$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$Let $D\subset\mathbb{C}$ be the complex unit disk. For a family of prestable curves over $D$ which is smooth over $D^* := D - 0$, but whose central fiber may have nodes, the Picard-Lefschetz formula describes the monodromy action of $\pi_1(D^*)$ on the homology of a smooth fiber in terms of vanishing cycles. I'm looking for an analogous formula in the case where the central fiber may have stacky structure.

Precisely, let $\cD := [D/\mu_n]$ be the stack quotient by the order $n$ cyclic group of rotations $\mu_n$. Thus $\cD$ is topologically also a unit disk, but whose center has a cyclic automorphism group of order $n$. The canonical map $D\rightarrow\cD$ is a universal cover. Let $f : \cX\rightarrow\cD$ be a family of prestable curves -- by this I mean that if $f_D : X := \cX_D\rightarrow D$ denotes the base change of $f$ to $D$, then $f_D$ is a family of prestable curves in the usual sense (i.e., fibers have at worst nodal singularities). In particular, there is an action of $\mu_n$ on $X := \cX_D$, $f_D$ is equivariant for this action, and $\cX \cong [X/\mu_n]$ (so the only stackiness for $\cX$ is along the central fiber). We further assume that $X$ is smooth over $\mathbb{C}$ (i.e. $X$ is a complex manifold), and the fibers of $X$ over $D^* - 0$ are smooth. The central fiber $X_0$ may have multiple nodes.

Let $\cD^* := \cD - 0$, $\cX^* := \cX - \cX_0$, and $b\in\cD^*$. Then $\cX^*\rightarrow\cD^*$ is a topologically trivial fibration over a punctured disk, so it is natural to ask about the monodromy action of $\pi_1(\cD^*,b)$ on $H_1(\cX_b)$. If $b'\in D^* := D - 0$ is a lift of $b$, then $H_1(X_{b'})\cong H_1(\cX_b)$ canonically, and the Picard-Lefschetz formula describes the monodromy action of $\pi_1(D^*,b')$ on $H_1(X_{b'}) = H_1(\cX_b)$ in terms of intersections with vanishing cycles. In particular, if $\gamma\in\pi_1(\cD^*,b)$ is a generator, then applying Picard-Lefschetz to $f_D : X\rightarrow D$ gives the action of $\gamma^n$.

Is there a "Picard-Lefschetz formula" for the monodromy action of $\gamma$? If possible I'd like the answer to be in terms of vanishing cycles and possibly information about the action of $\mu_n$ on $X_0$ and the tangent spaces in $X$ of points in $X_0$. I'm happy to assume that $\mu_n$ acts freely on the nodes of $X_0$.

(Edited to assume the nodes have trivial stabilizers)

$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$(This is a slight rephrasing and modification of the original question)

Let $D\subset\mathbb{C}$ be the complex unit disk. Let $X$ be a compact complex manifold, fibered over $D$ in relative dimension 1. Assume the fibers above $D^* := D - 0$ are smooth, and the fiber above 0 has at worst ordinary double points (nodes) as singularities. For any $\eta\in D^*$, associated to each node $x\in X_0$ is a vanishing cycle $v_x\in H_1(X_\eta)$, and the Picard-Lefschetz formula describes precisely the monodromy action of a generator $\gamma\in\pi_1(D^*,\eta)$ on $H_1(X_\eta)$ in terms of the vanishing cycles $v_x$. In particular, if $\varphi$ is the monodromy representation, then $\varphi(\gamma)$ is unipotent.

Let $\mu_n$ be a cyclic group of order $n$ acting by rotations on $D$. Suppose that this action on $D$ lifts to an action on $X$ (in particular it acts on $X_0$). Let $X^* := X - X_0$, then $X^*/\mu_n\rightarrow D^*/\mu_n$ is a family of smooth curves over a punctured disk, and if $\eta'$ denotes the image of $\eta$, then we may again ask about the monodromy action of a generator $\gamma'\in\pi_1(D^*/\mu_n,\eta')$ on $H_1(X_\eta)$. Abusing notation, again let $\varphi$ denote the monodromy representation. Let $\varphi(\gamma') = SU$ (say, viewed in $\text{GL}(H_1(X_\eta,\mathbb{C}))$ be the Jordan-Chevalley decomposition, so $S$ commutes with $U$, $S$ is semisimple, and $U$ is unipotent.

By comparison with the situation over $D$, we know that $$\varphi(\gamma'^n) = \varphi(\gamma)$$ which forces $S^n = 1$ and $U^n = \varphi(\gamma)$, so $U$ is the unique unipotent $n$th root of $\varphi(\gamma)$. 

If $\mu_n$ acts freely on the nodes, then $X/\mu_n$ is again a manifold, so applying Picard-Lefschetz directly to $X/\mu_n$, we find that $\varphi(\gamma')$ is unipotent, so $S = 1$.

My question concerns the case where $\mu_n$ fixes the nodes, and also preserves the branches at every node. In this case, how can we relate the semisimple part $S$ of $\varphi(\gamma')$ to information about how $\mu_n$ acts on $X_0$?

$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$Let $D\subset\mathbb{C}$ be the complex unit disk. For a family of prestable curves over $D$ which is smooth over $D^* := D - 0$, but whose central fiber may have nodes, the Picard-Lefschetz formula describes the monodromy action of $\pi_1(D^*)$ on the homology of a smooth fiber in terms of vanishing cycles. I'm looking for an analogous formula in the case where the central fiber may have stacky structure.

Precisely, let $\cD := [D/\mu_n]$ be the stack quotient by the order $n$ cyclic group of rotations $\mu_n$. Thus $\cD$ is topologically also a unit disk, but whose center has a cyclic automorphism group of order $n$. The canonical map $D\rightarrow\cD$ is a universal cover. Let $f : \cX\rightarrow\cD$ be a family of prestable curves -- by this I mean that if $f_D : X := \cX_D\rightarrow D$ denotes the base change of $f$ to $D$, then $f_D$ is a family of prestable curves in the usual sense (i.e., fibers have at worst nodal singularities). In particular, there is an action of $\mu_n$ on $X := \cX_D$, $f_D$ is equivariant for this action, and $\cX \cong [X/\mu_n]$ (so the only stackiness for $\cX$ is along the central fiber). We further assume that $X$ is smooth over $\mathbb{C}$ (i.e. $X$ is a complex manifold), and the fibers of $X$ over $D^* - 0$ are smooth. The central fiber $X_0$ may have multiple nodes.

Let $\cD^* := \cD - 0$, $\cX^* := \cX - \cX_0$, and $b\in\cD^*$. Then $\cX^*\rightarrow\cD^*$ is a topologically trivial fibration over a punctured disk, so it is natural to ask about the monodromy action of $\pi_1(\cD^*,b)$ on $H_1(\cX_b)$. If $b'\in D^* := D - 0$ is a lift of $b$, then $H_1(X_{b'})\cong H_1(\cX_b)$ canonically, and the Picard-Lefschetz formula describes the monodromy action of $\pi_1(D^*,b')$ on $H_1(X_{b'}) = H_1(\cX_b)$ in terms of intersections with vanishing cycles. In particular, if $\gamma\in\pi_1(\cD^*,b)$ is a generator, then applying Picard-Lefschetz to $f_D : X\rightarrow D$ gives the action of $\gamma^n$.

Is there a "Picard-Lefschetz formula" for the monodromy action of $\gamma$? If possible I'd like the answer to be in terms of vanishing cycles and possibly information about the action of $\mu_n$ on $X_0$ and the tangent spaces in $X$ of points in $X_0$. I'm happy to assume that the action on $X_0$ fixes every node pointwise.

$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$Let $D\subset\mathbb{C}$ be the complex unit disk. For a family of prestable curves over $D$ which is smooth over $D^* := D - 0$, but whose central fiber may have nodes, the Picard-Lefschetz formula describes the monodromy action of $\pi_1(D^*)$ on the homology of a smooth fiber in terms of vanishing cycles. I'm looking for an analogous formula in the case where the central fiber may have stacky structure.

Precisely, let $\cD := [D/\mu_n]$ be the stack quotient by the order $n$ cyclic group of rotations $\mu_n$. Thus $\cD$ is topologically also a unit disk, but whose center has a cyclic automorphism group of order $n$. The canonical map $D\rightarrow\cD$ is a universal cover. Let $f : \cX\rightarrow\cD$ be a family of prestable curves -- by this I mean that if $f_D : X := \cX_D\rightarrow D$ denotes the base change of $f$ to $D$, then $f_D$ is a family of prestable curves in the usual sense (i.e., fibers have at worst nodal singularities). In particular, there is an action of $\mu_n$ on $X := \cX_D$, $f_D$ is equivariant for this action, and $\cX \cong [X/\mu_n]$ (so the only stackiness for $\cX$ is along the central fiber). We further assume that $X$ is smooth over $\mathbb{C}$ (i.e. $X$ is a complex manifold), and the fibers of $X$ over $D^* - 0$ are smooth. The central fiber $X_0$ may have multiple nodes.

Let $\cD^* := \cD - 0$, $\cX^* := \cX - \cX_0$, and $b\in\cD^*$. Then $\cX^*\rightarrow\cD^*$ is a topologically trivial fibration over a punctured disk, so it is natural to ask about the monodromy action of $\pi_1(\cD^*,b)$ on $H_1(\cX_b)$. If $b'\in D^* := D - 0$ is a lift of $b$, then $H_1(X_{b'})\cong H_1(\cX_b)$ canonically, and the Picard-Lefschetz formula describes the monodromy action of $\pi_1(D^*,b')$ on $H_1(X_{b'}) = H_1(\cX_b)$ in terms of intersections with vanishing cycles. In particular, if $\gamma\in\pi_1(\cD^*,b)$ is a generator, then applying Picard-Lefschetz to $f_D : X\rightarrow D$ gives the action of $\gamma^n$.

Is there a "Picard-Lefschetz formula" for the monodromy action of $\gamma$? If possible I'd like the answer to be in terms of vanishing cycles and possibly information about the action of $\mu_n$ on $X_0$ and the tangent spaces in $X$ of points in $X_0$. I'm happy to assume that $\mu_n$ acts freely on the nodes of $X_0$.

(Edited to assume the nodes have trivial stabilizers)