not admit any de-Sitter solution and therefore calls for inclusion of corrections to the leading order. The relevant correction terms can be attributed to two different sources. One of these, being instanton correction which at large volume of the internal four-fold yields an additional term to the Superpotential

    $$\label{correct1}
    { W_{inst}} = P(z_i) e^{2 \pi i \rho}.$$

    While $P(z_i)$ is the holomorphic one-loop determinant, the leading order in the exponential originates from the action of Euclidean $D3$-branes wrapping on a four-cycle. With all other moduli fixed, they can be integrated out to produce an effective superpotential for $\rho$ alone.

    The origin of the other correction is described as follows. In type II-B string theory compactified on a Calabi-Yau fourfold a stack of $N$ number of D7 branes wrapping on 4-cycles in the compact manifold gives rise to non-abelian gauge groups. The effective low-energy $\mathcal{N}=1$ supersymmetric theory undergoes gluino condensation resulting into a superpotential,

    $$\label{superpotcorrec}
     W_{gauge}=\Lambda_{N}^3=A e^{\frac{2 \pi i \rho}{N}}.$$

    Here $\Lambda_{N}$ is the dynamical scale of gauge theory. The coefficient $A$ can be determined by the energy scale below which Supersymmetric QCD is valid. Plugging $\rho = i T$ in the superpotential along with a logarithmic $T$ dependence in the 