3.2 Toric geometry
3.2.2 CalabiYaus as hypersurfaces in toric varieties
3.2 Toric geometry 39 D1
D2 D3
(a) Toric diagram of C3/(Z2×Z2).
D1
D2 D3
E3
E1 E2
(b) Rened toric diagram.
D1
D2 D3
E3
E1 E2
(c) One possible triangula-tion of the toric diagram.
Figure 3.4: Toric diagram of C3/(Z2 ×Z2), its subdivision via the inclusion of new vectorswi corresponding to the exceptional divisorsEi, and one of its possible triangula-tions.
D1
D2 D3
E3
E1
E2
(a) Triangulation E1. D1
D2 D3
E3
E1
E2
(b) Triangulation E2. D1
D2 D3
E3
E1
E2
(c) Triangulation E3. D1
D2 D3
E3
E1
E2
(d) Triangulation S.
Figure 3.5: The four possible triangulations for the toric diagram of C3/(Z2×Z2).
This nishes the procedure of rening the fan. Now we need to subdivide it. Here we encounter for the rst time a situation where the choice is ambiguous: there are 4 possible choices of triangulations (or equivalently for the StanleyReissner ideal) which lead to a maximal renement. The four possible choices are given in gure 3.5. As we shall see later in section 5.3, the choice of triangulation inuences the massless matter spectrum: opping the curves represented by lines in gure 3.5 makes some states massive while others become massless.
K with the canonical bundle. For Ω to be globally dened and non-vanishing on the CalabiYau, we need this bundle to be trivial, O(K) =O. It can be shown that for a toric variety XΣ
KXΣ =−X
i
Di (3.17)
where the Di are the divisors associated with the edges of the fan Σ (they could be ordinary or exceptional divisors). In order to obtain a compact CalabiYau, we start from a toric variety and take a compact hypersurface within the variety. The hypersur-face itself will be given in terms of the vanishing locus of a homogeneous polynomial in the coordinates of the toric variety. The point is to choose this hypersurface X such that the canonical bundle is trivial onX. In this way the hypersurface will be CY even though the ambient space itself is not. In order to know how to choose the polynomial, we have to know how to calculate the canonical bundle or the rst Chern class1 on CY hypersurface. Let us denote the ambient space by A and the CY hypersurface by X.
We decompose the tangent space at each pointp ofX into a direct sum of the tangent and the normal bundle of X:
Tp,A|X =Tp,X ⊕Np,X. (3.18) Using short exact sequences introduced in section 2.2, this can be written as
0→TX →TA|X →NX →0, (3.19) where the vertical bar means restriction to the hypersurface X. This is known as the adjunction formula. It implies for the total Chern class
c(TX) = c(TA)/c(NX), (3.20) with NX =OA(X)|X. Using the adjunction formula together with
c(L1⊕ L2 ⊕. . .⊕ LN) =
N
Y
i=1
c(Li) =
N
Y
i=1
(1 +c1(Li)), (3.21) we can calculate all Chern classes of line bundles from (3.20). To illustrate the procedure we consider an example.
Example (Hypersurface in CPN) Hypersurfaces in CPN provide a very simple class of examples. As usual, we denote the N + 1 coordinates of the ambient space by zi. As we have already seen, all the coordinates correspond to linear equivalent divisors, which we denote by H with associated line bundle O(1). In this ambient space, we choose a degree k polynomialP which describes the submanifoldX. Since this divisor will be a formal linear combination of the divisors corresponding to thezi in homology,
1One has to be careful since there are examples like the Enriques surface where the rst Chern class is zero but the canonical bundle is non-vanishing but pure torsion (K=Z2 for the Enriques). These surfaces thus do not have a non-vanishing volume form and thus we do not call them CY.
3.2 Toric geometry 41 X will be linear equivalent to kH. The associated line bundle is hence O(k). We thus nd
c(TX) =c0(TX) +c1(TX) +. . .=c(TA)/c(NX) =
"N+1 Y
i=1
(1 +H)
#
/[1 +kH]
= [(1 + (N + 1)H+. . .)]·[1−kH+k2H2−. . .].
(3.22)
The rst Chern class is a two-form and thus represented by the term linear in H. We obtainc1(TX) = (N+1−k)H, which means that the rst Chern class vanishes precisely for a hypersurface given in terms of a degree k = N + 1 polynomial. Thus a simple way to construct compact CY manifolds in d dimensions is to start with CPd+1 and consider a degreed+ 2 polynomial. For d= 1, we obtain the cubic in CP2, which is a torus. For d = 2, we get the quartic in CP3 which is a K3, and for d = 3, we get the (well-known) quintic, which is a CY threefold.
The next question is how to incorporate the notion of submanifolds into toric geometry.
In the case of toric resolutions of orbifolds, this can be done in an indirect way via the introduction of inherited divisors and auxiliary polyhedra [55, 56]. The idea is to resolve the orbifold xed points and xed tori locally, which are then of the form C3/ZN and C2/ZN, respectively, using the procedure outlined in 3.2. After the local resolution, the various resolved xed points are glued into a T6/ZN with the appropriate xed point structure. This is done by specifying how the local coordinates zi of Cn glue together across the dierent patches, for which we introduce the so-called inherited divisors Ri, where i = 1,2,3 labels the three tori. Their names stem from the fact that they are inherited from the torus. We take zi as the torus coordinates and denote the position of the σth xed point locus in the ith torus by zi,σxed. This means that the ordinary divisors are placed at the positions Di,σ ={zi =zi,σxed}. Away from the singularity, we dene for factorizable orbifolds the inherited divisors via
Ri ={zi =c6=zi,σxed}. (3.23) However, since the ith torus is folded by the orbifold action of the orderNi, the Ri are obtained by taking the union over points which are identied, Ri = SNi
k=1{zi = kc}
with=e2πi/Ni. In the limitc→zi,σxed, theRithus correspond toNi copies ofDi,σ, thus Ri ∼NiDi,σ. After introducing the exceptional divisors to resolve the singularities, the linear equivalence relations are of the schematic form
Ri ∼NiDi,σ+X
k
Ek,σ, (3.24)
where i = 1,2,3 labels the three tori, k labels the exceptional divisors, and σ labels the xed points. As we can see, the linear equivalences relate local resolutions from dierent xed points, which reects the gluing. Note, however, that the whole procedure is done without explicitly mentioning hypersurfaces. We will see in section 3.3 how to write toroidal orbifolds as (complete) intersections in toric ambient spaces and how the hypersurface equations give rise to the inherited divisors used above to describe the
(a) Triangulation E1. (b) Triangulation E2. (c) Triangulation E3. (d) Triangulation S.
Figure 3.6: The auxiliary polyhedra for the four possible triangulations ofT6/(Z2×Z2).
gluing. Using (3.24), the ordinary divisors can be expressed in terms of the inherited and exceptional divisors, which form a divisor basis.
A further complication arises if there are singularities on the torus that are mapped onto each other on the orbifold via the orbifold action. In this case, one has introduce new divisors which are the sum of the divisors over the equivalent xed points, and (3.24) is altered accordingly. This complication also does not arise when treating the resolutions within the GLSM framework.
Example (T6/Z2 ×Z2) We will not go into the details of the global resolution process, which can be found in [60], but only outline the procedure. First, we introduce a label i = 1,2,3 for each two-torus and xed point labels α, β, γ = 1,2,3,4 labeling the xed point position in the three tori. Furthermore, we introduce a label k = 1,2,3 which labels the twisted sectors θ1, θ2, θ1θ2 of the orbifold, respectively. We follow the convention that the kth twisted sector leaves the kth torus xed. The local resolutions have already been discussed in a previous example, and the corresponding toric dia-grams can be found in gure 3.5. Roughly, the vectors corresponding to the inherited divisors Ri are introduced in the opposite direction of those corresponding to the ordi-nary divisors Di,σ. The resulting auxiliary polyhedra are given in gure 3.6. Note how the local resolution appears as a face in the polyhedron describing the global gluing.
From the polyhedron, we obtain the linear equivalences R1 ∼2D1,α+X
γ
E2,αγ +X
β
E3,αβ, R2 ∼2D2,β+X
γ
E1,βγ+X
α
E3,αβ, R3 ∼2D3,γ+X
β
E1,βγ+X
α
E2,αγ. (3.25)
Using these equations to express the ordinary divisors D in terms of the exceptional divisors E and the inherited divisors R yields a basis of divisors.