# Why does this allegedly Hermitian Kähler metric have non-zero diagonal terms?

In string theory, the Kähler potential of Kähler moduli (e.g. - the volume of a Calabi-Yau manifold) is given by (see, for instance, Becker, Becker, Schwarz: "String Theory and M Theory" p. 498)

$$K = -3\log[-\mathrm{i}(\rho-\bar{\rho})]$$

There are addition irrelevant terms (not coupled to $\rho$) which I am neglecting.

The Kähler metric is Hermitian and is given by,

$$G_{a\bar{b}}= \partial_a\bar{\partial_b}K$$

By the definition of a Hermitian metric $G_{\rho \rho}$ should be 0. But for the given potential it isn't. Why?

• Related question by OP: physics.stackexchange.com/q/177153/2451 – Qmechanic Apr 20 '15 at 12:59
• Looking at a Hermitian matrix I see no reason why $G_{\rho\rho}$ should be zero, rather, it should be real. Why do you claim that it should be zero by definition? – ACuriousMind Apr 20 '15 at 14:58
• A hermitian matrix by definition has $G_{aa}$ as 0. See the proof in Nakahara or any complex manifold book. – sol0invictus Apr 22 '15 at 14:09

Using the Dolbeault bigrading, the (2,0) and (0,2) components of the Kähler metric $g_{zz}=0$ and $g_{\bar{z}\bar{z}}=0$ do indeed vanish, respectively. In particular, the formula $$g_{z\bar{z}}=\partial_{z}\partial_{\bar{z}}K$$ for the mixed (1,1) components does not generalize to the (2,0) and (0,2) components.