# Describing Calabi–Yau 3-fold

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: https://arxiv.org/abs/1007.4847 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: 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! • In the linked article for$N=1, D=4$SUGRA, I don't think it says that the Kähler potential is given by$\kappa = \phi_i\phi^{i\ast}$, only that this is a particularly simple example, that is worked out as a toy model. – doetoe Sep 9 '19 at 16:53 • Look at the article carefully pls. What do you mean by I don’t think! ! It’s will known in 4d supergravity models , the Kähler potential is given in terms of scalars!!! – Dr. phy Sep 9 '19 at 17:09 • Yes, always, not only in 4D. You claim however that it is given by$\kappa = \phi_i\phi^{i\ast}$. The authors only say "A simple form of K [...] which will be used in the next subsection as a toy model, is given by$K = \phi_i\phi^{i\ast}\$. – doetoe Sep 9 '19 at 17:23
• @doetoe, you don’t understand the question as a whole!! – Dr. phy Sep 9 '19 at 21:22