RESPONS ABILID AD ADMI NISTR ATIVA OCM A, ODECM A, CNM

An important issue is the study of the stability of these strings. Although supersymmetry ensures that these configurations are solutions to the equations of motion, a priori, there is nothing to prevent them from decaying into a different configuration. As we already mentioned in chapter 1, the stabilitiy of BPS cosmic string solutions has been investigated recently in the context of three and four dimensional N = 1 supergravity in [99] [100] [101]. In these papers it was proven that these solutions are stable against all sorts of perturbations. However the present situation is more subtle since a complete analysis requires the study to be done in the full N = 2 supergravity theory. In particular, it is not clear if the cosmic strings would survive a perturbation of the truncated fields. Moreover, these analyses do not prevent the presence of zero modes which, if they are excited, can also lead to the disappearance of the strings. The cosmic string solution we have presented in this chapter has one of such zero modes, the value of the axion-dilaton field.

The constant value of the axion-dilaton field is not fixed by the BPS equa- tions nor by the scalar potential. The mass per unit length of the string is also independent of the value of the axion-dilaton field (7.5.26). The dilaton fixes the overall length scale of the configuration in the following sense. There are two natural lengths in the solution given by the inverse of the masses of the Higgs and the gauge field, and they are both functions of the dilaton field:

m2_W ∝ − 1 ImS, m 2 Φ∝ − 1 ImS, (7.5.27)

so that the corresponding length scales are

l2_W ∝ −ImS, l2_Φ∝ −ImS. (7.5.28) Suppose we have a solution to the BPS equations given by the profile functions f(r), Wθ(r), C(r) andρ. Then it is easy to check that the functions

f(λr),Wθ(λr),C(λr)/λandρ−2 log(λ) also satisfy the BPS equations for any

real λ >0. From here it is obvious that the value of the dilaton determines the length scales in the transverse direction to the string, in particular the core radius.

Supersymmetric cosmic strings in N = 2 supergravity.

This situation looks similar to the case of semilocal strings [155], where there is also a one parameter family of solutions with equal energy and different core radii. In that case finite energy perturbations can excite the zero mode connecting solutions within the same family, leading to the spread of the magnetic flux and eventually to the disappearance of the strings. An instability of this type was also found in a class of BPS cosmic strings solutions which appear in aN = 1 supersymmetric model proposed by Blanco-Pilladoet al. [59] to describe the last stage of a brane-antibrane inflation. We review this analysis, published in [72], in appendix B.

This is not going to occur in our model. In order to go from one solution to a different one, the dilaton has to change its value everywhere in the plane transverse to the string. The kinetic energy needed in order to excite the value of the dilaton globally diverges, and this implies that, once the system has chosen a given value for the dilaton, finite energy perturbations cannot drive the system to a solution with a different value ofS. The radius of the string will remain unchanged.

7.6

Discussion

As a generalization of the work done in [73], in this chapter we have enlarged the family ofN = 2 supergravity actions which allow the embedding of N = 1 supergravity actions containing aD-term potential and a constant FI term. We have extended the result of [73] to a class of special geometries more familiar in compactifications of string theory. We are using here a “very special K¨ahler geometry” characterized by a cubic prepotential, instead of the minimal special geometry used in [73]. To be specific we take the special manifold to be:

ST[2, n]≡ SU(1,1) U(1) ×

SO(2, n) SO(2)×SO(n),

in the Calabi-Visentini basis (7.2.8), which is related to the cubic prepotential by a symplectic rotation [139, 152].

This choice of special geometry has two important consequences. An axion- dilaton field,S =a−ieρ, is present in the reducedN = 1 theory after truncation from N = 2. Moreover, it is possible to define a gauging for which the scalar potential is bounded from below. However, it has a runaway dependence on the dilaton:

V ∝e−ρ.

As an application, we have shown how to construct a half-BPS cosmic string solution from aN = 2 supergravity action inD= 4. Following [73] we have used a string ansatz compatible with a consistent truncation fromN = 2 to N = 1.

7.6. Discussion

In order to obtain the scalar potential we have gauged the same isometry used in [73]. We have found that the BPS equations imply that the axion-dilaton has to be simultaneously holomorphic and anti-holomorphic, which can only be satisfied if it is a constant:

S= Constant, ImS <0.

Despite the runaway behavior of the potential, we have proved that all the string solutions have the same energy per unit length, regardless of the value of the dilaton, and it is given by the Gibbons-Hawking surface term [96, 73]. The value of the dilaton fixes the masses of the Higgs and the gauge field and, hence, also the radius of the string. We have argued that the system cannot evolve between two solutions with different values of the dilaton, since this would require an infi- nite amount of energy. Thus, once the strings are formed their radii remain fixed. Observations of the timing of milisecond pulsars give the constraintµ_{string .}

2×10−7 _{[51]. However this constraint depends on the specific model used to}

calculate it, what leads to a considerable uncertainty. This implies for our model that the FI term has to satisfy:

0< π|m|η_.2×10−7,

where the lower bound is coming from the study of Minkowski vacua in section 7.3.

APPENDIX

A

Conventions.

In this thesis we use the units c=~ = 1. Thus, the Plank massmpl is simply

given by

m−_pl2≡ G

~c

=G, (A.0.1)

whereGis the Newtons constant. In general we will work in units of the reduced Plank massMp,

M_p−2= 8πG= 1, (A.0.2)

except in applications to cosmology where we might choose to keep Mp or G

explicitly for clarity.

The use of indices is summarized in table A.1.

A.1

Space-time conventions

We choose the Minkowski space-time metricηmnto have mostly positive Lorentz

signature

ηmn= Diag(−1,1,1,1). (A.1.1)

Conventions.

Space-time indices

µ 0, . . . ,3 space-time coordinates, withx0_{for the time}

m 0, . . . ,3 local frame

Supersymmetric truncations inN = 1 supergravity (Chapt. 2-5)

I 1, . . . , nC chiral multiplets

α 1, . . . , nh truncated chiral multiplets

i 1, . . . , nl surviving chiral multiplets

a 1, . . . , nV vector multiplets

˜

a 1, . . . ,˜nV truncated vector multiplets

N = 2 supergravity (Chapt. 6-7)

s 1, . . . , nH hypermultiplets

X 1, . . . ,4nH scalar fields in hypermultiplets

A 1, . . . ,2nH spinors in hypermultiplets

Λ 0, . . . , nV vector multiplets

α 1, . . . , nV scalar fields and spinors in vector multiplets

i 1,2 SU(2)

x 1,2,3 triplet of SU(2)

Table A.1– Summary of indices.

Defining the space-time vielbein as

gµν =eµmηmneνn, (A.1.2)

the spin connection and the Levi-Civita connection are given by

ωµmn(e) = 2eν[m∂[µeν]n]−eν[men]σeµp∂νeσp, Γρµν = 12g ρλ _2∂ (µgν)λ−∂λgµν . (A.1.3)

Both formulations are equivalent due to the constraint

∇µeνm=∂µeνm+ωµmn(e)eνn−Γρµνeρm= 0. (A.1.4)

The covariant derivative of a vector reads

∇µkm=∂µkm+ωmnµ kn. (A.1.5)

The Ricci tensor can be obtained using the following formulae

Rµνmn = 2∂[µων]mn(e) + 2ω[µmp(e)ων]pn(e), Rµνρσ = Rρσmneµmeνn= 2∂[ρΓ µ σ]ν+ 2Γ µ τ[ρΓ τ σ]ν, Rµν = Rµρnmenρeνm, R=gµνRµν. (A.1.6) (A.1.7)