3.2 Toric geometry
3.2.3 Intersection numbers
(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.
3.2 Toric geometry 43
hhhhhhhhhhhh
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Int(S1S2S3) Triangulation E1" E2" E3" S"
E1,βγE2,αγE3,αβ 0 0 0 1
E1,βγE2,αγ2 , E1,βγE3,αβ2 −2 0 0 −1 E2,αγE1,βγ2 , E2,αγE3,αβ2 0 −2 0 −1 E3,αβE1,βγ2 , E3,αβE2,αγ2 0 0 −2 −1
E1,βγ3 0 8 8 4
E2,αγ3 8 0 8 4
E3,αβ3 8 8 0 4
R1R2R3 2
R1E1,βγ2 , R2E2,αγ2 , R3E3,αβ2 −2
Table 3.1: The upper part gives the intersection numbers when using the same trian-gulation at all 64 xed points of the T6/(Z2 ×Z2) blowup. The lower part gives the triangulation-independent intersection numbers.
of triangulated toric diagrams and patch gluings. This gives us everything we need to determine the last topological piece of information needed for the description of the heterotic supergravity approximation on CY manifolds: the intersection numbers of the divisors. The term intersection numbers can be taken literally when dierent divisors are involved: it counts the number of points in which d divisors intersect (in homology). However, using linear equivalence relations, one can use the intersection numbers of distinct divisors to also calculate self-intersections of divisors, which have a less straightforward geometric interpretation. In particular, they can be fractional (for non-compact varieties) or negative (for exceptional divisors). Roughly, self-intersections can be thought of as the intersection of a submanifold with its slightly perturbed copy. A negative self-intersection then signals that a divisor cannot be moved. The fractional intersection numbers signal that by taking the local (non-compact) case we are neglecting contributions from the global gluing.
The intersection numbers involving only distinct divisors can be simply read o from the toric diagram. In the non-compact case, the intersection number of all divisors which form a cone together with the origin have intersection number 1. Since the cones depend on the choice of the triangulation, we see that the intersection numbers become a function of the StanleyReissner ideal as well. In the compact case, the prescription is similar, only that the intersection number is weighted by the volume of the cone. Let us illustrate this in examples.
Example (CP2) We have constructed the fan in gure 3.1. We denote the three divisors associated with the vi byDi. Since we have three 2-dimensional cones, we nd the intersection numbers D1D2 = D1D3 = D2D3 = 1. From the linear equivalences, we know that D1 ∼D2 ∼D3 =H. Thus we have the intersection ring H2 = 1.
Example (T6/(Z2×Z2)) As alluded to before, in this case the StanleyReissner ideal is not unique and hence the intersection numbers depend on the triangulation.
Let us carry out the analysis in the case of the symmetric triangulation. We nd the
triple intersection numbers D1,αE2,αγE3,αβ = 1. Note that the index α has to be the same on the ordinary and exceptional divisors. This ensures that each local resolution is glued into the correct xed point. The intersections for D2E1E3 and D3E1E2 can be read o in a similar way. For the intersection involving the inherited divisors, we nd e.g. R1R2R3 = 2, where the multiplicity arises from the volume factor in the auxiliary polyhedron. The intersection numbers of the form DiDjRkcan be worked out similarly (see [60] for the detailed procedure). Next, one uses the linear equivalence relations (3.25) in order to obtain a divisor basis containing the three inherited divisorsRiand the 48exceptional divisorsEk,σ. Intersections involving the same divisor multiple times can be calculated by starting from expressions likeRiDjDk, replacing the ordinary divisors and solving the linear system of equations. While tedious due to the huge amount of equations, this can be done rather easily with the help of a computer. For the cases where the same triangulation is used at all xed points, we give the intersection numbers in table 3.1 We have now all topological information we need in order to build MSSM-like models on the resolved T6/Z2×Z2 orbifold.
In summary, to collect all geometric data needed for the description of a heterotic string model on a resolved CY orbifold, one proceeds as follows:
Construct the fan of toric variety under investigation,
Introduce exceptional divisors and choose the StanleyReissner ideal such that the singularities are resolved,
Introduce the inherited gluing divisors and construct the auxiliary polyhedra,
Use the vectors of the toric diagram to read o the linear equivalence relations,
Read o the intersection numbers and calculate the self-intersections using the linear equivalence relations.
Now that we have a basis of divisors, we can construct fundamental objects like Chern classes and Kähler forms by expanding them in terms of a basis of divisors Si or rather their Poincaré-dual forms. In the case of the Kähler form, we expand
J =
h1,1
X
i=1
diSi. (3.26)
The scalars di are called Kähler moduli. Using the Kähler form, we can dene the volume of curves Ci, divisors Di, and the entire CalabiYauX as
vol(Ci) = Z
Ci
J , vol(Di) = 1 2
Z
Di
J2, vol(X) = 1 6
Z
X
J3. (3.27) These integrals can be evaluated using the intersection numbers and they will lead to linear, quadratic, and cubic terms in the Kähler moduli, respectively. In a sensible theory, these volumes should all be larger than zero, which restricts the relative size of the Kähler moduli. The range of Kähler moduli for which the volumes are positive forms a cone, the so-called Kähler cone. Since the volume depends on the intersection numbers
3.2 Toric geometry 45 and thus on the SR ideal, we get dierent Kähler cones for dierent triangulation choices. These Kähler cones can be connected via op transitions: if the volume of some curve becomes negative, this signals that one crossed the boundary between two Kähler cones. So the theory can be made sense of by using a dierent SR ideal, in which the intersection numbers changed such that for the given Kähler moduli values all volumes are positive again. It is also useful to dene the dual of the Kähler cone, the so-called Mori cone. This is the cone of all eective curves, i.e. the cone in which all curves have a non-negative intersection with all divisors. Let us illustrate the ops again using the example of C3/(Z2 ×Z2) (the story does not change when going to the compact case, since the only triangulation dependence comes from the xed point resolution).
Example (C3/(Z2×Z2)) The four possible triangulations are given in gure 3.4.
Let us discuss for example how to get from triangulation E1 to triangulation S. In the E1 triangulation, the divisors D1 and E1 intersect in a curve C11. This curve is not present in the S triangulation. Instead, we nd a curve C23, corresponding to the intersection E2E3. First, we expand the Kähler form as J = −b1E1 −b2E2 −b3E3. The minus signs are of course convention, which we choose here in analogy with the compact case, where the volumes of the Ei and of X are positive for bi > 0. Let us look at the curves. In triangulation E1, we nd
vol(C11) = Z
C11
J = Z
X
D1E1J = Z
X
−b1D1E1E1−b2D1E1E2−b3D1E1E3
= 1b1−1b2−1b3, vol(C23) =
Z
C23
J = Z
X
E2E3J = Z
X
−b1E1E2E3−b2E2E3E2−b3E2E3E3
= 0b1+ 0b2+ 0b3 = 0,
(3.28)
where the coecients are the triple intersection numbers which we read o from the toric diagram and (in the case of self-intersections) calculate from the linear equivalence rela-tions0∼2Di+Ej+Ek with all indices dierent. We see that the curveC11 has positive volume ifb1 > b2+b3 while the curveC23has a vanishing volume (i.e. it does not exist).
Thus the Kähler cone for triangulation E1 is given by {b1, b2, b3 ∈R+ | b1 > b2+b3}. Let us see what happens if we make the volume of C11 negative by taking bi > 0 but b1 < b2 +b3. We can still make sense out of this if we at the same time change from triangulation E1 to triangulation S. In this triangulation, one nds
vol(C11) = Z
C11
J = Z
X
D1E1J = Z
X
−b1D1E1E1−b2D1E1E2−b3D1E1E3
= 0b1+ 0b2+ 0b3 = 0, vol(C23) =
Z
C23
J = Z
X
E2E3J = Z
X
−b1E1E2E3−b2E2E3E2−b3E2E3E3
=−1b1+ 1b2+ 1b3.
(3.29)
So indeed we nd that the volume of C11 vanishes identically and furthermore that precisely in this case the volume of C23 becomes positive. Hence for this choice of the values of the Kähler moduli, we have left the Kähler cone of triangulation E1 and entered the Kähler cone of triangulation S.