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 $(E',\theta')$?