Computational Tools for Cohomology of Toric Varieties

In this review, novel non-standard techniques for the computation of cohomology classes on toric varieties are summarized. After an introduction of the basic definitions and properties of toric geometry, we discuss a specific computational algorithm for the determination of the dimension of line-bundle valued cohomology groups on toric varieties. Applications to the computation of chiral massless matter spectra in string compactifications are discussed and, using the software package cohomCalg, its utility is highlighted on a new target space dual pair of (0,2) heterotic string models.Comment: 17 pages, 4 tables; prepared for the special issue "Computational Algebraic Geometry in String and Gauge Theory" of Advances in High Energy Physics, cohomCalg implementation available at http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg

Cohomology of Line Bundles: A Computational Algorithm

We present an algorithm for computing line bundle valued cohomology classes over toric varieties. This is the basic starting point for computing massless modes in both heterotic and Type IIB/F-theory compactifications, where the manifolds of interest are complete intersections of hypersurfaces in toric varieties supporting additional vector bundles.Comment: 11 pages, 1 figure, 2 tables; v2: typos and references corrected; v3: proof-related statements updated, cohomCalg implementation available at http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg

Cohomology of Line Bundles: Applications

Massless modes of both heterotic and Type II string compactifications on compact manifolds are determined by vector bundle valued cohomology classes. Various applications of our recent algorithm for the computation of line bundle valued cohomology classes over toric varieties are presented. For the heterotic string, the prime examples are so-called monad constructions on Calabi-Yau manifolds. In the context of Type II orientifolds, one often needs to compute equivariant cohomology for line bundles, necessitating us to generalize our algorithm to this case. Moreover, we exemplify that the different terms in Batyrev's formula and its generalizations can be given a one-to-one cohomological interpretation. This paper is considered the third in the row of arXiv:1003.5217 and arXiv:1006.2392.Comment: 56 pages, 8 tables, cohomCalg incl. Koszul extension available at http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg

Calabi-Yau Manifolds with Large Volume Vacua

We describe an efficient, construction independent, algorithmic test to determine whether Calabi--Yau threefolds admit a structure compatible with the Large Volume moduli stabilization scenario of type IIB superstring theory. Using the algorithm, we scan complete intersection and toric hypersurface Calabi-Yau threefolds with $2 \leq h^{1,1} \le 4$ and deduce that 418 among 4434 manifolds have a Large Volume Limit with a single large four-cycle. We describe major extensions to this survey, which are currently underway.Comment: 4 page

On Instanton Effects in F-theory

We revisit the issue of M5-brane instanton corrections to the superpotential in F-theory compactifications on elliptically fibered Calabi-Yau fourfolds. Elaborating on concrete geometries, we compare the instanton zero modes for non-perturbative F-theory models with the zero modes in their perturbative Sen limit. The fermionic matter zero modes localized on the intersection of the instanton with the space-time filling D7-branes show up in a geometric way in F-theory. Methods for their computation are developed and, not surprisingly, exceptional gauge group structures do appear. Finally, quite intriguing geometrical aspects of the one-loop determinant are discussed.Comment: 52 pages, 8 figures, 13 tables; v2: extended discussion of matter zero modes, refs added; v3: sections 3.3 + 4.1 restructure

F-theory uplifts and GUTs

We study the F-theory uplift of Type IIB orientifold models on compact Calabi-Yau threefolds containing divisors which are del Pezzo surfaces. We consider two examples defined via del Pezzo transitions of the quintic. The first model has an orientifold projection leading to two disjoint O7-planes and the second involution acts via an exchange of two del Pezzo surfaces. The two uplifted fourfolds are generically singular with minimal gauge enhancements over a divisor and, respectively, a curve in the non-Fano base. We study possible further degenerations of the elliptic fiber leading to F-theory GUT models based on subgroups of E8.Comment: 28 pages, 5 tables; v2: typos removed, minor correction