Novel nonstandard 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

The computation of certain cohomology groups is a critical technical step in string model building, relevant, for example, in determining the (chiral) zero-mode spectrum or parts of the effective four-dimensional theory, like the Yukawa coupling. Common methods often try to relate the computation at hand via a chain of isomorphisms back to known results in order to avoid most of the cumbersome computations from the ground up. Spectral sequences are the established technique to deal with such problems, but often end up to become laborious rather quickly. Having reasonable efficient algorithms to one's avail is therefore a vital requirement to make progress.

Supersymmetry in four dimensions puts strong restrictions on the geometries admissible for string compactifications. In the absence of additional background fluxes (besides a gauge flux), this leads to the class of Calabi-Yau manifolds, where of particular interest for

the monad construction, which naturally arises in the (

the spectral cover construction, which gives stable holomorphic vector bundles with structure group SU

the construction via extensions, which is the natural counterpart of brane recombinations.

All these three constructions have in common that they involve line bundles in one way or the other. For instance, the monad is defined via sequences of the Whitney sums of line bundles, whereas the

Using a simple yet powerful algorithm, we can compute the line-bundle-valued cohomology dimensions

This paper is organized as follows. In Section

One of the most important aspects of toric geometry is the ability to understand it in purely combinatorial terms, which is ideally suited to be handled by computers (see [

The homogeneous coordinates

Given the GLSM charges and the Stanley-Reisner ideal to identify the geometric phase, the toric variety

The combinatorial perspective on toric geometry mentioned at the start is formulated in terms of toric fans, cones, and triangulations. In this language a geometric phase corresponds to a triangulation of a certain set of lattice vectors

Given a toric variety

The geometric input data for the computational algorithm presented below are the GLSM charges

More precisely, negative integer exponents are only admissible for those coordinates that are contained in subsets of the Stanley-Reisner ideal generators. The most economic way is therefore to determine in a first step the set of square-free monomials

The multiplicity factors are defined by the dimensions of an intermediate relative homology. Let

The construction of the relative complex

The homology group dimensions

After computing the multiplicity factors

In order to show the working algorithm in detail, we consider the del Pezzo-1 surface. Its toric data is summarized in Table

Toric data for the del Pezzo-1 surface.

Vertices of the polyhedron/fan | Coords. | GLSM charges | Divisor class | |||

1 | 0 | |||||

1 | 0 | |||||

1 | 1 | |||||

0 | 1 |

Intersection form:

All the aforementioned steps involved in the computation of the cohomology have been conveniently implemented in a high-performance cross-platform package called

Due to the explicit form of the relevant monomials that are counted by the algorithm, one can consider a rather simple generalization that also takes the action of finite groups into account [

The so-called equivariant structure uplifts the action on the base geometry to the bundle and preserves the group structure. In fact, for a generic group

The choice of an equivariant structure provides the means of how the finite group acts on the relevant monomials (

This powerful generalization of the algorithm allows for instance to compute the untwisted matter spectrum in heterotic orbifold models or (parts of) the instanton zero mode spectrum for the Euclidean D-brane instantons in Type II orientifold models (see [

In most string theory applications, the geometries of interest are not toric varieties by themselves, but rather defined as subspaces thereof. These are defined as complete intersections of hypersurfaces of certain degrees. In order to relate the cohomology of the toric variety

To make this paper self-contained and because it has been implemented in the

then allows to relate the cohomology of the toric variety

Given a more generic case of several (mutually transverse) hypersurfaces

Before we come to a concrete application in heterotic string model building, let us present the construction of holomorphic vector bundles via the so-called monad. Such a structure directly arises in the (

Given the GLSM charges defined in (

The (0, 2) GLSM generalizes this in the sense that the bundle the left-moving world-sheet fermions couple to is not any longer the tangent bundle of the Calabi-Yau, but a more general holomorphic (stable) vector bundle

Now let us show all this for concrete heterotic (

In order to obtain the number of zero modes in different representations of the GUT group, we have to calculate the cohomology classes of bundles involving the holomorphic vector bundle [

Correlation between zero modes in representations of the GUT group

Number of zero modes in reps. of | 1 | |||||
---|---|---|---|---|---|---|

| 248 | |||||

| ||||||

The moduli appearing in such a framework are given by possible deformations of the Calabi-Yau manifold, which are counted by the Hodge numbers

In the following we give an example of a pair of heterotic models which are related by the so-called target space duality [

Let us start with an example in which we can already see most of the structure but which is not too involved. Consider

Toric data for the smooth (

Coordinate GLSM charges | Hypers. degrees | |||||||||

0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |

1 | 1 | 1 | 1 | 2 | 2 | 2 | 0 | 3 | 3 | 4 |

Toric data for the dual

Coordinate GLSM charges | Hypersurf. degrees | ||||||||||||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |

0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |

1 | 1 | 1 | 1 | 2 | 2 | 2 | 0 | 0 | 0 | 3 | 3 | 2 | 2 |

This is a manifestation of a so far only very poorly understood perturbative (in

So far most implementations of computational methods in string model building have been based on toric geometry [

The computational tool reviewed in this paper can also be applied to situations where other packages fail. As explained, the powerful algorithm for the determination of the dimensions of line-bundle-valued cohomology classes is taylor made for dealing also with general complete intersection and for (

Of course, also the algorithm implementation

Note that there is also the Macaulay 2 package [

The authors would like to thank Helmut Roschy for his contributions to the original work presented in this paper.