Dr Andrei Constantin

Abstract:

We present closed form expressions for the ranks of all cohomology groups of holomorphic line bundles on several Calabi‐Yau threefolds realised as complete intersections in products of projective spaces. The formulae have been obtained by systematising and extrapolating concrete calculations and they have been checked computationally. Although the intermediate calculations often involve laborious computations of ranks of Leray maps in the Koszul spectral sequence, the final results for cohomology follow a simple pattern. The space of line bundles can be divided into several different regions, and in each such region the ranks of all cohomology groups can be expressed as polynomials in the line bundle integers of degree at most three. The number of regions increases and case distinctions become more complicated for manifolds with a larger Picard number. We also find explicit cohomology formulae for several non‐simply connected Calabi‐Yau threefolds realised as quotients by freely acting discrete symmetries. More cases may be systematically handled by machine learning algorithms.

Abstract:

We derive an approximate analytic relation between the number of consistent heterotic Calabi-Yau compactifications of string theory with the exact charged matter content of the standard model of particle physics and the topological data of the internal manifold: the former scaling exponentially with the number of Kähler parameters. This is done by an estimate of the number of solutions to a set of Diophantine equations representing constraints satisfied by any consistent heterotic string vacuum with three chiral massless families, and has been computationally checked to hold for complete intersection Calabi-Yau threefolds (CICYs) with up to seven Kähler parameters. When extrapolated to the entire CICY list, the relation gives ∼10 23 string theory standard models; for the class of Calabi-Yau hypersurfaces in toric varieties, it gives ∼10 723 standard models.

Abstract:

We propose an analytic method to calculate the matter field Kähler metric in heterotic compactifications on smooth Calabi-Yau three-folds with Abelian internal gauge fields. The matter field Kähler metric determines the normalisations of the N = 1 chiral superfields, which enter the computation of the physical Yukawa couplings. We first derive the general formula for this Kähler metric by a dimensional reduction of the relevant supergravity theory and find that its T-moduli dependence can be determined in general. It turns out that, due to large internal gauge flux, the remaining integrals localise around certain points on the compactification manifold and can, hence, be calculated approximately without precise knowledge of the Ricci-flat Calabi-Yau metric. In a final step, we show how this local result can be expressed in terms of the global moduli of the Calabi-Yau manifold. The method is illustrated for the family of Calabi-Yau hypersurfaces embedded in ℙ1× ℙ3and we obtain an explicit result for the matter field Kähler metric in this case.

Abstract:

We present a general method for computing Hodge numbers for Calabi-Yau manifolds realised as discrete quotients of complete intersections in products of projective spaces. The method relies on the computation of equivariant cohomologies and is illustrated for several explicit examples. In this way, we compute the Hodge numbers for all discrete quotients obtained in Braun’s classification.

Abstract:

We analyze Yukawa unification in the context of E8×E8 heterotic Calabi-Yau models which rely on breaking to a grand unified theory (GUT) via a nonflat gauge bundle and subsequent Wilson line breaking to the standard model. Our focus is on underlying GUT theories with gauge group SU(5) or SO(10). We provide a detailed analysis of the fact that, in contrast to traditional field theory GUTs, the underlying GUT symmetry of these models does not enforce Yukawa unification. Using this formalism, we present various scenarios where Yukawa unification can occur as a consequence of additional symmetries. These additional symmetries arise naturally in some heterotic constructions, and we present an explicit heterotic line bundle model which realizes one of these scenarios.