REACCIONES ANTÍGENO-ANTICUERPO ERITROCITARIAS

polyhedra. The intersection numbers of the Calabi–Yau are

R1R2R3 = 6, R3E2,βE4,β = 1, E1,βγE2,βE4,β = 1, R2E3,γ2 =−2, R3E2,β2 =−2, R3E4,β2 =−2, E_1,βγ3 = 6, E_2,β3 = 8, E_3,γ3 = 8, E_4,β3 = 8, E1,βγE2,β2 =−2, E1,βγE3,γ2 =−2, E1,βγE4,β2 =−2, E 2 2,βE4,β =−2. (4.16)

Intersection numbers which contain no factor of E2,β are halved for the orientifold. If

the intersection number contains one factor of E2,β, it remains the same. If two (three)

factors E2,β are present, the number on the Calabi–Yau is multiplied by a factor of two

(four). This leads to the following modified triple intersection numbers:

R1R2R3 = 3, R3E2,βE4,β = 1, E1,βγE2,βE4,β = 1, R2E3,γ2 =−1, R3E2,β2 =−4, R3E4,β2 =−1, E_1,βγ3 = 3, E_2,β3 = 32, E_3,γ3 = 4, E_4,β3 = 4, E1,βγE2,β2 =−4, E1,βγE3,γ2 =−1, E1,βγE4,β2 =−1, E 2 2,βE4,β =−4. (4.17)

4.5

Global O–plane configuration and tadpole can-

cellation

Those of the 64 O3–planes on the cover which are located away from the locations of the resolved patches resulting from the global involution remain the same in the orbifold of the resolved manifold. They are untouched by the process of resolving the singularities and the resulting modified local orientifold actions. The O3–plane solutions which coincide with orbifold fixed sets are replaced by the solutions of the corresponding resolved patch. The total number of O3–planes on the resolved orbifold quotient is obtained by counting the equivalence classes ofO3–planes under the orbifold group and replacing those classes which coincide with resolved patches by theO3–plane solutions on these patches. TheO3–plane solutions are also reflected in the intersection ring. Take for example the solution {z1 = y1 = y3 = 0} given in (1) of (4.9). The corresponding intersection number isD1E1E3 = 1₂, indicating theZ2–singularity at the

intersection point. Thus fractional intersection numbers are indicative of the presence of O3–planes. O3–planes which are located away from the fixed points and do not lie in the D–planes are reflected in the intersection numbers with the inherited divisors

Ri, see for example T6/Z3 discussed in Appendix B.1.6. If on the other hand theO3–

planes lie in an O7–plane, their intersection numbers do not become fractional, since the effect of the orientifold involution is already captured by the O7–plane.

Since each O7–plane induces −8 units of D7–brane charge, a stack of 8 coincident D7–branes must be placed on top of each divisor fixed under the combination of the

involution and the scaling action. Each such stack therefore carries an SO(8) gauge group.

For the D3–brane charge, the case is a bit more involved. The contribution from the O3–planes is

Q3(O3) =−

1

4×nO3, (4.18) where nO3 denotes the number of O3–planes. The D7–branes also contribute to the

D3–tadpole: Q3(D7) =− X a nD7,aχ(Sa) 24 , (4.19)

where nD7,a denotes the number of D7–branes in the stack located on the divisor Sa.

As we have seen, the Sa can be local D–divisors as well as exceptional divisors. The

last contribution to the D3–brane tadpole comes from the O7–planes:

Q3(O7) =−

X

a

χ(Sa)

6 . (4.20)

So the total D3–brane charge that must be cancelled is

Q3,tot=− nO3 4 − X a (nD7,a+ 4)χ(Sa) 24 . (4.21)

These are the values for the orientifold quotient, in the double cover this value must be multiplied by two.

4.5.1

Example B:

T

/_Z

on

SU(2)×SU(6)

In total, there are seven O7–planes. The 64 O3–planes which are the fixed points of

I6 on the covering space fall into 16 equivalence classes: The four O3–planes on the

line z1 _{= z}2 _{= 0 are invariant, the remaining 12 on the plane z}2 _{= 0 fall into four}

equivalence classes with three elements each. The remaining 48 O3–planes lie in eight orbits of length six. The four O3–planes on the line z1 = z2 = 0 coincide with the patches on the line z1 _=z2 _{= 0 and disappear, since there are no O3–solutions on the}

local patches. The remaining O3–planes all lie in theO7–planes wrapped on theD3, γ

and are therefore not visible in the intersection numbers. In total, we are left with 12

O3–planes in the resolved orientifold quotient.

Now we treat the problem of tadpole cancellation. On top of theO7–planes we place eight D7–branes to cancel the D7–tadpole locally. This gives rise to a stack of D7– branes with gauge groupSO(8) on each of the divisorsD1, D3,γ, E2,β. The contribution

to the D3–tadpole from each O7/SO(8)–stack is −12₂₄χ(S). In the triangulation a), E2

has the topology of _F0, therefore χ(E2) = 4. D1 has the topology of F0 blown up in

4.5 Global O–plane configuration and tadpole cancellation 67

surface blown up in nine points, so χ(D3) = 4 + 9 = 13. The total D3–tadpole from

all O7/SO(8)–stacks is therefore

Q3(O7/D7) =− 1 2 X β=1,2 χ(E2,β) +χ(D1) + 4 X γ=1 χ(D3) ! =−38. (4.22)

The D3–tadpole coming from the (non–exotic) O3–planes themselves is

Q3(O3−) =−1₄nO3 =−3. (4.23)

Part II

Applications in String

Phenomenology

Chapter 5

Preliminaries

In this section, the preliminary knowledge on type IIB flux compactifications which is needed in later sections is presented.

5.1

The type

IIB

low energy effective action

We briefly recall the massless spectrum and the 10D low energy effective action of type IIB string theory. The massless Neveu-Schwarz–Neveu-Schwarz sector consists of the scalar dilaton φ, the metric or graviton Gµν and the anti-symmetric tensor

Bµν. The Ramond-Ramond sector consists of the even formsC0, C2, C4 etc. The field

strengths of the form field Cn is denoted by Fn+1, the field strength of Bµν isH3. The

10–dimensional low energy effective action describing the bosonic massless degrees of freedom of the type IIB superstring is [27]

SIIB = 1 2κ2₁₀ Z d10x(−GE)1/2 RE − ∂µS ∂µS 2 (ImS)2 − Mij 2 F i 3 ·F j 3 − 1 4|Fe5| 2 − ij 8κ2 10 Z C4∧F3i ∧F j 3, (5.1)

with κ10 = 2π gse−φ10 the ten-dimensional gravitational coupling (gs the string cou-

pling), GE,µν =e−φ/2Gµν the metric in the Einstein frame,RE the Ricci-tensor in the

Einstein frame, S =C0+i e−φ10 the complexified dilaton, Mij = 1 ImS |S|2 _−ReS −ReS 1 , F3 = H3 F3 . (5.2) e

F5 =F5 − 1₂C2∧H3+ ₂1B2 ∧F3 is the self-dual 5-form field strength, i.e. ∗Fe₅ =Fe₅.

In this form, the effective action is manifestly SL(2,_R) invariant:

S0 = a S+b c S+d, F 0 3 = d c b a H3 F3 , Fe₅0 =Fe₅, G0_E,µν =G_E,µν, (5.3) with a, b, c, d∈_R and ad−bc= 1. 71

Now, the surplus 6 dimensions are being compactified on a Calabi–Yau manifold

X. For the metric we use the following block-diagonal ansatz:

ds2 =g_µν(4)(x)dxµdxν +g(6)_mn(y)dymdyn, (5.4) where gµν(4)(x) is the four–dimensional Minkowski metric and gmn(6)(y) is the metric of

the internal Calabi–Yau space. As explained in Section 2.5, a Calabi–Yau manifold has a moduli space which consists ofh(1,1)_{(X) K¨ahler moduliT}i _andh(2,1)_{(X) complex}

structure moduli Ui_{. In addition, there is the complex dilaton field S. The parameter}

space of S is locally spanned by the coset

MS =

SU(1,1)

U(1) . (5.5)

Furthermore, we have e−φ10 _=e−φ4 _Vol(X)−1/2_{, with Vol(X) the volume of the com-}

pactification manifold X and φ4 the dilaton in four dimensions.

Without D–brane moduli, locally the closed string moduli space M is a direct product of the complex dilaton field S, the K¨ahler MK and complex structure moduli M=MS⊗ MK⊗ MCS. (5.6)

All factors are special K¨ahler manifolds on which a K¨ahler potential can be defined. Before introducing the orientifold projection and D-branes, the theory has N = 2 supersymmetry in four dimensions, afterwards onlyN = 1. AnN = 1 supersymmetric theory is completely described by three quantities: The holomorphic superpotential, the K¨ahler potential, and the gauge kinetic function.