# Complex conjugation of bundle-valued forms

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.

Never mind, it's obvious that the bundle metric $$h_{ab}$$ identifies $$V \sim V^*$$ and so the integrand is indeed a scalar.