We study the topological properties of Calabi-Yau threefold hypersurfaces at large h 1,1 . We obtain two million threefolds X by triangulating polytopes from the Kreuzer-Skarke list, including all polytopes with 240 ≤ h 1,1 ≤ 491. We show that the Kähler cone of X is very narrow at large h 1,1 , 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 (h 1,1 ) p , with 3 p 7 depending on the type of cycle in question. Instantons from branes wrapping these cycles are thus highly suppressed.

Several recent works [1][2][3] have claimed that the Weak Gravity Conjecture (WGC) excludes super-Planckian displacements of axion fields, and hence large-field axion inflation, in the absence of monodromy. We argue that in theories with N 1 axions, super-Planckian axion diameters D are readily allowed by the WGC. We clarify the nontrivial relationship between the kinetic matrix K -unambiguously defined by its form in a Minkowski-reduced basis -and the diameter of the axion fundamental domain, emphasizing that in general the diameter is not solely determined by the eigenvalues f 2 1 ≤ . . . ≤ f 2 N of K: the orientations of the eigenvectors with respect to the identifications imposed by instantons must be incorporated. In particular, even if one were to impose the condition f N < M pl , this would imply neither D < M pl nor D < √ N M pl . We then estimate the actions of instantons that fulfill the WGC. The leading instanton action is bounded from below by S ≥ SM pl /f N , with S a fixed constant, but in the universal limit S S √ N M pl /f N . Thus, having f N > M pl does not immediately imply the existence of unsuppressed higher harmonic contributions to the potential. Finally, we argue that in effective axion-gravity theories, the zero-form version of the WGC can be satisfied by gravitational instantons that make negligible contributions to the potential.

We study universality of geometric gauge sectors in the string landscape in the context of Ftheory compactifications. A finite time construction algorithm is presented for 4 3 × 2.96 × 10 755 Ftheory geometries that are connected by a network of topological transitions in a connected moduli space. High probability geometric assumptions uncover universal structures in the ensemble without explicitly constructing it. For example, non-Higgsable clusters of seven-branes with intricate gauge sectors occur with probability above 1 − 1.01 × 10 −755 , and the geometric gauge group rank is above 160 with probability .999995. In the latter case there are at least 10 E8 factors, the structure of which fixes the gauge groups on certain nearby seven-branes. Visible sectors may arise from E6 or SU (3) seven-branes, which occur in certain random samples with probability ≃ 1 200.

We propose a scenario for realizing super-Planckian axion decay constants in Calabi-Yau orientifolds of type IIB string theory, leading to large-field inflation. Our construction is a simple embedding in string theory of the mechanism of Kim, Nilles, and Peloso, in which a large effective decay constant arises from alignment of two smaller decay constants. The key ingredient is gaugino condensation on magnetized or multiply-wound D7-branes. We argue that, under very mild assumptions about the topology of the Calabi-Yau, there are controllable points in moduli space with large effective decay constants.

We argue that super-Planckian diameters of axion fundamental domains can arise in Calabi-Yau compactifications of string theory. In a theory with N axions θ i , the fundamental domain is a polytope defined by the periodicities of the axions, via constraints of the form −π < Q i j θ j < π. We compute the diameter of the fundamental domain in terms of the eigenvalues f 2 1 ≤ . . . ≤ f 2 N of the metric on field space, and also, crucially, the largest eigenvalue of (QQ ) −1 . At large N , QQ approaches a Wishart matrix, due to universality, and we show that the diameter is at least N f N , exceeding the naive Pythagorean range by a factor > √ N . This result is robust in the presence of P > N constraints, while for P = N the diameter is further enhanced by eigenvector delocalization to N 3/2 f N . We directly verify our results in explicit Calabi-Yau compactifications of type IIB string theory. In the classic example with h 1,1 = 51 where parametrically controlled moduli stabilization was demonstrated by Denef et al. in [1], the largest metric eigenvalue obeys f N ≈ 0.013M pl . The random matrix analysis then predicts, and we exhibit, axion diameters ≈ M pl for the precise vacuum parameters found in [1]. Our results provide a framework for pursuing large-field axion inflation in well-understood flux vacua.

We perform an extensive analysis of the statistics of axion masses and interactions in compactifications of type IIB string theory, and we show that black hole superradiance excludes some regions of Calabi-Yau moduli space. Regardless of the cosmological model, a theory with an axion whose mass falls in a superradiant band can be probed by the measured properties of astrophysical black holes, unless the axion self-interaction is large enough to disrupt formation of a condensate. We study a large ensemble of compactifications on Calabi-Yau hypersurfaces, with 1 ≤ h 1,1 ≤ 491 closed string axions, and determine whether the superradiance conditions on the masses and self-interactions are fulfilled. The axion mass spectrum is largely determined by the Kähler parameters, for mild assumptions about the contributing instantons, and takes a nearly-universal form when h 1,1 ≫ 1. When the Kähler moduli are taken at the tip of the stretched Kähler cone, the fraction of geometries excluded initially grows with h 1,1, to a maximum of ≈ 0.5 at h 1,1 ≈ 160, and then falls for larger h 1,1. Further inside the Kähler cone, the superradiance constraints are far weaker, but for h 1,1 ≫ 100 the decay constants are so small that these geometries may be in tension with astrophysical bounds, depending on the realization of the Standard Model.

We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 N =2 superconformal field theories. Only a finite rational set of scaling dimensions is allowed. It is determined by using information about the global topology of the locus of metric singularities on the Coulomb branch, the special Kähler geometry near those singularities, and electric-magnetic duality monodromies along orbits of the U(1) R symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classification of Coulomb branch geometries at all ranks.

Abstract:We study the statistics of the metric on Kähler moduli space in compactifications of string theory on Calabi-Yau hypersurfaces in toric varieties. We find striking hierarchies in the eigenvalues of the metric and of the Riemann curvature contribution to the Hessian matrix: both spectra display heavy tails. The curvature contribution to the Hessian is non-positive, suggesting a reduced probability of metastability compared to cases in which the derivatives of the Kähler potential are uncorrelated. To facilitate our analysis, we have developed a novel triangulation algorithm that allows efficient study of hypersurfaces with h 1,1 as large as 25, which is difficult using algorithms internal to packages such as Sage. Our results serve as input for statistical studies of the vacuum structure in flux compactifications, and of the distribution of axion decay constants in string theory.