Duality and Supersymmetry Constraints on the Weak Gravity Conjecture

Positivity bounds coming from consistency of UV scattering amplitudes are in general insufficient to prove the weak gravity conjecture for theories beyond Einstein-Maxwell. Additional ingredients about the UV may be necessary to exclude those regions of parameter space which are na\"ively in conflict with the predictions of the weak gravity conjecture. In this paper we explore the consequences of imposing additional symmetries inherited from the UV theory on higher-derivative operators for Einstein-Maxwell-dilaton-axion theory. Using black hole thermodynamics, for a preserved SL($2,\mathbb{R}$) symmetry we find that the weak gravity conjecture then does follow from positivity bounds. For a preserved O($d,d;\mathbb{R}$) symmetry we find a simple condition on the two Wilson coefficients which ensures the positivity of corrections to the charge-to-mass ratio and that follows from the null energy condition alone. We find that imposing supersymmetry on top of either of these symmetries gives corrections which vanish identically, as expected for BPS states.


Introduction
The weak gravity conjecture (WGC) posits the existence of a state with charge larger than its mass, in appropriate units [1]. In its mild from, it is enough to have the extremality bound of charged black holes be less stringent when corrections to the classical action are included [2][3][4][5][6][7][8][9][10][11][12][13]. While the mild form alone does not display the full predictive power for phenomenology 1 , when combined with additional UV properties such as modular invariance [29][30][31] it can lead to stronger forms of the WGC such as the sublattice WGC [29,30] and the tower WGC [32]. For this reason, considerable efforts have been devoted toward a proof of the mild form of the WGC as a basis of the web of WGCs.
An important issue in this context is to identify consistency conditions necessary to demonstrate the conjecture. At low energies, corrections to the black hole extremality bound may be captured by higher-derivative operators, so that the mild form of the WGC follows if the effective couplings satisfy a certain inequality [2]. Then, it is natural to expect that positivity bounds [33] which follow from consistency of UV scattering amplitudes may play a crucial role in demonstrating the conjecture. It is indeed the case in Einstein-Maxwell theory under reasonable assumptions [4,5] (see also [32,34,35] for attempts to use positivity bounds to constrain the charged particle spectrum at low energies).
However, there are some low-energy effective theories for which positivity bounds are not sufficient on their own to prove the positivity of corrections to the charge-to-mass ratio of extremal black holes. This occurs, for example, in the Einstein-Maxwell-dilaton theory, where the term ∂ µ φ∂ ν φF µρ F ν ρ (1.1) contributes to the charge-to-mass ratio in a way directly at odds with the WGC once positivity bounds are accounted for [7]. Of course, such a term never exists in isolation and presumably this negative contribution never dominates over other (positive) contributions. As discussed in [7], for a few hand-picked choices of UV completion one can show that indeed this puzzling term is never problematic. A similar scenario is present for the Einstein-axion-dilaton theory, where, as discussed recently in [36], there are regions of parameter space in which the axion weak gravity conjecture is violated, even when positivity bounds are taken into account. There imposing extra structure on the UV theory, namely an SL(2, R) symmetry respected by the higherderivative terms, greatly constrains the form of the corrections and ensures the axion WGC follows from the positivity bounds.
In this paper we consider the implications of imposing such additional structures on the UV theory for the WGC as applied to extremal black holes in Einstein-Maxwell-dilatonaxion (EMda) theory (see, e.g., [7,9,10] for previous discussions of the mild form of the WGC in the presence of a dilaton). Extra symmetries in the effective action which descend from the UV theory, when combined with either scattering positivity bounds or null energy conditions, are then strong enough to demonstrate the WGC for this system. In particular, we will work with an SL(2, R) symmetry and an O(d, d; R) symmetry: both can be present in the two-derivative EMda action and in 4D effective string actions these correspond to S-and T-duality, respectively [37][38][39]. We also study implications of N ≥ 2 supersymmetry in these setups. We find that the puzzling terms mentioned earlier are helpful to make corrections to extremality identically zero even in the presence of nontrivial higher derivative operators, as expected for BPS states.
This paper is organized as follows: in section 2 we recall the realizations of the SL(2, R) and O(d, d; R) symmetries for the two-derivative EMda action and impose these symmetries on the higher-derivative terms; in section 3 we present the leading order, dyonic solutions; in section 4 we use black hole thermodynamics to compute the corrections to the extremal charge-to-mass ratio, and we conclude in section 5. Throughout we use reduced Planck units: 8πG N = 1.

Symmetries of the Low-Energy Effective Action
Let us begin by recalling the two-derivative action for EMda theory: we will work with two U(1)s for simplicity. In discussing the two symmetries it is useful to go back-and-forth both between string and Einstein frame and between axion and 2-form field. Start in string frame, where H = dB − A a ∧ F a and the index a = 1, 2 is summed over. Here we have introduced the short-hand G · H ≡ G µ···σ H µ···σ and G 2 ≡ G · G for antisymmetric tensors G and H of the same rank. Going to string frame via g µν → e 2φ g µν gives Dualizing to an axion is accomplished via Integrating out θ reproduces (2.2), while integrating out H gives H = e 4φ dθ and where F µν = 1 2 √ −g µνρσ F ρσ ( 0123 = − 0123 = +1).

SL(2, R)
The SL(2, R) symmetry is best discussed in Einstein frame, where by defining τ = θ +ie −2φ and the action becomes The SL(2, R) symmetry acts nonlinearly on the fields as 2 where a b c d ∈ SL(2, R) , (2.8) and is present at the level of the equations of motion. Electric and magnetic charges are defined by and using (2.7), these transform under SL(2, R) according to (2.10) 2 It is worth noting that the transformations of the fields are altered in the presence of higher-derivative terms in the action: for example, the α1111 term in (2.11) induces These ensure that the equations of motion are invariant under the SL(2, R) transformation at O(α).
We will make use of the rescaled charges Q a = 4πq a and P a = 4πp a as well.
A complete set of SL(2, R)-preserving 3 , four-derivative operators may be written as where E 2 = Riem 2 − 4 Ric 2 + R 2 is a total derivative in 4D. For the action to be real the Wilson coefficients must be real and have the following symmetries: All-told there are 12 real parameters controlling the Wilson coefficients of equation (2.11).
As mentioned in the introduction, without the imposed symmetry there are far more allowed terms and with regions of parameter space in conflict with the WGC that are not ruled out by positivity bounds. Note that the coefficient of the previously-noted term ∂φ∂φF F is now related by the SL(2, R) symmetry to, among others, the coefficient of (∂φ) 2 (F 2 ).

O(d, d; R)
The O(d, d; R) symmetry is best discussed in string frame with the 3-form H.
The SL(2, R) symmetry is known to be broken to SL(2, Z) due to non-perturbative effects. Imposing this less restrictive symmetry on the four-derivative action allows for a wider range of terms, e.g. r j(τ ) (Im τ ) 2 (F −2 )(F +2 ) and r j(τ ) (Im τ ) 4 G4(τ )(F +2 ) 2 , where r j(τ ) is any rational function of the j-invariant and G4 is the Eisenstein series of weight four. However, being non-perturbatively generated such terms will be highly suppressed. 4 With k gauge fields in 4+d dimensions the symmetry is enhanced to O(d, d+k; R). We will not consider this extension here.
In going to Einstein frame and dualizing H → θ, one finds (using tree-level equations of motion) (2.20) We have omitted the higher-derivative terms involving the axion because they all vanish for the solution of section 3.

Leading-Order Solution
The higher-derivative operators discussed in the previous section will induce corrections to extremal black holes; for the thermodynamic arguments of section 4, we need only the uncorrected black hole solution in order to compute the leading corrections to the extremality condition. We will consider dyonic black holes, where the solutions are regular even in the extremal limit, with constant axion for ease of calculation. This can be arranged via an appropriate SL(2, R) transformation, but does not represent the most general O(d, d; R) solution. Such solutions are given by The physical charges are Q = 4πq and P = 4πp, and the constants κ a > 0 are determined by The inner and outer horizons are located at r = 0 and r = 2ξ, respectively, so that extremality corresponds to ξ → 0. From the metric we may read off the mass, temperature and entropy: For large enough charges the black hole is large and the curvatures are small at the outer horizon, even at extremality. This ensures that the derivative expansion is under control.

Higher-Derivative Corrections via Thermodynamics
We will leverage black hole thermodynamics to compute corrections to the charge-to-mass ratio of extremal black holes, and so begin by recalling the key ingredients to this procedure. See [45] and [7] for more complete discussions of these ideas. The full four-derivative action can be written as where I 0 is the two-derivative action and ∆I all denotes higher-derivative terms with their corresponding Wilson coefficients, collectively denoted by α. The contribution I ∂ contains all boundary terms and is required for a well-defined variational principle: the details of its form will not be relevant to our discussion. The Gibbons-Hawking-York term contributes M 2T to the action [46], and the boundary terms associated with the other terms in the bulk action vanish in the infinite-volume limit. One may evaluate the Euclidean action to find the free energy G in the grand canonical ensemble, given by with Φ a = (−A a t )| r=2ξ − (−A a t )| r=∞ the electric potentials at the outer horizon and Ψ a the analogous magnetic potential. Via straightforward thermodynamic manipulations one may find the mass as a function of the charges and temperature in the canonical ensemble.
The strength of this approach lies in there being no need to find solutions to the perturbed equations of motion. Namely, the Euclidean action may be reliably evaluated to O(α) in the grand canonical ensemble using only the leading-order solution 5 : The dynamical fields are collectively denoted by X, and may even include fields which vanish at O(α 0 ).

Leading-Order Thermodynamics
For the solution of (3.1), we discuss briefly the determination of the black hole's thermodynamic properties via the Euclidean action. This will give the leading behavior, on top of which we compute corrections due to the four-derivative terms. Evaluating the two-derivative Euclidean action for (3.1) leads to from which it follows that in the grand canonical and canonical ensembles we have (4.5) We have used w q = w(QT, P T ) and w p = w(P T, QT ) in the canonical ensemble, with w(y, z) being the root of the quintic with a small y, z > 0 expansion which begins This root is identified by its giving a positive mass which remains finite for T → 0. The other branches of solutions correspond to different signs for q, p or to thermodynamically unstable configurations. The expressions above exactly match those found by reading off from the metric in equation (3.3) upon eliminating κ 1 , κ 2 and ξ in favor of Q, P and T .

SL(2, R)
For the sake of example we present briefly the results for the α 1111 term in (2.11) (other corrections have a similar form). In the grand canonical ensemble, we find In the canonical ensemble, we find (4.14) . Taking the T → 0 limit at fixed charges gives the corrected extremal charge-to-mass ratio:  Each contribution is shown in figure 1 separately. There the invariance of z ext under the preserved electromagnetic duality (Q ↔ P and 1 ↔ 2) is more clearly seen.
One may obtain positivity bounds on the coefficients present in z ext by considering scattering amplitudes around the background g µν = η µν , τ = i and A a = 0, such as appears in the asymptotic region of the black holes considered here. For M(s, t = 0) a crossing-symmetric forward amplitude, we have [33] [ (4.17) The contour C encircles the origin and is deformed to two integrations along cuts beginning at ±s 0 : the contributions at infinity are dropped, having assumed the Froissart bound. By the optical theorem the imaginary part of the crossing-symmetric amplitude M is positive, showing that the coefficient of s 2 in M is also positive.
Assuming graviton exchange is subdominant, one has the following forward scattering amplitudes: From these we may read off that α 1 + α 2 ≥ 0 and α 11 , α 22 ≤ 0, so that their corresponding contributions to z ext are each positive. For the four-photon amplitudes, the crossingsymmetric combinations must be positive for all real u a , v a . In particular, choosing u = (1, 0), v = (0, 1) shows that α 1212 ≥ 0, and u = (1, x), v = (1, −x) gives for all real x. The choice is enough to conclude that z ext ≥ 1 in equation (4.16).

O(d, d; R)
For those theories with a preserved O(d, d; R) symmetry the gravitational four-derivative terms are always of the same order as those involving the gauge fields. Without a hierarchy among the Wilson coefficients, we do not have positivity bounds from scattering amplitudes at our disposal. However, since the O(d, d; R) symmetry is far more constraining than SL(2, R), the four-derivative terms are controlled by only two undetermined coefficients and the interplay between positive and negative contributions to z ext is nearly fixed. It is necessary to make connection between the gauge fields discussed in section 2.2 and the gauge fields solution of section 3. That is, we should identify A 1,2 with components of A M in such a way that  The extremal charge-to-mass ratio may be written as where (any term not present vanishes). The individual terms are (see figure 2) 1120p 4 q 2 F 1 1, 6; 10; 1 − q p (4.27) 24p(p + 8q) 2 F 1 1, 6; 8; 1 − q p (4.29) . (4.38) Requiring that z ext ≥ 1 then amounts to 2α ± β ≥ 0, or 2α ≥ |β|. As we now show, this follows from the null energy condition alone. Also this requires a positive coefficient of the Gauss-Bonnet term, which agrees with other considerations such as string theory examples and entropy considerations.
On any background and for any null vector k µ , the null energy condition requires where the stress tensor T µν has contributions from both the two-and four-derivative terms in the action: The T µν terms are explicitly of order α, but there are also implicit O(α) corrections from evaluating T (2) µν on the corrected background. For our purposes it will suffice to work with the spherically-symmetric background with p = q, so that φ, θ = O(α) and many terms drop out of T (2) µν and T (4) µν . With the choice the leading contribution vanishes: (4.42) The last equality follows from the spherical symmetry of the corrected solution, for which g µν remains diagonal and only F a tr and F a ϑϕ are potentially nonzero. Thus on this background only the explicit O(α) terms of T (4) µν can possibly give nonzero contribution to T µν k µ k ν : (4.43) These terms we may simply evaluate on the leading-order solution of equation (3.1) with p = q and κ 2 = κ 1 . In contracting with k µ the only contribution which does not vanish is from the last term in T RF F µν , which leads to That is, the null energy condition requires that RF F have a negative coefficient and so 2α ∓ β ≥ 0, which coincides exactly with the WGC bound.

Supersymmetry
We can also ask what restrictions supersymmetry places on top of the two symmetries considered above. With N ≥ 2 supersymmetry we expect there to be no correction to the charge-to-mass ratio of BPS states, much like was found for quantum corrections to the WGC in [6]. For the O(d, d; R) case this is easy to check: the heterotic string has (α, β) = ( α 16 , − α 8 ), so that β = −2α and the corrections are The top sign corresponds to the choice in equation (4.23) which places the gauge fields in the N = 4 gravity multiplet giving z ext = 1 as expected, while the lower sign places the gauge fields in vector multiplets giving positive corrections to z ext .
For SL(2, R) we may gain some insight by using relations between scattering amplitudes for fields in the same supermultiplet. In particular, with φ ± = φ ± iθ the scalar helicity states in an N = 2 vector multiplet, SUSY requires (4.46) so that the correction to z ext proportional to α 1 + α 2 must identically vanish 6 . If A 1 is in the N = 2 gravity multiplet and A 2 is in the vector multiplet with φ, θ, then But the right-hand side has an s 2 term proportional to α 1111 while the left-hand side has no s 2 term generated by the higher-derivative terms, and so α 1111 must vanish. Similarly, with now only the right-hand side having an s 2 term proportional to α 11 , so that it must be that α 11 = 0. All told, only α 2222 and α 22 are nonzero, giving positive correction to z ext only when the vector multiplet photon is charged.

Discussion
The Einstein-Maxwell-dilaton-axion theory has a large number of possible four-derivative terms which correct the action in the effective action framework. The Wilson coefficients which control the relative sizes and signs of these terms may be partially constrained by appealing to scattering positivity bounds, but there remain allowed regions of the parameter space in which corrections to extremal black holes are at odds with the expectations of the WGC. Rather than working with a particular UV theory in order to determine more finely the form of the higher-derivative corrections, one may consider the implications of using symmetries inherited from the UV as an intermediate assumption.
The leading EMda action can have both an SL(2, R) and underlying O(d, d; R) symmetry. Imposing each of these individually on the action restricts the allowed higher-derivative terms to a small handful which then succumb to other considerations. We have shown that after having imposed SL(2, R) symmetry, positivity bounds are then enough to demonstrate the WGC in general. In imposing the O(d, d; R) symmetry we have found that the WGC requires a relationship between α and β that follows from the null energy condition alone. Whether the null (or other) energy condition implies the mild form of the WGC in more general settings is a question worth further exploration.
Imposing SUSY on top of these two symmetries, we have found that the extremal charge-to-mass ratio is not corrected when black holes carry the gravity multiplet photon charge alone, as expected for BPS states. Importantly, the (always) negative contributions from terms such as ∂φ∂φF F are vital in canceling the positive contributions from other terms. It would be interesting, however, to consider what mileage one can get from considering the implications of SUSY alone on the four-derivative terms, again as an intermediate assumption on the UV theory. We leave this interesting question to future work.