In the book String Theory and M-Theory by K. Becker, M. Becker and J.H. Schwarz:

  1. Why is the potential for moduli given by eq (10.168): $$\tag{10.168 }V(T,K) ~=~ \frac1{4\mathcal{V}^3} \Big( \int_{CY_4} F \wedge \star F - \frac16 \chi T_{M2} \Big)?$$
    Maybe the answer of this question is trivial, but I cannot see how $T$ and $K$ enter this picture and get $V(T,K)$ as their potential.

  2. How can one arrives at eq (10.181): $$\tag{10.181} V~=~e^{\mathcal{K}} \Big( G^{a \bar{b}} \mathcal{D}_a W \mathcal{D}_\bar{b} \bar{W} - 3|W|^2 \Big)?$$

  • $\begingroup$ For 2) I find some hints in the book : Supergravity : Formulae (17.45),(17.84),(17.128),(17.129),(17.131) (17.134). It is totally beyond my skills, but the idea is to make a link between embedding Kalhler manifold with conformal symmetry , with projective Kahler manifolds for physical fields (Superpoincaré gravity). So the scalar potential is calculated from the metrics and the superpotential on the embedding manifold, which have relations with $K$ and $W$. $\endgroup$ – Trimok Aug 14 '13 at 18:28

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