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We study Calabi‐Yau threefolds with large Hodge numbers by constructing and counting triangulations of reflexive polytopes. By counting points in the associated secondary polytopes, we show that the number of fine, regular, star triangulations of polytopes in the Kreuzer‐Skarke list is bounded above by 14,111494≈10928. Adapting a result of Anclin on triangulations of lattice polygons, we obtain a bound on the number of triangulations of each 2‐face of each polytope in the list. In this way we prove that the number of topologically inequivalent Calabi‐Yau hypersurfaces arising from the Kreuzer‐Skarke list is bounded above by 10428. We introduce efficient algorithms for constructing representative ensembles of Calabi‐Yau hypersurfaces, including the extremal case h1,1=491, and we study the distributions of topological and physical data therein. Finally, we demonstrate that neural networks can accurately predict these data once the triangulation is encoded in terms of the secondary polytope. The authors study Calabi‐Yau threefolds with large Hodge numbers by constructing and counting triangulations of reflexive polytopes. By counting points in the associated secondary polytopes, it is shown that the number of fine, regular, star triangulations of polytopes in the Kreuzer‐Skarke list is bounded above. Adapting a result of Anclin on triangulations of lattice polygons, one obtains a bound on the number of triangulations of each 2‐face of each polytope in the list. In this way it is proven that the number of topologically inequivalent Calabi‐Yau hypersurfaces arising from the Kreuzer‐Skarke list is bounded above. Efficient algorithms for constructing representative ensembles of Calabi‐Yau hypersurfaces are introduced, including the extremal case h1,1=491. Finally, it will be demonstrated that neural networks can accurately predict these data once the triangulation is encoded in terms of the secondary polytope. [ABSTRACT FROM AUTHOR]

We present a method to construct the extended Kähler cone of any Calabi-Yau threefold by using Gopakumar-Vafa invariants to identify all geometric phases that are related by flops or Weyl reflections. In this way we obtain the Kähler moduli spaces of all favorable Calabi-Yau threefold hypersurfaces with h1,1 ≤ 4, including toric and non-toric phases. In this setting we perform an explicit test of the Weak Gravity Conjecture by using the Gopakumar-Vafa invariants to count BPS states. All of our examples satisfy the tower/sublattice WGC, and in fact they even satisfy the stronger lattice WGC. [ABSTRACT FROM AUTHOR]

We critically assess the twin prospects of describing the observed universe in string theory, and using cosmological experiments to probe string theory. For the purposes of this short review, we focus on the limitations imposed by our incomplete understanding of string theory. After presenting an array of significant obstacles, we indicate a few areas that may admit theoretical progress in the near future. [ABSTRACT FROM AUTHOR]

We provide a pedagogical overview of inflation in string theory. We focus upon the sensitivity of inflation to Planck-scale physics, which we argue provides both the primary motivation and the central theoretical challenge for the subject. We illustrate these issues through two case studies of inflationary scenarios in string theory: warped D-brane inflation and axion monodromy inflation. Finally, we indicate how future observations can test scenarios of inflation in string theory. [ABSTRACT FROM AUTHOR]

We study Euclidean D3-branes wrapping divisors D in Calabi-Yau orientifold compactifications of type IIB string theory. Witten’s counting of fermion zero modes in terms of the cohomology of the structure sheaf O D applies when D is smooth, but we argue that effective divisors of Calabi-Yau threefolds typically have singularities along rational curves. We generalize the counting of fermion zero modes to such singular divisors, in terms of the cohomology of the structure sheaf O D ¯ of the normalization D ¯ of D. We establish this by detailing compactifications in which the singularities can be unwound by passing through flop transitions, giving a physical incarnation of the normalization process. Analytically continuing the superpotential through the flops, we find that singular divisors whose normalizations are rigid can contribute to the superpotential: specifically, h + • O D ¯ = 1 0 0 and h − • O D ¯ = 0 0 0 give a sufficient condition for a contribution. The examples that we present feature infinitely many isomorphic geometric phases, with corresponding infinite-order monodromy groups Γ. We use the action of Γ on effective divisors to determine the exact effective cones, which have infinitely many generators. The resulting nonperturbative superpotentials are Jacobi theta functions, whose modular symmetries suggest the existence of strong-weak coupling dualities involving inversion of divisor volumes. [ABSTRACT FROM AUTHOR]

We construct supersymmetric AdS4 vacua of type IIB string theory in compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. We first find explicit orientifolds and quantized fluxes for which the superpotential takes the form proposed by Kachru, Kallosh, Linde, and Trivedi. Given very mild assumptions on the numerical values of the Pfaffians, these compactifications admit vacua in which all moduli are stabilized at weak string coupling. By computing high-degree Gopakumar-Vafa invariants we give strong evidence that the α′ expansion is likewise well-controlled. We find extremely small cosmological constants, with magnitude < 10−123 in Planck units. The compactifications are large, but not exponentially so, and hence these vacua manifest hierarchical scale-separation, with the AdS length exceeding the Kaluza-Klein length by a factor of a googol. [ABSTRACT FROM AUTHOR]

We analyze the de Sitter construction of [1] using ten-dimensional supergravity, finding exact agreement with the four-dimensional effective theory. Starting from the fermionic couplings in the D7-brane action, we derive the ten-dimensional stress-energy due to gaugino condensation on D7-branes. We demonstrate that upon including this stress-energy, as well as that due to anti-D3-branes, the ten-dimensional equations of motion require the four-dimensional curvature to take precisely the value determined by the four-dimensional effective theory of [1]. [ABSTRACT FROM AUTHOR]

We prove a formula for the Hodge numbers of square‐free divisors of Calabi‐Yau threefold hypersurfaces in toric varieties. Euclidean branes wrapping divisors affect the vacuum structure of Calabi‐Yau compactifications of type IIB string theory, M‐theory, and F‐theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor D requires counting fermion zero modes, which depend on the Hodge numbers hi(OD). Suppose that X is a smooth Calabi‐Yau threefold hypersurface in a toric variety V, and let D be the restriction to X of a square‐free divisor of V. We give a formula for hi(OD) in terms of combinatorial data. Moreover, we construct a CW complex PD such that hi(OD)=hi(PD). We describe an efficient algorithm that makes possible for the first time the computation of sheaf cohomology for such divisors at large h1, 1. As an illustration we compute the Hodge numbers of a class of divisors in a threefold with h1,1=491. Our results are a step toward a systematic computation of Euclidean brane superpotentials in Calabi‐Yau hypersurfaces. A formula for the Hodge numbers of square‐free divisors of Calabi‐Yau threefold hypersurfaces in toric varieties will be proven. Euclidean branes wrapping divisors affect the vacuum structure of Calabi‐Yau compactifications of type IIB string theory, M‐theory, and F‐theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor D requires counting fermion zero modes, which depend on the Hodge numbers hi(OD). Suppose that X is a smooth Calabi‐Yau threefold hypersurface in a toric variety V, and let D be the restriction to X of a square‐free divisor of V. The authors give a formula for hi(OD) in terms of combinatorial data. Moreover, a CW complex PD is constructed such that hi(OD).=hi(PD). An efficient algorithm will be described that makes possible for the first time the computation of sheaf cohomology for such divisors at large h1;1. As an illustration it follows a compution of the Hodge numbers of a class of divisors in a threefold with h1;1=491. The results are a step toward a systematic computation of Euclidean brane superpotentials in Calabi‐Yau hypersurfaces. [ABSTRACT FROM AUTHOR]

We introduce a method for finding flux vacua of type IIB string theory in which the flux superpotential is exponentially small and at the same time one or more complex structure moduli are stabilized exponentially near to conifold points. The authors introduce a method for finding flux vacua of type IIB string theory in which the flux superpotential is exponentially small and at the same time one or more complex structure moduli are stabilized exponentially near to conifold points. [ABSTRACT FROM AUTHOR]

We develop methods for resummation of instanton lattice series. Using these tools, we investigate the consequences of the Weak Gravity Conjecture for large-field axion inflation. We find that the Sublattice Weak Gravity Conjecture implies a constraint on the volume of the axion fundamental domain. However, we also identify conditions under which alignment and clockwork constructions, and a new variant of N -flation that we devise, can evade this constraint. We conclude that some classes of low-energy effective theories of large-field axion inflation are consistent with the strongest proposed form of the Weak Gravity Conjecture, while others are not. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show that the strong CP problem is solved in a large class of compactifications of string theory. The Peccei-Quinn mechanism solves the strong CP problem if the CP-breaking effects of the ultraviolet completion of gravity and of QCD are small compared to the CP-preserving axion potential generated by low-energy QCD instantons. We characterize both classes of effects. To understand quantum gravitational effects, we consider an ensemble of flux compactifications of type IIB string theory on orientifolds of Calabi-Yau hypersurfaces in the geometric regime, taking a simple model of QCD on D7-branes. We show that the D-brane instanton contribution to the neutron electric dipole moment falls exponentially in N4, with N the number of axions. In particular, this contribution is negligible in all models in our ensemble with N > 17. We interpret this result as a consequence of large N effects in the geometry that create hierarchies in instanton actions and also suppress the ultraviolet cutoff. We also compute the CP breaking due to high-energy instantons in QCD. In the absence of vectorlike pairs, we find contributions to the neutron electric dipole moment that are not excluded, but that could be accessible to future experiments if the scale of supersymmetry breaking is sufficiently low. The existence of vectorlike pairs can lead to a larger dipole moment. Finally, we show that a significant fraction of models are allowed by standard cosmological and astrophysical constraints. [ABSTRACT FROM AUTHOR]

We construct a supersymmetric vacuum of string theory in which the magnitude of the vacuum energy is <10-123 in Planck units. The mechanism that we introduce generates a well-controlled exponential hierarchy from an appropriate choice of the quantized parameters in the superpotential. Our approach provides a new angle on the cosmological constant problem. [ABSTRACT FROM AUTHOR]

We describe a method for finding flux vacua of type IIB string theory in which the Gukov-Vafa-Witten superpotential is exponentially small. We present an example with W0≈2×10-8 on an orientifold of a Calabi-Yau hypersurface with (h1,1,h2,1)=(2,272), at large complex structure and weak string coupling. [ABSTRACT FROM AUTHOR]