Question

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 $\xi = \frac{1}{\sqrt{2}}(y^{2i-1} + iy^{2i})$

Answer 1

First of all, I think that the Hodge $\ast$-operator takes $(p,q)$-forms to $(D-q,D-p)$-forms. You seem to be talking about the composition of the Hodge $\ast$-operator with complex conjugation.

I will not do the calculation for you, but I will outline it and perhaps you can tell us later which of the expressions is indeed the correct one. (I'm not being purposefully cruel: it's just that I don't work with $\varepsilon$ tensors and hence don't know the answer off the top of my head.)

Let $\theta^a$ be a basis of the $(1,0)$-forms with their complex conjugates $\bar\theta^a$ a basis for the $(0,1)$-form, relative to which the hermitian metric takes the form $$h = \sum_a (\theta^a \otimes \bar\theta^a + \bar\theta^a \otimes \theta^a).$$ If we define the $e^j$ by $$\theta^a = \frac1{\sqrt{2}} ( e^{2a-1} + i e^{2a} )$$ then we see that the $e^j$ are orthonormal with respect to the riemannian metric.

The Hodge $\ast$-operator is easy to define on monomials $e^I = e^{i_1} \wedge \cdots \wedge e^{i_n}$, where $I = (i_1,\cdots,i_n)$ is a multi-index with $1\leq i_1 < \cdots < i_n \leq 2D$. If $I_c=(i_{n+1},\cdots,i_{2D})$ is the complementary multi-index, then $$\ast e^I = (-)^\sigma e^{I_c}$$ where $(-)^\sigma$ is the sign of the permutation $$\sigma = \begin{pmatrix} i_1 & i_2 & \cdots & i_{2D} \cr 1 & 2 & \cdots & 2D \end{pmatrix}.$$

Your mission, Ygor, should you decide to accept it, is to see what $\ast$ does on a monomial $$\theta^{a_1} \wedge \cdots \wedge \theta^{a_p} \wedge \bar\theta^{b_1} \wedge \cdots \wedge \bar\theta^{b_q}.$$

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Perhaps a simpler way is to check for which prefactor the inner product on $(p,q)$-forms defined by $$\alpha \wedge \ast \bar\beta = \left<\alpha,\beta\right> \operatorname{dvol}$$ is the expected one.