Recall that for a given Riemann surface $\Sigma$ Hitchin's self-duality equation consists of a complex rank $r$ vector bundle $E$ (with degree 0 for simplicity), a connection $d_A: \Omega^k(\Sigma, E) \rightarrow \Omega^{k+1}(\Sigma, E)$ along with a Higgs field $\Phi \in \Omega^{(1,0)}(\Sigma, {\rm End}(E))$. The equation itself reads: \begin{equation} \begin{aligned} & F_A + \left[\Phi, \Phi^*\right] = 0\\ & {\bar \partial}_A \Phi = 0 \end{aligned} \end{equation}

There are two ways of viewing this equation:

- Fix a Hermitian metric $h$ on $E$ and think of the equation as the one for a pair $(A, \Phi)$ modulo complex gauge group $G_{\mathbb{C}}$ transformation; In this way the $(0,1)$ part of the connection under the complex structure $J$ of the Riemann surface gives a holomorphic structure of $E$: ${\bar\partial}_E = {\bar \partial}_A$.
- Fix the holomorphic structure ${\bar \partial}_E$, then the Hitchin's equation can be viewed as an equation for the Hermitian metric $h$ and Higgs field $\Phi$. Then there is a unique Chern connection, which is unitary and compatible with the holomorphic structure.

**Question**: what is the transformation between these two conventions explicitly? More specifically, is there any canonical mapping between gauge orbits of solutions in two conventions?

For instance, in the first point of view we take $h$ to be identity matrix (sometimes called unitary gauge), then we have the moduli space of solutions; in the second point of view we have solutions $(h, \Phi)$ where $h$ is in general different from identity. How does one go from one convention to the other by some explicit transformation?