Let $\Sigma$ be a Riemann surface (not necessarily compact), and $x_1, \cdots, x_k$ a set of points on $\Sigma$. Let $n_1, \cdots, n_k$ be a sequence of integers, each of which is $\geq 2$, and such that $\chi(\Sigma)+\sum_i(n_i-1)<0$. Then the uniformization theorem (due to McOwen, Troyanov, Hitchin)[edit: apparently the correct attribution is to E. Picard; see comment below] says that there is a unique singular hyperbolic structure on $\Sigma$, where there are conical singularities at $x_i$ with angle $2\pi n_i$, for each $i$. This uniformization corresponds to a representation
$$\rho: \pi_1(\Sigma)\rightarrow PSL_2(\mathbb{R});$$
note that such a representation does not lie in the Teichmüller component of the $PSL_2(\mathbb{R})$-character variety of $\pi_1(\Sigma)$, as this would correspond to the smooth hyperbolic metric on $\Sigma$. For a reference, see the chapter by Goldman in Problems on Mapping Class Groups and Related Topics (page 218), though if there is any error in the above it is due to me.
Question: can we describe the Higgs bundle corresponding to $\rho$ under the Simpson correspondence explicitly?
In the case of a smooth metric, the description is as follows. We take a compactification $\bar{\Sigma}$ of $\Sigma$ by adding points $q_1, \cdots q_s$. Then we have the divisor $D=q_1+\cdots q_s$, the Higgs bundle has the form $L\oplus L^{-1}$ where $L$ is a line bundle with $L^{\otimes 2}\simeq \Omega^1(D)$, and the Higgs field is the tautological isomorphism $\theta: L\rightarrow L^{-1}\otimes \Omega^1(D)$. In the case of conical singularities at the $x_i$'s it seems to me that the Higgs field must have zeroes at $x_i$ with multiplicity $n_i-1$. I would be happy to understand the case of a single cone singularity with cone angle $4\pi$, for example.
Such a description must be known to Hitchin (if it exists), but looking at Hitchin's paper I couldn't seem to find what I was looking for.
