I am looking at the following Riemann surface (let's call it $M$),
\begin{equation}
y^n=\frac{(x-x\_1)(x-x\_3)}{(x-x\_2)(x-x\_4)}
\end{equation}
which is a Riemann surface of genus $n-1$. It can be thought of as a quotient of the complex plane by a Schottky group $\Sigma$,
\begin{equation}
M\cong\mathbb{C}'/\Sigma
\end{equation}
where $\mathbb{C}'$ is the domain of discontinuity of $\Sigma$. Then, we can look at the covering,
\begin{equation}
\pi\_{\Sigma}:\mathbb{C}'\rightarrow \mathbb{C}'/{\Sigma}
\end{equation}
which takes $z$ to $[z]$. The inverse map $w=\pi^{-1}\_{\Sigma}$ is multivalued on $\mathbb{C}'$,
\begin{equation}
w\sim\gamma w
\end{equation}
where $\gamma\in\Sigma$. However, the Schwarzian derivative of $w$ in some coordinate patch $z$ is single valued,
\begin{equation}
\{w,z\}=\frac{w'''}{w'}-\frac{3}{2}\left(\frac{w''}{w'}\right)^2.
\end{equation}
On overlapping coordinate patches, it transforms like,
\begin{equation}
\{w',z'\}=\left(\frac{dz}{dz'}\right)^2\{w,z\}+\{z',z\}
\end{equation}
which means it is a projective connection on $M$. Also, near the $z\_i$s, since we can use the coordinate, $y^n\sim(z-z\_i)$,
\begin{equation}
\{w',y\}\sim\{z,y\}=-\frac{1}{2}\frac{n^2-1}{y^2}+\cdots.
\end{equation}
According to [Faulkner - The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT](https://arxiv.org/abs/1303.7221), these properties are enough to fix the Schwarzian derivative up to $3g-3$ unknown accessory parameters,
\begin{equation}
\{w,z\}=\Delta\left(\sum\_{i=1}^4\frac{1}{(z-z\_i)^2}+\frac{-z\_3+z\_1+z\_2+z\_4-2z}{(z-z\_1)(z-z\_2)(z-z\_4)}\right)+\sum\_{s=1}^{3(n-2)}p\_s\omega\_s
\end{equation}
where $\omega$s are a basis for the quadratic differentials on $M$ and $\Delta=1/2(1-1/n^2)$. I understand that we can add any linear combination of quadratic differntials on $M$ to the form of the Schwarzian because they transform uniformly, but the first term in the brackets is not clear to me. It should transform like a projective connection. Does anyone have a justification for including the first term multiplying $\Delta$?



Schwarzian derivative, accessory parameters, projective connections
First of all, you should take care about which coordinates you take. Namely, Faulkner uses the coordinate $x$ given by the defining equation $$y^n=\frac{(x-x\_1)(x-x\_2)}{(x-x\_3)(x-x\_4)}$$ as a branched projective structure (he denotes this by $z$ in his paper!). He then computes the Schwarzian derivative of $w$ - the Schottky uniformisation - with respect to $x$. As $x$ is a well-defined branched projective structure $${w,x}(dx)^2$$ is a well-defined meromorphic differential on $M$, having prescribed singularities at $x\_1,\dots,x\_4$. He derives in equation (4.10) in his paper that the quadratic residues must be $\Delta$. This is well-known since Mandelbaum.


The second term in the brackets is needed as ${w,x}(dx)^2$ has no pole over $x=\infty.$ It might look a bit unnatural as it has no pole at $z\_3$, but in fact $\frac{(dx)^2}{(x-x\_1)\dots(x-x\_4)}$ is a holomorphic quadratic differential.


