Symmetries of Calabi-Yau prepotentials with isomorphic flops
Calabi-Yau threefolds with infinitely many flops to isomorphic manifolds have an extended Kähler cone made up from an infinite number of individual Kähler cones. These cones are related by reflection symmetries across flop walls. We study the implications of this cone structure for mirror symmetry, by considering the instanton part of the prepotential in Calabi-Yau threefolds. We show that such isomorphic flops across facets of the Kähler cone boundary give rise to symmetry groups isomorphic to Coxeter groups. In the dual Mori cone, non-flopping curve classes that are identified under these groups have the same Gopakumar-Vafa invariants. This leads to instanton prepotentials invariant under Coxeter groups, which we make manifest by introducing appropriate invariant functions. For some cases, these functions can be expressed in terms of theta functions whose appearance can be linked to an elliptic fibration structure of the Calabi-Yau manifold.
Flops for complete intersection Calabi-Yau threefolds
Abstract:We study flops of Calabi-Yau threefolds realised as Kähler-favourable complete intersections in products of projective spaces (CICYs) and identify two different types. The existence and the type of the flops can be recognised from the configuration matrix of the CICY, which also allows for constructing such examples. The first type corresponds to rows containing only 1s and 0s, while the second type corresponds to rows containing a single entry of 2, followed by 1s and 0s. We give explicit descriptions for the manifolds obtained after the flop and show that the second type of flop always leads to isomorphic manifolds, while the first type in general leads to non-isomorphic flops. The singular manifolds involved in the flops are determinantal varieties in the first case and more complicated in the second case. We also discuss manifolds admitting an infinite chain of flops and show how to identify these from the configuration matrix. Finally, we point out how to construct the divisor images and Picard group isomorphisms under both types of flops.
Cosmic inflation and genetic algorithms
Large classes of standard single-field slow-roll inflationary models consistent with the required number of e-folds, the current bounds on the spectral index of scalar perturbations, the tensor-to-scalar ratio, and the scale of inflation can be efficiently constructed using genetic algorithms. The setup is modular and can be easily adapted to include further phenomenological constraints. A semi-comprehensive search for sextic polynomial potentials results in O (300,000) viable models for inflation. The analysis of this dataset reveals a preference for models with a tensor-to-scalar ratio in the range 0.0001 ≤ r ≤ 0.0004. We also consider potentials that involve cosine and exponential terms. In the last part we explore more complex methods of search relying on reinforcement learning and genetic programming. While reinforcement learning proves more difficult to use in this context, the genetic programming approach has the potential to uncover a multitude of viable inflationary models with new functional forms.
Numerical metrics for complete intersection and Kreuzer–Skarke Calabi–Yau manifolds
We introduce neural networks (NNs) to compute numerical Ricci-flat Calabi–Yau (CY) metrics for complete intersection and Kreuzer–Skarke (KS) CY manifolds at any point in Kähler and complex structure moduli space, and introduce the package cymetric which provides computation realizations of these techniques. In particular, we develop and computationally realize methods for point-sampling on these manifolds. The training for the NNs is carried out subject to a custom loss function. The Kähler class is fixed by adding to the loss a component which enforces the slopes of certain line bundles to match with topological computations. Our methods are applied to various manifolds, including the quintic manifold, the bi-cubic manifold and a KS manifold with Picard number two. We show that volumes and line bundle slopes can be reliably computed from the resulting Ricci-flat metrics. We also apply our results to compute an approximate Hermitian–Yang–Mills connection on a specific line bundle on the bi-cubic.