# Computational Tools for Cohomology of Toric Varieties

Authors: Ralph Blumenhagen; Benjamin Jurke; Thorsten Rahn
Journal: Advances in High Energy Physics
 (2011)

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


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## Body

## 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 for𝒩=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 then-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 dimensionshi(X;LX)=dimHi(X;LX) 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 Section2 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,…,xn as well as R projective relations (2.1)(x1,…,xn)~(λrQ1(r)x1,…,λrQn(r)xn)forλr∈C×.
The Qi(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 coordinatesxi become 𝒩= (2,2) chiral superfields in the GLSM picture, and the Fayet-Iliopoulos parameters ξr of the Abelian symmetries can be interpreted as the Kähler parameters of the geometric space. This parameter space of ξ⃗=(ξ1,…,ξR) 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 (2.2)Sρ={xρ1,xρ2,…,xρ|Sρ|}forρ=1,…,N
which are not allowed to vanish simultaneously. Note that such a collection is often written in product form; that is, the square-free monomial xρ1xρ2⋯xρ|𝒮ρ| refers exactly to the same set. All those sets form the Stanley-Reisner ideal (2.3)SR(X)=〈S1,…,SN〉,
which can be equivalently used to uniquely specify a geometric phase. Note that the Stanley-Reisner ideal is Alexander dual to the irrelevant ideal BΣ used in the mathematical literature.Given the GLSM charges and the Stanley-Reisner ideal to identify the geometric phase, the toric varietyX of dimension d=n-R can be described as the coset space (2.4)X=(Cn-Z)(C×)R,
where Z is the set of removed points specified by SR(X) via (2.5)Z=⋃ρ=1N{xρ1=xρ2=⋯=xρ|Sρ|=0}.
This set Z can be understood as the toric generalization of the removed origin in a projective space ℂℙn=(ℂn+1-{0})/ℂ×, as the Stanley-Reisner ideal for ℂℙn 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 vectorsνi that span the fan ΣX. The GLSM charges Qi(r) reappear in the form of R linear relations (2.6)∑i=1nQi(r)νi=0forr=1,…,R.
By associating the lattice vectors νi to the homogeneous coordinates xi, 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 ΣX.


## 3. Dimensions of Line-Bundle-Valued Cohomology Groups

Given a toric varietyX and a line bundle LX, a frequent issue is to compute the LX-valued cohomology group dimensions hi(X;LX) for i=0,…,dimX. After a couple of preliminary observations in [6, 7], in [8] a complete novel algorithm for the determination of hi(X;LX) 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 chargesQi(r) and the Stanley-Reisner ideal generators 𝒮1,…,𝒮N. 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 LX=𝒪X(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 monomials𝒬 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 𝔥i(𝒬) that specifies to which cohomology group's dimension hi(X;𝒪X(D)) the number of monomials 𝒩D(𝒬) with GLSM charge D contributes. The cohomology group dimension formula can be summarized as (3.1)dim⁡Hi(X;OX(D))=∑Qhi(Q)︷multiplicityfactor⋅ND(Q)︸numberofmonomials,
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 𝔥i(𝒬) and 𝒩D(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 (3.2)Sρ:={Sρ1,…,Sρk}⊂{S1,…,SN}
of generators, 𝒬(Sρ) be the square-free monomial that arises from the union of all coordinates in each generator 𝒮ρi of the subset.The construction of the relative complexΓ𝒬, from which 𝔥i(𝒬) is defined, goes as follows. From the full simplex on [N]={1,…,N}, extract only those subsets ρ⊂[N] with 𝒬(Sρ)=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 Fk-1(𝒬) of the complex Γ𝒬, that is, (3.3)Fk(Q):={ρ⊂[N]:|ρ|=k+1Q(Sρ)=Q}.
Furthermore, let ℂFk(𝒬) be the complex vector space with basis vectors eρ for ρ∈Fk(𝒬). The relative complex (3.4)ΓQ:0⟶FN-1(Q)⟶ϕN-1⋯⟶ϕ1F0(Q)⟶ϕ0F-1(Q)⟶0,
where F-1(𝒬):={∅} is a face of dimension −1, is then specified by the chain mappings (3.5)ϕk:Fk(Q)⟶Fk-1(Q),eρ⟼∑s∈ρsign⁡(s,ρ)eρ-{s}.
A basis vector eρ-{s} vanishes if ρ with the element s removed is not contained in Γ𝒬. 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(3.6)hi(Q):=dim⁡H|Q|-i-1(ΓQ)
of the relabeled complex then provide the multiplicity factors that determine to which cohomology group Hi(X;𝒪X(D)) the monomials associated to 𝒬 contribute. It should be emphasized that the 𝔥i(𝒬) depend only on the geometry (the Stanley-Reisner ideal) of the toric variety X and not on the line bundle 𝒪X(D), that is, the multiplicity factors only have to be computed once for each geometry.


### 3.2. Counting Monomials

After computing the multiplicity factors𝔥i(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 xi and the specific line bundle 𝒪X(D). Let 𝒬 again be a square-free monomial. In order to simplify the notation, let I=(i1,…,ik,…,in) be an index relabeling such that the product of the first k coordinates gives 𝒬=xi1⋯xik. Then one considers monomials of the form (3.7)RQ(x1,…,xn):=(xi1)-1-a(xi2)-1-b⋯(xik)-1-c(xik+1)d⋯(xin)e=T(xik+1,…,xin)xi1⋯xik⋅W(xi1,…,xik),
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 (3.8)ND(Q):=dim⁡{RQ:deg⁡GLSM(RQ)=D},
which counts the number of relevant monomials that have the same GLSM degree as the divisor D that specified the line bundle LX=𝒪X(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 Table1 for the reader’s convenience. The two Stanley-Reisner ideal generators yield four possible combinations that become relevant in the computation, namely, (3.9)Q=1,x1x2,x3x4,x1x2x3x4.
The computation of the multiplicity factors for those square-free monomials leads to (3.10)C0(1)={{∅}},C1(x1x2)={{S1}},C1(x3x4)={{S2}},C2(x1x2x3x4)={{S1,S2}},
and all other spaces ℭi(𝒬) vanishing. After computing the homology, this leads to the following contributions of the monomials (3.7) to the cohomology groups: (3.11)H0(dP1;O(m,n)):T(x1,x2,x3,x4),H1(dP1;O(m,n)):T(x3,x4)x1x2⋅W(x1,x2),T(x1,x2)x3x4⋅W(x3,x4),H2(dP1;O(m,n)):1x1x2x3x4⋅W(x1,x2,x3,x4).
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. degGLSM(1/x1x2)=(-2,0) is no good either, but degGLSM(1/x3x4)=(-1,-2) fits perfectly, such that there is a sole contribution (3.12)1x3x4⇝h•(dP1;O(-1,-2))=(0,1,0).Table 1
Toric data for the del Pezzo-1 surface.
Vertices of the polyhedron/fanCoords.GLSM chargesDivisor classQ1Q2ν1=(-1,-1)x110Hν2=(1,0)x210Hν3=(0,1)x311H+Xν4=(0,-1)x401XIntersection form:HX-X2.SR(dP1)=〈x1x2,x3x4〉=〈𝒮1,𝒮2〉.All the aforementioned steps involved in the computation of the cohomology have been conveniently implemented in a high-performance cross-platform package calledcohomCalg [11].


## 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 (3.2)Sρ:={Sρ1,…,Sρk}⊂{S1,…,SN}
of generators, 𝒬(Sρ) be the square-free monomial that arises from the union of all coordinates in each generator 𝒮ρi of the subset.The construction of the relative complexΓ𝒬, from which 𝔥i(𝒬) is defined, goes as follows. From the full simplex on [N]={1,…,N}, extract only those subsets ρ⊂[N] with 𝒬(Sρ)=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 Fk-1(𝒬) of the complex Γ𝒬, that is, (3.3)Fk(Q):={ρ⊂[N]:|ρ|=k+1Q(Sρ)=Q}.
Furthermore, let ℂFk(𝒬) be the complex vector space with basis vectors eρ for ρ∈Fk(𝒬). The relative complex (3.4)ΓQ:0⟶FN-1(Q)⟶ϕN-1⋯⟶ϕ1F0(Q)⟶ϕ0F-1(Q)⟶0,
where F-1(𝒬):={∅} is a face of dimension −1, is then specified by the chain mappings (3.5)ϕk:Fk(Q)⟶Fk-1(Q),eρ⟼∑s∈ρsign⁡(s,ρ)eρ-{s}.
A basis vector eρ-{s} vanishes if ρ with the element s removed is not contained in Γ𝒬. 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(3.6)hi(Q):=dim⁡H|Q|-i-1(ΓQ)
of the relabeled complex then provide the multiplicity factors that determine to which cohomology group Hi(X;𝒪X(D)) the monomials associated to 𝒬 contribute. It should be emphasized that the 𝔥i(𝒬) depend only on the geometry (the Stanley-Reisner ideal) of the toric variety X and not on the line bundle 𝒪X(D), that is, the multiplicity factors only have to be computed once for each geometry.


## 3.2. Counting Monomials

After computing the multiplicity factors𝔥i(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 xi and the specific line bundle 𝒪X(D). Let 𝒬 again be a square-free monomial. In order to simplify the notation, let I=(i1,…,ik,…,in) be an index relabeling such that the product of the first k coordinates gives 𝒬=xi1⋯xik. Then one considers monomials of the form (3.7)RQ(x1,…,xn):=(xi1)-1-a(xi2)-1-b⋯(xik)-1-c(xik+1)d⋯(xin)e=T(xik+1,…,xin)xi1⋯xik⋅W(xi1,…,xik),
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 (3.8)ND(Q):=dim⁡{RQ:deg⁡GLSM(RQ)=D},
which counts the number of relevant monomials that have the same GLSM degree as the divisor D that specified the line bundle LX=𝒪X(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 Table1 for the reader’s convenience. The two Stanley-Reisner ideal generators yield four possible combinations that become relevant in the computation, namely, (3.9)Q=1,x1x2,x3x4,x1x2x3x4.
The computation of the multiplicity factors for those square-free monomials leads to (3.10)C0(1)={{∅}},C1(x1x2)={{S1}},C1(x3x4)={{S2}},C2(x1x2x3x4)={{S1,S2}},
and all other spaces ℭi(𝒬) vanishing. After computing the homology, this leads to the following contributions of the monomials (3.7) to the cohomology groups: (3.11)H0(dP1;O(m,n)):T(x1,x2,x3,x4),H1(dP1;O(m,n)):T(x3,x4)x1x2⋅W(x1,x2),T(x1,x2)x3x4⋅W(x3,x4),H2(dP1;O(m,n)):1x1x2x3x4⋅W(x1,x2,x3,x4).
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. degGLSM(1/x1x2)=(-2,0) is no good either, but degGLSM(1/x3x4)=(-1,-2) fits perfectly, such that there is a sole contribution (3.12)1x3x4⇝h•(dP1;O(-1,-2))=(0,1,0).Table 1
Toric data for the del Pezzo-1 surface.
Vertices of the polyhedron/fanCoords.GLSM chargesDivisor classQ1Q2ν1=(-1,-1)x110Hν2=(1,0)x210Hν3=(0,1)x311H+Xν4=(0,-1)x401XIntersection form:HX-X2.SR(dP1)=〈x1x2,x3x4〉=〈𝒮1,𝒮2〉.All the aforementioned steps involved in the computation of the cohomology have been conveniently implemented in a high-performance cross-platform package calledcohomCalg [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 (4.1)Hi(X)=Hinvi(X)⨁Hnon-invi(X)
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 groupG, each group element g induces an involution mapping g:X→X on the base geometry and has a corresponding uplift ϕg: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 ϕg∘ϕh=ϕgh holds such that the mapping g↦ϕg 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 𝒪X(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 (4.3)g1:(x1,x2,x3)⟼(αx1,α2x2,x3)forα:=13=e2πi/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: (4.4)1u14u2u3︸g1→1,1u1u24u3︸g1→1,1u1u2u34︸g1→1,1u13u22u3︸g1→α,1u13u2u32︸g1→α2,1u12u23u3︸g1→α2,1u1u23u32︸g1→α,1u12u2u33︸g1→α,1u1u22u33︸g1→α2,1u12u22u32︸g1→1,︸h2(CP2;O(-6))=(4inv,3α,3α2),
such that hinv•(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 varietyX to the cohomology of a subspace, the Koszul sequence is used.To make this paper self-contained and because it has been implemented in thecohomCalg 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 𝒪X(D), such that Z(σ)≅D. This induces a mapping 𝒪X→𝒪X(D) and its dual 𝒪X(-D)↪𝒪X, the latter of which can be shown to be injective. Given an effective divisor (5.1)D:=∑iaiHi⊂X,
where all ai≥0, there is a short exact sequence (5.2)0⟶OX(-D)↪OX↠OD⟶0,
called the Koszul sequence. Here 𝒪D is the quotient of the sheaf 𝒪X of holomorphic functions on X by all holomorphic functions vanishing at least to order ai along the irreducible hypersurface Hi⊂X. This allows to treat 𝒪D as the structure sheaf on the divisor D, which effectively identifies the sheaf cohomology Hi(X;𝒪D) with Hi(D;𝒪D). 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 (5.3)0⟶OX(T-D)↪OX(T)↠OD(T)⟶0
that is obtained by tensoring (5.2) with the line bundle 𝒪X(T). The induced long exact cohomology sequence (5.4)0⟶H0(X;OX(T-D))⟶H0(X;OX(T))⟶H0(D;OD(T))⟶H1(X;OX(T-D))⟶H1(X;OX(T))⟶H1(D;OD(T))⟶H2(X;OX(T-D))⟶H2(X;OX(T))⟶H2(D;OD(T))⟶⋯then allows to relate the cohomology of the toric varietyX directly to the cohomology of the hypersurface.Given a more generic case of several (mutually transverse) hypersurfaces{S1,…,Sl}, one can compute the cohomology on the complete intersection via the generalized Koszul sequence(5.5)0⟶OX(-∑j=1lSj+D)⟶⋯⟶⨁i1<i2OX(-Si1-Si2+D)⟶⨁i1OX(-Si1+D)⟶OX(D)⟶OS(D)⟶0.
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 ℐk: (5.6)0⟶OX(-∑j=1lSj+D)⟶⨁i1<⋯<il-1OX(-∑j=1l-1Sij+D)↠I1⟶0,0⟶I1↪⨁i1<⋯<il-2OX(-∑j=1l-2Sij+D)↠I2⟶0,⋮0⟶Il-2↪⨁i1OX(-Si1+D)↠Il-1⟶0,0⟶Il-1↪OX(D)↠OS(D)⟶0.
The individually induced long exact sequences of cohomology can then be used for the step-wise computation of H•(S;𝒪S(D)), which is the cohomology on the complete intersection S=⋂i=1lSi.


## 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 TS=Ker(f)/Im(g) of the sequence (6.1)0⟶OS⊕R︷oneOsforeachPicardgenerator↪g⨁i=1nOS(Qi)︸onebundlewiththeGLSMchargesforeachcoordinate↠f⨁j=1lOS(Sj)︷onebundlewiththedegreeforeachhypersurface⟶0,
where the individual line bundles are restricted to the complete intersection S=⋂i=1lSi. 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 hi(S;TS), 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 bundleV, which is analogously defined via a sequence of the Whitney sums of line bundles (6.2)0⟶OS⊕RV↪g⨁a=1δOS(Na)↠g⨁l=1λOS(Ml)⟶0.
The rank is rk(V)=δ-λ-RV. The charges Na and Ml have to satisfy the anomaly cancellation conditions (6.3)∑aNa(α)=∑lMl(α),∀α,∑lMl(α)Ml(β)-∑aNa(α)Na(β)=∑jSj(α)Sj(β)-∑iQi(α)Qi(β),∀α,β,
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 hi(S;ΛkV).


## 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 groupH.
Number of zero modes in reps. ofH1hS1(V)hS1(V*)hS1(Λ2V)hS1(Λ2V*)hS1(V⊗V*)E8248↓↓SU(3)×E6(1,78)⊕ (3,27)⊕ (3¯,27¯)⊕  (8,1)SU(4)×SO(10)(1,45)⊕ (4,16)⊕ (4¯,16¯)⊕ (6,10)⊕ (15,1)SU(5)×SU(5)(1,24)⊕ (5,10¯)⊕ (5¯,10)⊕ (10,5)⊕ (10¯,5¯)⊕ (24,1)The moduli appearing in such a framework are given by possible deformations of the Calabi-Yau manifold, which are counted by the Hodge numbers(7.1)h2,1(S),h1,1(S)
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 (7.2)H1(S;End(V))≅H1(S;V*⨂V),
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 (Ma,Va)=(Ma,TMa) while the second one, referred to as (Mb,Vb), 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(7.3)V1,1,1,1,2,2,2[3,4,3]↠P1,1,1,1,2,2,26[3,4,3].
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: (7.5)hMa•(Va)=(0,68,2,0),hMa1,1+hMa2,1+hMa1(End(Va))=2+68+140=210,
where the reader should keep in mind that in this case Va=TMa 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: (7.7)hMb•(Vb)=(0,68,2,0),hMb1,1+hMb2,1+hMb1(End(Vb))=3+51+156=210.
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 Ma.
Coordinate GLSM chargesHypers. degrees0000111111211112220334Table 4
Toric data for the dual(0,2) model 3-fold geometry Mb.
Coordinate GLSM chargesHypersurf. degrees000000001100110000111100111111112220003322This is a manifestation of a so far only very poorly understood perturbative (ings) 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 implementationcohomCalg has its limitations. First, in situations where the number of the Picard generators (projective relations, reflected by h1,1) 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].



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*Source: 152749-2011-08-25.xml*