Consider a holomorphic vector bundle $V$ over an $n$_dim complex manifold $M$. I am interested in the notion of an inner product between complex bundle-valued forms $\phi^a, \psi^b \in \Omega^{(p,q)}(M; V)$, where $a,b$ denote bundle indices.
The standard notion of inner product between ordinary complex differential forms $\phi, \psi \in \Omega^{(p,q)}(M)$ is:
$$ \left\langle \phi, \psi \right\rangle = \int_M \phi \wedge \star \bar{\psi} $$
Where $\star \bar{\psi} \in \Omega^{(n-p, n-q)}$ so that the integrand is a scalar density. But for bundle-valued forms, assuming the following inner product with some fibre metric $h_{ab}$:
$$ \left\langle \phi, \psi \right\rangle = \int_M h_{a\bar{b}} \phi^a \wedge \star \bar{\psi}^{\bar{b}} $$
how is it possible for the integrand to be a scalar? I assume:
- $\bar{\psi}$ is a section of $\bar{V} \otimes \bigwedge^{(q,p)} T^*M$.
- $\star \bar{\psi}$ is a section of $\bar{V} \otimes \bigwedge^{(n-p,n-q)} T^*M$.
It seems in order for the integrand to be a scalar, you would need either the complex conjugation operation $\bar{\cdot}$ or the Hodging operation $\star \cdot$ to somehow change the $V$ tensor factor to the dual tensor factor $V^*$, although I don't see how this can be justified - if anyone could provide an explanation or point to literature on this I'd be very grateful.
