Pruebas eléctricas

Recall that in the DBI action and Chern-Simons action, the field strength F_aappears together with the pull-backs of B2 fields as a gauge invariant Fa= i(`2sFa+ i2πı∗B). From now one let

us set `2_s = 1. Then given a stack of NaD-branes, which typically hosts the U (N ) gauge theory,

we decompose the background value of the field strength F_a as Fa= T0(Fa0+ iı

∗_{B) +}X

i

TiFa(i), (1.123)

where T0 denotes the unity element 1 in U (Na) representing the diagonal subgroup U (1)aand

Ti are the traceless abelian elements of SU (Na). In terms of the line bundles Lia, one has

c1(L0a) = −1 2π(F 0 a + iı∗B) ∈ H2(Da), c1(Lia) = −1 2πF i a∈ H2(Da). (1.124)

Turning on the gauge flux will break the gauge group U (N_a) to its subgroup which commutes to the U (1)i. For example turning on the Fa(0) will breaks U (Na) to SU (Na) while turning on the

F_ai breaks SU (Na) to its commuting ones, For example, if the Fai coincides with the hypercharge

generator in SU (5), then one has the breaking SU (5) → SU (3) × Su(2) × U (1)_i.

As an a side remark, following [57], we should notice that there is an important fact for constructing the gauge flux c1(La) from the relative cohomology group, namely a non-trivial

c1(La) can be trivial in H2(X3). To see this, recall that the D7-branes divisor Da defines a

inclusion ı : Da→ X3 which can further induce the pushforward map ı∗and the pullback map ı∗

in the corresponding homology group and its Poincaré dual cohomology group respectively, as ı∗ : H2(Da) → H2(X3); ı∗ : H2(X3) → H2(Da). (1.125)

Note that the pullback map ı∗ induces a long sequence on the cohomology group involving the relative cohomology group H2(X3, Da). The relevant fact for us is that there are non-trivial

part of H2(Da) which is trivial in H2(X3). In other words, one can split

La= ı∗(La) ⊗ Lnona , (1.126)

where the part ı∗(La) denote the pullback of the line bundle La defined in the X3. As a

consequence, the two parts, as divisors, have a vanishing intersection on Da, namely

Z

Da

c1(ı∗(La)) ∧ c1(Lnona ) = 0. (1.127)

1.8. Orientifold Compactifications of Type II Strings

In this section, we will mainly discuss consequences when introducing spacetime-filling D-brane in Type II Calabi-Yau compactifications.

As we learned from the Gauss law in electrodynamics, the total charges in a compact manifold have to be vanishing. D-branes, as higher dimensional extended objects, also carry the RR charges. Based on the same reason, such RR charges in a compact manifold such as the Calabi- Yau manifolds have to vanish. The similar argument can also applied to the D-branes tension (As we mentioned, D-branes tensions can be viewed as NSNS charges). Such considerations are dubbed as tadpole cancellation, which we will discuss more together with other consistency conditions in 1.10.2. Hence in a compact Calabi-Yau manifold, one needs to add certain sources

with negative RR charges and negative tension in order to be consistent 26. Fortunately, in string theory, such objects exist, known as orientifold planes or simply O-planes.

1.8.1. Orientifold planes

The orientifold planes typically arise when gauging a discrete Z2 symmetry, where such Z2

symmetry O in type II string theories typically contains a world-sheet parity Ω and a Z2involution

symmetry σ acting on 10D spacetime M coordinates 27 Xµ and reverses the orientation of the strings [58]. Then an Op-plane can be introduced as the fix-point locus of the involution symmetry σ from the spacetime perspective. More precisely, the involution σ transforms28

σ : Xm(y, ¯y) ↔ −Xm(y, ¯y); m = p + 1, ...., 9 (1.128) and an Op-plane resides at the fixed plane of this σ and extends in the (X0, X1, ...., Xp) directions.

In perturbative type II string theories, orientifold planes, like D-branes, also couple to massless closed string modes and carry RR charges, as well as breaking half of bulk supersymmetries. However, unlike D-branes, they have fixed negative tensions and hence are not dynamical (at least in the perturbative strings) in a sense that they cannot fluctuate and therefore cannot carry degrees of freedom29. Based on their dimension of spatial space, we distinguish them as Op-planes in the same sense as Dp-branes. Similarly, Op-planes are stable in type IIA with ps being even and in Type IIB with ps being odd. As mentioned above, a Op-plane carries charge under the RR fields Cp+1, and this couplings are also captured by the Chern-Simons action,

which is given : SOp = −Qpµp 2 Z Op X 2p ı∗C2p s L(1₄TO7) L(1₄NO7) . (1.129)

where |Q_p| = 2p−4 30 _{and L denotes the Hirzebruch L-genus. The coupling to C}

p+1is then −Qpµp 2 Z Op Cp+1. (1.130)

Note that there is opposite sign comparing with the Dp-branes, which is exactly the reason we are introducing them. As they are not dynamical and no open strings attaching them, thereby

26_{Of course, such two conditions can be satisfied separately. For example, one can add anti D-brane to cancel}

the RR tadpole and invoking other mechanisms to tackle with the issue with the NSNS tadpoles. However, the anti Dp-branes need to be fixed at certain loci otherwise they would approach to Dp-branes by attraction and eventually will annihilate, inspired by the interaction between a particle and its anti-particle, and would be not wanted. One famous example with such stabilized anti-Dp-branes is the anti D3-branes, sitting in the deeply warped region on CY, in the KKLT setting with an effort to construct dS spaces from String theory compactification. However, such cases needs extra and maybe subtle conditions, which makes this option not so economic.

27

Note in Calabi-Yau compactification, it shall act trivially on Minkowski spaces and thus the O-planes are space-time filling.

28

Here we rewrite the 10D coordinates in terms real ones Xs.

29

Since a fluctuating negative object necessarily has negative-norm states

30

Note that Op-planes do not always carry negative RR charge. We usually denote the Op-planes with negative planes as O−p-planes and O+p-planes with positive RR charges. But they do always carry negative tensions. Unless specified, we will omit the superscript ± and assume they are with negative RR charges in this chapter.

1.8. Orientifold Compactifications of Type II Strings

no massless gauge fields lives on their world-volume. The DBI-like action for Op-planes reads So = 2p−4To

Z

dp+1ξe−φ(p−det(g)), (1.131)

where g := det(g_ab) denotes the determinant of the inducing metric gab from the spacetime

metric g_µν to the Op-plane world-volume and the tension T_o = QpTp. Such action is essentially

the higher dimensional generalization of Nambu-Goto action.

1.8.2. Type IIB O3/O7 orientifold compactifications

Now we are in the position to introduce Calabi-Yau orientifold compactifications. Before we go to the specific Type IIB O3/O7 orientifold compactifications, let us say a bit more on general aspects of the above Z2 action O on Calabi-Yau compactifications, dubbed orientifold

action. In order for such action to preserve supersymmetries, there are certain constraints. In particular, the involution σ must be isometric and (anti) holomorphic [59]. To be more precise, the involution 31 σ∗ must be anti-holomorphic (σ∗J_nm = −J_nm) in type IIA and holomorphic σJ_nm= J_nm in type IIB, respectively. Further the unique holomorphic (n, 0)-from Ω₃ must also be an eigenform of σ∗ in type IIB, namely σ∗Ωn= ±Ωn. In type IIA, it acts on the holomorphic

n-form as σ∗Ωn= e2πθΩ¯n, where θ is some phase, typically set to 0 by redefining Ωn.

In order to preserve same supercharges as the one preserved by spacetime-filling D-branes, it turns out that in the presence of D3/D7-brane systems in type IIB Calabi-Yau compactifications, the orientifold action O should be taken as

Ω(−1)FL_{σ, Ω}

n= −σ∗Ωn, (1.133)

while in the presence of D5/D9-brane systems, it is taken as

Ωσ, Ωn= σ∗Ωn. (1.134)

And in the presence of D6-brane systems in Type IIA Calabi-Yau compactifications, it’s taken as

Ωσ, Ωn= σ∗Ω¯n, (1.135)

where FL is the spacetime left-mover fermion number. Now having said only odd-valued p for

Op-planes stay stable in Type IIB and even-valued for Type IIA, one might wonder can all allowed values for p coexist in the same Calabi-Yau compactification. To see that one needs to look at the involution action on Ωn in the vicinity of O-planes. For definiteness, let us work

on a Calabi-Yau threefold X3 . Assuming nearby the Op-planes, the holomorphic top form Ωn

locally takes the form as Ω_n= dy1∧ dy2_{∧ · · · ∧ dy}3_{. Now the action reads}

σ∗(dy1∧ dy2∧ · · · ∧ dy3)|Op = ±(dy1∧ dy2∧ · · · ∧ dy3) (1.136)

as at the fixed locus Op either an even or an odd number of yi have to change the sign. For an even number, i.e. σ∗(Ωn) = Ωn, the internal fixed locus is 2 or 6 dimensions. And an

31_{Here σ}∗

denotes the pullback of σ for the Dolbeault cohomology of Xn. The involution σ induces an eigenspace

splitting of the Dolbeault cohomology groups

Hp,q(Xn) = H+p,q(Xn) ⊕ H−p,q(Xn) (1.132)

odd number, the internal fixed locus is 0 or 4 dimensions. This is exactly the dimension of the D-branes systems. In other words, the orientifold action O generating the Op-planes is coincident with the one for Dp-branes. Hence in orientifold compactification, with the same spacetime-filling dimensional Op-planes and Dp-branes, one can in principle cancel the tadpole issue and preserve certain supersymmetries in low-dimensional effective theories. Otherwise, the supersymmetries will be totally broken if one use the Op-planes to solve the RR tadpole issue. In the rest, we will always stick to the first situation.

From here we will focus on orientifold compactification of Type IIB string theory with O7/O3- planes in Calabi-Yau compactifications with generic dimensions. For our purposes, we take a look on how the the 10D Type IIB supergravity fields transform under the orientifold projection O := Ωp(−1)Fσ. In order to enter the low dimensional effective action, the Type IIB various

fields should survive, i.e. they should transform even under the orientifold projection. We listed the type IIB supergravity fields under the Ω_p(−1)F in 1.4, those who transform even (odd) under Ωp(−1)F should be required to transform even (odd) under the the involution σ.

even odd

Ωp φ, gµν, C2 C0, B2, C4

(−1)F _{φ, g}

µν, B2 C0, C2, C4

Table 1.4.: The various Type IIB supergravity fields transformations under the world-sheet parity Ωp and fermion number sign (−1)F.

even odd

σ φ, gµν, C0, C4 C2, B2

Table 1.5.: The various Type IIB supergravity fields transformations under the involution σ.

For more details on the explicit dimensional reductions and low-energy effective theory action, we refer to references, for examples [53,60,61] with Calabi-Yau three-folds. Before moving to next step, we would like to make a crucial comment. Recall we have said that D-branes and O-planes, as dynamic objects with tensions, should typically have back-reactions on supergravity bulk fields (together with the background fluxes), if there are not on top of each others. Let us first talk about back-reactions on metric gµν. Thus it implies that the geometry of internal spaces

would be changed after introducing D-branes and O-planes, and thus the internal spaces might not be Calabi-Yau manifolds. Indeed, it turns out in our cases that after the involution action σ, the geometry Bn:= Xn/σ, dubbed downstairs geometry, with respect to the upstair geometry:

Xn, is not a Calabi-Yau space anymore, but a Kähler manifold. The reason is that in such

compactification, the background fluxes typically need to be turned on for consistency conditions and such fluxes will be obstructions for the internal space being a Calabi-Yau, i.e. contributing some torsions and obstructing J and Ω_n to be integrable. However, one of specialties of Type IIB O3/O7 orientifold compactifications is that the Kähler manifold B_n is a so-called conformal Calabi-Yau, which means that the metric has related a Calabi-Yau one by a conformal factor, namely:

ds2(Bn) = e−2Ads2(Xn), (1.137)