Paper

Minimal Surfaces and Weak Gravity

McAllister, Liam, Demirtas, Mehmet, Long, Cody, Stillman, Mike

2019

Overview

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.

• Publisher: Cornell University Library, arXiv.org
• Subject: Gravitation / Homology / Hyperplanes / Hyperspaces / Instantons / Mathematical analysis / Minimal surfaces / String theory / Upper bounds