Prof Andre Lukas

Abstract:

We study a class of supersymmetric Froggatt-Nielsen (FN) models with multiple U(1) symmetries and Standard Model (SM) singlets inspired by heterotic string compactifications on Calabi-Yau threefolds. The string-theoretic origin imposes a particular charge pattern on the SM fields and FN singlets, dividing the latter into perturbative and non-perturbative types. Employing systematic and heuristic search strategies, such as genetic algorithms, we identify charge assignments and singlet VEVs that replicate the observed mass and mixing hierarchies in the quark sector, and subsequently refine the Yukawa matrix coefficients to accurately match the observed values for the Higgs VEV, the quark and charged lepton masses and the CKM matrix. This bottom-up approach complements top-down string constructions and our results demonstrate that string FN models possess a sufficiently rich structure to account for flavour physics. On the other hand, the limited number of distinct viable charge patterns identified here indicates that flavour physics imposes tight constraints on string theory models, adding new constraints on particle spectra that are essential for achieving a realistic phenomenology.

Abstract:

The diffeomorphism class of simply connected smooth Calabi-Yau threefolds with torsion-free cohomology is determined via certain basic topological invariants: the Hodge numbers, the triple intersection form, and the second Chern class. In the present paper, we shed some light on this classification by placing bounds on the number of diffeomorphism classes present in the set of smooth Calabi-Yau threefolds constructed from the Kreuzer-Skarke (KS) list of reflexive polytopes up to Picard number six. The main difficulty arises from the comparison of triple intersection numbers and divisor integrals of the second Chern class up to basis transformations. By using certain basis-independent invariants, some of which appear here for the first time, we are able to place lower bounds on the number of classes. Upper bounds are obtained by explicitly identifying basis transformations, using constraints related to the index of line bundles. Extrapolating our results, we conjecture that the favorable entries of the KS list of reflexive polytopes lead to some (Formula presented.) diffeomorphically distinct Calabi-Yau threefolds.

Abstract:

Calabi–Yau manifolds can be obtained as hypersurfaces in toric varieties built from reflexive polytopes. We generate reflexive polytopes in various dimensions using a genetic algorithm. As a proof of principle, we demonstrate that our algorithm reproduces the full set of reflexive polytopes in two and three dimensions, and in four dimensions with a small number of vertices and points. Motivated by this result, we construct five-dimensional reflexive polytopes with the lowest number of vertices and points. By calculating the normal form of the polytopes, we establish that many of these are not in existing datasets and therefore give rise to new Calabi–Yau four-folds. In some instances, the Hodge numbers we compute are new as well.