Deformation of Hitchin-Simpson correspondence Let $X$ be a compact Riemann surface, Hitchin-Simpson correspondence over $X$ says that irreducible representations of $\pi_1(X)$ one-to-one correspondend  to stable Higgs bundles with vanishing Chern clssess over $X$.
Now suppose we fix a representation $\rho$  of $\pi_1(X)$,  then it  corressponds to a  stable Higgs bundle $(E,\theta)$ over $X$.  Fix the underlying differential surface of $X$, but change the complex structure  and obtain a  new Riemann surface $X'$. Still  by the Hitchin-Simpson correspondence over $X'$, the irredubible representation $\rho$ corresponds to a new Higgs bundle $(E',\theta')$ on $X'$. 
My Question is  how describe the change $(E,\theta)\to (E',\theta')$ by the change $X\to X'$. For example we may just assume $E= L\oplus L^{-1}$,   and $\theta: L\cong L^{-1}\otimes \Omega_{X}$, where $L^2= \Omega_X$. Then what is    $(E',\theta')$?
 A: I cannot say anything about the general case, but I think your example was considered already by Hitchin in his Self-duality paper (§11). Let me add some
comments because Hitchin somehow has an opposite point of view than asked in your question:
First note, that the solution of the self-duality eq's corresponding to the stable Higgs pair $(E,\theta)$ gives a flat connection which determines the uniformization of
the given Riemann surface $\Sigma$: Its monodromy lies in $PSL(2,\mathbb R),$ and the hermitian metric on $E$ gives a constant curvature metric by restricting it to $L$ or $L^*.$ If you  vary the Higgs field $\theta$ by adding a holomorphic quadratic differential, you get a new metric of constant curvature, which determines a different Riemann surface structure $\tilde\Sigma$ on your smooth surface. Moreover, you know that your new solution of the self duality eq's comes from a Higgs pair $(\tilde E,\tilde\theta)$ of the same form as $(E;\theta)$ but for $\tilde \Sigma$ instead of $\Sigma.$ By identifying quadratic differentials with (real) tangent vectors to the Teichmüller space and going over to limits, you should get the answer to your question in that particular easy example.
