content_image/368569.jpg content_image/368572.jpg content_image/368570.jpg content_image/368571.jpg content_image/368573.jpg content_image/368574.jpg content_image/368575.jpg
with M, N, = 0 , ... 4 and κ 2 = M − 3 P , where M$_{P}$ is the 5 d reduced Planck mass, and Λ denotes the five-dimensional cosmological constant. We now review the results of [12]. 

## A. The flux vacua 

To begin, we shall constrain the magnitude of the scalar field to lie at | Φ | = η , leading to the effective action 

<img src='content_image/368569.jpg'>

where θ is the phase of Φ. The equations of motion for this model are 

where 

<img src='content_image/368572.jpg'>

<img src='content_image/368570.jpg'>

<img src='content_image/368571.jpg'>

is the energy momentum tensor. We will look for a solution of the form 

<img src='content_image/368573.jpg'>

where µ, ν = 0 , 1 , 2 , 3 denote the 4 d coordinates, and the compact extra dimension is pa- rameterized by the coordinate 0 ≤ y < 2 π . We are interested in the case 

<img src='content_image/368574.jpg'>

i.e., in solutions with the extra dimension stabilized at a constant circumference 2 πL . We shall also require that the four-dimensional slices are described by a spacetime of maximal symmetry whose scalar curvature is given by R (4) = 12 H $^{2}$, with H 2 negative for the anti-de Sitter (AdS) case. 

The solutions to the equations of motion for the scalar field Eq. (9) which are compatible with maximal spacetime symmetry are given by 

<img src='content_image/368575.jpg'>

The change of axion phase θ around the compact dimension must be an integer multiple of 2 π , and hence the various flux vacua are parameterized by the integer n . 

