Computational Tools for Cohomology of Toric Varieties

Abstract

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 cohomCalg, its utility is highlighted on a new target space dual pair of (0,2) heterotic string models.

1. Introduction

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 𝒩=1 supersymmetry are the Calabi-Yau threefolds and fourfolds. Due to the Atiyah-Singer index theorem, chirality is realized by also turning on a nontrivial gauge background, which can be understood as the curvature of a nontrivial holomorphic vector bundle on the manifold. The majority of known Calabi-Yau manifolds are based on toric geometry. In particular, they are constructed as complete intersections of hypersurfaces in toric varieties. The vector bundle can then be described by different methods, where the three mostly used ones are (1)the monad construction, which naturally arises in the (0,2) gauged linear sigma model, (2)the spectral cover construction, which gives stable holomorphic vector bundles with structure group SU(𝑛) on elliptically fibered Calabi-Yau threefolds, (3)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 𝑛-fold spectral cover is equipped in addition with a nontrivial line bundle on it, which via the Fourier-Mukai transform gives an SU(𝑛) vector bundle on the Calabi-Yau manifold. The basis starting point of every cohomology computation is therefore the knowledge of line-bundle-valued cohomology classes on the ambient toric variety.

Using a simple yet powerful algorithm, we can compute the line-bundle-valued cohomology dimensions ℎ^𝑖(𝑋;𝐿_𝑋)=dim𝐻^𝑖(𝑋;𝐿_𝑋) for any toric variety based on the information contained in the Stanley-Reisner ideal. The Koszul complex then allows to relate the cohomology on the toric variety to the cohomology of a hypersurface or complete intersection. The particular form of the algorithm also allows to easily deal with finite group actions on such geometries, that is, to consider orbifold spaces and twisted string states.

This paper is organized as follows. In Section 2 some basics of toric geometry are introduced, including the Stanley-Reisner ideal and toric fans. Section 3 introduces the computational algorithm for cohomology group dimensions of toric varieties that will be used throughout this paper. Section 4 shows how a finite group action and the resulting quotient space can be handled. In Section 5 the Koszul sequence is introduced, which allows to relate the ambient variety's cohomology to the cohomology of hypersurfaces and complete intersections. Monad bundle constructions and the Euler sequence are introduced in Section 6. In Section 7 we show an example of how to compute the data for a (2,2) model that is dual to a (0,2) model. The paper closes in Section 8 with a brief outlook on potential further applications and developments.

2. Toric Varieties

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 [1–4] for introductions into the subject). Toric geometry is also directly related to gauged linear 𝜎-models (GLSMs) in physics [5]. On a more basic notion, a toric variety is a generalization of a projective space, which consists of a set of homogeneous coordinates 𝑥₁,…,𝑥_𝑛 as well as 𝑅 projective relations 𝑥₁,…,𝑥_𝑛∼𝜆^𝑄1(𝑟)_𝑟𝑥₁,…,𝜆^{𝑄𝑛(𝑟)}_𝑟𝑥_𝑛for𝜆_𝑟∈ℂ^×. (2.1) The 𝑄_𝑖^(𝑟) for 𝑟=1,…,𝑅 and 𝑖=1,…,𝑛 are GLSM charges, that is, the Abelian 𝑈(1) charges in the associated GLSM, and corresponding to the projective weights. In direct comparison to projective spaces, toric varieties can be characterized as arising due to the usage of multiple projective relations instead of just a single one. The special case of a projective space therefore corresponds to 𝑅=1 in the above notation.

The homogeneous coordinates 𝑥_𝑖 become 𝒩= (2,2) chiral superfields in the GLSM picture, and the Fayet-Iliopoulos parameters 𝜉_𝑟 of the Abelian symmetries can be interpreted as the Kähler parameters of the geometric space. This parameter space of 𝜉=(𝜉₁,…,𝜉_𝑅) is then split into 𝑅-dimensional cones due to the vanishing of the D-terms associated to the GLSM. Within each cone the D-flatness condition can be solved and the cones correspond to the geometrical Kähler cones. Each such cone is often referred to as a geometric phase and can be fully characterized by a set of collections of coordinates 𝒮_𝜌=𝑥_𝜌1,𝑥_𝜌2,…,𝑥_{𝜌𝜌||𝒮}for𝜌=1,…,𝑁 (2.2) which are not allowed to vanish simultaneously. Note that such a collection is often written in product form; that is, the square-free monomial 𝑥_𝜌1𝑥_𝜌2⋯𝑥_{𝜌𝜌||𝒮} refers exactly to the same set. All those sets form the Stanley-Reisner ideal SR(𝑋)=𝒮₁,…,𝒮_𝑁, (2.3) which can be equivalently used to uniquely specify a geometric phase. Note that the Stanley-Reisner ideal is Alexander dual to the irrelevant ideal 𝐵_Σ used in the mathematical literature.

Given the GLSM charges and the Stanley-Reisner ideal to identify the geometric phase, the toric variety 𝑋 of dimension 𝑑=𝑛−𝑅 can be described as the coset space ℂ𝑋=^𝑛−𝑍ℂ^×𝑅, (2.4) where 𝑍 is the set of removed points specified by SR(𝑋) via 𝑍=^𝑁_𝜌=1𝑥_𝜌1=𝑥_𝜌2=⋯=𝑥_{𝜌𝜌||𝒮}=0. (2.5) This set 𝑍 can be understood as the toric generalization of the removed origin in a projective space ℂℙ^𝑛=(ℂ^𝑛+1−{0})/ℂ^×, as the Stanley-Reisner ideal for ℂℙ^𝑛 is just the collection of all coordinates.

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 𝜈_𝑖 that span the fan Σ_𝑋. The GLSM charges 𝑄_𝑖^(𝑟) reappear in the form of 𝑅 linear relations ^𝑛_𝑖=1𝑄_𝑖^(𝑟)𝜈_𝑖=0for𝑟=1,…,𝑅. (2.6) By associating the lattice vectors 𝜈_𝑖 to the homogeneous coordinates 𝑥_𝑖, it becomes obvious that the linear relations (2.6) between the lattice vectors encode the projective equivalences (2.1) between the homogeneous coordinates. In the language of fans the Stanley-Reisner ideal consists of all square-free monomials whose coordinates are not contained in any cone of the toric fan Σ_𝑋.

3. Dimensions of Line-Bundle-Valued Cohomology Groups

Given a toric variety 𝑋 and a line bundle 𝐿_𝑋, a frequent issue is to compute the 𝐿_𝑋-valued cohomology group dimensions ℎ^𝑖(𝑋;𝐿_𝑋) for 𝑖=0,…,dim𝑋. After a couple of preliminary observations in [6, 7], in [8] a complete novel algorithm for the determination of ℎ^𝑖(𝑋;𝐿_𝑋) was presented. This was subsequently proven in [9] and independently in [10].

The geometric input data for the computational algorithm presented below are the GLSM charges 𝑄_𝑖^(𝑟) and the Stanley-Reisner ideal generators 𝒮₁,…,𝒮_𝑁. The basic idea of the algorithm is to count the number of monomials, where the total GLSM charge is equal to the divisor class of 𝐷, which is the divisor that specifies the line bundle 𝐿_𝑋=𝒪_𝑋(𝐷). The form of those monomials is highly restricted by the Stanley-Reisner ideal, that is, the simpler the structure of SR(𝑋), the easier the computation.

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 𝒬 that arise from unions of the coordinates in any subset of SR(𝑋). Each 𝒬 gives a set of coordinates with negative exponents, and to each 𝒬 there is an associated weighting factor 𝔥_𝑖(𝒬) that specifies to which cohomology group's dimension ℎ^𝑖(𝑋;𝒪_𝑋(𝐷)) the number of monomials 𝒩_𝐷(𝒬) with GLSM charge 𝐷 contributes. The cohomology group dimension formula can be summarized as dim𝐻^𝑖𝑋;𝒪_𝑋(𝐷)=𝔥_𝑖(𝒬)_𝒬^{multiplicityfactor0𝑥0𝑓𝑒37}⋅𝒩_𝐷(𝒬)_{numberofmonomials}, (3.1) where the sum ranges over all square-free monomials that can be obtained from unions of Stanley-Reisner ideal generators. In the remainder of this section, both 𝔥_𝑖(𝒬) and 𝒩_𝐷(𝒬) will be properly defined.

3.1. Computation of Multiplicity Factors

The multiplicity factors are defined by the dimensions of an intermediate relative homology. Let [𝑁]∶={1,…,𝑁} be a set of indices for the 𝑁 square-free monomials that generate the Stanley-Reisner ideal. Then let, for each subset 𝑆_𝜌∶=𝒮_𝜌1,…,𝒮_𝜌𝑘⊂𝒮₁,…,𝒮_𝑁 (3.2) of generators, 𝒬(𝑆_𝜌) be the square-free monomial that arises from the union of all coordinates in each generator 𝒮_𝜌𝑖 of the subset.

The construction of the relative complex Γ^𝒬, from which 𝔥_𝑖(𝒬) is defined, goes as follows. From the full simplex on [𝑁]={1,…,𝑁}, extract only those subsets 𝜌⊂[𝑁] with 𝒬(𝑆_𝜌)=𝒬; that is, one considers all possible combinations of the Stanley-Reisner ideal generators whose coordinates unify to the same square-free monomial 𝒬. For some fixed |𝜌|=𝑘, this then defines the set of (𝑘−1)-dimensional faces 𝐹_𝑘−1(𝒬) of the complex Γ^𝒬, that is, 𝐹_𝑘(𝒬)[𝑁]∶𝜌𝒬𝑆∶=𝜌⊂=𝑘+1_𝜌=𝒬. (3.3) Furthermore, let ℂ^{𝐹𝑘(𝒬)} be the complex vector space with basis vectors 𝑒_𝜌 for 𝜌∈𝐹_𝑘(𝒬). The relative complex Γ^𝒬∶0⟶𝐹_𝑁−1(𝒬)⟶^𝜙𝑁−1⋯⟶^𝜙1𝐹₀(𝒬)⟶^𝜙0𝐹₋₁(𝒬)⟶0, (3.4) where 𝐹₋₁(𝒬)∶={∅} is a face of dimension −1, is then specified by the chain mappings 𝜙_𝑘∶𝐹_𝑘(𝒬)⟶𝐹_𝑘−1(𝒬),𝑒_𝜌⟼_𝑠∈𝜌sign𝑠,𝜌𝑒_{𝜌−{𝑠}}. (3.5) A basis vector 𝑒_{𝜌−{𝑠}} vanishes if 𝜌 with the element 𝑠 removed is not contained in Γ^𝒬. Furthermore, the signum is defined by sign(𝑠,𝜌)∶=(−1)^ℓ−1 when 𝑠 is the ℓth element of 𝜌⊂[𝑁]={1,…,𝑁} when written in increasing order.

The homology group dimensions 𝔥_𝑖(𝒬)∶=dim𝐻_{|𝒬|−𝑖−1}Γ^𝒬 (3.6) of the relabeled complex then provide the multiplicity factors that determine to which cohomology group 𝐻^𝑖(𝑋;𝒪_𝑋(𝐷)) the monomials associated to 𝒬 contribute. It should be emphasized that the 𝔥_𝑖(𝒬) depend only on the geometry (the Stanley-Reisner ideal) of the toric variety 𝑋 and not on the line bundle 𝒪_𝑋(𝐷), that is, the multiplicity factors only have to be computed once for each geometry.

3.2. Counting Monomials

After computing the multiplicity factors 𝔥_𝑖(𝒬), it remains to count the number of relevant monomials. This second part of the algorithm depends on the GLSM charges of the homogeneous coordinates 𝑥_𝑖 and the specific line bundle 𝒪_𝑋(𝐷). Let 𝒬 again be a square-free monomial. In order to simplify the notation, let 𝐼=(𝑖₁,…,𝑖_𝑘,…,𝑖_𝑛) be an index relabeling such that the product of the first 𝑘 coordinates gives 𝒬=𝑥_𝑖1⋯𝑥_𝑖𝑘. Then one considers monomials of the form 𝑅^𝒬𝑥₁,…,𝑥_𝑛∶=𝑥_𝑖1^−1−𝑎𝑥_𝑖2^−1−𝑏⋯𝑥_𝑖𝑘^−1−𝑐𝑥_𝑖𝑘+1^𝑑⋯𝑥_𝑖𝑛^𝑒=𝑇𝑥_𝑖𝑘+1,…,𝑥_𝑖𝑛𝑥_𝑖1⋯𝑥_𝑖𝑘⋅𝑊𝑥_𝑖1,…,𝑥_𝑖𝑘, (3.7) where 𝑇 and 𝑊 are monomials (not necessarily square-free) as well as exponents 𝑎,𝑏,𝑐,𝑑,𝑒∈ℕ∪{0}. One obviously finds the coordinates of the square-free monomial 𝒬 in the denominator, whereas their complements are in the numerator. Based on the particular form of the relevant monomials define 𝒩_𝐷(𝒬)∶=dim𝑅^𝒬∶deg_GLSM𝑅^𝒬=𝐷, (3.8) which counts the number of relevant monomials that have the same GLSM degree as the divisor 𝐷 that specified the line bundle 𝐿_𝑋=𝒪_𝑋(𝐷).

3.3. A Step by Step Example: del Pezzo-1 Surface

In order to show the working algorithm in detail, we consider the del Pezzo-1 surface. Its toric data is summarized in Table 1 for the reader’s convenience. The two Stanley-Reisner ideal generators yield four possible combinations that become relevant in the computation, namely, 𝒬=1,𝑥₁𝑥₂,𝑥₃𝑥₄,𝑥₁𝑥₂𝑥₃𝑥₄. (3.9) The computation of the multiplicity factors for those square-free monomials leads to ℭ₀(1)={{∅}},ℭ₁𝑥₁𝑥₂=𝒮₁,ℭ₁𝑥₃𝑥₄=𝒮₂,ℭ₂𝑥₁𝑥₂𝑥₃𝑥₄=𝒮₁,𝒮₂, (3.10) and all other spaces ℭ_𝑖(𝒬) vanishing. After computing the homology, this leads to the following contributions of the monomials (3.7) to the cohomology groups: 𝐻⁰𝑑𝑃₁;𝒪(𝑚,𝑛)∶𝑇𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝐻¹𝑑𝑃₁()∶𝑇𝑥;𝒪𝑚,𝑛³,𝑥₄𝑥₁𝑥₂⋅𝑊𝑥₁,𝑥₂,𝑇𝑥₁,𝑥₂𝑥₃𝑥₄⋅𝑊𝑥₃,𝑥₄,𝐻²𝑑𝑃₁()∶1;𝒪𝑚,𝑛𝑥₁𝑥₂𝑥₃𝑥₄⋅𝑊𝑥₁,𝑥₂,𝑥₃,𝑥₄. (3.11) Consider computing ℎ^•(𝑑𝑃₁;𝒪(−1,−2)). Since all GLSM charges are positive, there is no contribution to ℎ⁰. Likewise, the denominator monomial of the ℎ² contribution already has the GLSM charge (3, 2), which “overshoots” the target values and therefore also gives no contribution. deg_GLSM(1/𝑥₁𝑥₂)=(−2,0) is no good either, but deg_GLSM(1/𝑥₃𝑥₄)=(−1,−2) fits perfectly, such that there is a sole contribution 1𝑥₃𝑥₄⇝ℎ^•𝑑𝑃₁;𝒪(−1,−2)=(0,1,0). (3.12)

Table 1 

Toric data for the del Pezzo-1 surface.

All the aforementioned steps involved in the computation of the cohomology have been conveniently implemented in a high-performance cross-platform package called cohomCalg [11].

4. Equivariant Cohomology for Finite Group Actions

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 [12, 13]. In orientifold and orbifold settings, the internal part of the space-time is usually specified by a discrete symmetry acting on the “upstairs” geometry. This then induces a corresponding splitting of the cohomology groups 𝐻^𝑖(𝑋)=𝐻^𝑖_inv(𝑋)𝐻^𝑖_non-inv(𝑋) (4.1) as the generating 𝑝-cycles can be either invariant or noninvariant under the symmetry. It is also necessary to specify the induced action on the bundle defined on the upstairs geometry.

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 𝐺, each group element 𝑔 induces an involution mapping 𝑔∶𝑋→𝑋 on the base geometry and has a corresponding uplift 𝜙_𝑔∶𝑉→𝑉 that has to be compatible with the bundle structure. This makes the diagram (4.2) commutative, and the 𝐺-structure 𝑉 is called an equivariant structure if it preserves the group structure, that is, if 𝜙_𝑔∘𝜙_ℎ=𝜙_𝑔ℎ holds such that the mapping 𝑔↦𝜙_𝑔 is a group homomorphism.

The choice of an equivariant structure provides the means of how the finite group acts on the relevant monomials (3.7) counted by the algorithm. For a given line bundle 𝒪_𝑋(𝐷), one then has to check for all monomials whether or not they are invariant under the induced action. Consider, for example, the bundle 𝒪(−6) on ℂℙ² and the ℤ₃ action 𝑔₁∶𝑥₁,𝑥₂,𝑥₃⟼𝛼𝑥₁,𝛼²𝑥₂,𝑥₃for𝛼∶=√³1=e^2𝜋i/3 (4.3) on the base coordinates. The same action is used for the monomials, and thus it defines the equivariant structure. The relevant monomials for the algorithm then pick up the following values from the involution: 1𝑢⁴₁𝑢₂𝑢_3𝑔1→1,1𝑢₁𝑢⁴₂𝑢_3𝑔1→1,1𝑢₁𝑢₂𝑢⁴_3𝑔1→1,1𝑢³₁𝑢²₂𝑢_{3𝑔1→𝛼},1𝑢³₁𝑢₂𝑢²_{3𝑔1→𝛼2},1𝑢²₁𝑢³₂𝑢_{3𝑔1→𝛼2},1𝑢₁𝑢³₂𝑢²_{3𝑔1→𝛼},1𝑢²₁𝑢₂𝑢³_{3𝑔1→𝛼},1𝑢₁𝑢²₂𝑢³_{3𝑔1→𝛼2},1𝑢²₁𝑢²₂𝑢²_{3𝑔1→1ℎ2ℂℙ2();𝒪−6=(4inv,3𝛼,3𝛼2)},, (4.4) such that ℎ^•_inv(ℂℙ²;𝒪(−6))=(0,0,4) follows. This gives the cohomology of the quotient space ℂℙ²/ℤ₃ as defined by the action in (4.3).

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 [14] for concrete applications).

5. The Koszul Complex

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 𝑋 to the cohomology of a subspace, the Koszul sequence is used.

To make this paper self-contained and because it has been implemented in the cohomCalg Koszul extension package, let us briefly describe how this works. Let 𝐷⊂𝑋 be an irreducible hypersurface, and let 0≠𝜎∈𝐻⁰(𝑋;𝒪(𝐷)) be a global nonzero section of 𝒪_𝑋(𝐷), such that 𝑍(𝜎)≅𝐷. This induces a mapping 𝒪_𝑋→𝒪_𝑋(𝐷) and its dual 𝒪_𝑋(−𝐷)↪𝒪_𝑋, the latter of which can be shown to be injective. Given an effective divisor 𝐷∶=_𝑖𝑎_𝑖𝐻_𝑖⊂𝑋, (5.1) where all 𝑎_𝑖≥0, there is a short exact sequence 0⟶𝒪_𝑋(−𝐷)↪𝒪_𝑋↠𝒪_𝐷⟶0, (5.2) called the Koszul sequence. Here 𝒪_𝐷 is the quotient of the sheaf 𝒪_𝑋 of holomorphic functions on 𝑋 by all holomorphic functions vanishing at least to order 𝑎_𝑖 along the irreducible hypersurface 𝐻_𝑖⊂𝑋. This allows to treat 𝒪_𝐷 as the structure sheaf on the divisor 𝐷, which effectively identifies the sheaf cohomology 𝐻^𝑖(𝑋;𝒪_𝐷) with 𝐻^𝑖(𝐷;𝒪_𝐷). A proper definition of the involved mappings, which become quite laborious to work out explicitly, can be found in [15]. In addition of the plain Koszul sequence (5.2), there is also a twisted variant 0⟶𝒪_𝑋(𝑇−𝐷)↪𝒪_𝑋(𝑇)↠𝒪_𝐷(𝑇)⟶0 (5.3) that is obtained by tensoring (5.2) with the line bundle 𝒪_𝑋(𝑇). The induced long exact cohomology sequence 0⟶𝐻⁰𝑋;𝒪_𝑋(𝑇−𝐷)⟶𝐻⁰𝑋;𝒪_𝑋(𝑇)⟶𝐻⁰𝐷;𝒪_𝐷(𝑇)⟶𝐻¹𝑋;𝒪_𝑋(𝑇−𝐷)⟶𝐻¹𝑋;𝒪_𝑋(𝑇)⟶𝐻¹𝐷;𝒪_𝐷(𝑇)⟶𝐻²𝑋;𝒪_𝑋(𝑇−𝐷)⟶𝐻²𝑋;𝒪_𝑋(𝑇)⟶𝐻²𝐷;𝒪_𝐷(𝑇)⟶⋯ (5.4)

then allows to relate the cohomology of the toric variety 𝑋 directly to the cohomology of the hypersurface.

Given a more generic case of several (mutually transverse) hypersurfaces {𝑆₁,…,𝑆_𝑙}, one can compute the cohomology on the complete intersection via the generalized Koszul sequence0⟶𝒪_𝑋−^𝑙_𝑗=1𝑆_𝑗+𝐷⟶⋯⟶_𝑖1<𝑖2𝒪_𝑋−𝑆_𝑖1−𝑆_𝑖2+𝐷⟶_𝑖1𝒪_𝑋−𝑆_𝑖1+𝐷⟶𝒪_𝑋(𝐷)⟶𝒪_𝑆(𝐷)⟶0. (5.5) In contrast to the hypersurface sequence, this is no longer a short exact sequence and hence does not give rise to a long exact sequence in cohomology. One way to proceed is via the technique of spectral sequences, which inductively allows one to compute the wanted cohomology classes on the complete intersection. However, for our implementation, we decided to take a different approach. We break down this long sequence (5.5) into several short exact sequences using several auxiliary sheaves ℐ_𝑘: 0⟶𝒪_𝑋−^𝑙_𝑗=1𝑆_𝑗+𝐷⟶_{𝑖1<⋯<𝑖𝑙−1}𝒪_𝑋−^𝑙−1_𝑗=1𝑆_𝑖𝑗+𝐷↠ℐ₁⟶0,0⟶ℐ₁↪_{𝑖1<⋯<𝑖𝑙−2}𝒪_𝑋−^𝑙−2_𝑗=1𝑆_𝑖𝑗+𝐷↠ℐ₂⋮⟶0,0⟶ℐ_𝑙−2↪_𝑖1𝒪_𝑋−𝑆_𝑖1+𝐷↠ℐ_𝑙−1⟶0,0⟶ℐ_𝑙−1↪𝒪_𝑋(𝐷)↠𝒪_𝑆(𝐷)⟶0. (5.6) The individually induced long exact sequences of cohomology can then be used for the step-wise computation of 𝐻^•(𝑆;𝒪_𝑆(𝐷)), which is the cohomology on the complete intersection 𝑆=⋂^𝑙_𝑖=1𝑆_𝑖.

6. Monad Construction of Vector Bundles

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 (0,2) GLSM description and can be regarded as a generalization of the tangent bundle of a complete intersection in a toric variety.

Given the GLSM charges defined in (2.1), the tangent bundle can be defined as the quotient 𝑇_𝑆=Ker(𝑓)/Im(𝑔) of the sequence 0⟶𝒪^{one𝒪sforeachPicardgenerator0𝑥0𝑓𝑒37}_𝑆^𝑔⊕𝑅↪^𝑛_𝑖=1𝒪_𝑆𝑄↠_{𝑖onebundlewiththeGLSMchargesforeachcoordinate}^{𝑓onebundlewiththedegreeforeachhypersurface𝑙0𝑥0𝑓𝑒37}_𝑗=1𝒪_𝑆(𝑆_𝑗)⟶0, (6.1) where the individual line bundles are restricted to the complete intersection 𝑆=⋂^𝑙_𝑖=1𝑆_𝑖. The rank of the resulting vector bundle is given by rk(𝑇)=𝑛−𝑙−𝑅. Using the methods presented so far, it is clear that they allow to compute the dimensions of the cohomology classes ℎ^𝑖(𝑆;𝑇_𝑆), where the initial input data for the set of long exact sequences are the line-bundle-valued cohomology classes on the ambient toric variety.

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 𝑉, which is analogously defined via a sequence of the Whitney sums of line bundles 0⟶𝒪↪^⊕𝑅𝑉_𝑆^𝑔𝛿_𝑎=1𝒪_𝑆𝑁↠_𝑎^𝑔𝜆_𝑙=1𝒪_𝑆𝑀_𝑙⟶0. (6.2) The rank is rk(𝑉)=𝛿−𝜆−𝑅_𝑉. The charges 𝑁_𝑎 and 𝑀_𝑙 have to satisfy the anomaly cancellation conditions _𝑎𝑁_𝑎^(𝛼)=_𝑙𝑀_𝑙^(𝛼),∀𝛼,_𝑙𝑀_𝑙^(𝛼)𝑀_𝑙^(𝛽)−_𝑎𝑁_𝑎^(𝛼)𝑁_𝑎^(𝛽)=_𝑗𝑆_𝑗^(𝛼)𝑆_𝑗^(𝛽)−_𝑖𝑄_𝑖^(𝛼)𝑄_𝑖^(𝛽),∀𝛼,𝛽, (6.3) where 1≤𝛼, 𝛽≤𝑅 denote the components corresponding to the 𝑈(1) actions in the GLSM. The most delicate issue for such constructions is the proof of 𝜇-stability. However, it should be clear that besides that the monad construction provides a large set of heterotic (0,2) backgrounds and that the methods described so far are indeed taylor-made for the determination of the zero-mode spectrum, which is given by the dimensions of vector bundle valued cohomology classes ℎ^𝑖(𝑆;Λ^𝑘𝑉).

7. A (2,2) Model Dual to a (0,2) Model

Now let us show all this for concrete heterotic (0,2) models, for which we first recall a couple of issues. The theory is naturally equipped with an 𝐸₈×𝐸₈ gauge theory. One of these 𝐸₈’s may be taken to be invisible to the real world, and hence only one 𝐸₈ remains. The holomorphic vector bundle now is endowed with a certain structure group 𝐺 which breaks this 𝐸₈ down to some GUT group. The remaining GUT group is then simply the commutant of 𝐺 in 𝐸₈. Depending on what kind of GUT group we are interested in, we may choose the structure group 𝐺 to be either SU(3), SU(4), or SU(5) breaking 𝐸₈ down to 𝐸₆, SO(10) or SU(5), respectively.

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 [16]. The precise correlation of vector bundle cohomology and zero modes for all three GUT groups are given in Table 2 (for a nice review on the particle spectrum of heterotic theories, see, i.e., [17]).

Table 2 

Correlation between zero modes in representations of the GUT group 𝐻.

The moduli appearing in such a framework are given by possible deformations of the Calabi-Yau manifold, which are counted by the Hodge numbers ℎ^2,1(𝑆),ℎ^1,1(𝑆) (7.1) and by possible deformations of the bundle, that is, the bundle moduli, which are counted by the dimension of the cohomology of the endomorphism bundle End(𝑉) of 𝑉. Furthermore one can show that 𝐻¹(𝑆;End(𝑉))≅𝐻¹𝑆;𝑉^∗𝑉, (7.2) which simplifies its determination. In case of the standard embedding, the vector bundle is simply the tangent bundle and hence has SU(3) structure and gauge group 𝐸₆. Many vector bundles can be constructed using monads, by defining the vector bundle to be the cohomology of the complex (6.2). Using only this complex, it is possible to construct bundles with the structure groups shown in Table 2, and hence computing all these cohomologies simply boils down to the computation of line bundle cohomology on the complete intersection. This on the other hand can be related, using the Koszul sequence (5.5), to the cohomology of line bundles on the ambient toric variety.

In the following we give an example of a pair of heterotic models which are related by the so-called target space duality [7, 18, 19] and were derived in [20]. The first of those will be a (2,2) model (𝑀_𝑎,𝑉_𝑎)=(𝑀_𝑎,𝑇_𝑀𝑎) while the second one, referred to as (𝑀_𝑏,𝑉_𝑏), is of type (0,2) equipped with an SU(3)-bundle which is assumed to be stable.

Let us start with an example in which we can already see most of the structure but which is not too involved. Consider 𝑉_{1,1,1,1,2,2,2}[3,4,3]↠ℙ⁶_{1,1,1,1,2,2,2}[3,4,3]. (7.3) Since this configuration is singular we have to resolve it by introducing a new coordinate. This yields the smooth configuration shown in Table 3, leading to the following monad for the tangent bundle: (7.4) where the Koszul sequence (5.3) has to be applied as well. Using cohomCalg Koszul extension, we can obtain the number of zero modes of the chiral spectrum in this model as well as the dimension of the moduli space: ℎ^•_𝑀𝑎𝑉_𝑎=(),ℎ0,68,2,0_𝑀1,1𝑎+ℎ_𝑀^2,1_𝑎+ℎ¹_𝑀𝑎End𝑉_𝑎=2+68+140=210, (7.5) where the reader should keep in mind that in this case 𝑉_𝑎=𝑇_𝑀𝑎 is just the tangent bundle. The dual (0,2) model geometry can then be determined to be the data in Table 4, and its monad is specified by the sequence (7.6) This configuration satisfies conditions (6.3), and we obtain the following topological data: ℎ^•_𝑀𝑏𝑉_𝑏=(),ℎ0,68,2,0_𝑀1,1𝑏+ℎ_𝑀^2,1_𝑏+ℎ¹_𝑀𝑏End𝑉_𝑏=3+51+156=210. (7.7) Comparing to the data (7.5) we can see that the number of zero modes in the chiral spectrum does not change, and even though the individual Hodge numbers as well as their sum are both different, the dimension of the full moduli space stays the same.

Table 3 

Toric data for the smooth (2,2) model 3-fold geometry 𝑀_𝑎.

Table 4 

Toric data for the dual (0,2) model 3-fold geometry 𝑀_𝑏.

This is a manifestation of a so far only very poorly understood perturbative (in 𝑔_𝑠) target space duality in the configuration space of heterotic string compactifications with 𝒩=1 supersymmetry in four dimensions.

8. Outlook

So far most implementations of computational methods in string model building have been based on toric geometry [21] and in particular on the combinatorial formulas of Batyrev and Borisov [22–24]. Of course there are also general software tools for algebraic geometry like [25–27]. Clearly, these are very powerful but also have their limitations. First, they only apply in the (2,2) case, where the vector bundle is identified with the tangent bundle. Second, for complete intersections the combinatorial formulas hold only for the so-called nef-partitions which ensure that the corresponding polytopes representing the space are reflexive.

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 (0,2) models, where the vector bundle is defined via line bundles, for example, the monad construction or the spectral cover construction.

Of course, also the algorithm implementation cohomCalg has its limitations. First, in situations where the number of the Picard generators (projective relations, reflected by ℎ^1,1) becomes large (about the order of ten), the computations become too involved and the program too time consuming. A second drawback is the exponential growth of the computing time with the number of the Stanley-Reisner ideal generators, which at the moment takes several hours for about 40 generators. Third, if there are not enough zeros in the many intermediate long exact sequences, the result is not unique and consequently one has to determine the kernel image of maps by hand.

Note that there is also the Macaulay 2 package [3] which can be used as an alternative to the algorithm presented herein. Preliminary testing indicates that it seems to be able to handle geometries of a high Picard rank and huge numbers of Stanley-Reisner ideal generators, but for simple geometries, cohomCalg appears to be faster. Further study is necessary to fully evaluate strengths and weaknesses for the two algorithms implemented in [28] and the algorithm described in Section 3. Also see [29, prop.  4.1].

Acknowledgment

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