Abstract
Positivity bounds coming from consistency of UV scattering amplitudes are not always sufficient 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 naïvely 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, ℝ) symmetry we find that the weak gravity conjecture then does follow from positivity bounds. For a preserved O(d, d; 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.
Highlights
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
For a preserved SL(2, R) symmetry we find that the weak gravity conjecture does follow from positivity bounds
For a preserved O(d, d; 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
Summary
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. The SL(2, R) symmetry is best discussed in Einstein frame, where by defining τ = θ +. The O(d, d; R) symmetry is best discussed in string frame with the 3-form H. The O(d, d; R)-symmetric four-derivative terms we discuss below are invariant under O(d , d ; R) when reducing to even lower dimensions. There are only two independent four-derivative terms which respect the O(d , d ; R) symmetry [45], which with the above choice for internal scalars read. In going to Einstein frame and dualizing H → θ, one finds (using tree-level equations of motion). We have omitted the higher-derivative terms involving the axion because they all vanish for the solution of section 3
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