The Positive Binding Conjecture is a proposed formulation of the Weak Gravity Conjecture appropriate to Anti de-Sitter (AdS) space. It proposes that in a consistent gravitational theory, with a $U(1)$ gauge symmetry, there must exist a charged particle with non-negative self-binding energy. In order to formulate this as a constraint on a given effective theory, we calculate the self-binding energy for a charged particle in AdS$_4$ and AdS$_5$. In particular, we allow it to couple to an additional scalar field of arbitrary mass. Unlike the flat-space case, even when the scalar field is massive it contributes significantly to the binding energy, and therefore is an essential component of the conjecture. We give analytic expressions for the self-binding energy for the cases when the scalar field is massless and when it saturates the Breitenlohner-Freedman (BF) bound. We show that the massless case reproduces the flat-space expressions in the large AdS radius limit, and that the BF bound case leads to vanishing total self-binding energy for BPS particles. For other masses of the scalar we give numerical expressions for its contribution to the self-binding energy.
I will also talk about recent works on axionic Euclidean wormholes if there is enough time.
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