validity of perturbation theory, so one has to go beyond perturbation theory to provide further checks of the conjecture. On the other hand, if the first correction is negative, then the spectrum is not convex up to some q 0 going as a negative power of imply that the first non-BPS operator will obey it (and obviously all the BPS operators do), and in particular the leading perturbative correction will be consistent with the conjecture (assuming that the charge of O is not parameterically large). Thus one has to go beyond perturbation theory to check the full conjecture in these cases.
Note that in theories with such bulk duals, the spectrum naively contains also small extremal black holes (with a size much smaller than the AdS radius). There are then two options. In many cases (and in particular in supersymmetric examples) the extremal black holes are BPS (and saturate the weak gravity conjecture in the flat space limit); in this case the small extremal black holes and the multi-particle BPS states all have a linear spectrum ∆(q) with the same slope. The other option allowed by the weak gravity conjecture in flat space is that the extremal black holes may have a lower charge-to-mass ratio than some light charged particle. In this case if we consider the extremal black hole states, they would not obey the convexity condition (1.1). However, precisely in this case the extremal black holes are also unstable towards decay to light particles, so the lowest-energy states with a specific charge would have energies much smaller than the small extremal black holes, avoiding a contradiction with the conjecture.
Note also that our conjecture about convexity of the spectrum as a function of the charge is not related as far as we can see to the convexity of the spectrum as a function of the spin (at least for large spin), which was analyzed in [34]. In particular, in the case of spin, the twist spectrum goes to a constant in the large spin limit, while in the case of charge, in generic theories ∆(q) grows faster than linearly at large q. It would be interesting to investigate the spectrum as a function of both the charge and the spin, but this is beyond the scope of the current paper (the results of [35] should be useful for this).
We have restricted our conjecture to d > 2, so that it does not apply to two-dimensional CFTs. It is possible to find CFTs in two dimensions which have an integer parameter N , such that for large N , q 0 needs to be taken to scale like N . 7 Note that for two-dimensional CFTs the dual three dimensional bulk physics is very different from higher dimensions, since gravity is not dynamical and gauge fields can be massive. Note that when the lightest charged operator is a scalar, the spectrum in the free theory is linear in the charge, so the first perturbative correction has to take a specific sign for the perturbative spectrum to be convex. On the other hand, when the light charged operators are fermions, the spectrum is convex (growing faster than linearly) already in the free theory, so we do not have non-trivial tests of our conjecture in such cases (except when the number of fermion fields is very large, as in large N gauge theories, such that the faster-than-linear growth of the free fermion theory sets in only at large charges).

The U (1) model in 4 − dimensions
The simplest tests of the conjecture, which we will start from, can be performed in gen- holds. However, we should note that strictly speaking the theories with small are not unitary CFTs by themselves [36], and the extrapolation to = 1 is not under control. So these examples do not provide rigorous tests of the conjecture, but it seems that if we ignore the non-unitarity at non-integer dimensions (which is related to specific operators which vanish as → 0), the conjecture does non-trivially hold in all of these cases.
The simplest non-trivial CFT of this type that we can consider is the U (1) = O(2) model at the Wilson-Fisher fixed point [37]. We follow here the analysis of the anomalous dimension in [38]. The Euclidean action is given by In d = 4 − dimensions there is a Wilson-Fisher fixed point (when the mass is fine-tuned to zero) at a coupling λ (4π) 2 = 5 The dimension of φ n in this model is [38] ∆ This gives which satisfies (3.3) at leading order in . 8

The quartic O(N ) model in 4 − dimensions
More generally, we can consider the O(N ) model. Here we will follow the analysis of [42,43].
The Euclidean Lagrangian is with i = 1, · · · , N . In d = 4 − dimensions the fixed point is at The global group here is non-Abelian, so we need to use the general conjecture (1.2); in particular φ itself is in the fundamental representation of O(N ), and we can test the conjecture where r 0 is taken to be this representation.
In particular, we can consider the operators ϕ n , where ϕ ≡ φ 1 + iφ 2 ; these lie in the n'th symmetric product of the fundamental representation of O(N ). Their dimension is given 8 Note that the expression (5.3) is derived assuming a small expansion parameter n 1, and then the first term in (5.4) always dominates. For large n the theory goes over to the large charge regime discussed in the previous section, which is convex. The spectrum remains convex [39] throughout the continuous interpolation between these two regimes [38] (see also [40,41]). by [43] We therefore have γ n 1 ,n 2 = 2n 1 n 2 N + 8 − n 1 n 2 (N + 8) 3 2 48n 1 + 48n 2 (5.8) and so (3.3) is satisfied at leading order in . fine-tuned to zero), we will follow the analysis of [44,45]. The Euclidean Lagrangian is
This is consistent with (3.3) for all values of n 1 and n 2 (including n 1 = n 2 = 1).

A simple supersymmetric theory
Next, consider the supersymmetric theory with four supercharges, a single chiral superfield Φ, and a superpotential W = gΦ 3 . to be integers, so we should consider three times the U (1) R charge. Note that already in the free limit of this theory (say in d = 4), we can see that the lowest-dimension operator with charge 1, which isψ, has dimension 3/2, so ∆(1) = 3/2, while the lowest-dimension operator with charge 2 is φ which has dimension 1, so ∆(2) = 1. This is a simple example where in the phrasing of the conjecture (1.1) we cannot take q 0 = 1 but we must take q 0 = 2.
For finite coupling (as in the d = 3 fixed point), the operator Φ is a BPS operator so its dimension (and those of φ and ψ) is the same as in the free theory, but the higher Φ n are not BPS operators, so they have positive anomalous dimensions (by the BPS bound).
As discussed in the previous section, the BPS bound means that our conjecture is always satisfied at leading perturbative order in supersymmetric theories, as long as there is at least one BPS operator charged under the group G 0 we are considering (with a charge that is not parameterically large; note that this operator will always carry some R-charge).
We do not know how to compute the anomalous dimensions of φ n for d = 3 (beyond the knowledge that ∆(φ n ) > n∆(φ)). If these can be computed (perhaps by numerical bootstrap methods) they could provide nice tests of our convexity conjecture (e.g. by confirming that We can take the Lagrangian of the d = 4 theory and consider it also in (4− ) dimensions, keeping four-component fermions. The theory is no longer supersymmetric, but it still has just the U (1) R symmetry, and it has a weakly coupled fixed point for g that is controllable in the -expansion, and we can ask if our conjecture it still satisfied. At leading order in the expansion the computation of anomalous dimensions (as a function of g) is the same as in d = 4 where the theory is supersymmetric, so we know that φ has no anomalous dimension (at order g 2 ∝ ) while φ 2 has a positive anomalous dimension. On general grounds, as above, the anomalous dimension of φ n would be proportional to n(n − 1) (as in the previous sections where we had one-loop anomalous dimensions), and we know from the n = 2 case that the coefficient is positive. Thus, our conjecture holds for these theories, but only when we take q 0 = 2.

The quartic
It is possible to formally calculate the dimension also in d = 5 (where the non-trivial fixed point is a UV fixed point rather than an IR fixed point), and this gives [47] This leads to γ n 1 ,n 2 < 0, violating the conjecture. Similarly, one can calculate in 6 − dimensions for σ|φ| 2 theories (which are believed to flow to IR fixed points related to the UV fixed points mentioned above) in the expansion, and this gives [48,49] γ n 1 ,n 2 = −264 which is again in contradiction with the conjecture.
However, in d > 4 (and in particular in d = 5) it is believed that the CFTs appearing in the computations above are non-unitary, related to the non-boundedness of the scalar potential when we flow to these fixed points from well-defined UV theories (see, for example, [47,48,50]  was done in particular in [51][52][53][54], whose results we use 10 . In perturbation theory, the monopoles are specified by Goddard-Nuyts-Olive (GNO) charges, associated to the Cartan of U (N c ), such that the sum of these charges is the charge under U (1) top . The monopole dimension does depend in general on the specific GNO charges, 10 We thank Shai Chester for suggesting this example.
but at large N f this dependence drops out, and the dimension is determined just by the The monopoles also transform as non-trivial representations of SU (N f ), with different dimensional representations for different q's. This comes from the quantization of fermion zero modes in the monopole background. To agree with previous sections we rescale the monopole charge by a factor of 2 compared to the literature, such that it is an integer. In this normalization, the monopoles with q = 1 transform in some SU (N f ) representation with N f /2 boxes (note that N f must be even due to the parity anomaly), and all these representations are degenerate at leading order in 1/N f . Monopoles with higher q's transform in q'th-products of these representations. We can thus take r 0 in the formulation of The results for the monopole dimension ∆ q for charge q are given in table 1 taken from [53] (rescaling the charges by a factor of 2). It is simple to check that they are indeed convex at large N f , and so they satisfy the conjecture (1.1). Table 1: The first few monopole operator dimensions ∆ q for monopole charges q in the U (N c ) gauge theory in 3 dimensions with N f fermions in the fundamental, taken from [53].
Note that for these monopole operators, as well as for the ones mentioned in the next two subsections and in [55][56][57], the large q expansion works quite well all the way down to q = 1 (this was first observed in the context of the expansion of their dimensions in [58]).
The convexity of the spectrum naturally follows from this behavior. However, note that this statement about the validity of the large q expansion is not true for the other examples we discuss in this paper. Presumably it is related to the fact that the dimensions of the lowest-charge monopole operators are parameterically large (of order N f ) in the limit where we can compute them, while the other low-charge operators we discuss have dimensions of order one, and to the fact that the gap to the massive states in the effective theory of the Nambu-Goldstone bosons (expanded around the monopole states) is not small even for q = 1.

U (N c ) Chern-Simons theories in 3 dimensions with fermions
The theories above can be easily generalized by adding a Chern-Simons level k, as long as There is a simple expression for the dimension ∆ q for large κ 1, where κ = k N f (but not necessarily large q), which is [54]  This is convex and satisfies (1.1).
For κ between 0 and 1 2 , the spectrum is independent of κ, at leading order in N f [54]. Therefore, the leading order behaviour is as in table 1. For κ > 1 2 which is not very large, the expression for the dimensions is explicit but complicated; however, the spectrum remains convex as a function of q [54]. Therefore, in any calculable regime, the theory has a convex spectrum satisfying (1.1).
Even though the results above were derived for U (1) gauge group, the same results apply at leading order in 1/N f also for U (N c ).

U (1) Chern-Simons theories in 3 dimensions with scalars
Next, we consider scalar versions of three dimensional QED, involving U (1) gauge theories with N f charged complex scalars, at Chern-Simons level k [54,59,60]. There are two types of such theories which flow to CFTs in the IR (for large enough N f ), those with quartic terms, which are commonly known as the CP N f −1 model, and those where the quartic terms are tuned to zero, which we will refer to as the tri-critical model.
In this case the large κ limit again behaves as in (5.19), but with a different positive prefactor, ∆ q 1 √ 2 (qκ) 3/2 , and so the spectrum is convex. For small κ, there is no degeneracy between different values of κ as there was in the fermionic case, and the spectrum becomes more convex monotonically for increasing κ [54].
In this case the simplest monopoles are SU (N f ) singlets for κ = 0, while otherwise they sit in specific SU (N f ) representations that were analyzed in [54]. For the case of κ = 0 the spectrum for the CP N f and tri-critical models at large N f is given in table 2 as taken from [59,61]. This is seen to be convex in both cases and so satisfies (1.1). One can check [61] that the conjecture holds also for non-zero κ.
q CP N f tri-critical 1 0.125 0.097 2 0.311 0.226 3 0.544 0.384 4 0.816 0.567 5 1.121 0.771 Table 2: Table showing the coefficient of the leading term in the 1/N f expansion, namely the large N f limit of ∆(q)/N f , for the dimension of the first few monopoles, for the CP N f model [53] and the tri-critical model [61]. When we have scalars, we must also consider φ 4 couplings that are generated by the renormalization group flow, and we need to consider fixed points for these couplings in addition to the gauge coupling. This question was analyzed in [65] (see also [66]), where they took N s scalars and a scalar potential of the form where we can take all the ψ's and all theψ's to have the same flavor index (these operators vanish for n > 2N c , requiring adding derivatives to obtain higher-charge operators, but in the large N c limit this is not relevant for our considerations). In the case of N s scalars the global symmetry is SU (N s ) × U (1) B . We can consider again meson operators, of type

Banks-Zaks fixed point in 4 dimensions
where now we choose them to transform in symmetric products of the adjoint representation of SU (N s ). In particular we can choose, for instance, all φ's to have index 1 and all φ * 's to have index 2, and we will implicitly assume this below.
Consider first the scalar case. We are interested in calculating the difference in dimension between the one-meson and two-meson operators At leading order in perturbation theory this is sufficient to determine convexity of the spectrum.
There are two types of one-loop contributions to the scalar-meson anomalous dimensions.
One contribution comes from gluon exchanges, and it turns out that this contribution exactly vanishes when we consider the combination (5.23). The reason for this cancellation between the 1-meson and 2-meson anomalous dimensions can be simply understood as follows. To calculate the 2-meson anomalous dimension we consider the correlator 12 cancel when computing differences such as (5.23). Next, there are those diagrams where the gluon is exchanged between scalars in the same meson, so say between φ * l 1 and φ k 1 . Those will also cancel with the 1-meson contributions when we take the difference in dimensions (5.23).
The other two types of diagrams are where the gluon is exchanged between the mesons, so say between φ * l 1 and φ * l 2 . In one set of diagrams the gluon is exchanged between two φ's (or two φ * 's), and in the other it is exchanged between a φ and a φ * . The two possibilities are shown in figure 1.
When computing the 2-meson diagrams in figure 1, it is easy to see that there is a sign difference between (the logarithmic divergence of) the two diagrams in figure 1 (while their absolute value is equal). This is essentially because in the diagram on the left the two vertices contribute (2q + w) µ (2p − w) ν , while in the diagram on the right the two vertices contribute (2q + w) µ (2p + w) ν , while all other factors are the same (and only the w 2 term from the vertices contributes to the logarithmic divergence). So, diagrams where the gluon is exchanged between φ * and φ contribute to the anomalous dimension with opposite sign to those where the gluon is exchanged between φ and φ (or φ * and φ * ). Therefore, the diagrams cancel between themselves, leading to an overall vanishing contribution. It is easy to see that the same cancellation occurs in any computation of γ n 1 ,n 2 .
Thus, the only one-loop contribution to γ n 1 ,n 2 comes from the φ 4 couplings (5.20). Note that like the previous contributions, this contribution is independent of the number of fermions. It is not difficult to compute the contribution of the couplings above to (5.23), and we find that it is proportional to (f +h), with a positive coefficient. It was already noted in [65] that this combination is positive in all the fixed points, since this is related also to the positivity of the scalar potential. Thus, this example is consistent with our conjecture.
We can also consider other gauge+scalar fixed points where one can compute multi-meson dimensions perturbatively, and we will similarly find that the gluon exchange amplitudes cancel at one-loop order. In particular this applies also to the 3d fixed points for SU (N c ) (or U (N c )) theories with N f scalars in the fundamental representation, where perturbation theory is valid for large N f . In this case (where some anomalous dimension computations were performed for fermionic theories in [67]) there are no φ 4 couplings (which are fine-tuned to zero), so the one-loop multi-meson anomalous dimensions cancel, and we need to go to two-loop order to check our conjecture.
In the case of fermions, the situation is more subtle, since we have several different meson operators of different spins in the same bi-fundamental flavor representation (and with the same classical dimension), and even more operators (including several operators of the same spin) when we look at two-meson operators. For instance we can consider ψγ µ ψψγ µ ψ ,ψγ µν ψψγ µν ψ , ... (5.25) which will mix with the two-scalar-meson operators. Testing the conjecture then requires keeping track, at each level n, of the lightest operator, while carefully taking the mixing into account. This is a complicated task, even at the 1-loop level, and so we leave it for future work. This issue arises both for the 4d Banks-Zaks fixed point, and for 3d fixed points (where the fermion has only two spin indices rather than four, but still there are several different meson and multi-meson operators with the same classical dimension and the same flavor representations, complicating the computation).

Summary
In this paper we pointed out that the natural formulation of the Weak Gravity Conjecture in AdS space is in terms of the self-binding energy of a charged state. This has a simple CFT dual, which is that there should exist a charged operator such that its products with itself have a non-negative anomalous dimension. This formulation generalizes naturally to all CFTs, even without weakly-curved gravitational duals, in the form of the conjectures that we presented in the introduction.
In this paper we performed some simple preliminary tests of the conjecture, and we found that it is always satisfied. It would be interesting to perform more tests of the conjecture, in various additional CFTs where operator dimensions can be computed (for instance, one could look at those fixed points found in [68] which are unitary). It would also be interesting to study various generalizations of the conjecture, for instance to non-relativistic CFTs, or to operators living on boundaries or defects in CFTs.
Assuming that the conjecture is correct, it would be very interesting to try to derive it. One option is to use conformal bootstrap methods to show that the spectrum must always be convex. Since we presented an example where q 0 needs to be larger than one for the conjecture to hold, it is not completely clear how to do this; but it is possible that convexity always holds for scalar charged operators, and this seems like the simplest case to try to prove by bootstrap methods. Note that just by conformal bootstrap methods one cannot disprove the conjecture, since generalized free fields trivially obey the conjecture and are consistent with the bootstrap conditions (it may be possible to numerically find solutions to the bootstrap equations where the conjecture would not hold, but one would still need to prove that these correspond to full-fledged CFTs). Note also that it seems model [69,70], even though the full CFT is believed to be non-unitary.).