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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.

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

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