El oficial-educador: dirigente de conductas, encargado del deber cívico y

Plugging (4.2.35) into (4.2.36) and integrating over the internal space, while keeping χ4D

constant on it, one obtains the following value for the gaugino mass

m1/2 = i eAR M e −2Φ_Ω∧H 2R_MdvolMe−2Φ = ie AR M e −2Φ_Ω∧G 4R_MdvolMe−2Φ , (4.2.37)

where in the last step we have used the condition G3,0 _{= 0 again. We thus see that the}

gaugino mass equals the gravitino mass (4.2.33)

m1/2 = m3/2 . (4.2.38)

As we will see in section 4.4.2, this result has a very simple four-dimensional interpretation.

4.2.3

Conditions on the curvature

Let us now discuss the conditions on the curvature (4.2.29) that arise at orderα0 from the minimization of the potential piece (4.1.6b). First of all, note that

R+ijkl = R−klij − (dH)ijkl . (4.2.39) So, by using the BI (2.2.5) we get the relation

R+ijkl = R−klij + O(α0) . (4.2.40) Hence, in the scalar potential the terms (4.2.29) can be rewritten as

ΩijkRij− = 0, JijRij− = 0 (4.2.41)

up to O(α02_{). These conditions can be rephrased by saying that the internal spinor η} +

specifying the SU(3) structure should be covariantly constant with respect to the torsion- full covariant derivative∇−_i . From (4.2.12a) and (4.2.21b), we know that this is not the case in the torsional SUSY-breaking backgrounds of subsection 4.2.1. However, let us assume that the SUSY-breaking is mild, so that ∇−_i η+ ∼ O(α0β), with 0 < β ≤ 1. Roughly

speaking, this would mean that both equations in (4.2.29) are violated at O(α0β). In particular, by using (4.1.8b), the curvature squared term in (4.1.6b) would be of O(α02β), and so negligible in our approximation forβ ≥1/2. Under this condition, the full potential would be extremized at our level of accuracy.

We can make this argument more concrete. From (4.2.12a) and (4.2.21b) we have

∇−_i η+ = −

i

4Sijγ j_η∗

+ . (4.2.42)

Taking into account (4.2.22) and the condition |W2|2 = 24|W1|2, we get qualitatively

∇−_i η+ ∼ W1γiη∗+. The torsion class W1 has the dimension of mass and defines a dimen-

sionless SUSY-breaking length scale LSB (measured in string units) through

W1 ∼ (lsLSB)−1 (4.2.43)

Then, takinggij ∼l2sL2KK, withLKK being the KK length measured in string units we have

∇−_i η+ ∼ LKKL−SB1. Furthermore, by introducing the four-dimensional KK-scale MKK =

eA_/(l

sLKK) and recalling (4.2.32), we can restate (4.2.43) in a more physical way

m3/2 ∼ MKKLKKL−SB1 . (4.2.44)

One has mild SUSY-breaking, which can be seen as spontaneous from the four-dimensional point of view, when m3/2 MKK. This condition corresponds to

LKK

LSB

1 . (4.2.45)

This also means that the violation of SUSY in terms of spinors is small∇−_i η+1.

Lets try to parametrize this relation in terms of LKK. In the regime of the validity of

a supergravity approximation all compactification scales should be larger than the string length which is guaranteed if

LKK > 1 = α0 4π2_l2 s = 4π 2_l2 s α0 . (4.2.46)

Demanding that∇−_i η+is of orderα0β (measured in string length, as∇−i η+is dimensionless)

yields by (4.2.46)

∇−_i η+ ∈

L−_KK2β, L2_KKβ . (4.2.47) For minimal SUSY-breaking one should choose ∇−_i η+ ∼L

−2β

KK, which is even smaller then

O(α0β_{). Together with our former estimate ∇}−

i η+ ∼ LKKL−SB1 we get a relation between

the KK length and the SUSY-breaking length

LSB ∼ L2KKβ+1 . (4.2.48)

We can consider these issues also in a bit more detail. The curvature terms (4.2.41) can be rewritten by using (4.2.42) and the formula

[∇−_i ,∇−_j]η+ = 1 4R−klijγ kl_η + . (4.2.49) They read Ji1i2R i1i2 − j1j2 = 2P lk_S k[j1S ∗ j2]l, (4.2.50a) Ωj1i1i2R i1i2 − j2j3 = 4i Pj1 k_∇− [j2Sj3]k + 1 2Ωj1 kl_S k[j2S ∗ j3]l, (4.2.50b)

wherePij is defined in (3.2.5) and projects onto holomorphic indices of the almost complex structure. Then, we have the following curvature squared terms contributing to (4.1.6b)

|JijRij−|2 ∼ |W1|4, (4.2.51a)

|ΩijkR ij

−|2 ∼ |∂W1|2+|W12∂W1|+|W1|4

∼ (lsLKK)−2|W1|2+ (lsLKK)−1|W1|3+|W1|4 . (4.2.51b)

By using (4.2.43), the dimensionless contribution to the O(α0) equations of motions asso- ciated to the curvature terms in (4.1.6b) can be approximated as

(EoM)_O(α0₎ ∼ gijls2 |ΩyR−|2+|JyR−|2 ∼ 1 L2 KK hL_KK LSB 2 + L_KK LSB 3 + L_KK LSB 4i . (4.2.52)

Note that from gij one gets an extra factor of l2sL2KK. We have separated an overall factor

ofL−_KK2, which gives a leading factor ofO(α0), while the terms in squared brackets provides a further suppression because of (4.2.45). In order to make the correction (4.2.52) of order

α02, for example, and therefore safely negligible at ourO(α0) approximation, (4.2.52) should scale at least like L−_KK4. Then, one would have to demand LSB =L2KK, and thus β = 1/2.

On the other hand, one could further relax this condition, depending on the details of the background. If for example in (4.2.51b) one finds|∂W1|.|W1|2, then it is enough to take

LSB =L 3/2

KK, i.e. β = 1/4.

We conclude that the curvature sector of the O(α0)-correction restricts possible so- lutions such that only mild supersymmetry breaking is allowed. The mass scale of the SUSY-breaking has to be well below the compactification scale and hence the breaking of supersymmetry can be regarded as spontaneous from the four-dimensional perspective. The more severe restriction that we encountered in this section arose from the order α00

part of the potential. We will analyze possible solutions in section 4.4 and 4.5. But before we come to this, we will present another view of DWSB coming from the perspective of calibrations.