# 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 $(E',\theta')$?

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.

• I am only familiar with Simspon's paper, not with Hitchin's paper. Can you explain it in detail? Expecially the sentence "which determines a different Riemann surface structure $\bar{\Sigma}$ on your smooth surface" and " going over to limits, you should get the answer to your question in that particular easy example."
– Lan
May 7, 2014 at 15:29
• Do you means that $E'=L\oplus L^{-1}$, with $\theta': L\cong L^{-1}\otimes \Omega_{\bar{\Sigma}}$ and $\theta': L^{-1}\to L\otimes \Omega_{\bar{\Sigma}}$ is determined by the quadratic differntial $\alpha$ which is the difference of complex structure $\bar{\Sigma}-\Sigma$?
– Lan
May 7, 2014 at 15:34
• 1.in every conformal class of metrics on a cpt. or. surface of genus $g\geq2$, there is a unique metric of constant curvature -4. Moreover, every metric which is not compatible with a given complex structure on your smooth surface gives a different Riemann surface structure on the surface. May 8, 2014 at 7:45
• 2. You should try to compute what happens to first order in $t$ when you look at the solution to the self-duality equations corresponding to $t\alpha.$ This gives you a section $\bar\alpha\in\Gamma(\Sigma,\bar K K^{-1})$ (w.r.t. the hyperbolic metric) which can be considered as the tangent vector given by the variation of Riemann surface structures. May 8, 2014 at 7:50
• 3. A holomorphic quadratic differential $\alpha$ determines in a natural way a holomorphic Higgs field $\Psi_\alpha\colon L\oplus L^*\to KL\oplus KL^*$ whenever $L^2=K.$ Then you add this Higgs field onto the Higgs field $\theta.$ May 8, 2014 at 7:51