Recently I have been reading a number of mathematical physics articles in which different definitions are given for the Hodge dual on a hermitian manifold. For the Hodge operator that transforms a (p,q)-form into a (D-p,D-q)-form I came across the following definition.

$\ast \omega_{p,q} = \frac{(-1)^{\frac{1}{2}D(D+1)}i^D}{p! q! (D-p)!(D-q)!} \bar{\omega}_{i_1 \ldots i_q \bar{i}_1 \ldots \bar{i}_p} \epsilon^{i_1\ldots i_q}{}_{\bar{j}_{q+1} \ldots \bar{j}_D} \epsilon^{\bar{i}_1\ldots \bar{i}_p}{}_{j_{p+1} \ldots j_D} d\xi^{j_{p+1}} \wedge \ldots \wedge d\xi^{j_D} \wedge d\bar{\xi}^{\bar{j}_{q+1}} \wedge \ldots \wedge d \bar{\xi}^{\bar{j}_D}$.

Another definition that I encountered was similar, but had a different prefactor: $\frac{(-1)^{\frac{1}{2} D(D-1) + (D-p)q}i^D}{p! q! (D-p)!(D-q)!}$.

My question is, which prefactor is correct? Moreover, what is the reason for the $i^D$ term? (I also encountered a third definition which has the second prefactor, but without the $i^D$ term)

I hope anyone could help out (despite al the indices :P)

Ygor

PS. D denotes the complex dimension, and $\xi = \frac{1}{\sqrt{2}}(y^{2i-1} + iy^{2i})$