Minimal surfaces and weak gravity

Mehmet Demirtas,Mike Stillman,Cody Long,Liam Mcallister

doi:10.1007/jhep03(2020)021

Mehmet Demirtas, Mike Stillman + Show 2 more

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https://doi.org/10.1007/jhep03(2020)021

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Abstract

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 [Σ] ∈ H4(X, ℝ) represented by a union Σ∪ of holomorphic and antiholomorphic cycles. The instanton form of the WGC applied to the axion charge [Σ] implies an upper bound on the action of a non-BPS Euclidean D3-brane wrapping the minimal-volume representative Σmin of [Σ]. We give an explicit example of an orientifold X of a hypersurface in a toric variety, and a hyperplane H ⊂ H4(X, ℝ), such that for any [Σ] ∈ H that satisfies the WGC, the minimal volume obeys Vol (Σmin) ≪ Vol(Σ∪): 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 Σmin are then more important than would be predicted from a study of BPS instantons wrapping the separate components of Σ∪. Our analysis hinges on a novel computation of effective divisors in X that are not inherited from effective divisors of the toric variety.

Highlights

When X is a Kahler threefold, the computation of Vol(Σmin) is almost trivial for a special set of classes [Σ] ∈ H4(X, Z): if [Σ] can be represented by an effective divisor ΣE, ΣE is calibrated by the Kahler form J, and is absolutely volume-minimizing in its class, with volume
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
Given a homology class [Σ] ∈ Hp(M, Z), 0 < p < n, what representative of [Σ] has minimal volume? A crucial subtlety is that the volume functional may not attain a minimum on any smooth representative of [Σ]

Summary

An orientifold where weak gravity implies large recombination

We give an explicit example of an orientifold X of a Calabi-Yau threefold hypersurface, and a hyperplane H ⊂ H4(X, Z), such that for any [Σ] ∈ H, the ratio rmΣin defined in (2.11) can be made arbitrarily large by a choice of the Kahler form J.10. To show that the WGC implies large recombination, we need to find a non-effective divisor class [Σ] ∈ H4(X, Z), and a point in the Kahler cone of X for which rmΣin is large. To find such a non-effective divisor class and characterize its piecewise calibrated representatives, we must first identify the effective and anti-effective divisor classes on X. As [Σ(k)] is non-effective, we express it as a sum of effective and anti-effective divisor classes, and construct a piecewise-calibrated representative Σ(∪k):. It follows that the sub-Lattice WGC requires large recombination in our example

Discussion

A Geometry of the example

B Effective divisors in the example

Cohomology of line bundles on toric varieties

The explicit example