\node (B0) at (2,1.5) {$\Db(\coh\X_1)$};
\node (B4) at (4.75,1.5) {$\scrM(\delta+\bnabla)$};
\node (B5) at (8,1.5) {$\Db(\coh\X_2)$};
\draw[->] (A0) -- node[above] {$\scriptstyle\Phi_N^{d-1}$}(A5);
\draw[->] (B0) -- node[above] {$\scriptstyle\res^{-1}$}(B4);
\draw[->] (B4) -- node[above] {$\scriptstyle\res$}(B5);
\draw[->] (B0) -- node[left] {$\scriptstyle\RHom_{}(\scrV_{\delta},-)$}(A0);
\draw[->] (B5) -- node[right] {$\scriptstyle\RHom_{}(\scrV_{\delta'},-)$}(A5);
\end{tikzpicture}
\]
\begin{proof}
Proposition \ref{composition mut}\,(2) gives an isomorphism
\[
\RHom(\scrV_{\delta'},-)\cong \Phi_N^{d-1}\circ\RHom(\scrV_{\delta},-)
\]
of functors from $\Db(\coh \X_2)$ to $\Db(\fmod\Lambda_{\delta'})$. Therefore the result follows from the following isomorphisms
\begin{align*}
\RHom(\scrV_{\delta'},-)\circ \res^{-1}\circ \res&\cong\Phi_N^{d-1}\circ\RHom(\scrV_{\delta},-)\circ \res^{-1}\circ \res\\
&\cong\Phi_N^{d-1}\circ\RHom(\scrT_{\delta},-)\circ \res\\
\end{align*}
where the second and  third isomorphisms follow from \eqref{res tilt}.
\end{proof}
\end{cor}

\begin{rem}
With the same notations as in Proposition \ref{composition mut}, put $\X' \defeq [X^{\rm ss}(-\ell)/G]$.
The composition
\begin{align*} \phi_{\delta, \delta'} \colon
\Db(\coh\X) \xrightarrow{\res^{-1}} \scrM(\delta+\bnabla) \xrightarrow{\res} \Db(\coh\X') \xrightarrow{\res^{-1}} \scrM(\delta'+\bnabla) \xrightarrow{\res} \Db(\coh\X)
\end{align*}
gives a natural autoequivalence of $\Db(\coh\X)$ associated to the wall-crossing.
Similarly, there is the autoequivalence of $\Db(\coh\X')$ that is given by
\begin{align*} \phi'_{\delta', \delta} \colon
\Db(\coh\X') \xrightarrow{\res^{-1}} \scrM(\delta' +\bnabla) \xrightarrow{\res} \Db(\coh\X) \xrightarrow{\res^{-1}} \scrM(\delta+\bnabla) \xrightarrow{\res} \Db(\coh\X').
\end{align*}

Note that Proposition \ref{composition mut} also implies that
$M_{\delta'} \cong \mu_{M_{\delta\cap\delta'}}^{-d_{F}^++1}(M_{\delta})$.
Then Proposition \ref{composition mut} and Corollary \ref{HomMMP4.2 toric}  yield the following commutative diagram

\[
\begin{tikzpicture}[xscale=1.3]
\node (A0) at (3,0) {$\Db(\fmod\Lambda_{\delta})$};
\node (A5) at (5,0) {$\Db(\fmod\Lambda_{\delta'})$};
\node (A10) at (7,0) {$\Db(\fmod\Lambda_{\delta})$};
\node (B4) at (3,1.5) {$\Db(\coh\X)$};
\node (B8) at (5,1.5) {$\Db(\coh\X)$};
\node (B10) at (7,1.5) {$\Db(\coh\X)$};
\node (C0) at (3, -1.5) {$\Db(\coh\X')$};
\node (C1) at (5, -1.5) {$\Db(\coh\X')$};
\node (C2) at (7, -1.5) {$\Db(\coh\X')$,};

\draw[double equal sign distance] (B4) -- (B8);
\draw[double equal sign distance] (C1) -- (C2);
\draw[->] (A0) -- node[above] {$\scriptstyle\Phi_N^{d-1}$}(A5);
\draw[->] (A5) -- node[below] {$\scriptstyle\Phi_N^{d^*-1}$}(A10);

\draw[->] (B8) -- node[above] {$\scriptstyle\phi_{\delta, \delta'}$}(B10);
\draw[->] (C0) -- node[below] {$\scriptstyle\phi'_{\delta', \delta}$}(C1);

\draw[->] (B4) -- node[left] {$\scriptstyle\RHom_{}(\scrV_{\delta},-)$}(A0);
\draw[->] (B8) -- node[right] {$\scriptstyle\RHom_{}(\scrV_{\delta'},-)$}(A5);
\draw[->] (B10) -- node[right] {$\scriptstyle\RHom_{}(\scrV_{\delta},-)$}(A10);

\draw[->] (C0) -- node[left] {$\scriptstyle\RHom_{}(\scrV_{\delta},-)$}(A0);
\draw[->] (C1) -- node[left] {$\scriptstyle\RHom_{}(\scrV_{\delta'},-)$}(A5);
\draw[->] (C2) -- node[right] {$\scriptstyle\RHom_{}(\scrV_{\delta},-)$}(A10);

\end{tikzpicture}
\]
where $d \defeq d_F^+$ and $d^* \defeq d_{F^*}^+$.
Thus, under the identifications of categories
\[ \Db(\coh\X) \xrightarrow{\RHom_{}(\scrV_{\delta},-)} \Db(\fmod\Lambda_{\delta}) \xleftarrow{\RHom_{}(\scrV_{\delta},-)} \Db(\coh\X'), \]
the natural autoequivalences $\phi_{\delta, \delta'}$ and $\phi'_{\delta', \delta}$ correspond to $\Phi_N^{d + d^* -1} \in \Auteq \Db(\fmod\Lambda_{\delta})$.

In some example including threefold flops (cf.~[40]) and certain higher dimensional flops (cf.~[8, 9]),
it is observed that a natural autoequivalence associated to the geometry of a contraction (eg. spherical twists along the structure sheaf of the flopping locus) can be recovered as the autoequivalence associated to the periodicity of mutations of an NCCR.
The diagram above should be regarded as a version of that phenomenon in the toric GIT setting.
\end{rem}

\subsection{Descending to toric hyperK\"{a}hler varieties}
This section studies a class of varieties so called toric hyperK\"{a}hler.
A quick reminder for the geometry of toric hyperK\"{a}hler varieties is given in the following, but for more details see \cite[Section 3]{svdbhypertoric}.

Let $X$ be a symplectic representation  of an algebraic torus $G = T$.
Then $X$ admits a canonical moment map $\upmu \colon X \to \mathfrak{t}^*$, where $\mathfrak{t} \defeq \mathrm{Lie}(T)$.
Throughout let us assume that the action of $T$ on $X$ is faithful, which gives that $\upmu$ is flat and surjective.

For $\xi \in \mathfrak{t}^*$ and generic $\ell \in \M_{\bR}$, the quotient stack $[\upmu^{-1}(\xi)^{\mathrm{ss}}(\ell)/T]$ is smooth Deligne-Mumford, and
the natural morphism
$\varphi^{\xi} \colon [\upmu^{-1}(\xi)^{\mathrm{ss}}(\ell)/T] \to \upmu^{-1}(\xi)/\!\!/T$
gives a stacky crepant resolution of the singular symplectic variety $\upmu^{-1}(\xi)/\!\!/T$.
Algebraic varieties of the form $\upmu^{-1}(\xi)^{\mathrm{ss}}(\ell)/\!\!/T$
are called \textit{toric hyperK\"{a}hler varieites} or \textit{hypertoric varieties} in the literature.

For the inclusion $i \colon [\upmu^{-1}(\xi)^{\mathrm{ss}}(\ell)/T] \hookrightarrow [X^{\mathrm{ss}}(\ell)/T]$, [33] proved the following.

\begin{thm}
The restriction $i^*\cV$ of any tilting bundle $\cV$ on $[X^{\mathrm{ss}}(\ell)/T]$ to $[\upmu^{-1}(\xi)^{\mathrm{ss}}(\ell)/T]$ remains to be tilting.
\end{thm}

This holds basically because, in the toric setting, the stack $[\upmu^{-1}(\xi)^{\mathrm{ss}}(\ell)/T]$ is cut out from $[X^{\mathrm{ss}}(\ell)/T]$ by a regular sequence.
Note that, in [33], the theorem is claimed only for tilting bundles constructed by \cite{{hl-s}}, but the proof does work for any tilting bundles.

With the same notation as in Proposition \ref{composition mut}, put
\[ \text{$\scrV_{\delta}^\xi \defeq i^* \scrV_{\delta}$, $\cN^{\xi} \defeq i^*\cN$, $\Lambda_{\delta}^{\xi} \defeq \End(\scrV_{\delta}^\xi)$, $M_{\delta}^\xi \defeq \varphi^\xi_* \scrV_{\delta}^\xi$, and $N^{\xi} \defeq \varphi^{\xi}_* \cN^{\xi}$.} \]

Since an exact sequence of locally free sheaves is locally split, its arbitrary restriction remains to be exact.
Thus all the arguments in the proof of Proposition \ref{composition mut} still work when one restricts everything to $[\upmu^{-1}(\xi)^{\mathrm{ss}}(\ell)/T]$.
This observation gives the following.

\begin{thm}
The same statement as in Theorem \ref{mutation MMA toric}\,(1), (2) and \ref{composition mut}\,(1), (3) holds for
and $N^{\xi}$.
\end{thm}

This theorem generalises results in \cite[Section 5.2.2]{hara17} about Mukai flops to arbitrary toric hyperK\"{a}hler varieties.
Note that Proposition \ref{composition mut}\,(2) does not hold in general for toric hyperK\"{a}hler varieties. If the singular locus of $\upmu^{-1}(\xi)/\!/ T$ has codimension at least $3$, the same statement as in Proposition \ref{composition mut}\,(2) holds.
For more details, see Remark \ref{counterexample for 2.21}.

\section{Applications to Calabi-Yau complete intersections}
\label{section: CICY}

Let $a_1,\hdots, a_n>0$ be  positive integers.
%such that any subset of $n-1$ integers have no common factor.
Write $S\defeq \bC[x_1,\hdots,x_n]$ for the $\bZ$-graded polynomial ring with $\deg(x_i)=a_i$. Let $d_1,\hdots,d_r$ be positive integers such that $\sum_{i=1}^rd_i=\sum_{i=1}^n a_i$.
Let  $f_1,\hdots,f_r\in S$ be homogeneous elements with $\deg(f_i)=d_i$ such that these define a Calabi-Yau complete intersection
\[
Y\defeq \bigcap_{i=1}^r\{f_i=0\}\subset {\bf P}(a_1,\hdots,a_n)
\]
in the weighted projective stack ${\bf P}(a_1,\hdots,a_n)\defeq [\bA^n\backslash \{{\bf 0}\}/\bG_m]$.
This section shows that a certain fundamental group action on $\Db(\coh Y)$ can be interpreted in terms of Iyama--Wemyss mutations via noncommutative matrix factorizations.
See Appendix \ref{mf appendix} for fundamental definitions and  properties of  matrix factorizations.

Set  $G_i\defeq \bG_m$ for $i=1,2$,  and   define a 2-dimensional torus $G\defeq G_1\times G_2$. Let
\[
X\defeq\sfk^{n+1}_{\bf x}\oplus \sfk^{r+1}_{\bf y}
\]
be the direct sum of  vector spaces with coordinate ${\bf x}=(x_0,\hdots,x_n)$ and ${\bf y}=( y_0,\hdots,y_r )$.
Consider a $G$-action on $X$  defined by
\begin{align*}
G\times X&\ni (s,t)\times \bigl((x_0,\hdots,x_n),(y_0,\hdots,y_r)\bigr)\\&\mapsto
\bigl((sx_0, s^{a_1}x_1,\hdots,s^{a_n}x_n), (s^{-1}ty_0, s^{-d_1}ty_1,\hdots, s^{-d_r}ty_r)\bigr)\in G\times X.
\end{align*}
The $G$-representation $X$ is not quasi-symmetric, but it is a generic quasi-symmetric representation with respect to the $G_1$-action.
Then the associated hyperplane arrangement $\scrH$ in $\M_{\bR}=\Hom(G_1,\bG_m)_{\bR}\cong \bR$ is
\[
\scrH=\begin{cases}
\bZ+(1/2)& \mbox{if } \sum_{i=1}^n a_i \mbox{ is even}.\\
\bZ & \mbox{if } \sum_{i=1}^n a_i \mbox{ is odd}.
\end{cases}
\]

Consider a $G_1$-invariant and $G_2$-semi-invariant regular function
\[
W\defeq x_0y_0+f_1y_1+\cdots+f_ry_r
\]
on $X$, which induces a regular function on $[X/G_1]$. Then $\bigl([X/G_1],\chi,W\bigr)^{G_2}$ is a gauged Landau-Ginzburg model, where $\chi\colon G_2\to \bG_m$ is the identity map.
Define a locally free sheaf $\cE$ on ${\bf P}(1,a_1,\hdots,a_n)$ by
\[\cE\defeq\cO(-1)\oplus\cO(-d_1)\oplus\cdots\oplus\cO(-d_r).\]
Since $Y$ is the zero locus of  a regular section $s\defeq(x_0,f_1,\hdots,f_r)\in \Gamma({\bf P}(1,a_1,\hdots,a_n),\cE^{*})^{G_2}$,
by Kn\"orrer periodicity [20, 29, 15], there is an equivalence
\begin{equation}\label{derived knorrer}
\Db(\coh Y)\cong \Dcoh_{G_2}({\rm V}(\cE),\chi,W),
\end{equation}
where ${\rm V}(\cE)$ is the vector bundle associated to $\cE$, and $G_2$-action on ${\rm V}(\cE)$ is induced from the $G_2$-equivariant structure of the $G_2$-equivariant locally free sheaf $\cE(\chi)$ on the stack ${\bf P}(1,a_1,\hdots,a_n)$ with the trivial $G_2$-action.
Denote by $\ell\colon G_1\to \bG_m$ the identity map, which is a generic element in  $\M_{\bR}$, and for this $\ell$, set $\X_{\pm}\defeq [X^{\rm ss}(\pm\ell)/G_1]$. Then there is an isomorphism ${\rm V}(\cE)\cong \X_{+}$.

For $\delta\in \M_{\bR}\backslash \scrH$,
a {\it magic window} $\scrM(\delta+\bnabla,W)\subset \Dcoh_{G_2}([X/G_1],\chi,W)$ is defined by
\[
\scrM(\delta+\bnabla,W)\defeq \left\l E_1\to E_0\to E_1(\chi)\relmiddle|  \Res^{G_2}(E_i)\in \scrM(\delta+\bnabla)\right\r,
\]
where $\Res^{G_2}\colon \coh_{G_2}[X/G_1]\to \coh[X/G_1]$ is the restriction functor. Then a version of Theorem \ref{hl-s} holds for $\scrM(\delta+\bnabla,W)$ by \cite[Section 6.3]{hl-s} or [2]:
\begin{equation}\label{mf window}
\res\defeq i^*\colon \scrM(\delta+\bnabla,W)\simto \Dcoh_{G_2}(\X_{\pm},\chi,W).
\end{equation}
Therefore, \eqref{derived knorrer} and \eqref{mf window} gives an equivalence
\begin{equation}\label{knorrer}
\Db(\coh Y)\cong \scrM(\delta+\bnabla,W).
\end{equation}
If $[\delta]\xrightarrow{\ell}[\delta']\in \Pi_1(\bC\backslash\scrH_{\bC})$ is a positive labeled arrow,
\[
\uprho\left([\delta]\xrightarrow{\ell}[\delta']\right)
\defeq \scrM(\delta+\bnabla,W)\xrightarrow{\res} \Dcoh_{G_2} (\X_+,\chi,W)\xrightarrow{\res^{-1}}\scrM(\delta'+\bnabla,W),\]
and if $[\delta]\xrightarrow{-\ell}[\delta']\in \Pi_1(\bC\backslash\scrH_{\bC})$ is a positive labeled arrow,
\[
\uprho\left([\delta]\xrightarrow{-\ell}[\delta']\right)
\defeq \scrM(\delta+\bnabla,W)\xrightarrow{\res} \Dcoh_{G_2} (\X_-,\chi,W)\xrightarrow{\res^{-1}}\scrM(\delta'+\bnabla,W).\]
Then these assignment  define a fundamental groupoid action of $\Pi_1(\bC\backslash\scrH_{\bC})$ on magic windows \cite[Corollary 6.11]{hl-s}, and in particular there is a fundamental group action
\begin{equation}\label{fund action 1}
\uprho\colon \pi_1(\bC\backslash\scrH_{\bC})\to \Auteq \scrM(\delta+\bnabla,W)\simto\Auteq \Db(\coh Y)
\end{equation}
on $\Db(\coh Y)$ via the equivalence \eqref{knorrer}.
By an identical manner as the action \eqref{ext fund action},
it is possible to extend the action \eqref{fund action 1} to the action of $\pi_1\bigl((\bC\backslash\scrH_{\bC})/\bZ\bigr)$ \cite[Corollary 6.11]{hl-s}.
Thus, since  $(\bC\backslash\scrH_{\bC})/\bZ\cong \bP^1\backslash \{0,1,\infty\}$,  there is a group action
\begin{equation}\label{fund action 2}
\widetilde{\uprho}\colon \pi_1\bigl(\bP^1\backslash \{0,1,\infty\}\bigr)\to \Auteq \Db(\coh Y).
\end{equation}

Since  $W\in R\defeq\sfk[X]^{G_1}$ and $W\in \Gamma(\Spec R,\cO(\chi))^{G_2}$,  it defines a noncommutative gauged LG model $(\Lambda_{\delta},\chi,W)^{G_2}$ for any $\delta\in \M_{\bR}\backslash \scrH$.
Since the abelian category $\Qcoh_{G_2}[X/G_1]\cong \Qcoh_G X$ has finite injective resolutions, the functor
\[\Hom_{[X/G_1]}(\scrT_{\delta},-)\colon \coh_{G_2}([X/G_1],\chi,W)\to\fmod_{G_2}(\Lambda_{\delta},\chi,W)\] defines the right derived functor
\begin{equation}\label{tilt mf}
\RHom_{[X/G]}(\scrT_{\delta},-)\colon \Dcoh_{G_2}([X/G_1],\chi,W)\to\Dmod_{G_2}(\Lambda_{\delta},\chi,W).
\end{equation}
Similarly, the functor $\Hom_{\X_+}(\scrV_{\delta},-)\colon \coh_{G_2}(\X_+,\chi,W)\to \fmod_{G_2}(\Lambda_{\delta},\chi,W)$ induces its right derived functor
\begin{equation}\label{tilt mf 2}
\RHom_{\X_+}(\scrV_{\delta},-)\colon \Dcoh_{G_2}(\X_+,\chi,W)\to\Dmod_{G_2}(\Lambda_{\delta},\chi,W).
\end{equation}
\begin{prop}
The restriction of \eqref{tilt mf}
\begin{equation}\label{window nccr}
\RHom_{[X/G]}(\scrT_{\delta},-)\colon \scrM(\delta+\bnabla,W)\simto\Dmod_{G_2}(\Lambda_{\delta},\chi,W)
\end{equation}
to a magic window and the functor \eqref{tilt mf 2}
\[
\RHom_{\X_+}(\scrV_{\delta},-)\colon \Dcoh_{G_2}(\X_+,\chi,W)\simto\Dmod_{G_2}(\Lambda_{\delta},\chi,W)
\]
are  equivalences.
\begin{proof}
Since $\res\colon \scrM(\delta+\bnabla,W)\simto\Dcoh_{G_2}(\X_+,\chi,W)$ is an equivalence  and there is  a natural isomorphism $\RHom_{[X/G]}(\scrT_{\delta},-)\cong\RHom_{\X_+}(\scrV_{\delta},-)\circ\res$ of functors, it is enough to prove that the functor  \eqref{tilt mf 2} is an equivalence. Let $T_{\delta}\defeq \bigoplus_{\chi\in\scrC_{\delta}}\cO(\chi)\in \coh_GX$, and set $V_{\delta}\defeq i^*(T_{\delta})\in \coh_{G}X^{\rm ss}(\ell)$.
Then $V_{\delta}$ corresponds to $\scrV_{\delta}$ via a natural equivalence $\coh_G X^{\rm ss}(\ell)\cong \coh_{G_2}\X_+$, and isomorphisms $\Lambda_{\delta}\cong\End_{\X_+}(\scrV_{\delta})\cong\End_{X^{\rm ss}(\ell)}(V_{\delta})^{G_1}$ hold.
Therefore there exists the following commutative diagram:
\[
\begin{tikzpicture}[xscale=1.3]
\node (A0) at (2,0) {$\Dcoh_G(X^{\rm ss}(\ell),1\times \chi,W)$};
\node (A5) at (8,0) {$\Dmod_{G_2}(\Lambda_{\delta},\chi,W)$};
\node (B4) at (2,2) {$\Dcoh_{G_2}(\X_+,\chi,W)$};
\draw[->] (A0) -- node[above] {$\scriptstyle\RHom(V_{\delta},-)^{G_1}$}(A5);
\draw[->] (B4) -- node[left] {$\cong$}(A0);
\draw[->] (B4) -- node[above] {\scriptsize \eqref{tilt mf 2}}(A5);
\end{tikzpicture}
\]
Since the bottom functor is an equivalence by Theorem \ref{H3 thm}, so is \eqref{tilt mf 2}.
\end{proof}
\end{prop}

Let $(\delta,\delta')$ be an adjacent pair in $\bR\,\backslash \scrH$, and let $N\defeq M_{\delta\cap\delta'}$. Then for any $i\geq 0$ the algebra $\Lambda_{i}\defeq\End(\mu_N^{-i}(M_{\delta}))$ is a $G_2$-equivariant finite $R$-algebra. By Proposition \ref{composition mut}\,(2), the $G_2$-equivariant $\Lambda_i$-module $T_i\defeq\Hom_R(\mu_N^{-i}(M_{\delta}),\mu_N^{-i-1}(M_{\delta}))$ is a tilting module, and it defines the mutation functor
\Phi_N=\RHom(T_i,-)\colon\Db(\fmod\Lambda_i)\simto \Db(\fmod\Lambda_{i+1}).
Furthermore, since $T_i$ is a $G_2$-tilting module in the sense of Definition \ref{eq tilt},  the mutation functor
\[
\Phi_N\defeq\RHom(T_i,-)\colon \Dmod_{G_2}(\Lambda_i,\chi,W)\to\Dmod_{G_2}(\Lambda_{i+1},\chi,W)
\]
for derived factorization categories is an equivalence by Theorem \ref{H3 thm}.

The following shows that the equivalences of magic windows generating the group action \eqref{fund action 1} correspond to mutation functors between noncommutative matrix factorizations.
\begin{thm}\label{window shift}
Let $\delta,\delta'\in \bR \,\backslash\scrH$ be an adjacent pair with $\delta<\delta'$, and write $N\defeq M_{\delta\cap\delta'}$. Then the following diagram of equivalences commutes.
\[
\begin{tikzpicture}[xscale=1.3]
\node (A0) at (1,0) {$\Dmod_{G_2}(\Lambda_{\delta},\chi,W)$};
\node (A5) at (5,0) {$\Dmod_{G_2}(\Lambda_{\delta'},\chi,W)$};
\node (A9) at (9,0) {$\Dmod_{G_2}(\Lambda_{\delta},\chi,W)$};
\node (B0) at (1,1.5) {$\scrM(\delta+\bnabla,W)$};
\node (B5) at (5,1.5) {$\scrM(\delta'+\bnabla,W)$};
\node (B9) at (9,1.5) {$\scrM(\delta+\bnabla,W)$};
\draw[->] (A0) -- node[above] {$\scriptstyle\Phi_N^{n}$}(A5);
\draw[->] (A5) -- node[above] {$\scriptstyle\Phi_N^{r}$}(A9);
\draw[->] (B0) -- node[above] {$\scriptstyle\uprho\bigl([\delta]\xrightarrow{\ell}[\delta']\bigr)$}(B5);
\draw[->] (B5) -- node[above] {$\scriptstyle\uprho\bigl([\delta']\xrightarrow{-\ell}[\delta]\bigr)$}(B9);
\draw[->] (B0) -- node[left] {$\scriptstyle\RHom_{}(\scrT_{\delta},-)$}(A0);
\draw[->] (B5) -- node[right] {$\scriptstyle\RHom_{}(\scrT_{\delta'},-)$}(A5);
\draw[->] (B9) -- node[right] {$\scriptstyle\RHom_{}(\scrT_{\delta},-)$}(A9);
\end{tikzpicture}
\]
\begin{proof}
We only show that the left square commutes, since the  commutativity of the right one follows from a similar argument. Consider the following diagram
\[
\begin{tikzpicture}[xscale=1.3]
\node (A0) at (1,0) {$\Dmod_{G_2}(\Lambda_{\delta},\chi,W)$};
\node (A5) at (7,0) {$\Dmod_{G_2}(\Lambda_{\delta'},\chi,W)$.};
\node (B0) at (1,3) {$\scrM(\delta+\bnabla,W)$};
\node (B5) at (7,3) {$\scrM(\delta'+\bnabla,W)$};
\node (C) at (4,1.5)
\draw[->] (A0) -- node[above] {$\scriptstyle\Phi_N^{n}$}(A5);
\draw[->] (B0) -- node[above] {$\scriptstyle\uprho\bigl([\delta]\xrightarrow{\ell}[\delta']\bigr)$}(B5);
\draw[->] (B0) -- node[left] {$\scriptstyle\RHom_{}(\scrT_{\delta},-)$}(A0);
\draw[->] (B5) -- node[right] {$\scriptstyle\RHom_{}(\scrT_{\delta'},-)$}(A5);
\draw[->] (B0) -- node[above] {$\res$}(C);
\draw[->] (B5) -- node[above] {$\res$}(C);
\draw[->] (C) -- node[left] {$\scriptstyle\RHom(\scrV_{\delta},-)$\\}(A0);
\draw[->] (C) -- node[right] {$\scriptstyle\RHom(\scrV_{\delta'},-)$\\}(A5);
\end{tikzpicture}
\]
Then the top triangle commutes by the definition of the map $\uprho$, and the left and right triangles are commutative since $\scrV_{\delta}=\res(\scrT_{\delta})$. Thus it is enough to show that the bottom triangle commutes. But this follows from Proposition \ref{composition mut}\,(2) and \cite[Lemma 4.10]{bdfik}.
\end{proof}
\end{thm}

Since $Y$ is Calabi--Yau,  a line bundle $\cL$ on $Y$ is a spherical object in $\Db(\coh Y)$, and it defines the {\it spherical twist} autoequivalence
\[
\ST_{\cL}\colon \Db(\coh Y)\simto \Db(\coh Y).
\]
The following shows that the spherical twist associated to a line bundle of the form $\cO_Y(m)$ corresponds to the compositions of Iyama--Wemyss mutations via noncommutative matrix factorizations.

\begin{cor}\label{cor:spherical twist}
For $m\in\bZ$, put $\delta\defeq m+(\alpha/2)+1\in\bR\backslash\scrH$, and  set $N\defeq M_{\delta-1\cap\delta}$. Then the  diagram
\[
\begin{tikzpicture}[xscale=1.3]
\node (A0) at (1,0) {$\Dmod_{G_2}(\Lambda_{\delta},\chi,W)$};
\node (A5) at (5,0) {$\Dmod_{G_2}(\Lambda_{\delta},\chi,W)$};
\node (B0) at (1,1.5) {$\Db(\coh Y)$};
\node (B5) at (5,1.5) {$\Db(\coh Y)$};
\draw[->] (A0) -- node[above] {$\scriptstyle\Phi_N^{n+r}$}(A5);
\draw[->] (B0) -- node[above] {$\scriptstyle\ST_{\cO(m)}$}(B5);
\draw[->] (B0) -- node[left] {$\cong$}(A0);
\draw[->] (B5) -- node[right] {$\cong$}(A5);
\end{tikzpicture}
\]
commutes, where the vertical equivalences are the compositions of \eqref{knorrer} and \eqref{window nccr}.
\end{cor}

\vspace{4mm}
This corollary can be deduced from Theorem \ref{window shift} and a result in [12]. Consider one parameter subgroups $\lambda_{\pm}\colon  \bG_m\to G$ defined by $\lambda_{\pm}(t)\defeq (t^{\pm1},1)$, and define
\[
S_{\pm}\defeq\left\{x\in X\relmiddle| \lim_{t\to 0}\lambda_{\pm}(t)x=0\right\}.
\]
Then $X^{\rm ss}_{\pm}\defeq X\backslash S_{\pm}= X^{\rm ss}(\mp\ell)$ and $E\defeq X^{\rm ss}_-\cap S_+= (\sfk^{n+1}\backslash\{0\})\times \{0\}$ holds, and the fixed locus $Z$ of $\lambda_{-}$-action on $X$ equals to the origin $\{0\}\cong\Spec \sfk$. In particular, there are natural equivalences
\begin{align}
\coh_GX^{\rm ss}_-&\cong \coh_{G_2}{\rm V}(\cE)\label{X=V}\\
\coh_GE&\cong \coh_{G_2}\bfP(1,a_1,\hdots,a_n)\label{E=P},
\end{align}
and, in what follows, we tacitly use these identifications.
Write $\chi_{a,b}\colon G\to \bG_{m}$ for the character defined by $\chi_{a,b}(s,t)\defeq s^at^b$. Fix an integer $w\in \bZ$. We define
\[
\Dcoh_G(\Spec \sfk,\chi_{0,1},0)_w
\]
to be the thick subcategory of $\Dcoh_G(\Spec \sfk,\chi_{0,1},0)$ generated by factorizations such that the $(\lambda_{-})$-weights of the components is $w$, and consider the subcategory
\[
\cG_w^-\defeq \left\{(F_1\to F_0\to F_1(\chi_{0,1}))\relmiddle| \mbox{ $(\lambda_-)$-weight of  $F_i|_{Z}$ lies in $[w,w+\eta^-)$ } \right\}
\]
of $\Dcoh_G(X,\chi_{0,1},W)$, where $\eta^-$ is the $(\lambda_-)$-weight of $\det(N_{S_-}X)$.
Then $\Dcoh_G(\Spec \sfk,\chi_{0,1},0)_w$ is generated by an exceptional object $\bigl(0\to \cO(\chi_{-w,0})\to0\bigr)$,  $\eta^-=\alpha+1$ holds, and
\begin{align*}
\cG_w^-&=\left\{(F_1\to F_0\to F_1(\chi_{0,1}))\relmiddle| \mbox{ $G_1$-weights of  $F_i|_{Z}$ lies in $(-w-\eta^-,-w]$ } \right\}\\
\end{align*}
where $\alpha\defeq\sum_{i=1}^na_i$.
Consider an equivalence
\[
\psi_w\colon\Dcoh_G(X^{\rm ss}_-,\chi_{0,1},W)\simto\Dcoh_G(X^{\rm ss}_+,\chi_{0,1},W)
\]
defined by the composition
\Dcoh_G(X^{\rm ss}_-,\chi_{0,1},W)\xrightarrow{\res^{-1}}\cG_{w}^-\xrightarrow{\res}\Dcoh_G(X^{\rm ss}_+,\chi_{0,1},W).
Then the {\it window shift autoequivalence} is defined to be the composition
$\Phi_w\defeq \psi_{w+1}^{-1}\circ\psi_w\in \Auteq\Dcoh_G(X^{\rm ss}_-,\chi_{0,1},W)$.
\[\cK_w\defeq j_*\pi^*\bigl(0\to \cO(\chi_{-w,0})\to0\bigr)\in \Dcoh_G(X^{\rm ss}_-,\chi_{0,1},W),\]
where $\pi\colon E\to \Spec \sfk$ is a natural projection, and $j\colon E\to X^{\rm ss}_-$ is the zero section.

\begin{lem}[{\cite[Section 3.1]{hl-sh}\footnote[1]{Although [12] only discusses complexes, there are similar functors and semi-orthogonal decompositions for matrix factorizations by [2], and so a similar argument as in [12] works in our setting.}}]\label{spherical lem 1}
For each $F\in \Dcoh_G(X^{\rm ss}_-,\chi_{0,1},W)$, there is a functorial exact triangle
\begin{equation}
\RHom(\cK_w,F)\otimes \cK_w\to F\to \Phi_w(F)\to\RHom(\cE_w,F[1])\otimes \cE_w.
\end{equation}
\end{lem}

\vspace{3mm}
Write $\bfP\defeq\bfP(1,a_1,\hdots,a_n)$ for simplicity, and  denote by ${\rm V}(\cE)|_Y$  the pull-back of the vector bundle $q\colon{\rm V}(\cE)\to \bfP$ by the closed immersion $i_Y\colon Y\hookto \bfP$. Consider the following diagram
\[
\begin{tikzpicture}[xscale=1.3]
\node (A0) at (1,1.5) {${\rm V}(\cE)|_Y$};
\node (A5) at (4,1.5) {${\rm V}(\cE)$};
\node (B0) at (1,0) {$Y$};
\node (B5) at (4,0) {$\bfP$,};
\draw[->] (A0) -- node[above] {$\scriptstyle i$}(A5);
\draw[->] (B0) -- node[above] {$\scriptstyle i_Y$}(B5);
\draw[->] (B0) edge[bend left] node [left] {$\scriptstyle j_Y$}(A0);
\draw[->] (A0) edge[bend left] node [right] {$\scriptstyle p$}(B0);
\draw[->] (B5) edge[bend left] node [left] {$\scriptstyle j$}(A5);
\draw[->] (A5) edge[bend left] node [right] {$\scriptstyle q$}(B5);
\end{tikzpicture}
\]
where $p$ is the natural projection, $i$ is the natural closed immersion, and $j$ and $j_Y$ are the zero sections. Note that $j\colon \bfP=[E/G_1]\hookto {\rm V}(\cE)=[X_-^{\rm ss}/G_1]$ is induced from $j\colon E\hookto X^{\rm ss}_-$, and so we use the same notation.
By Kn\"orrer periodicity, the functor
\begin{equation}\label{knorrer 2}
i_*p^*\colon \Db(\coh Y)\simto \Dcoh_G(X^{\rm ss}_-,\chi_{0,1},W)
\end{equation}
is an equivalence.
\begin{lem}
There is an  isomorphism
\[\bigl(0\to j_*\cO_E(\chi_{-w,0})\to0\bigr)\cong  i_*p^*\cO_Y(-w-\alpha-1)[r+1]\]
in $\Dcoh_G(X^{\rm ss}_-,\chi_{0,1},W)$.
\begin{proof}
Recall that $s=(x_0,f_1,\hdots,f_r)\in \Gamma({\bf P},\cE^{*})^{G_2}$, and set  $t\defeq (y_0,\hdots,y_r)\in\Gamma(\bfP,\cE)^{G_2}$. Consider the following $G$-equivariant locally free sheaf on $E$:
\[
\widetilde{\cE}\defeq \cO(\chi_{-1,0})\oplus\cO(\chi_{-d_1,0})\oplus\cdots\oplus \cO(\chi_{-d_r,0}).
\]
Then $\widetilde{\cE}$ corresponds to $\cE$ via \eqref{E=P}, and pulling-back the sections $s$ and $t$ by the projection $q\colon X^{\rm ss}_-\to E$ defines $G$-invariant sections $q^*(s)\in \Gamma(X^{\rm ss}_-,(q^*\widetilde{\cE})^*)^G$ and $q^*(t)\in \Gamma(X^{\rm ss}_-,q^*\widetilde{\cE}(\chi_{0,1}))^G$.
Since $E\cong Z_{q^*(t)^*}$, there are   isomorphisms
\begin{align}
\bigl(0\to j_*\cO_E(\chi_{-w,0})\to0\bigr)&\cong\bigl(0\to \cO_{Z_{q^*(s)}}(\chi_{-w,0})\to0\bigr)\otimes \det(q^*\widetilde{\cE})[r+1]\\
&\cong \bigl(0\to \cO_{Z_{q^*(s)}}(\chi_{-w-\alpha-1,0})\to0\bigr)[r+1]\\
&\cong \bigl(0\to i_*p^*\cO_{Y}(-w-\alpha-1)\to0\bigr)[r+1]
\end{align}
where the first isomorphism follows from Lemma \ref{koszul lem}. This finishes the proof.
\end{proof}
\end{lem}

The following is a generalization of \cite[Theorem 8.5]{ko}, which we prove by a similar argument as in loc. cit.

\begin{lem}\label{lem for cor}
The following diagram commutes.
\[
\begin{tikzpicture}[xscale=1.3]
\node (A0) at (1,0) {$\Dcoh_G(X^{\rm ss}_-,\chi_{0,1},W)$};
\node (A5) at (5,0) {$\Dcoh_G(X^{\rm ss}_-,\chi_{0,1},W)$};
\node (B0) at (1,1.5) {$\Db(\coh Y)$};
\node (B5) at (5,1.5) {$\Db(\coh Y)$};
\draw[->] (A0) -- node[above] {$\scriptstyle\Phi_{w}$}(A5);
\draw[->] (B0) -- node[above] {$\scriptstyle\ST_{\cO_Y(-w-\alpha-1)}$}(B5);
\draw[->] (B0) -- node[left] {$\scriptstyle i_*p^*$}(A0);
\draw[->] (B5) -- node[right] {$\scriptstyle i_*p^*$}(A5);
\end{tikzpicture}
\]
\begin{proof}
Let $F\in \Db(\coh Y)$. By Lemma \ref{spherical lem 1}, there is a functorial exact triangle
\begin{equation}
\RHom(\cK_w,i_*p^*F)\otimes \cK_w\to i_*p^*F\to \Phi_w(i_*p^*F)\to\RHom(\cE_w,i_*p^*F[1])\otimes \cE_w.
\end{equation}
Thus it is enough to show that there is a natural isomorphism
\begin{equation}\label{cor isom}
\RHom(\cK_w,i_*p^*F)\otimes \cK_w\cong \RHom(\cO_Y(-w-\alpha-1), F)\otimes i_*p^*\cO_Y(-w-\alpha-1).
\end{equation}
By Lemma \ref{lem for cor}, there is an isomorphism
\[
\cK_w=\bigl(0\to j_*\cO_E(\chi_{-w,0})0\bigr)\cong i_*p^*\cO_Y(-w-\alpha-1)[r+1],
\]
and this implies the required isomorphism \eqref{cor isom} since $i_*p^*$ is an equivalence.
\end{proof}
\end{lem}

\begin{proof}[Proof of Corollary \ref{cor:spherical twist}] For simplicity, write
\[
\cD_{\pm}\defeq\Dcoh_G(X_{\pm}^{\rm ss},\chi_{0,1},W).
\]
By Lemma \ref{lem for cor}, the spherical twist $\ST_{\cO_Y(m)}$ corresponds to
\Phi_{(-m-\alpha-1)}\colon \cD_-\simto\cD_-
via \eqref{knorrer 2}. By the definition of $\Phi_{(-m-\alpha-1)}$, the following diagram commutes:
\begin{equation}\label{diagram for cor}
\begin{tikzpicture}[xscale=1.0]
\node (A1) at (1,0) {$\cG^-_{-m-\alpha-1}$};
\node (A3) at (3,0) {$\cD_+$};
\node (A5) at (5,0) {$\cG^-_{-m-\alpha}$};
\node (A7) at (7,0) {$\cD_-$};
\node (A9) at (9,0) {$\cG^-_{-m-\alpha-1}$};
\node (B1) at (1,1.5) {$\cD_-$};
\node (B9) at (9,1.5) {$\cD_-$};
\draw[->] (A1) -- node[above] {$\scriptstyle\res$}(A3);
\draw[->] (A3) -- node[above] {$\scriptstyle\res^{-1}$}(A5);
\draw[->] (A5) -- node[above] {$\scriptstyle\res$}(A7);
\draw[->] (A7) -- node[above] {$\scriptstyle\res^{-1}$}(A9);
\draw[->] (B1) -- node[left] {$\scriptstyle\res^{-1}$}(A1);
\draw[->] (B9) -- node[right] {$\scriptstyle\res^{-1}$}(A9);
\draw[->] (B1) -- node[above] {$\scriptstyle\Phi_{(-m-\alpha-1)}$}(B9);
\end{tikzpicture}
\end{equation}
Since $\cG^-_{-m-\alpha-1}=\scrM(\delta+\bnabla)$, $\cG^-_{-m-\alpha}=\scrM(\delta-1+\bnabla)$, and $\cD_{\pm}=\Dcoh_{G_2}(\X_{\mp},\chi,W)$, the composition of the bottom equivalences in the above diagram is nothing but the composition
\[
\begin{tikzpicture}[xscale=1.3]
\node (B0) at (1,0) {$\scrM(\delta+\bnabla,W)$};
\node (B5) at (5,0) {$\scrM(\delta-1+\bnabla,W)$};
\node (B9) at (9,0) {$\scrM(\delta+\bnabla,W)$.};
\draw[->] (B0) -- node[above] {$\scriptstyle\uprho\bigl([\delta]\xrightarrow{-\ell}[\delta-1]\bigr)$}(B5);
\draw[->] (B5) -- node[above] {$\scriptstyle\uprho\bigl([\delta-1]\xrightarrow{\ell}[\delta]\bigr)$}(B9);
\end{tikzpicture}
\]
Therefore, the assertion follows from Theorem \ref{window shift}.
\end{proof}

\appendix

\section{Matrix factorizations}\label{mf appendix}
This appendix recalls definitions and fundamental properties of derived factorization categories.
See  [28, 1, 3, 14, 16] for more details.

\subsection{Definitions} Let $\X$ be an algebraic stack, and let $G$ be an algebraic group acting on $\X$.
A {\it gauged Landau-Ginzburg (=LG) model} is  data
\[
\]
such that $\cA$ is a $G$-equivariant $\X$-algebra (in the sense of \cite[Section 3]{H3}), $\chi\colon G\to \bG_m$ is a character of $G$ and $W\in \Gamma(\X,\cO(\chi))^G$.
If $\cA$ is noncommutative, the data is called a {\it noncommutative} gauged LG model.
If $\cA=\cO_{\X}$, write $(\X,\chi,W)^G$ for the gauged LG model, and if $\X=\Spec R$ and $A\defeq\Gamma(\X,\cA)$, let $(A,\chi,W)^G$ denote the gauged LG model.

\begin{dfn}
Let $(\cA,\chi,W)^G$ be a gauged LG model. A {\it factorization} of $(\cA,\chi,W)^G$ is a sequence
\[
\]
where each $E_i$ is a $G$-equivariant coherent $\cA$-module, $E_1(\chi)\defeq E_1\otimes\cO(\chi)$ and each $\varphi_i$ is a $G$-homomorphism such that  $\varphi_0\circ\varphi_1=\id_{E_1}\otimes W$ and $\varphi_1(\chi)\circ\varphi_0=\id_{E_0}\otimes W$.
\end{dfn}

Let $E$ and $F$ be factorizations of $(\cA,\chi,W)^G$. A {\it homomorphism} from $E$ to $F$ is a pair $\alpha=(\alpha_0,\alpha_1)$ of $G$-homomorphism $\alpha_i\colon E_i\to F_i$ such that the following diagram is commutative:
\[\xymatrix{
Two homomorphisms $\alpha,\beta\colon E\to F$  is said to be {\it homotopy equivalent}, denoted by $\alpha\sim\beta$, if  there exist two homomorphisms
$h_0:E_0\rightarrow F_1\hspace{3mm}{\rm and}\hspace{3mm}h_1:E_1(\chi)\rightarrow F_0$
of $G$-equivariant $\cA$-modules such that $\alpha_0-\beta_0=\varphi_1^Fh_0+h_1\varphi_0^E$ and $\alpha_1(\chi)-\beta_1(\chi)=\varphi_0^Fh_1+(h_0(\chi))(\varphi_1^E(\chi))$.

\begin{dfn}
\[
\coh_G(\cA,\chi,W)
\]
denote the category of factorizations of $(\cA,\chi,W)^G$, and its {\it homotopy category}
\[\Kcoh_G(\cA,\chi,W)\]
is defined to be the category such that its objects are the same as $\coh_G(\cA,\chi,W)$ and homomorphisms are defined by
\Hom(E,F)\defeq\Hom_{\coh_G(\cA,\chi,W)}(E,F)/\sim
\end{dfn}

The homotopy category $\Kcoh_G(\cA,\chi,W)$ has a natural  structure of a triangulated category.
For a bounded complex $E^{\bullet}$ in $\coh_G(\cA,\chi,W)$,
its {\it totalization} $\Tot(E^{\bullet})\in\coh_G(\cA,\chi,W)$ can be defined as an analogy of totalizations of double complexes.
We say that a factorization $E$ is {\it acyclic} if there is a bounded acyclic complex $A^{\bullet}$ in $\coh_G(\cA,\chi,W)$  such that $E\cong \Tot(A^{\bullet})$ in $\Kcoh_G(\cA,\chi,W)$.
\[
\Acoh_G(\cA,\chi,W)\subseteq \Kcoh_G(\cA,\chi,W)
\]
for the smallest thick subcategory of $\Kcoh_G(\cA,\chi,W)$ containing all acyclic factorizations.

\begin{dfn}
The {\it derived factorization category} of $(\cA,\chi,W)^G$ is defined by
\[
\Dcoh_G(\cA,\chi,W)\defeq \Kcoh_G(\cA,\chi,W)/\Acoh_G(\cA,\chi,W).
\]
If $\cA=\cO_{\X}$, write $\Dcoh_G(\X,\chi,W)$, and if $\X=\Spec R$ and $A=\Gamma(\X,\cA)$, write $
\Dmod_G(A,\chi,W)$ for the derived factorization categories.
Similar alternative notations are used for $\coh_G(\cA,\chi,W)$ and $\Kcoh_G(\cA,\chi,W)$.
\end{dfn}

\subsection{Equivariant tilting theory} Each normal subgroup $H\subseteq G$ associates a natural functor
\[
\Res^G_H\colon \coh_G\cA\to \coh_H\cA,
\]
which is called  the {\it restriction functor}.
The rest of this section recalls equivariant tilting theory for factorizations developed in [16] (see also [27]).

\begin{dfn}\label{eq tilt} A $G$-equivariant coherent $\cA$-module $\cT\in\coh_G\cA$ is called {\it $(G,H)$-tilting module} if $\Res^G_H(\cT)$ is a tilting object in $\D(\Qcoh_H\cA)$. We say that $\cT$ is {\it $G$-tilting module} if it is $(G,\{1\})$-tilting module.
\end{dfn}

Let $H\subseteq G$ be a normal subgroup, and let $Y=\Spec R$ be an affine variety with a $G$-action such that the $H$-action on $Y$ is trivial.
Then there is an induced $G/H$-action on $Y$.
Let $X$ be a variety with a $G$-action, and let $f\colon X\to Y$ be a $G$-equivariant morphism such that $f_*(\cO_X)\cong R$ and the induced morphism $[f/H]\colon [X/H]\to \Spec R$ is proper.
Let $\cA$ be a $G$-equivariant coherent $X$-algebra, and let  $\cT\in \coh_GX$.
Then its endmorphism algebra $\Lambda_{\cT}\defeq \End(\cT)$ is a $G$-equivariant coherent $R$-algebra,
and its $H$-invairant part $\Lambda_{\cT}^H$ is a $G/H$-equivariant coherent $R$-algebra.
Thus there is a left exact functor
\begin{equation}\label{tilt functor}
\Hom(\cT,-)^H\colon \coh_G\cA\to \fmod_{G/H}\Lambda_{\cT}^H.
\end{equation}
Let $\chi\colon G/H\to \bG_m$ be a character of $G/H$, and set $\widehat{\chi}\defeq \chi\circ p\colon G\to\bG_m$, where $p\colon G\to G/H$ is the natural projection.  Let $W\in \Gamma(Y,\cO(\chi))^{G/H}=\Gamma(Y,\cO(\widehat{\chi}))^{G}$. Then these define  gauged LG models
\[
(\cA,\widehat{\chi},W)^G \hspace{3mm}\mbox{ and }\hspace{3mm} (\Lambda_{\cT}^H,\chi,W)^{G/H},
\]
where $W$ in the left LG model denotes $f^*W$ by abuse of notation, and  the functor \eqref{tilt functor}  defines the right derived functor
\begin{equation}\label{mf tilt}
\Hom(\cT,-)^H\colon \coh_G(\cA,\widehat{\chi},W)\to \fmod_{G/H}(\Lambda_{\cT}^H,\chi,W).
\end{equation}
If the abelian category $\Qcoh_G\cA$ has finite injective resolutions, the functor \eqref{mf tilt} induces the right derived fucntor
\[
\RHom(\cT,-)^H\colon \Dcoh_G(\cA,\widehat{\chi},W)\simto \Dmod_{G/H}(\Lambda_{\cT},\chi,W).
\]
The following shows an equivariant and factorization version of a tilting equivalence.

\begin{thm}[{\cite[Theorem 4.7]{H3}}]\label{H3 thm} Let $\cT$ be a $(G,H)$-tilting module. Assume that $\Qcoh_G\cA$ has finite injective resolutions,  $G/H$ is reductive and $\Lambda_{\cT}^H$ is of finite global dimension. Then the right derived functor
\[
\RHom(\cT,-)^H\colon \Dcoh_G(\cA,\widehat{\chi},W)\simto \Dmod_{G/H}(\Lambda_{\cT}^H,\chi,W).
\]
of \eqref{mf tilt} is an equivalence.
\end{thm}

\subsection{Koszul factorizations} Let $(X,\chi,W)^G$ be a gauged LG model such that $X$ is a smooth variety. Let $\cE$ be a $G$-equivariant locally free sheaf on $X$ of rank $r$, and let $s\in \Gamma(X,\cE^{*})^G$ and $t\in \Gamma(X,\cE(\chi))^G$ be $G$-invariant sections such that $t\circ s=W\cdot \id_{\cE}$ and $s(\chi)\circ t=W$. The {\it Koszul factorization} $K(s,t)\in \coh_G(X,\chi,W)$ associated to $s$ and $t$ is defined by
\[
K(s,t)\defeq\left(\bigoplus_{n\in \bZ}(\bigwedge^{2n+1}\cE)(\chi^n)\xrightarrow{\varphi_1}\bigoplus_{n\in \bZ}(\bigwedge^{2n}\cE)(\chi^n)\xrightarrow{\varphi_0}\bigoplus_{n\in \bZ}(\bigwedge^{2n+1}\cE)(\chi^{n+1})\right),
\]
where $\varphi_i\defeq t\wedge(-)\oplus s\vee(-)$. Denote by $Z_s\subset X$ the zero scheme of $s$, and write $i\colon Z_{s}\hookto X$ for the closed immersion. Then the relative canonical bundle $\omega_i\defeq \omega_{Z_s}\otimes (\omega^*_X|_{Z_s})$ is isomorphic to $\det(\cE^*)$.
The following is a fundamental property of Koszul factoizations.
\begin{lem}[{\cite[Proposition 3.20]{bfk}\footnote[1]{There is a typo in the latter assertion in loc. cit.}}] \label{koszul lem} Assume that the sections $s$ and $t^*$ are regular. Then
there are isomorphisms
\begin{align*}
K(s,t)\cong \bigl(0\to \cO_{Z_s}\to0\bigr)
\cong \bigl(0\to \cO_{Z_{t^*}}\to0\bigr)\otimes \omega_i[-r].
\end{align*}
in $\Dcoh_G(X,\chi,W)$.
\end{lem}

\section{List of Notations}\label{notation appendix}
\noindent
\renewcommand{\arraystretch}{1.5}
\begin{longtable}[l]{p{2.3cm} p{8.25cm} p{3cm}}

$w\ast \chi$
$\ast$-action of the Weyl group $W$ on $\M_\bR$
p.\pageref{not: ast action}, line \lineref{line not: ast action}.
\\

weights of a quasi-symmetric representation $X$
p.\pageref{not: upbata}, line \lineref{line not: upbata}.
\\

$\defeq \sum_{\upbeta_i\in \scrW_{F}^+}\upbeta_i$
p.\pageref{not: upbeta_F^+}, line \lineref{line not: upbeta_F^+}.
\\

the groupoid associated to $\scrH^W$
p.\pageref{not: gamma gpd}, line \lineref{line not: gamma gpd}
\\

NCCR associated to $\delta$.
p.\pageref{end alg}, line \lineref{line not: NCCR}.
\\

a right mutation of $M$ at $N$
p.\pageref{not: mutation}, line \lineref{line not: mutation}.
\\

a left mutation of $M$ at $N$
p.\pageref{not: mutation}, line \lineref{line not: mutation}.
\\

$\defeq (\chi + \upbeta_{F_{\chi}(\delta_0)}^+)^+$
p.\pageref{not: mudeltadelta}, line \lineref{line not: mudeltadelta}
\\

$\defeq (1/2)\sum_{\alpha\in \Phi^+}\alpha$
Section \ref{notation}
\\

$\defeq\left\{\sum_{i=1}^da_i\upbeta_i\middle| a_i\in[-1,0]\right\}$
p.\pageref{not: bisigma}, line \lineref{line not: bsigma}.
\\

the set of roots of $G$
Section \ref{notation}
\\

the set of positive and negative roots of $G$
Section \ref{notation}
\\

the equivalence associated to the mutation at $N$
p.\pageref{not: mutation equiv}, line \lineref{line not: mutation equiv}.
\\

the dominant element in $W \ast \chi$
p.\pageref{not: dom chi}, line \lineref{line not: dom chi}
\\

$\defeq \left\{\chi\in\M_{\bR}\relmiddle| -\eta_{\lambda}/2\leq \l\chi,\lambda\r\leq \eta_{\lambda}/2 \mbox{ for all $\lambda\in \N$}\right\}$
p.\pageref{not: bnabla}, line \lineref{line not: bnabla}.
\\

p.\pageref{dual facet}, line \lineref{line not: dual facet}.
\\

$\defeq\{ B\in\scrS(P)\mid \mbox{$B\cap P$ is a facet of $P$}\}$
p.\pageref{not: B(P) and F(P)}, line \lineref{line not: B(P) and F(P)}.
\\

$\defeq (\delta+\bnabla)\cap \M^+$
p.\pageref{x delta}, line \lineref{line x delta}.
\\

$\defeq\left\{ \chi \in \scrC_{\delta}\relmiddle| F=F_{\chi}(\delta_0)\right\}$
p.\pageref{not: C^F_delta}, line \lineref{line not: C^F_delta}.
\\

$\defeq \#\scrW_F^+$
p.\pageref{not: d_F^+}, line \lineref{line not: d_F^+}.
\\

the set of right exchanges
p.\pageref{not:set of right exchanges}, line \lineref{line not:set of right exchanges}.
\\

the set of left exchanges
p.\pageref{not:set of left exchanges}, line \lineref{line not:set of left exchanges}.
\\

p.\pageref{dual facet}, line \lineref{line not: dual facet}.
\\

p.\pageref{not: F(P)}, line \lineref{line not: F(P)}.
\\

$\defeq\{\mbox{codimension $k$ face of $P$}\}$
p.\pageref{not: B(P) and F(P)}, line \lineref{line not: B(P) and F(P)}.
\\

$\scrF_+(P), \scrF_+^k(P)$
the sets of dominant faces
p.\pageref{not: dom faces}, line \lineref{line not: dom faces}.
\\

$\defeq\left\{ F\in \scrF_+(\delta_0+(1/2)\bsigma)\relmiddle| \exists \chi\in \scrC_{\delta}, F=F_{\chi}(\delta_0)\right\}$
p.\pageref{not: Fdeltadelta}, line \lineref{line not: Fdeltadelta}
\\

connected reductive group
Section \ref{notation}
\\

a set of  hyperplanes
p.\pageref{not: scrH}, line \lineref{line not: scrH}.
\\

$(\M^W)$-periodic hyperplane arrangement
p.\pageref{not: H^W}, line \lineref{line not: H^W}.
\\

the set of separating hyperplanes
p.\pageref{not: sep hyp set}, line \lineref{line not: sep hyp set}.
\\

$\defeq \bigoplus_{\chi\in \scrC_{\delta}^F}\left( \bigoplus_{{\upbeta\in{\tiny \gwedge}^*\scrW_F^+}} M((\chi+\upbeta)^+)\right)$
p.\pageref{not: LN}, line \lineref{line not: LN}
\\

magic window subcategory
p.\pageref{not: magic window}, line \lineref{line not: magic window}.
\\

$\defeq \left( V(\chi) \otimes_{\Bbbk} \Bbbk[X] \right)^G$
p.\pageref{not: MCM ass to chi}, line \lineref{line not: MCM ass to chi}.
\\

p.\pageref{not: M_delta}, line \lineref{ line not: M_delta}.
\\

$\defeq\bigoplus_{\chi\in \scrC_{\delta}^F}M(\chi)$
p.\pageref{not: M^F_delta}, line \lineref{line not: M^F_delta}.
\\

$\defeq \bigoplus_{\chi\in \scrC_{\delta}\cap \scrC_{\delta'}}M(\chi)$
p.\pageref{not: M_common}, line \lineref{line not: M_common}
\\

the character lattice of $T$
Section \ref{notation}
\\

$(\M_{\bR}^{W})_{\rm gen}$
$\defeq \M_{\bR}^W\backslash\bigcup_{A\in\scrA}\bigl(A\cap \M_{\bR}^{W}\bigr)$
p.\pageref{not: M_R^W}, line \lineref{ line not: M_R^W}.
\\

$\defeq \bigoplus_{\chi\in \scrC_{\delta}\backslash \scrC^F_{\delta}}M(\chi)\cong M_{\delta}/M_{\delta}^F$
p.\pageref{not: LN}, line \lineref{line not: LN}
\\

fixed maximal torus of $G$
Section \ref{notation}
\\

$\defeq \bigoplus_{\chi\in\scrC_{\delta}}V_X(\chi)\in\coh[X/G]$
p.\pageref{tilting bdl T}, line \lineref{line: tilting bdl T}.
\\

$\defeq \bigoplus_{\chi\in\scrC_{\delta}}V_{\X}(\chi) \in \coh \X$
p.\pageref{tilting bdl}, line \lineref{line: tilting bdl}.
\\

$\defeq\left\{\upbeta_i\relmiddle|\mbox{$\l\upbeta_i,\lambda\r>0$ for some $\lambda\in\scrL_F$}\right\}$
p.\pageref{not: W_F}, line \lineref{line not: W_F}.
\\

$\defeq\left\{\upbeta_i\relmiddle|\mbox{$\l\upbeta_i,\lambda\r=0$ for all $\lambda\in\scrL_F$}\right\}$
p.\pageref{not: W_F}, line \lineref{line not: W_F}.
\\

$\defeq[X^{\rm ss}(\ell)/G]$
p.\pageref{not: GIT stack}, line \lineref{line not: GIT stack}.

\end{longtable}
\renewcommand{\arraystretch}{1}

\begin{thebibliography}{999FGH}

[1] M.~Ballard, D.~Favero and L.~Katzarkov, {\it A category of kernels for equivariant factorizations and its implications for Hodge theory}. Publ. Math. Inst. Hautes $\acute{\rm E}$tudes Sci. {\bf 120}, 1--111 (2014).

[2]  M.~Ballard, D.~Favero and L.~Katzarkov, {\it Variation of geometric invariant theory quotients and derived categories}, J. Reine Angew. Math. {\bf 746}, 235--303 (2019).

[3] M.~Ballard, D.~Deliu, D.~Favero, M.~U.~Isik, and L.~Katzarkov, {\it Resolutions in factorization categories}. Adv. Math. {\bf 295}, 195--249 (2016).

[4] D.~Bergh, V.~A.~Lunts, O.~M.~Schn\"{u}rer1, {\it Geometricity for derived categories of algebraic stacks}, Selecta Math. (N.S.) {\bf 22} (2016), no. 4, 2535--2568.

[5] T.~Bridgeland, {\it Flops and derived categories}, Invent.\ Math.\ {\bf 147} (2002), no.~3, 613--632.

[6] W.~Bruns and J.~Herzog, {\it Cohen-Macaulay rings}. Cambridge Studies in Advanced Mathematics, {\bf 39}.

[7]
J.-C.~Chen, \emph{Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities}, J. Differential Geom. \textbf{61} (2002), no. 2, 227--261.

[8] W.~Hara, {\it Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops}.
Adv. Math.{\bf 318}(2017), 355--410.

[9] W.~Hara, {\it On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists}.
Matematiche (Catania) {\bf 77}(2022), no.2, 329--371.

[10] D.~Halpern-Leistner, {\it The derived category of a GIT quotient}. J. Amer. Math. Soc. {\bf 28} (2015), no. 3, 871--912.

[11] D.~Halpern-Leistner and S.~V.~Sam, {\it Combinatorial constructions of derived equivalences}. J. Amer. Math. Soc. {\bf 33}, no. 3, 871--912  (2020).

[12] D.~Halpern-Leistner and I.~Shipman, {\it Autoequivalences of derived categories via geometric invariant
theory.} Adv. Math. {\bf 303}, 1264--1299 (2016).

[13] A.~Higashitani and Y.~Nakajima,
{\it Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions}.
Selecta Math. (N.S.) {\bf 25}(2019), no.5, Paper No. 78, 25 pp.

[14] Y.~Hirano, {\it Equivalences of derived factorization categories of gauged Landau-Ginzburg models}. Adv. Math. {\bf 306}, 200--278 (2017).

[15] Y.~Hirano, {\it Derived Kn\"orrer periodicity and Orlov's theorem for gauged Landau-Ginzburg models}. Compos. Math. {\bf 153}, no. 5, 973--1007 (2017).

[16] Y.~Hirano, {\it Equivariant Tilting Modules, Pfaffian Varieties
and Noncommutative Matrix Factorizations
}. SIGMA Symmetry Integrability Geom. Methods  Appl.  {\bf 17}, Paper No. 055, 43 pp (2021).

[17] Y.~Hirano and M.~Wemyss, {\it Faithful actions from hyperplane arrangements}, Geom.\  Topol.\ \textbf{22} (2018), no.~6, 3395--3433.

[18] Y.~Hirano and M.~Wemyss, {\it Stability conditions for 3-fold flops}, arXiv:1907.09742.

[19] J.~Hall, D.~Rydh, {\it Perfect complexes on algebraic stacks}, Compos. Math. {\bf 153} (2017), no. 11, 2318--2367.

[20] M.~U.~Isik, {\it Equivalence of the derived category of a variety with a singularity category}, Int. Math. Res. Not. IMRN (2013), no. 12, 2787--2808.

[21] O.~Iyama and I.~Reiten, {\it Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras}. Am. J. Math. {\bf 130} (4), 1087--1149 (2008).

[22]  O.~Iyama and M.~Wemyss, {\it Maximal modifications and Auslander-Reiten duality for non-isolated singularities}. Invent.\ Math.\ \textbf{197} (2014), no.~3, 521--586.

[23]
O.~Iyama and M.~Wemyss, \emph{Tits cones intersections and applications}, \href{https://www.maths.gla.ac.uk/~mwemyss/MainFile_for_web.pdf}{\sf preprint}.

[24] Y.~Kawamata, {\it Flops connect minimal models}, Publ. RIMS, {\bf 44}, 419--423 (2008).

[25] N.~Koseki and G.~Ouchi, {\it Perverse schobers and Orlov equivalences}, Eur. J. Math. {\bf 9}, no. 2, Paper No. 32, 38 pp (2023).

[26] Y.~Nakajima,
{\it Mutations of splitting maximal modifying modules: the case of reflexive polygons}.
Int. Math. Res. Not. IMRN(2019), no.2, 470--550.

[27] C.~Okonek and A.~Teleman, {\it Graded tilting for gauged Landau-Ginzubrg models and geometric applications}. arXiv:1907.10099.

[28] L.~Positselski, {\it Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence}. Mem. Amer. Math. Soc. 212 (2011), no. 966.

[29] I.~Shipman, {\it A geometric approach to Orlov's theorem}, Compos. Math. {\bf 148}, no. 5, 1365-1389  (2012).

[30]\v{S}.~\v{S}penko and M.~Van den Bergh, {\it Non-commutative resolutions of quotient singularities for reductive groups}. Invent. Math. {\bf 210}, no. 1, 3--67 (2017).

[31] \v{S}.~\v{S}penko and M.~Van den Bergh,  {\it
Non-commutative crepant resolutions for some toric singularities I}.
Int. Math. Res. Not. IMRN(2020), no.21, 8120--8138.

[32] \v{S}.~\v{S}penko and M.~Van den Bergh,  {\it Non-commutative crepant resolutions for some toric singularities. II}. J. Noncommut. Geom. {\bf 14} (2020), no. 1, 73--103.

[33] \v{S}.~\v{S}penko and M.~Van den Bergh,  {\it Tilting bundles on hypertoric varieties}.
Int. Math. Res. Not. IMRN(2021), no.2, 1034--1042.

[34] \v{S}.~\v{S}penko and M.~Van den Bergh, J.-P.~Bell,
{\it On the noncommutative Bondal-Orlov conjecture for some toric varieties}.
Math. Z. {\bf 300}(2022), no.1, 1055--1068.

[35] The Stacks Project Authors, {\it Stacks Project}. \href{https://stacks.math.columbia.edu}{https://stacks.math.columbia.edu}

[36] C.~Teleman, {\it The quantization conjecture revisited}.  Ann. of Math. (2) {\bf 152} (2000),
no. 1, 1--43.

[37]
M.~Van den Bergh, \emph{Three-dimensional flops and noncommutative rings},
Duke Math.\ J.\ \textbf{122}  (2004), no.~3, 423--455.

[38] M.~Van den Bergh, {\it Non-commutative crepant resolutions.} The Legacy of Niels Henrik Abel, pp. 749--770. Springer, Berlin (2004)

[39] M.~Van den Bergh, {\it Non-commutative crepant resolutions, an overview}. arXiv:2207.09703.

[40]
M.~Wemyss, \emph{Flops and Clusters in the Homological Minimal Model Program}, Invent.\ Math.\ \textbf{211} (2018), no.~2, 435--521.

\end{thebibliography}

\end{document}