(8\p )+\ln (2)$ found in Ref. [@Lukas:2003dn]. From a similar calculation one finds that the fraction $\e_{(\t ,n)}$ by which the blow-up $(\t ,n)$ reduces the orbifold volume $V_0$ is given by $$\e_{(\t ,n)} = \frac{8}{3}\sum_{a=1}^3e^{-{\bf p}_{(\t ,a)}\cdot\bbe}\,
                u_{(\t ,n,a)}^2\; ,$$ where we have introduced the real parts $${\rm Re}(U^{i})=u_i$$ of the blow-up moduli.

The Kähler potential  relies on two approximations so we should discuss its range of validity in moduli space. First, we require all moduli to be larger than one, that is $$T^A\gg 1\; ,\qquad  U^i\gg 1\; ,\label{app1}$$ for the supergravity approximation underlying the calculation of $K$ to be valid. Second, the Kähler potential has been calculated to leading (quadratic) order in $U^i/T^A$ and terms of order four in these ratios have been neglected. Consequently, application of the Kähler potential  should be confined to the region of moduli space where the ratios $U^i/T^A$ are smaller than one, or more precisely, where $$\e_{(\t ,n)} \ll 1\; . \label{app2}$$ From the above interpretation of $\e_{(\t ,n)}$ this means that the volume taken away by the blow-ups should be small compared to the volume of the orbifold. In subsequent calculations, we will consistently apply this approximation in that we neglect higher 