String Theory and M-Theory

V(σ)
V0
10.3 Moduli stabilization 505
σ
0
Fig. 10.7. Form of the potential as a function of the radial modulus. The values of
the potential and the size depend on the values used for A, a and W0. The figure
displays a minimum at which the potential is negative leading to an AdS vacuum.
has a minimum that is given by
V0 = − a2A2 e
6σ0
−2aσ0 . (10.191)
Here σ0 is the value of σ in the radial modulus ρ = iσ at the minimum of the
potential. Since this potential is negative, the only maximally symmetric
space-time allowed by such a supersymmetric compactification is AdS spacetime.
One can break supersymmetry explicitly by adding anti-D3-branes. This
gives an additional term in the scalar potential of the form
σ
δV = D
, (10.192)
σ2 where D is proportional to the number of anti-D3-branes.
It can be chosen to make the vacuum energy density positive, so that a
compactification to dS space becomes possible. Including the anti-D3-brane
contribution results in the scalar potential
V (σ) = aAe−aσ
2σ 2
1
3 σaAe−aσ + W0 + Ae −aσ
+ D
. (10.193)
σ2 The form of V (σ) is displayed in Fig. 10.8. It shows that a vacuum with a
positive cosmological constant can be obtained. Strictly speaking, the vacua
obtained in this way are only metastable. However, the lifetime could be
extremely long.