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We show that the Weak Gravity Conjecture (WGC) implies a nontrivial upper bound on the volumes of the minimal-volume cycles in certain homology classes that admit no calibrated representatives. In compactification of type IIB string theory on an orientifold $X$ of a Calabi-Yau threefold, we consider a homology class $[Sigma] in H_4(X,mathbb{Z})$ represented by a union $Sigma_{cup}$ of holomorphic and antiholomorphic cycles. The instanton form of the WGC applied to the axion charge $[Sigma]$ implies an upper bound on the action of a non-BPS Euclidean D3-brane wrapping the minimal-volume representative $Sigma_{mathrm{min}}$ of $[Sigma]$. We give an explicit example of an orientifold $X$ of a hypersurface in a toric variety, and a hyperplane $mathcal{H} subset H_4(X,mathbb{Z})$, such that for any $[Sigma] in H$ that satisfies the WGC, the minimal volume obeys $mathrm{Vol}(Sigma_{mathrm{min}}) ll mathrm{Vol}(Sigma_{cup})$: the holomorphic and antiholomorphic components recombine to form a much smaller cycle. In particular, the sub-Lattice WGC applied to $X$ implies large recombination, no matter how sparse the sublattice. Non-BPS instantons wrapping $Sigma_{mathrm{min}}$ are then more important than would be predicted from a study of BPS instantons wrapping the separate components of $Sigma_{cup}$. Our analysis hinges on a novel computation of effective divisors in $X$ that are not inherited from effective divisors of the toric variety.
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Minimal surfaces in Euclidean space provide examples of possible non-compact horizon geometries and topologies in asymptotically flat space-time. On the other hand, the existence of limiting surfaces in the space-time provides a simple mechanism for making these configurations compact. Limiting surfaces appear naturally in a given space-time by making minimal surfaces rotate but they are also inherent to plane wave or de Sitter space-times in which case minimal surfaces can be static and compact. We use the blackfold approach in order to scan for possible black hole horizon geometries and topologies in asymptotically flat, plane wave and de Sitter space-times. In the process we uncover several new configurations, such as black helicoids and catenoids, some of which have an asymptotically flat counterpart. In particular, we find that the ultraspinning regime of singly-spinning Myers-Perry black holes, described in terms of the simplest minimal surface (the plane), can be obtained as a limit of a black helicoid, suggesting that these two families of black holes are connected. We also show that minimal surfaces embedded in spheres rather than Euclidean space can be used to construct static compact horizons in asymptotically de Sitter space-times.
The axionic weak gravity conjecture predicts the existence of instantons whose actions are less than their charges in appropriate units. We show that the conjecture is satisfied for the axion-dilaton-gravity system if we assume duality constraints on the higher derivative corrections in addition to positivity bounds which follow from unitarity, analyticity, and locality of UV scattering amplitudes. On the other hand, the conjecture does not follow if we assume the positivity bounds only. This presents an example where derivation of the weak gravity conjecture requires more detailed UV information than the consistency of scattering amplitudes.
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