Computing the monodromy of a local system $\mathcal{L}$ I was trying to learn a little bit about local systems and their monodromy. In the notes I'm following they define the monodromy of a local system in the following way:

Let $X$ be a topological space together with a local system $\mathcal{L}$. Given  $\gamma : I \to X$ a continuous path in $X$, the inverse image of this sheaf $\gamma^{-1} \mathcal{L}$ is a constant sheaf on $I=[0,1]$. The monodromy of $\mathcal{L}$ along $\gamma$ is the composition of the isomorphisms:
$\mathcal{L}_{\gamma(0)}\cong \mathcal{L}([0,1])\cong \mathcal{L}_{\gamma(1)}$

I want to perfom some explicit computations, but I don't know how should I start. The easy case I'm trying to do is $\mathcal{L}=\Gamma$ where $\Gamma$ is the sheaf of sections of n-sheeted connected covering space of $S^1$. My goal is to compute the monodromy of a system of differential (complex) equations with meromorphic coefficients. I guess this should be done with transition functions of the sheaf, but I don't know how to start even in this trivial example.
 A: $\newcommand{F}{\mathscr F} \newcommand{G}{\mathscr G}$I will take $\F$ to be $f_* \G$, where $f : S^1 \to S^1, z \mapsto z^n$ and $\G$ is the constant sheaf of stalk $\mathbb C$, i.e $\G(U) = \{s : U \to \mathbb C, s \text{ is locally constant }\}$. For a sheaf $G$ on a topological space $X$ and a continuous map $f : X \to Y$, the sheaf $f_*G$ is a sheaf on $Y$ defined by $f_*G(U) = G(f^{-1}(U))$. 
We will prove that $\F$ is a local system. Let $U = S^1 \backslash N, V = S^1 \backslash S$, where $N = i$ and $S = -i$. We want isomorphism of sheaves $\F(W) \to \Bbb C^n(W)$ for $W = U,V$. 
For $U$, we are looking at $\{ s : f^{-1}(U) \to \mathbb C, s \text{ is locally constant } \}$. $f^{-1}(U)$ is the disjoint union of the $n$ intervals $I_k = \{e^{i\theta} : \theta \in (\frac{\pi + 2k\pi}{2n}; \frac{\pi +2(k+2)\pi}{2n})\}$. We can write these sections $s_1, \dots, s_n$, $s_i : I_k \to \Bbb C$. This gives the desired isomorphism $ \F|_U \cong \Bbb C^n_U$.
In fact, same applies for $V$ : we are looking at $\F(V) = \{ s : f^{-1}(V) \to \mathbb C, s \text{ is locally constant } \}$. Again we can identify this  with locally constant functions $ t_r : J_r \to \Bbb C $ where $J_r = \{\frac{3\pi + 2k \pi}{2n}; \frac{3 \pi + 2(k+2) \pi}{2n}\}$.  
Now consider $(\lambda_1, \dots, \lambda_n) \in \F_1$. As $\F|_U$ is a constant sheaf, we have natural isomorphism $(\F|_U)_1 \cong (\F|_U)_{-1}$. This corresponds simply to $s_i(\theta) \mapsto s_i(\theta + \frac{\pi}{n})$. Now, since $-1 \in V$ we can use $(\F|_V)_{-1} \cong (\F|_U)_1$, and again this isomorphism is $t_r(\theta) \mapsto t_r(\theta + \frac{\pi}{n})$. Finally, we obtain that the composition of these isomorphism is simply the map $(\lambda_1, \dots, \lambda_n) \to (\lambda_n, \lambda_1, \dots, \lambda_{n-1})$. But this composition is exactly the one you wrote in the definition of monodromy, so we conclude that the monodromy representation of $\pi_1(S^1) \cong \Bbb Z$ is $\rho(1)(\lambda_1, \dots, \lambda_n) = (\lambda_n, \dots, \lambda_{n-1})$.
Edit : let me compute the sheaf cohomology of $\F$. This is computed by the complex $\mathbb C^n \overset{d}{\to} \mathbb C^n$ where $d(\lambda_1, \dots, \lambda_n) = (\lambda_n - \lambda_1, \lambda_1 - \lambda_2, \dots, \lambda_{n-1} - \lambda_n) = \rho - \rm id$, which has rank $n-1$. We have $H^0(S^1, \F) =  \ker d = \{ (t,t, \dots, t) : t \in \Bbb C \} \cong \Bbb C$ and $H^1(S^1, \F) = \rm coker$ $ d  \cong \Bbb C$.
