Abstract
We study the topological properties of Calabi-Yau threefold hypersurfaces at large h1,1. We obtain two million threefolds X by triangulating polytopes from the Kreuzer-Skarke list, including all polytopes with 240 ≤ h1,1 ≤ 491. We show that the Kähler cone of X is very narrow at large h1,1, and as a consequence, control of the α′ expansion in string compactifications on X is correlated with the presence of ultralight axions. If every effective curve has volume ≥ 1 in string units, then the typical volumes of irreducible effective curves and divisors, and of X itself, scale as (h1,1)p, with 3 ≲ p ≲ 7 depending on the type of cycle in question. Instantons from branes wrapping these cycles are thus highly suppressed.
Highlights
We study the topological properties of Calabi-Yau threefold hypersurfaces at large h1,1
We show that the Kahler cone of X is very narrow at large h1,1, and as a consequence, control of the α expansion in string compactifications on X is correlated with the presence of ultralight axions
We avoid the term ‘ultralight’ when speaking of the far lighter axions found here, with m 10−33 eV; these we instead call ‘massless’, even though strictly speaking their masses are negligibly small, not zero. 16As explained in section 6, in a small fraction of cases we cannot exclude the possibility of super-Planckian radii, but neither can we prove that all curves in X have positive volume in these cases
Summary
Let X be a projective algebraic variety of complex dimension n. A Weil divisor D on X is a finite formal sum of irreducible codimension-one subvarieties DA,. The divisor D is called effective if the nA are all nonnegative. We define the effective cone Eff(X) to be the convex cone in H2n−2(X, R) spanned by the classes of effective divisors. The relevance of the effective cone is that a Euclidean D3-brane wrapping a divisor. D in an orientifold of a Calabi-Yau threefold X can contribute to the superpotential only if D is effective. Effective divisors consist of finite collections of irreducible holomorphic hypersurfaces, each of which can support BPS D-branes
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