Background:
In Calabi-Yau 3-fold, the Kähler metric is given in terms of the Kähler potential $\kappa$ :
$$ g_{i\bar{j}} = \partial_i \partial_{\bar{j}} \kappa,$$
where $i, \bar{j} = 1,2,3 $ ( the Holomorphic coordinates) . While the complex structure metric is given by:
$$ G_{a\bar{b}} = \partial_a \partial_{\bar{b}} \kappa,$$
where $ a, \bar{b} = 1, …, h_{2,1} $, where $ h_{2,1} $ is the number of complex structure moduli.
See for instance here. The Question:
first if I in 5d, $N=2$ supergravity, for instance, (after dimensional reduction from 11d SUGRA), can I find an explicit form for the Kähler potential?
Can I after that calculate $G_{a\bar{b}}$ and $ g_{i\bar{j}}$? And what about the degrees of freedom?
I mean for example, $N=1$, $D= 4$, SUGRA. See for instance: In this paper, the Kähler potential is given by
$$ \kappa = \phi_i \phi^{i *} $$
and the Kähler metric is given by:
$$ g_{i j^*} = \frac{\partial^2 \kappa}{\partial \phi_i \partial \phi_{j^*}},$$
where $\phi_i$ are scalar fields.
Any help appreciated!