A system of Hodge bundles is a direct sum of holomorphic vector bundles $E = \oplus_{p+q=n} E^{p,q}$ with a morphism $\theta : E^{p,q} \rightarrow E^{p-1,q+1} \otimes \Omega_X^1$ such that $\theta^2 = 0$. In Simpson's paper in 1988 he mentioned that "in particular a system of Hodge bundles is a Higgs bundles with action of the group $A = U(1) \times U(1)$. A metric on a system of Hodge bundles means a metric on the Higgs bundle $E$, preserved by $A$. It is therefore a direction sum of metrics on the bundles $E^{p,q}$ and we can look at the indefinite Hermitian form ......" What I don't understand is what the group action means? The group $U(1)$ is $\{ z \in \mathbb{C} : |z|=1 \}$ from my perspective. How does this group induces the relation between Higgs bundle and Hodge bundle?
