51ÁÔÆæÈë¿Ú

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.

Abstract:

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.