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where φ$_{i}$ are complex scalar fields and the phase χ is a real field constrained by 

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The field configuration in Eq. (367) leads to an effective Lagrangian for the flat direction fields φ$_{i}$ , 

<img src='content_image/359643.jpg'>

where D$_{µ}$ is a gauge covariant derivative that reduces to the partial derivative when the gauge fields vanish, P$_{1}$ is the projection operator along Φ and P$_{2}$ along Ψ, where 

and the corresponding equation of motion 

where R = √$_{Φ}$$_{†}$$_{Φ. We are interested in the background dynamics where all the fields are}$ homogeneous in time, and for simplicity we study only the radial motion, such that Φ = R ˆ e$_{Φ}$ , where ˙ ˆ e$_{Φ}$ = 0 (the dot denotes derivative w.r.t time). Then the equation of motion simplifies to 

<img src='content_image/359644.jpg'>

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A notable feature is that the fields have non-minimal kinetic terms, since the field manifold defined by the flat direction is curved, actually a hyperbolic manifold. This results into the usual equation of motion for one scalar field with a potential for the radial mode except for the factor 1 / 2, which makes the potential effectively flatter in this direction. This can be traced to the square root nature of H$_{u}$ in Eq. (367). 

As far as our example of LH$_{u}$ is concerned there are only three families which we can account for. The flattest MSSM direction, QuQue , is lifted by n = 9 superpotential operator, QuQuQuH$_{d}$ee . The flat direction QuQue is an 18 complex dimensional manifold, Ref. [407]. The largest D-flat direction is only 37 complex dimensional [407]. One can imagine a larger representation which will have larger number of D -flat directions which 

