In this short subsection, we define (noncommutative) gauged Landau–Ginzburg
models, and its derived factorization categories.

######  Definition 4.6.

We call the data $(\mathcal{A},L,w)^{G}$ a gauged Landau–Ginzburg model when
$G$ is an algebraic group acting on a scheme $X$, $\mathcal{A}$ is a
$G$-equivariant coherent $X$-algebra, $L$ is a $G$-equivariant line bundle on
$X$, and $w\in\Gamma(X,L)^{G}$ is a $G$-invariant global section of $L$. A
gauged Landau–Ginzburg model $(\mathcal{A},L,w)^{G}$ is said to be
noncommutative, if the algebra $\mathcal{A}$ is not commutative. We write
$(X,L,w)^{G}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(\mathcal{O}_{X},L,w)^{G}$,
and for a character $\chi\colon G\to\mathbb{G}_{m}$ of $G$ we write
$(\mathcal{A},\chi,w)^{G}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(\mathcal{A},\mathcal{O}(\chi),w)^{G}$.

If $(\mathcal{A},L,w)^{G}$ is a gauged LG model, we have the triple

$\displaystyle\big{(}\operatorname{Qcoh}_{G}\mathcal{A},L\otimes_{\mathcal{O}_{X}}(-),w\big{)}$

as in (2.1), where
$L\otimes_{\mathcal{O}_{X}}(-)\colon\operatorname{Qcoh}_{G}\mathcal{A}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Qcoh}_{G}\mathcal{A}$
is the tensor product with $L$, and $w\colon\operatorname{id}\to
L\otimes_{\mathcal{O}_{X}}(-)$ is the functor morphism defined by the
multiplication by $w$. Then we define

$\displaystyle\operatorname{Qcoh}_{G}(\mathcal{A},L,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{Fact}\big{(}\operatorname{Qcoh}_{G}\mathcal{A},L\otimes_{\mathcal{O}_{X}}(-),w\big{)},$
$\displaystyle\operatorname{D^{co}Qcoh}_{G}(\mathcal{A},L,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{D^{co}}\big{(}\operatorname{Qcoh}_{G}\mathcal{A},L\otimes_{\mathcal{O}_{X}}(-),w\big{)},$
$\displaystyle\operatorname{Dcoh}_{G}(\mathcal{A},L,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{D}\big{(}\operatorname{coh}_{G}\mathcal{A},L\otimes_{\mathcal{O}_{X}}(-),w\big{)},$
$\displaystyle\operatorname{DMF}_{G}(\mathcal{A},L,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{D}\big{(}\operatorname{lfr}_{G}\mathcal{A},L\otimes_{\mathcal{O}_{X}}(-),w\big{)},$

where $\operatorname{lfr}_{G}\mathcal{A}$ is the subcategory of
$\operatorname{coh}_{G}\mathcal{A}$ consisting of locally free
$\mathcal{A}$-modules. We call $\operatorname{Dcoh}_{G}(\mathcal{A},L,w)$ the
derived factorization category of $(\mathcal{A},L,w)^{G}$, and
$\operatorname{DMF}_{G}(\mathcal{A},L,w)$ the derived matrix factorization
category of $(\mathcal{A},L,w)^{G}$. If $X=\operatorname{Spec}R$ is an affine
scheme and $A\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Gamma(X,\mathcal{A})$
is the corresponding $G$-equivariant $R$-algebra, we define

$\displaystyle\operatorname{Mod}_{G}(A,L,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{Fact}\big{(}\operatorname{Mod}_{G}A,L\otimes_{R}(-),w\big{)},$
$\displaystyle\operatorname{D^{co}Mod}_{G}(A,L,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{D^{co}}\big{(}\operatorname{Mod}_{G}\mathcal{A},L\otimes_{R}(-),w\big{)},$
$\displaystyle\operatorname{Dmod}_{G}(A,L,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{D}\big{(}\operatorname{mod}_{G}A,L\otimes_{R}(-),w\big{)},$
$\displaystyle\operatorname{KMF}_{G}(A,L,w)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{K}\big{(}\operatorname{proj}_{G}A,L\otimes_{R}(-),w\big{)}.$

###  4.3 Equivariant tilting modules and factorizations

In this subsection, we apply the general result in Section 4.1 to the
following geometric setting.

Let the notation be the same as in Section 3.4. Let $\chi\colon
G/H\to\mathbb{G}_{m}$ be a character of $G/H$, and set
$\widehat{\chi}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\chi\circ
p\colon G\to\mathbb{G}_{m}$, where $p\colon G\to G/H$ is the natural
projection. Write for $\mathcal{O}_{R}(\chi)$ (resp.
$\mathcal{O}_{X}\big{(}\widehat{\chi}\big{)}$) the associated
$G/H$-equivariant invertible sheaf on $\operatorname{Spec}R$ (resp.
$G$-equivariant invertible sheaf on $X$). Take a $G/H$-invariant global
section $w_{R}\in\Gamma(\operatorname{Spec}R,\mathcal{O}_{R}(\chi))^{G/H}$ of
$\mathcal{O}_{R}(\chi)$, and set $w\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(f)^{*}_{p}w_{R}\in\Gamma\big{(}X,\mathcal{O}_{X}\big{(}\widehat{\chi}\big{)}\big{)}^{G}$,
where $(f)^{*}_{p}\colon\operatorname{Mod}_{G/H}R\to\operatorname{Qcoh}_{G}X$
is the pull-back by the $p$-equivariant morphism $f$. Then we have the gauged
LG models

$\displaystyle\big{(}\mathcal{A},\widehat{\chi},w\big{)}^{G}\qquad\text{and}\qquad\big{(}\Lambda^{H},\chi,w_{R}\big{)}^{G/H},$

and we define

$\displaystyle\operatorname{Kadd}_{G}\big{(}\mathcal{T},\widehat{\chi},w\big{)}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{K}\big{(}\operatorname{add}_{\widehat{\chi}}\mathcal{T},\mathcal{O}\big{(}\widehat{\chi}\big{)}\otimes_{\mathcal{O}_{X}}(-),w\big{)},$

where $\operatorname{add}_{\widehat{\chi}}\mathcal{T}$ is the subcategory of
$\operatorname{coh}_{G}\mathcal{A}$ defined by the following additive closure

$\displaystyle\operatorname{add}_{\widehat{\chi}}\mathcal{T}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{add}\big{\\{}\mathcal{O}\big{(}\widehat{\chi}^{n}\big{)}\otimes_{\mathcal{O}_{X}}\big{(}\mathcal{T},\uptheta^{\mathcal{T}}\big{)}\mid
n\in\mathbb{Z}\big{\\}}.$

Since the functors
$\operatorname{\sf{F}}\colon\operatorname{Qcoh}_{G}\mathcal{A}\to\operatorname{Mod}_{G/H}\Lambda^{H}$
and
$\operatorname{\sf{G}}\colon\operatorname{Mod}_{G/H}\Lambda^{H}\to\operatorname{Qcoh}_{G}\mathcal{A}$
are factored with respect to $(\widehat{\chi},w)$ and $(\chi,w_{R})$, they
induce the functors

$\displaystyle\operatorname{\sf{F}}\colon\
\operatorname{Qcoh}_{G}\big{(}\mathcal{A},\widehat{\chi},w\big{)}\to\operatorname{Mod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)},$
$\displaystyle\operatorname{\sf{G}}\colon\
\operatorname{Mod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)}\to\operatorname{Qcoh}_{G}\big{(}\mathcal{A},\widehat{\chi},w\big{)}.$

Since $\operatorname{Qcoh}_{G}\mathcal{A}$ has enough injectives, we have the
right derived functor

$\displaystyle\mathbb{R}\operatorname{\sf{F}}\colon\
\operatorname{D^{co}Qcoh}_{G}\big{(}\mathcal{A},\widehat{\chi},w\big{)}\to\operatorname{D^{co}Mod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)},$

and it restricts to the functor

$\displaystyle\mathbb{R}\operatorname{\sf{F}}\colon\
\operatorname{Dcoh}_{G}\big{(}\mathcal{A},\widehat{\chi},w\big{)}\to\operatorname{Dmod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)}.$

If $G/H$ is reductive and every $\Lambda^{H}$-module has a finite projective
resolution, then every finitely generated $G/H$-equivariant
$\Lambda^{H}$-module has a finite projective resolution in
$\operatorname{mod}_{G/H}\Lambda^{H}$ by Lemma 3.25. In this case, we have the
left derived functor

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\colon\
\operatorname{Dmod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)}\to\operatorname{Dcoh}_{G}\big{(}\mathcal{A},\widehat{\chi},w\big{)}.$

######  Theorem 4.7.

Assume that $G/H$ is reductive and $\Lambda^{H}$ is of finite global
dimension.

  * $1.$

If $\big{(}\mathcal{T},\uptheta^{\mathcal{T}}\big{)}$ is partial
$(G,H)$-tilting, then the functor

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\colon\
\operatorname{Dmod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)}\to\operatorname{Dcoh}_{G}\big{(}\mathcal{A},\widehat{\chi},w\big{)}$

is fully faithful, and it restricts to the equivalence

$\displaystyle\mathbb{L}\operatorname{\sf{G}}\colon\
\operatorname{Dmod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Kadd}_{G}\big{(}\mathcal{T},\widehat{\chi},w\big{)}.$

  * $2.$

If $\big{(}\mathcal{T},\uptheta^{\mathcal{T}}\big{)}$ is $(G,H)$-tilting and
every $\mathcal{M}\in\operatorname{Qcoh}\mathcal{A}$ has a finite injective
resolution in $\operatorname{Qcoh}\mathcal{A}$, then we have the equivalence

$\displaystyle\mathbb{R}\operatorname{\sf{F}}\colon\
\operatorname{Dcoh}_{G}\big{(}\mathcal{A},\widehat{\chi},w\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Dmod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)}.$

###### Proof.

By an identical argument as in the proof of Lemma 3.28, we see that the
restriction functors

$\displaystyle\operatorname{\sf Res}^{G}_{H}\colon\
\operatorname{Dcoh}_{G}\big{(}\mathcal{A},\widehat{\chi},w\big{)}\to\operatorname{Dcoh}_{H}(\mathcal{A},\chi,w),$
$\displaystyle\operatorname{\sf Res}\colon\
\operatorname{Dmod}_{G/H}\big{(}\Lambda^{H},\chi,w_{R}\big{)}\to\operatorname{Dmod}\big{(}\Lambda^{H},\operatorname{id},w_{R}\big{)}$

are faithful. Let
$\operatorname{\sf{F}}_{H}\colon\operatorname{coh}_{H}\mathcal{A}\to\operatorname{mod}\Lambda^{H}$
and
$\operatorname{\sf{G}}_{H}\colon\operatorname{mod}\Lambda^{H}\to\operatorname{coh}_{H}\mathcal{A}$
be the same functors as in the proof of Theorem 3.27, and recall that the
adjunction morphisms
$\Lambda^{H}\to\operatorname{\sf{F}}_{H}\operatorname{\sf{G}}_{H}\big{(}\Lambda^{H}\big{)}$
and
$\operatorname{\sf{G}}_{H}\operatorname{\sf{F}}_{H}(\mathcal{T}_{H})\to\mathcal{T}_{H}$
are isomorphisms. Then these functors define the functors

$\displaystyle\mathbb{R}\operatorname{\sf{F}}_{H}\colon\
\operatorname{Dcoh}_{H}\big{(}\mathcal{A},\widehat{\chi},w\big{)}\to\operatorname{Dmod}\big{(}\Lambda^{H},\chi,w_{R}\big{)},$
$\displaystyle\mathbb{L}\operatorname{\sf{G}}_{H}\colon\
\operatorname{Dmod}\big{(}\Lambda^{H},\chi,w_{R}\big{)}\to\operatorname{Dcoh}_{H}\big{(}\mathcal{A},\widehat{\chi},w\big{)},$

and if $\big{(}\mathcal{T},\uptheta^{\mathcal{T}}\big{)}$ is partial
$(G,H)$-tilting (resp. $(G,H)$-tilting), $\mathbb{L}\operatorname{\sf{G}}_{H}$
is fully faithful (resp. an equivalence) by Theorem 4.5. Hence the results
follow from Lemma 3.29. ∎

##  5 Linear sections of Pfaffian varieties
and noncommutative resolutions

In this section, we prove that derived categories of noncommutative
resolutions of linear sections of Pfaffian varieties are equivalent to the
derived factorization categories of noncommutative gauged LG models.

###  5.1 Noncommutative resolutions of Pfaffian varieties and its linear
sections

Following [16], we recall noncommutative resolutions of Pfaffian varieties and
its linear sections. Let $V$ be a vector space of dimension $v$. For an
integer $q$ with $0\leq 2q\leq v$, we have a Pfaffian variety

$\displaystyle\operatorname{Pf}_{q}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{\\{}x\in\raisebox{1.0pt}[0.0pt][0.0pt]{$\textstyle\bigwedge^{\raisebox{-1.0pt}[0.0pt][0.0pt]{\tiny\\!$2$\,}}$}V^{*}\mid\operatorname{\sf
rk}(x)\leq
2q\big{\\}}\subseteq\mathbb{P}\big{(}\raisebox{1.0pt}[0.0pt][0.0pt]{$\textstyle\bigwedge^{\raisebox{-1.0pt}[0.0pt][0.0pt]{\tiny\\!$2$\,}}$}V^{*}\big{)},$

and we denote its affine cone by

$\displaystyle\operatorname{Pf}_{q}^{\operatorname{\sf
aff}}\subseteq\raisebox{1.0pt}[0.0pt][0.0pt]{$\textstyle\bigwedge^{\raisebox{-1.0pt}[0.0pt][0.0pt]{\tiny\\!$2$\,}}$}V^{*}.$

Then we have $\dim\operatorname{Pf}_{q}^{\operatorname{\sf aff}}=q(2v-2q-1)$.
The Pfaffian variety $\operatorname{Pf}_{q}$ is smooth if and only if $q=1$,
where it defines a Grassmannian $\operatorname{Gr}(2,V)$, or $q=\lfloor
v/2\rfloor$, where it defines the whole space
$\mathbb{P}\big{(}\raisebox{1.0pt}[0.0pt][0.0pt]{$\textstyle\bigwedge^{\raisebox{-1.0pt}[0.0pt][0.0pt]{\tiny\\!$2$\,}}$}V^{*}\big{)}$.
In other cases, the singular locus is the subvairiety
$\operatorname{Pf}_{(q-1)}\subset\operatorname{Pf}_{q}$. Let
$L\subset\raisebox{1.0pt}[0.0pt][0.0pt]{$\textstyle\bigwedge^{\raisebox{-1.0pt}[0.0pt][0.0pt]{\tiny\\!$2$\,}}$}V^{*}$
be a subspace of codimension $c$ such that

$\displaystyle\operatorname{Pf}_{q}\\!|_{L}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{Pf}_{q}\cap\,\,\mathbb{P}(L)\neq\varnothing$

and $\operatorname{Pf}_{q}\\!|_{L}$ has the expected dimension
$\dim\operatorname{Pf}_{q}-c$. If
$c>\dim\operatorname{Sing}(\operatorname{Pf}_{q})=(q-1)(2v-2q+1)-1$, we can
take a generic $L$ so that $\operatorname{Pf}_{q}\\!|_{L}$ is smooth. If $c$
is smaller than the bounds, then $\operatorname{Pf}_{q}\\!|_{L}$ is never
smooth, and in this case the usual bounded derived category
$\operatorname{D^{b}}\big{(}\operatorname{coh}\operatorname{Pf}_{q}\\!|_{L}\big{)}$
does not behave well (for example, in the context of homological projective
duality [16]).

Let $(Q,\omega)$ be a symplectic vector space of dimension $2q$ with a
symplectic form
$\omega\in\raisebox{1.0pt}[0.0pt][0.0pt]{$\textstyle\bigwedge^{\raisebox{-1.0pt}[0.0pt][0.0pt]{\tiny\\!$2$\,}}$}Q^{\ast}$.
Let

$\displaystyle\operatorname{Sp}(Q)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\\{f\in\operatorname{GL}(Q)\mid
f^{*}\omega=\omega\\}$

be the symplectic group of $Q$, and

$\displaystyle\operatorname{GSp}(Q)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\\{f\in\operatorname{GL}(Q)\mid\exists\,t\in
k^{*}\mbox{ such that }f^{*}\omega=t\,\omega\\}$

the symplectic similitude group of $Q$. Then we have a short exact sequence

$1\to\operatorname{Sp}(Q)\hookrightarrow\operatorname{GSp}(Q)\twoheadrightarrow\mathbb{G}_{m}\to
1$ (5.1)

that induces a semi-direct product
$\operatorname{GSp}(Q)=\operatorname{Sp}(Q)\rtimes\mathbb{G}_{m}$, where
$\mathbb{G}_{m}\subset\operatorname{GSp}(Q)$ is the diagonal subgroup. Let us
set

$\displaystyle Y\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{Hom}_{k}(V,Q)$

and consider the following quotient stacks

$\displaystyle\mathcal{Y}^{\operatorname{\sf aff}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=[Y/\operatorname{Sp}(Q)]\qquad\text{and}\qquad\mathcal{Y}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=[Y/\operatorname{GSp}(Q)].$

Then the surjective morphism

$\displaystyle p\colon\quad Y\to\operatorname{Pf}_{q}^{\operatorname{\sf
aff}},\qquad\varphi\mapsto\varphi^{*}\omega$

induces the morphisms

$\displaystyle\pi^{\operatorname{\sf aff}}\colon\
\mathcal{Y}^{\operatorname{\sf
aff}}\to\operatorname{Pf}_{q}^{\operatorname{\sf
aff}}\qquad\text{and}\qquad\pi\colon\
\mathcal{Y}\to\big{[}\operatorname{Pf}_{q}^{\operatorname{\sf
aff}}/\mathbb{G}_{m}\big{]},$

and if we set $\mathcal{Y}^{\operatorname{ss}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\left[\left(Y\backslash\,p^{-1}(0)\right)/\operatorname{GSp}(Q)\right]$,
the morphism
$\pi\colon\mathcal{Y}\to\big{[}\operatorname{Pf}_{q}^{\operatorname{\sf
aff}}/\mathbb{G}_{m}\big{]}$ restricts to a (stacky) resolution

$\displaystyle\pi\colon\
\mathcal{Y}^{\operatorname{ss}}\to\operatorname{Pf}_{q}$

of the Pfaffian variety $\operatorname{Pf}_{q}$.

Recall that irreducible representations of $\operatorname{Sp}(Q)$ are indexed
by Young diagrams of height at most $q$. We denote by $Y_{q,s}$ the set of
Young diagrams of height at most $q$ and width at most $s\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\lfloor
v/2\rfloor-q$. Since vector bundles on $\mathcal{Y}^{\operatorname{\sf aff}}$
are nothing but $\operatorname{Sp}(Q)$-equivariant vector bundles on $Y$, each
representation of $\operatorname{Sp}(Q)$ defines a vector bundle on
$\mathcal{Y}^{\operatorname{\sf aff}}$. For each Young diagram $\gamma\in
Y_{q,s}$, we can choose a vector bundle
$\mathcal{V}_{\gamma}\in\operatorname{vect}\mathcal{Y}$ whose pull-back to
$\mathcal{Y}^{\operatorname{\sf aff}}$ by the natural projection
$\mathcal{Y}^{\operatorname{\sf aff}}\to\mathcal{Y}$ is the vector bundle
associated to $\gamma$ (see [16, Section 2.4]). Then the vector bundle

$\displaystyle\mathcal{V}:=\bigoplus_{\gamma\in
Y_{q,s}}\mathcal{V}_{\gamma}\in\operatorname{coh}\mathcal{Y}$

is a partial tilting bundle on $\mathcal{Y}$. Restricting $\mathcal{V}$ to the
open substack $\mathcal{Y}^{\operatorname{ss}}$ we obtain a partial tilting
bundle $\mathcal{V}^{\operatorname{ss}}$ on $\mathcal{Y}^{\operatorname{ss}}$,
and the ${\operatorname{Pf}_{q}}$-algebra

$\displaystyle\mathcal{B}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\pi_{*}\operatorname{{\mathcal{E}}nd}_{\mathcal{Y}^{\operatorname{ss}}}(\mathcal{V}^{\operatorname{ss}})$

is a noncommutative resolution of $\operatorname{Pf}_{q}$ by [18, Section 5].
By the noncommutative Bertini theorem [17], for a generic $L$ the restriction

$\displaystyle\mathcal{B}_{L}:=\mathcal{B}|_{\operatorname{Pf}_{q}\\!|_{L}}$

is a noncommutative resolution of the linear section
$\operatorname{Pf}_{q}\\!|_{L}$. If $\operatorname{Pf}_{q}\\!|_{L}$ is smooth,
the category $\operatorname{D^{b}}(\operatorname{coh}\mathcal{B}_{L})$ is
equivalent to
$\operatorname{D^{b}}(\operatorname{coh}\operatorname{Pf}_{q}\\!|_{L})$.

###  5.2 Noncommutative resolutions of Pfaffian varieties
and noncommutative gauged LG models

In this subsection, we prove that the derived category of a noncommutative
resolution of a linear section of a Pfaffian variety is equivalent to the
derived factorization category of a noncommutative gauged Landau–Ginzburg
model.

We use the same notation as in the previous section. We recall that the
category $\operatorname{D^{b}}(\operatorname{coh}\mathcal{B}_{L})$ is
equivalent to a full subcategory of the derived factorization category of a
gauged LG model. We write
$L^{\perp}\subset\raisebox{1.0pt}[0.0pt][0.0pt]{$\textstyle\bigwedge^{\raisebox{-1.0pt}[0.0pt][0.0pt]{\tiny\\!$2$\,}}$}V$
for the annihilator of $L$. We have a $\operatorname{GSp}(Q)$-action on the
product $Y\times L^{\perp}$ defined by

$\displaystyle\operatorname{GSp}(Q)\times\big{(}Y\times
L^{\perp}\big{)}\to\big{(}Y\times
L^{\perp}\big{)},\qquad(f,\varphi,x)\mapsto\big{(}f\circ\varphi,\sigma(f)^{-1}x\big{)},$

where $\sigma\colon\operatorname{GSp}(Q)\to\mathbb{G}_{m}$ is the surjection
in (5.1). Consider the following quotient stacks

$\displaystyle\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{[}Y\times
L^{\perp}/\operatorname{GSp}(Q)\big{]},$
$\displaystyle\mathcal{Y}^{\operatorname{ss}}\times_{\mathbb{G}_{m}}L^{\perp}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{[}\big{(}Y\backslash\,p^{-1}(0)\big{)}\times
L^{\perp}/\operatorname{GSp}(Q)\big{]}.$

The function $W\colon Y\times L^{\perp}\to\mathbb{C}$ defined by
$W(\varphi,x)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(\varphi^{*}\omega)(x)$
is $\operatorname{GSp}(Q)$-invariant, and thus it defines a potential

$\displaystyle w\colon\
\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp}\to\mathbb{C}.$

We define an additional $\mathbb{G}_{m}$-action, which is called an R-charge,
on $Y\times L^{\perp}$ by

$\displaystyle\mathbb{G}_{m}\times\big{(}Y\times
L^{\perp}\big{)}\to\big{(}Y\times
L^{\perp}\big{)},\qquad(t,\varphi,x)\mapsto(t\varphi,x).$

This $\mathbb{G}_{m}$-action commutes with the above
$\operatorname{GSp}(Q)$-action, and so we have an induced
$\mathbb{G}_{m}$-action on $\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp}$. Then
the potential
$w\colon\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp}\to\mathbb{C}$ is a semi-
invariant regular function with respect to the character
$\chi_{1}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{id}_{\mathbb{G}_{m}}\colon\mathbb{G}_{m}\to\mathbb{G}_{m}$.
Following [16], we define the B-brane category
$\operatorname{DB}\big{(}\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp},w\big{)}$
as follows:

We write $\operatorname{rep}\operatorname{GSp}(Q)$ for the category of finite
dimensional algebraic representations of $\operatorname{GSp}(Q)$, and we
define the full subcategory

$\displaystyle\mathcal{Y}_{q,s}\subset\operatorname{rep}\operatorname{GSp}(Q)$

consisting of irreducible representations whose restriction to the subgroup
$\operatorname{Sp}(Q)$ is an irreducible representation of
$\operatorname{Sp}(Q)$ corresponding to some Young diagram in $Y_{q,s}$. Since
the origin $0\in Y\times L^{\perp}$ is a fixed point of the above action from
$\operatorname{GSp}(Q)\times\mathbb{G}_{m}$, the restriction to the origin
defines a functor

$\displaystyle(-)|_{0}\colon\
\operatorname{Dcoh}_{\mathbb{G}_{m}}\big{(}\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp},\chi_{1},w\big{)}\to\operatorname{Dcoh}_{\mathbb{G}_{m}}\big{(}[0/\operatorname{GSp}(Q)],\chi_{1},0\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{D^{b}}(\operatorname{rep}\operatorname{GSp}(Q)),$

where the latter functor is a similarly equivalence as in [7, Proposition
2.14]. Then we define the category

$\displaystyle\operatorname{DB}\big{(}\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp},w\big{)}$

to be the full subcategory of
$\operatorname{Dcoh}_{\mathbb{G}_{m}}\big{(}\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp},\chi_{1},w\big{)}$
consisting of objects $F$ such that each cohomology $H^{i}(F|_{0})$ of the
restriction
$F|_{0}\in\operatorname{D^{b}}(\operatorname{rep}\operatorname{GSp}(Q))$ lies
in $\mathcal{Y}_{q,s}$. Restricting to the open substack
$\mathcal{Y}^{\operatorname{ss}}\times_{\mathbb{G}_{m}}L^{\perp}$, we have the
B-brane subcategory

$\displaystyle\operatorname{DB}\big{(}\mathcal{Y}^{\operatorname{ss}}\times_{\mathbb{G}_{m}}L^{\perp},w\big{)}\subset\operatorname{Dcoh}_{\mathbb{G}_{m}}\big{(}\mathcal{Y}^{\operatorname{ss}}\times_{\mathbb{G}_{m}}L^{\perp},\chi_{1},w\big{)},$

and by [16, Section 4.1] we have an equivalence

$\operatorname{D^{b}}(\operatorname{coh}\mathcal{B}_{L})\cong{\rm
DB}\big{(}\mathcal{Y}^{\rm ss}\times_{\mathbb{G}_{m}}L^{\perp},w\big{)}.$
(5.2)

For an interval $I\subset\mathbb{Z}$, we define a subcategory

$\displaystyle\mathcal{Y}_{q,s}^{I}\subseteq\mathcal{Y}_{q,s}$

of $\mathcal{Y}_{q,s}$ consisting of representations whose restriction to the
diagonal subgroup $\mathbb{G}_{m}\subset\operatorname{GSp}(Q)$ has weights in
$I$. This subcategory defines a full subcategory

$\displaystyle\operatorname{DB}\big{(}\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp},w\big{)}_{I}\subset\operatorname{DB}\big{(}\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp},w\big{)}$

consisting of objects $F$ such that each cohomology $H^{i}(F|_{0})$ of the
restriction $F|_{0}$ lies in $\mathcal{Y}_{q,s}^{I}$. By [16, Theorem 4.7] we
have an equivalence

${\rm DB}\big{(}\mathcal{Y}^{\rm
ss}\times_{\mathbb{G}_{m}}L^{\perp},w\big{)}\cong{\rm
DB}\big{(}\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp},w\big{)}_{[-qv,qv]}.$
(5.3)

We set

$\displaystyle\mathcal{T}:=\bigoplus_{\rho\in\mathcal{Y}^{[-qv,qv]}_{q,s}}\mathcal{T}_{\rho}\in\operatorname{coh}_{\operatorname{GSp}(Q)}Y\times
L^{\perp},$

where $\mathcal{T}_{\rho}$ is the $\operatorname{GSp}(Q)$-equivariant vector
bundle on $Y\times L^{\perp}$ associated to an irreducible representation
$\rho\in\mathcal{Y}^{[-qv,qv]}_{q,s}$, and choose a
$\mathbb{G}_{m}\times\operatorname{GSp}(Q)$-equivariant vector bundle

$\displaystyle\widetilde{\mathcal{T}}\in\operatorname{coh}_{\mathbb{G}_{m}\times\operatorname{GSp}(Q)}Y\times
L^{\perp}$

on $Y\times L^{\perp}$ such that $\operatorname{\sf
Res}^{\mathbb{G}_{m}\times\operatorname{GSp}(Q)}_{\operatorname{GSp}(Q)}\widetilde{\mathcal{T}}\cong\mathcal{T}$.
Then the natural functor

$\displaystyle\operatorname{Kadd}_{\mathbb{G}_{m}\times\operatorname{GSp}(Q)}\big{(}\widetilde{\mathcal{T}},\chi,w\big{)}\to\operatorname{Dcoh}_{\mathbb{G}_{m}}\big{(}\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp},\chi_{1},w\big{)}$

is fully faithful by Lemma 4.2, and by [16, Lemma 2.8] we have a natural
equivalence

${\rm
DB}\big{(}\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp},w\big{)}_{[-qv,qv]}\cong\operatorname{Kadd}_{\mathbb{G}_{m}\times\operatorname{GSp}(Q)}\big{(}\widetilde{\mathcal{T}},\chi,w\big{)}.$
(5.4)

Since $\mathcal{T}$ is a partial tilting object in
$\operatorname{Qcoh}_{\operatorname{GSp}(Q)}Y\times L^{\perp}$ by
construction, $\widetilde{\mathcal{T}}$ is a partial
$(\mathbb{G}_{m}\times\operatorname{GSp}(Q),\operatorname{GSp}(Q))$-tilting
bundle. If we set

$\displaystyle\Lambda\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\operatorname{End}_{\mathcal{Y}\times_{\mathbb{G}_{m}}L^{\perp}}(\mathcal{T})=\operatorname{End}_{Y\times
L^{\perp}}(\mathcal{T})^{\operatorname{GSp}(Q)},$

then $\Lambda$ is a $\mathbb{G}_{m}$-equivariant algebra over the affine
quotient $\big{(}Y\times L^{\perp}\big{)}/\operatorname{GSp}(Q)$. Moreover, by
the results in [18, Section 1.5] and similar arguments as in [18, Section 5],
we see that the algebra $\Lambda$ is a noncommutative resolution of
$\big{(}Y\times L^{\perp}\big{)}/\operatorname{GSp}(Q)$. By Theorem 4.7, we
have an equivalence

$\displaystyle\operatorname{Kadd}_{\mathbb{G}_{m}\times\operatorname{GSp}(Q)}\big{(}\widetilde{\mathcal{T}},\chi,w\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Dmod}_{\mathbb{G}_{m}}(\Lambda,\chi,w).$
(5.5)

Combining the equivalences (5.2), (5.3), (5.4) and (5.5) we have the
following:

######  Corollary 5.1.

Notation is the same as above. We have an exact equivalence

$\displaystyle\operatorname{D^{b}}(\operatorname{coh}\mathcal{B}_{L})\cong\operatorname{Dmod}_{\mathbb{G}_{m}}(\Lambda,\chi,w).$

In particular, if $\operatorname{Pf}_{q}\\!|_{L}$ is smooth, we have an
equivalence

$\displaystyle\operatorname{D^{b}}\big{(}\operatorname{coh}\operatorname{Pf}_{q}\\!|_{L}\big{)}\cong\operatorname{Dmod}_{\mathbb{G}_{m}}(\Lambda,\chi,w).$

######  Remark 5.2.

Let $T\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{\\{}\operatorname{diag}\big{(}z_{1},\ldots,z_{q},z_{0}z_{1}^{-1},\ldots,z_{0}z_{q}^{-1}\big{)}\mid
z_{i}\in\mathbb{G}_{m}\big{\\}}\subset\operatorname{GSp}(Q)$ be a standard
maximal torus of $\operatorname{GSp}(Q)$. Denote by $\chi_{i}\colon
T\to\mathbb{G}_{m}$ the character defined by

$\displaystyle(z_{0},z_{1},\ldots,z_{q})\mapsto z_{i}.$

Then the $T$-weights of $Y$ are $\\{\chi_{i},\chi_{0}-\chi_{i}\\}_{1\leq i\leq
q}$ with each weight occurring with multiplicity $v$, and $T$-weights of
$L^{\perp}$ is $-\chi_{0}$ with multiplicity $\dim L^{\perp}=c$. Thus, if
$c=qv$, the sum of all $T$-weights of $Y$ is trivial, and in this case the
category $\operatorname{D^{b}}(\operatorname{coh}\mathcal{B}_{L})$ is
Calabi–Yau of dimension $2qs-1$.

##  Appendix A Algebras over groupoids in algebraic spaces

In this appendix, we prove a generalization of Proposition 3.12. We freely use
the terminology and notation from [19], and categories fibered in groupoids
are defined over the big fppf site
$(\operatorname{Sch\\!}/S)_{\operatorname{fppf}}$ over a fixed base scheme
$S$. First, we recall some definitions from [19], where the main references
are Chapters Groupoids in Algebraic Spaces, Algebraic Stacks, and Sheaves on
Algebraic Stacks.

Let $p\colon\mathcal{X}\to(\operatorname{Sch\\!}/S)_{\operatorname{fppf}}$ be
a category fibered in groupoids. Recall that we have the induced fppf topology
on $\mathcal{X}$, where a family of morphisms $\\{x_{i}\to x\\}_{i}$ in
$\mathcal{X}$ is a covering of $x\in\mathcal{X}$ if the family $\\{p(x_{i})\to
p(x)\\}_{i}$ is an fppf cover of the scheme $p(x)$. We write
$\mathcal{X}_{\operatorname{fppf}}$ for this fppf site. Then we have the
structure sheaf
$\mathcal{O}_{\mathcal{X}}\colon\mathcal{X}_{\operatorname{fppf}}^{\operatorname{op}}\to(\operatorname{Rings})$
of $\mathcal{X}_{\operatorname{fppf}}$ defined by

$\displaystyle\mathcal{O}_{\mathcal{X}}(x)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Gamma\big{(}p(x),\mathcal{O}_{p(x)}\big{)},$

and thus we have the associated ringed site
$\big{(}\mathcal{X}_{\operatorname{fppf}},\mathcal{O}_{\mathcal{X}}\big{)}$.
An $\mathcal{O}_{\mathcal{X}}$-module
$\mathcal{F}\colon\mathcal{X}_{\operatorname{fppf}}^{\operatorname{op}}\to(\operatorname{Sets})$
is said to be quasi-coherent, if for any $x\in\mathcal{X}$ there is a covering
$\\{x_{i}\to x\\}$ of $x$ such that for each $x_{i}$ there is an exact
sequence of $\mathcal{O}_{\mathcal{X}/x_{i}}$-modules

$\displaystyle\bigoplus_{I}\mathcal{O}_{\mathcal{X}/x_{i}}\to\bigoplus_{J}\mathcal{O}_{\mathcal{X}/x_{i}}\to\mathcal{F}|_{x_{i}}\to
0,$

where
$\big{(}\mathcal{X}_{\operatorname{fppf}}/x_{i},\mathcal{O}_{\mathcal{X}/x_{i}}\big{)}$
is the localization of the ringed site
$\big{(}\mathcal{X}_{\operatorname{fppf}},\mathcal{O}_{\mathcal{X}}\big{)}$ at
$x_{i}\in\mathcal{X}$ and $\mathcal{F}|_{x_{i}}$ is the restriction of
$\mathcal{F}$ to
$\big{(}\mathcal{X}_{\operatorname{fppf}}/x_{i},\mathcal{O}_{\mathcal{X}/x_{i}}\big{)}$.
We denote by $\operatorname{Qcoh}\mathcal{X}$ the category of quasi-coherent
$\mathcal{O}_{\mathcal{X}}$-modules. If $\mathcal{X}$ is an algebraic stack,
$\operatorname{Qcoh}\mathcal{X}$ is equivalent to the category of quasi-
coherent sheaves on the lisse-étale site of $\mathcal{X}$ defined in [14,
Definition 9.1.14]. In particular, if $\mathcal{X}$ is an algebraic space,
$\operatorname{Qcoh}\mathcal{X}$ is equivalent to the category of quasi-
coherent sheaves on the small étale site of $\mathcal{X}$ defined in [14,
Definition 7.1.5], and if $\mathcal{X}$ is a scheme, it is equivalent to the
category of usual quasi-coherent sheaves on the small Zariski site of
$\mathcal{X}$. If $X$ is an algebraic space, we write $X_{\text{\rm\'{e}t}}$
(resp. $X_{\operatorname{fppf}}$) for the small étale site of $X$ (resp. the
big fppf site $(\operatorname{Sch\\!}/X)_{\operatorname{fppf}}$ of $X$), and
sheaves on $X$ means sheaves on $X_{\text{\rm\'{e}t}}$. Similarly, sheaves on
a scheme $X$ means sheaves on the small Zariski site of $X$.

######  Definition A.1.

An ${\mathcal{X}}$-algebra is a sheaf of (not necessarily commutative) rings
$\mathcal{A}$ on $\mathcal{X}_{\operatorname{fppf}}$ together with a morphism
of sheaves of rings

$\displaystyle\uprho\colon\ \mathcal{O}_{\mathcal{X}}\to\mathcal{A}$

such that the $\mathcal{O}_{\mathcal{X}}$-module $\mathcal{A}$ is quasi-
coherent and the image of $\uprho$ is in the center of $\mathcal{A}$, i.e.,
for any $x\in\mathcal{X}$ the image of
$\uprho(x)\colon\mathcal{O}_{\mathcal{X}}(x)\to\mathcal{A}(x)$ is contained in
the center of the ring $\mathcal{A}(x)$.

If $(\mathcal{A},\uprho)$ is an $\mathcal{X}$-algebra, we have the restriction
by $\uprho$

$\displaystyle(-)_{\uprho}\colon\quad\operatorname{Mod}\mathcal{A}\to\operatorname{Mod}\mathcal{O}_{\mathcal{X}},\qquad\mathcal{M}\mapsto\mathcal{M}_{\uprho},$

where $\operatorname{Mod}\mathcal{A}$ denotes the category of right
$\mathcal{A}$-modules.

######  Definition A.2.

A right $\mathcal{A}$-module $\mathcal{M}$ is said to be quasi-coherent if the
$\mathcal{O}_{\mathcal{X}}$-module $\mathcal{M}_{\uprho}$ is quasi-coherent.
We denote by

$\displaystyle\operatorname{Qcoh}\mathcal{A}$

the full subcategory of $\operatorname{Mod}\mathcal{A}$ consisting of quasi-
coherent right $\mathcal{A}$-modules.

######  Remark A.3.

If $\mathcal{X}$ is an algebraic space, we tacitly consider
$\mathcal{X}$-algebras and quasi-coherent modules over $\mathcal{X}$-algebras
as sheaves on the small étale site $\mathcal{X}_{\text{\rm\'{e}t}}$ instead of
big fppf sheaves as above.

Let $\mathcal{A}$ be an $\mathcal{X}$-algebra, and denote by
$\widetilde{\mathcal{X}}$ the stackification of $\mathcal{X}$. By [19, Lemma
12.1; 06WQ], we have an equivalence of topoi

$\operatorname{Sh}(\mathcal{X}_{\operatorname{fppf}})\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Sh}\big{(}\widetilde{\mathcal{X}}_{\operatorname{fppf}}\big{)},$
(A.1)

where $\operatorname{Sh}(-)$ denotes the category of sheaves on a site $(-)$.
For an object
$\mathcal{F}\in\operatorname{Sh}(\mathcal{X}_{\operatorname{fppf}})$, we
denote by
$\widetilde{\mathcal{F}}\in\operatorname{Sh}\big{(}\widetilde{\mathcal{X}}_{\operatorname{fppf}}\big{)}$
the image of $\mathcal{F}$ by the equivalence (A.1). By this equivalence the
ring object
$\mathcal{A}\in\operatorname{Sh}(\mathcal{X}_{\operatorname{fppf}})$ defines a
ring object
$\widetilde{\mathcal{A}}\in\operatorname{Sh}\big{(}\widetilde{\mathcal{X}}_{\operatorname{fppf}}\big{)}$
and a morphism

$\displaystyle\widetilde{\uprho}\colon\
\mathcal{O}_{\widetilde{\mathcal{X}}}\to\widetilde{\mathcal{A}}.$

By [19, Lemma 12.2; 06WR], the sheaf of rings $\widetilde{\mathcal{A}}$ is
quasi-coherent $\mathcal{O}_{\widetilde{\mathcal{X}}}$-module, and so
$\widetilde{\mathcal{A}}$ is an $\widetilde{\mathcal{X}}$-algebra.

######  Lemma A.4.

The equivalence of topoi
$\operatorname{Sh}(\mathcal{X}_{\operatorname{fppf}})\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Sh}\big{(}\widetilde{\mathcal{X}}_{\operatorname{fppf}}\big{)}$
induces the equivalence

$\displaystyle\operatorname{Mod}\mathcal{A}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Mod}\widetilde{\mathcal{A}}$

of right modules, and it restricts to the equivalence of quasi-coherent
modules

$\displaystyle\operatorname{Qcoh}\mathcal{A}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Qcoh}\widetilde{\mathcal{A}}.$

###### Proof.

The first assertion is obvious, and for the second statement it is enough to
show that a right $\mathcal{A}$-module $\mathcal{M}$ is quasi-coherent if and
only if the right $\widetilde{\mathcal{A}}$-module $\widetilde{\mathcal{M}}$
is quasi-coherent. But this follows from the following commutative diagram

${\operatorname{Mod}\mathcal{A}}$${\operatorname{Mod}\widetilde{\mathcal{A}}}$${\operatorname{Mod}\mathcal{O}_{\mathcal{X}}}$${\operatorname{Mod}\mathcal{O}_{\widetilde{\mathcal{X}}}}$$\scriptstyle{\widetilde{(-)}}$$\scriptstyle{(-)_{\uprho}}$$\scriptstyle{(-)_{\widetilde{\uprho}}}$$\scriptstyle{\widetilde{(-)}}$

and [19, Lemma 12.2; 06WR]. ∎

Let $\mathcal{G}=(U,R,s,t,c)$ be a groupoid in algebraic spaces over $S$ (see
[19, Definition 11.1; 043W] for the notation).

######  Definition A.5.

An algebra over $\mathcal{G}$, or $\mathcal{G}$-algebra, is a $U$-algebra
$\mathcal{A}\colon
U_{\text{\rm\'{e}t}}^{\operatorname{op}}\to(\operatorname{Rings})$ together
with an isomorphism

$\displaystyle\uptheta^{\mathcal{A}}\colon\
s^{*}\mathcal{A}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,t^{*}\mathcal{A}$

of $R$-algebras such that the equations

$p_{1}^{*}\uptheta^{\mathcal{A}}\circ
p_{2}^{*}\uptheta^{\mathcal{A}}=c^{*}\uptheta^{\mathcal{A}}\qquad\text{and}\qquad
e^{*}\uptheta^{\mathcal{A}}=\operatorname{id}_{\mathcal{A}}$ (A.2)

of morphisms of sheaves of rings hold, where $p_{i}\colon R\times_{s,U,t}R\to
R$ is the $i$-th projection and $e\colon U\to R$ is the identity.

######  Remark A.6.

Note that the structure sheaf $\mathcal{O}_{U}$ together with the canonical
isomorphism

$\displaystyle\uptheta^{\operatorname{can}}\colon\
s^{*}\mathcal{O}_{U}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,t^{*}\mathcal{O}_{U}$

defines a $\mathcal{G}$-algebra
$(\mathcal{O}_{U},\uptheta^{\operatorname{can}})$. If
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ is a $\mathcal{G}$-algebra,
since $\uptheta^{\mathcal{A}}$ is $\mathcal{O}_{R}$-linear, the diagram

${s^{*}\mathcal{O}_{U}}$${\mathcal{O}_{R}}$${t^{*}\mathcal{O}_{U}}$${s^{*}\mathcal{A}}$${t^{*}\mathcal{A}}$$\scriptstyle{\sim}$$\scriptstyle{s^{*}\uprho}$$\scriptstyle{\uptheta^{\operatorname{can}}}$$\scriptstyle{\sim}$$\scriptstyle{t^{*}\uprho}$$\scriptstyle{\uptheta^{\mathcal{A}}}$

is commutative, where the isomorphisms on the top level are natural
isomorphisms.

######  Definition A.7.

Notation is the same as above.

  * 1.

A quasi-coherent module over a $\mathcal{G}$-algebra
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ is a quasi-coherent right
$\mathcal{A}$-module $\mathcal{M}\colon
U^{\operatorname{op}}_{\text{\rm\'{e}t}}\to(\operatorname{Sets})$ together
with an isomorphism

$\displaystyle\uptheta^{\mathcal{M}}\colon\
s^{*}\mathcal{M}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,t^{*}\mathcal{M}$

of quasi-coherent right $s^{*}\mathcal{A}$-modules such that the similar
equations as (A.2) hold, where the right $t^{*}\mathcal{A}$-module
$t^{*}\mathcal{M}$ is considered as a right $s^{*}\mathcal{A}$-module via the
isomorphism $\uptheta^{\mathcal{A}}\colon
s^{*}\mathcal{A}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,t^{*}\mathcal{A}$
of $R$-algebras.

  * 2.

Let $\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ and
$\big{(}\mathcal{N},\uptheta^{\mathcal{N}}\big{)}$ be quasi-coherent modules
over a $\mathcal{G}$-algebra
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$. A morphism from
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ to
$\big{(}\mathcal{N},\uptheta^{\mathcal{N}}\big{)}$ is a morphism

$\displaystyle\varphi\colon\ \mathcal{M}\to\mathcal{N}$

of right $\mathcal{A}$-modules such that the diagram

${s^{*}\mathcal{M}}$${s^{*}\mathcal{N}}$${t^{*}\mathcal{M}}$${t^{*}\mathcal{N}}$$\scriptstyle{s^{*}\varphi}$$\scriptstyle{\uptheta^{\mathcal{M}}}$$\scriptstyle{\uptheta^{\mathcal{N}}}$$\scriptstyle{t^{*}\varphi}$

is commutative.

We denote by

$\displaystyle\operatorname{Qcoh}\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$

the category of quasi-coherent modules over a $\mathcal{G}$-algebra
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$.

######  Remark A.8.

We have a natural identification

$\displaystyle\operatorname{Qcoh}\big{(}\mathcal{O}_{U},\uptheta^{\operatorname{can}}\big{)}=\operatorname{Qcoh}(U,R,s,t,c),$

where $\operatorname{Qcoh}(U,R,s,t,c)$ denotes the category of quasi-coherent
modules on $(U,R,s,t,c)$ in the sense of [19, Definition 12.1; 0441].

Let $\mathcal{G}=(U,R,s,t,c)$ be a groupoid in algebraic spaces over $S$.
Following [19, 044O] we denote by $[U/_{p}R]$ the category fibered in
groupoids associated to $\mathcal{G}$, which is defined as follows: An object
of $[U/_{p}R]$ is a pair $(T,u)$ of $S$-scheme $T$ and a morphism $u\colon
T\to U$ of algebraic spaces over $S$, and a morphism
$(f,\varphi)\colon(T,u)\to(T^{\prime},u^{\prime})$ is defined to be a pair of
a morphism $f\colon T\to T^{\prime}$ of $S$-schemes and a morphism
$\varphi\colon T\to R$ of algebraic spaces over $S$ such that
$s\circ\varphi=u$ and $t\circ\varphi=u^{\prime}\circ f$. Then the functor

$\displaystyle[U/_{p}R]\to(\operatorname{Sch\\!}/S),\qquad(T,u)\mapsto T$

is a category fibered in groupoids. We denote by $[U/R]$ the stackification of
$[U/_{p}R]$, which is called the quotient stack of $\mathcal{G}$. We have the
natural morphism

$\alpha\colon\ (\operatorname{Sch\\!}/U)\to[U/_{p}R]$ (A.3)

defined by $\alpha(u\colon T\to U)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(T,u)$
and $\alpha(f\colon T\to T^{\prime})\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=(f,e\circ
u)$, and it defines the functor

$\displaystyle\alpha^{-1}\colon\
\operatorname{Sh}\big{(}[U/_{p}R]_{\operatorname{fppf}}\big{)}\to\operatorname{Sh}(U_{\operatorname{fppf}})$

defined by $\alpha^{-1}\mathcal{F}(u\colon T\to U)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathcal{F}(\alpha(u\colon
T\to U))=\mathcal{F}(T,u)$.

For a $\mathcal{G}$-algebra
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$, we define a sheaf of rings

$\displaystyle[\mathcal{A}/_{p}R]\colon\
[U/_{p}R]_{\operatorname{fppf}}^{\operatorname{op}}\to(\operatorname{Rings})$

on $[U/_{p}R]_{\operatorname{fppf}}$ as follows: For an object
$(T,u)\in[U/_{p}R]$ we set

$\displaystyle[\mathcal{A}/_{p}R](T,u)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Gamma(T,u^{*}\mathcal{A}),$

and for a morphism $(f,\varphi)\colon(T,u)\to(T^{\prime},u^{\prime})$ we
define a morphism of rings

$\displaystyle[\mathcal{A}/_{p}R](f,\varphi)\colon\
[\mathcal{A}/_{p}R](T^{\prime},u^{\prime})\to[\mathcal{A}/_{p}R](T,u)$

by the following commutative diagram:

${\Gamma(T^{\prime},u^{\prime*}\mathcal{A})}$${\Gamma(T,u^{*}\mathcal{A})}$${\Gamma(T,f^{*}u^{\prime*}\mathcal{A})}$${\Gamma(T,\varphi^{*}t^{*}\mathcal{A})}$${\Gamma(T,\varphi^{*}s^{*}\mathcal{A}).}$$\scriptstyle{[\mathcal{A}/_{p}R](f,\varphi)}$$\scriptstyle{f^{*}}$$\scriptstyle{\sim}$$\scriptstyle{\varphi^{*}{\uptheta^{\mathcal{A}}}}$$\scriptstyle{\sim}$

Similarly, for a quasi-coherent module
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}$ over a
$\mathcal{G}$-algebra $\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$, we
define the right $[\mathcal{A}/_{p}R]$-module

$\displaystyle[\mathcal{M}/_{p}R]\colon\
[U/_{p}R]_{\operatorname{fppf}}^{\operatorname{op}}\to(\operatorname{Sets})$

by $[\mathcal{M}/_{p}R](T,u)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Gamma(T,u^{*}\mathcal{M})$,
and this defines an additive functor

$[(-)/_{p}R]\colon\
\operatorname{Qcoh}\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}\to\operatorname{Mod}[\mathcal{A}/_{p}R].$
(A.4)

By the equality $[\mathcal{O}_{U}/_{p}R]=\mathcal{O}_{[U/_{p}R]}$ and Remark
A.6, the morphism $\uprho\colon\mathcal{O}_{U}\to\mathcal{A}$ defines a
morphism from $(\mathcal{O}_{U},\uptheta^{\operatorname{can}})$ to
$\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$ in
$\operatorname{Qcoh}\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$, and
thus we have the induced morphism

$\displaystyle[\uprho/_{p}R]\colon\
\mathcal{O}_{[U/_{p}R]}\to[\mathcal{A}/_{p}R]$

of sheaves of rings.

######  Lemma A.9.

Notation is the same as above.

  * $1.$

If
$(\mathcal{M},\uptheta^{\operatorname{can}})\in\operatorname{Qcoh}(\mathcal{O}_{U},\uptheta^{\operatorname{can}})$,
then $[\mathcal{M}/_{p}R]\in\operatorname{Qcoh}[U/_{p}R]$.

  * $2.$

The sheaf $[\mathcal{A}/_{p}R]$ together with the ring homomorphism
$[\uprho/_{p}R]$ is a $[U/_{p}R]$-algebra.

  * $3.$

If
$\big{(}\mathcal{M},\uptheta^{\mathcal{M}}\big{)}\in\operatorname{Qcoh}\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$,
the right $[\mathcal{A}/_{p}R]$-module $[\mathcal{M}/_{p}R]$ is quasi-
coherent.

###### Proof.

(1) Let $\\{a_{i}\colon A_{i}\to U\\}_{i\in I}$ be an affine étale covering of
$U$. Since $\mathcal{M}$ is quasi-coherent $\mathcal{O}_{U}$-module, for each
$i\in I$ there is an exact sequence

$\bigoplus_{J}\mathcal{O}_{A_{i}}\to\bigoplus_{K}\mathcal{O}_{A_{i}}\to
a_{i}^{*}\mathcal{M}\to 0.$ (A.5)

For any object $(T,u)\in[U/_{p}R]$, set $T_{i}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=A_{i}\times_{U}T$
and write $t_{i}\colon T_{i}\to T$ for the second projection. If we set
$u_{i}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=u\circ
t_{i}\colon T_{i}\to U$ and $\varphi_{i}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=e\circ
u_{i}\colon T_{i}\to R$, then

$\displaystyle\bigl{\\{}(t_{i},\varphi_{i})\colon(T_{i},u_{i})\to(T,u)\bigr{\\}}_{i\in
I}$

is a covering of $(T,u)\in[U/_{p}R]_{\operatorname{fppf}}$. Since the
localized ringed site

$\displaystyle\big{(}[U/_{p}R]_{\operatorname{fppf}}/(T_{i},u_{i}),\mathcal{O}_{[U/_{p}R]_{\operatorname{fppf}}/(T_{i},u_{i})}\big{)}$

is equivalent to the ringed site
$\big{(}({\operatorname{Sch\\!}/T_{i}})_{\operatorname{fppf}},\mathcal{O}_{({\operatorname{Sch\\!}/T_{i}})_{\operatorname{fppf}}}\big{)}$
by [19, Lemma 9.4; 06W9], the pull-back of the exact sequence (A.5) by the
first projection $T_{i}\to A_{i}$ gives rise to an exact sequence

$\displaystyle\bigoplus_{J}\mathcal{O}_{[U/_{p}R]_{\operatorname{fppf}}/(T_{i},u_{i})}\to\bigoplus_{K}\mathcal{O}_{[U/_{p}R]_{\operatorname{fppf}}/(T_{i},u_{i})}\to\mathcal{M}|_{(T_{i},u_{i})}\to
0.$

(2) Since the image of $\uprho\colon\mathcal{O}_{U}\to\mathcal{A}$ is
contained in the center of $\mathcal{A}$, the image of

$\displaystyle[\uprho/_{p}R]\colon\mathcal{O}_{[U/_{p}R]}\to[\mathcal{A}/_{p}R]$

is contained in the center of $[\mathcal{A}/_{p}R]$. The
$\mathcal{O}_{[U/_{p}R]}$-module $[\mathcal{A}/_{p}R]$ is quasi-coherent by
(1).

(3) This follows from (1). ∎

By Lemma A.9, the functor (A.4) defines an additive functor

$\displaystyle[(-)/_{p}R]\colon\
\operatorname{Qcoh}\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}\to\operatorname{Qcoh}[\mathcal{A}/_{p}R].$

######  Proposition A.10.

The functor
$[(-)/_{p}R]\colon\operatorname{Qcoh}\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}\to\operatorname{Qcoh}[\mathcal{A}/_{p}R]$
is an equivalence.

###### Proof.

This follows from a similar argument as in the proof of [19, Proposition 13.1;
06WT]. We construct a quasi-inverse of the functor $[(-)/_{p}R]$ as follows.

First, for a sheaf $\mathcal{F}$ on $[U/_{p}R]_{\operatorname{fppf}}$, we
define a sheaf $\mathcal{F}^{R}$ on the small étale site
$U_{\text{\rm\'{e}t}}$ by

$\displaystyle\mathcal{F}^{R}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{(}\alpha^{-1}\mathcal{F}\big{)}|_{U_{\text{\rm\'{e}t}}},$

where $\alpha\colon(\operatorname{Sch\\!}/U)\to[U/_{p}R]$ is the morphism in
(A.3). By construction we have
$\mathcal{O}_{[U/_{p}R]}^{R}=\mathcal{O}_{U_{\text{\rm\'{e}t}}}$. For each
étale morphism $(u\colon T\to U)\in U_{\text{\rm\'{e}t}}$, we have

$\displaystyle[\mathcal{A}/_{p}R]^{R}(u\colon T\to
U)=\Gamma(T,u^{*}\mathcal{A})=\mathcal{A}(u\colon T\to U).$

Thus we have a natural identification $[\mathcal{A}/_{p}R]^{R}=\mathcal{A}$,
and so if $\mathcal{M}$ is a quasi-coherent right
$[\mathcal{A}/_{p}R]$-module, the small étale sheaf $\mathcal{M}^{R}$ is a
quasi-coherent $\mathcal{A}$-module.

Next, we define a category $\\{U/_{p}R\\}$ containing the category $[U/_{p}R]$
by replacing $S$-schemes $T$ in the definition of $[U/_{p}R]$ with algebraic
spaces over $S$. If $(f,\varphi)\colon(W,u)\to(W^{\prime},u^{\prime})$ is a
morphism in $\\{U/_{p}R\\}$, this morphism induces a natural (functorial)
commutative diagram

${W}$${W^{\prime}}$${\left[U/_{p}R\right].}$$\scriptstyle{f}$$\scriptstyle{\alpha\circ
u}$$\scriptstyle{\alpha\circ u^{\prime}}$

Then, for any quasi-coherent right $[\mathcal{A}/_{p}R]$-module $\mathcal{M}$,
we have the composition

$\displaystyle(\alpha\circ u^{\prime})^{-1}\mathcal{M}\to
f_{*}f^{-1}\big{(}(\alpha\circ
u^{\prime})^{-1}\mathcal{M}\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,f_{*}(\alpha\circ
u)^{-1}\mathcal{M},$

where the first morphism is the adjunction morphism and the second isomorphism
follows from the above commutative diagram. By the adjunction $f^{-1}\dashv
f_{*}$, this composition map induces the morphism $f^{-1}(\alpha\circ
u^{\prime})^{-1}\mathcal{M}\to(\alpha\circ u)^{-1}\mathcal{M}$ and restricting
this morphism to $W_{\text{\rm\'{e}t}}$, we have the comparison map

$\displaystyle c_{(f,\varphi)}\colon\ f^{*}u^{\prime*}\mathcal{M}^{R}\to
u^{*}\mathcal{M}^{R}$

and by [19, Lemma 11.6; 06WK] this is an isomorphism.

Finally, we define a functor
$F\colon\operatorname{Qcoh}[\mathcal{A}/_{p}R]\to\operatorname{Qcoh}\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}$
by

$\displaystyle F(\mathcal{M})\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\big{(}\mathcal{M}^{R},\uptheta^{\mathcal{M}^{R}}\big{)},$

where $\uptheta^{\mathcal{M}^{R}}\colon
s^{*}\mathcal{M}^{R}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,t^{*}\mathcal{M}$
is defined to be the inverse of the comparison map

$\displaystyle c_{(\operatorname{id}_{R},\operatorname{id}_{R})}\colon\
t^{*}\mathcal{M}^{R}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,s^{*}\mathcal{M}^{R}$

induced by the morphism
$(\operatorname{id}_{R},\operatorname{id}_{R})\colon(R,s)\to(R,t)$ in
$\\{U/_{p}R\\}$. By construction, $F$ is a quasi-inverse of the functor
$[(-)/_{p}R]$. ∎

Write

$\displaystyle[\mathcal{A}/R]\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\widetilde{[\mathcal{A}/_{p}R]}$

for the sheaf of rings on the quotient stack $[U/R]$ corresponding to
$[\mathcal{A}/_{p}R]$ via the equivalence (A.1). By Lemma A.4 and Proposition
A.10, we have the following generalization of [19, Proposition 13.1; 06WT].

######  Proposition A.11.

We have an equivalence

$\displaystyle\operatorname{Qcoh}\big{(}\mathcal{A},\uptheta^{\mathcal{A}}\big{)}\,\,\,\,\raisebox{4.0pt}{$\sim$}{\kern-11.00008pt\longrightarrow}\,\operatorname{Qcoh}[\mathcal{A}/R].$

### Acknowledgements

The author is supported by JSPS KAKENHI 19K14502. Part of this work was done
during his stay at the Max Planck Institute for Mathematics in Bonn, in the
period from April to December 2019. The author gratefully acknowledge MPIM
Bonn for their support and hospitality. Furthermore, the author would like to
thank the referees for their careful reading and valuable suggestions.

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[figure]style=plain,subcapbesideposition=top

# Experimental violation of the Leggett-Garg inequality with a single-spin
system

Maimaitiyiming Tusun  CAS Key Laboratory of Microscale Magnetic Resonance and
School of Physical Sciences, University of Science and Technology of China,
Hefei 230026, China  CAS Center for Excellence in Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei 230026,
China  School of Physics and Electronic Engineering, Xinjiang Normal
University, Urumqi 830054, China  Wei Cheng  CAS Key Laboratory of Microscale
Magnetic Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Zihua Chai  CAS Key Laboratory of Microscale
Magnetic Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Yang Wu  CAS Key Laboratory of Microscale Magnetic
Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Ya Wang  CAS Key Laboratory of Microscale Magnetic
Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Xing Rong<EMAIL_ADDRESS>CAS Key Laboratory of
Microscale Magnetic Resonance and School of Physical Sciences, University of
Science and Technology of China, Hefei 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei 230026, China  Jiangfeng Du<EMAIL_ADDRESS>CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical
Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China

###### Abstract

Investigation the boundary between quantum mechanical description and
classical realistic view is of fundamental importance. The Leggett-Garg
inequality provides a criterion to distinguish between quantum systems and
classical systems, and can be used to prove the macroscopic superposition
state. A larger upper bound of the LG function can be obtained in a multi-
level system. Here, we present an experimental violation of the Leggett-Garg
inequality in a three-level system using nitrogen-vacancy center in diamond by
ideal negative result measurement. The experimental maximum value of Leggett-
Garg function is $K_{3}^{exp}=1.625\pm 0.022$ which exceeds the
L$\mathrm{\ddot{u}}$ders bound with a $5\sigma$ level of confidence.

The objects we can see with our naked eye are composed of a very large number
of atoms, and these atoms can be in a quantum superposition state. If the laws
of quantum mechanics are extrapolating to the laws satisfied by everyday
objects, there is inevitably the prospect of macroscopic coherence.
Schr$\mathrm{\ddot{o}}$dinger’s cat, simultaneously both dead and alive, is
considered to be one of the most characteristic examples of macroscopic
coherence. This situation runs counter to our intuitive understanding of how
the daily macroworld works.

Leggett and Garg gave a method to verify whether a given system can be in a
superposition of discrete statesLeggett_1985 . They proposed two assumptions
that the classical world certainly satisfiedLeggett_2002 ; Leggett_2008 : (i)
_macroscopic realism_ (MR)— that a system cannot be in a superposition of the
classically observable state and the measurement of the object will get a
certain value; (ii) _Non-invasive measurability_ (NIM)—that it is possible in
principle to determine the state of a system with arbitrarily precision
without destroying its subsequent evolution. Classical world meet the
requirements of these assumptions, but for quantum mechanics, neither of these
two assumptions can be satisfied. The Leggett-Garg inequality (LGI) is a test
of MR under a set of reasonable assumptions about the system, in particular a
version of NIM. The violation of the inequality leads to the conclusion that
either MR or the other assumptions is wrongMaroney_2014 . In this way, the LGI
provide a method to investigate the existence of macroscopic coherence and to
test the applicability of quantum mechanics as we scale from microscopic to
macroscopic worldLeggett_2002 .

Until recently it has been believed that the maximal violation of three-time
measurement LGI, which characterized the correlation between the results of
this measurements, is independent of the number of possible macroscopically
distinct states of the system. This is due to the fact that the measurements
are dichotomicKatiyar_2017 . However, Budroni and EmaryBudroni_2014  showed
that this is only true if the measurement follows the naive
L$\mathrm{\ddot{u}}$ders update ruleluder_1951 . According to
L$\mathrm{\ddot{u}}$ders rule, the state is updated as
$\rho\rightarrow\Pi_{\pm}\rho\Pi_{\pm}$ depending on the outcome of the
measurement, where $\Pi_{+}$ and $\Pi_{-}$ are two projectors on eigenspaces
associated with measurement results $Q_{+}=1$ and $Q_{-}=-1$. In a more
general case, one can observe greater violations by using a higher-dimensional
system. Therefore, striving for a larger violation value in a higher-
dimensional system is worth studying. The quantum upper bound of the LGI
depends on both the details of the measurement and the dimension of the
systemKatiyar_2017 . For the LGI experiment performed on the three-level
system, a higher theoretical quantum upper bound can be obtained than that on
the two-level systemPalacios-Laloy_2010 ; Katiyar_2013 ; Dressel_2011 . With
interest in the beyond quantum upper bound of the LGI, the NMR
systemKatiyar_2017  and single photonWang_2017  are used to mimic the three-
level system. There is also a work performed in a single three-level
systemGeorge_2013 . However, this work used the projective measurement which
is invasive measurement method and fails to convince a macrorealist. Achieving
the violation of the LGI which breaks through the L$\mathrm{\ddot{u}}$ders
bound in a single three-level system remains elusive. Here, we report an
experimental violation of LGI with the exceeding of L$\mathrm{\ddot{u}}$ders
bound in a real three-level system. Our experiment is performed in a single
nitrogen-vacancy (NV) center in diamond. The Ideal Negative Result Measurement
(INRM) method is implemented to avoid invasive measurementsKatiyar_2017 ;
Knee_2012 .

The LGI test the correlations of a single system measured at different times.
The most frequently encountered inequalities concern the n-measurement LG
stringsBudroni_2014

$\begin{split}K_{n}=\langle Q(t_{2})Q(t_{1})\rangle+\langle
Q(t_{3})Q(t_{2})\rangle+\langle Q(t_{4})Q(t_{3})\rangle\\\
+\cdot\cdot\cdot+\langle Q(t_{n})Q(t_{n-1})\rangle-\langle
Q(t_{n})Q(t_{1})\rangle.\end{split}$ (1)

Here, $Q$ is a macroscopic dichotomic variable with distinct outcomes $q$, at
some time $t_{i}$, the outcomes of these measurements are denoted as
$q_{i}^{(1)}=+1$ and $q_{i}^{(2)}=-1$, $n$ denotes the number of time steps.
In a macrorealistic system, the outcomes $q_{i}^{(l)}(l=1,2)$ represent the
two distinct sates of the system. The correlation function is $\langle
Q_{i}Q_{j}\rangle=\sum_{l,m}q_{i}^{(l)}q_{j}^{(m)}P(q_{i}^{(l)},q_{j}^{(m)})$,
where $P(q_{i}^{(l)},q_{j}^{(m)})$ is the joint probability to obtain $q_{i}$
and $q_{j}$ at times $t_{i}$ and $t_{j}$. The n-measurement LGIs are bounded
as follows. If $n\geq 3$ and odd then $-n\leq K_{n}\leq n-2$, and if $n\geq 4$
and even then $-(n-2)\leq K_{n}\leq n-2$Budroni_2014 . In the typical three-
time measurement LGI scenario as shown in Fig. 1(a), the LG function is given
byEmary_2014

$K_{3}=\langle Q(t_{2})Q(t_{1})\rangle+\langle Q(t_{3})Q(t_{2})\rangle-\langle
Q(t_{3})Q(t_{1})\rangle.$ (2)

More specifically, when the measurements performed to the system do not
disturb the system (NIM assumption) and the system is classical
(macrorealistic), the value of $K_{3}$ is bounded by $-3\leq K_{3}\leq 1$. On
the other hand, when the system is quantum, it is possible to choose the
evolution time between measurements in such a way that $K_{3}$ will go beyond
1, violating the LGI that $K_{3}\leq 1$Emary_2014 . Quantum mechanical
experiments are able to violate three-time measurement LGI up to 1.5 for a
qubit with dichotomic measurements $q_{i}^{(1,2)}=\pm 1$. More general systems
have the same bound when the measurements follow the L$\mathrm{\ddot{u}}$ders
update rule. However, if one relaxes the assumption that the measurement
follows the L$\mathrm{\ddot{u}}$ders update rule, the quantum bound on $K_{3}$
could be extended to a value that depends on the dimension of the
systemBudroni_2014 . For a three-level system, the measurement operators are
$\Pi_{1}=|1\rangle_{n}$$\langle 1|$, $\Pi_{0}=|0\rangle_{n}$$\langle 0|$ and
$\Pi_{-1}=|\textnormal{-}1\rangle_{n}$$\langle\textnormal{-}1|$ corresponding
to the three states of system. $\Pi_{1}$ and $\Pi_{0}$ are associated with
outcome $q^{(1)}=q^{(0)}=1$. $\Pi_{-1}$ is associated with outcome
$q^{(-1)}=-1$. The quantum state after the measurement is updated as
$\rho\rightarrow\Pi_{1}\rho\Pi_{1}+\Pi_{0}\rho\Pi_{0}$ or
$\Pi_{-1}\rho\Pi_{-1}$ depending on whether the measurement result is $+1$ or
$-1$. This protocol is different from the L$\rm{\ddot{u}}$ders rule and can
lead to the value $K_{3}=1.756$ in the ideal case.

Figure 1: (a) Three-time measurement of LG test, in this scenario, the
initialization can be regard as the first measurement. (b) General scheme for
the LG test with INRM. The state of the system is initialized to
$|\psi(t_{1})\rangle$, and the ancilla qubit is initialized to $|0\rangle$.
After first and second evolution of $U(\theta)$, the state of the system is
evolved to $|\psi(t_{2})\rangle$ and $|\psi(t_{3})\rangle$, corresponding to
observable dichotomic variables $Q(t_{2})$ and $Q(t_{3})$. INRM is implemented
by controlled gate (CG) and the postselecting the final state populations
corresponding to the non-flipping of the ancilla qubit.

Ideal negative result measurement is one of candidates to achieve non-invasive
measurement. Measurement process can be regarded as an interaction process
between the system to be measured and the apparatus. For instance, a detector
couples with one of system states, registering a ‘click’ if the system is in
the specific state, but it does not interact with the other states. If the
detector clicks, the result is discarded, and if it does not click, we keep
the result. In this way, we can infer system states without any interaction
between the system to be measured and the apparatus by keeping only the
negative (no click) results. In our scheme, the INRM scheme is realized by
controlled gates and the postselection of the ancilla qubit state as depicted
in Fig. 1(b). The ancillary qubit is initialized in the state $|0\rangle$. The
controlled gates flip the ancilla qubit except when the three-level system is
in one of the three states. Postselecting the situation that the ancilla qubit
has not been flipped, the non-invasiveness of the measurement can be
guaranteed.

Figure 2: (a) Atomic structure of NV center. (b) Energy levels of the NV
center in the ground state. The $m_{s}$ and $m_{I}$ are the quantum states of
electron spin and nuclear spin, respectively. MW1, MW2 and MW3 (blue arrows)
are three selective microwave pulses to drive corresponding electron spin
transitions. RF1 and RF2 (orange arrows) are two radio frequency pulses to
drive transitions of the nuclear spin. (c) Experimental pulses sequence to
realize INRM. The $\rho_{i}$ and $\rho_{fj}$, where $j=1,2,3,4$, are used to
denote the initial and final state in the experiments. Controlled gate (CG) is
realized through selective microwave pulses. Three-level Rabi rotation
$U(\theta)$ on nuclear spin is realized by applying RF1 and RF2
simultaneously.

Our experiment was implemented on an NV center in diamond. As depicted in Fig.
2(a), the NV center consists of a substitutional nitrogen atom with an
adjacent vacancy site in the diamond crystal lattice. The three energy levels
of 14N nuclear spin, $|1\rangle_{n}$, $|0\rangle_{n}$ and $|-1\rangle_{n}$,
make up the three-level system. Two NV electron spin energy levels,
$|1\rangle_{e}$ and $|0\rangle_{e}$, are chosen as an ancilla qubit. The
Hamiltonian of the NV center is
$H_{NV}=2\pi(DS_{z}^{2}+\omega_{e}S_{z}+QI_{z}^{2}+\omega_{n}I_{z}+AI_{z}S_{z})$Wu_2018
, where $S_{z}$ and $I_{z}$ are corresponding spin operators of the electron
and nuclear spin, $D=2.87\,\rm{GHz}$ is the zero-field splitting of electron
spin, $Q=-4.95\,\rm{MHz}$ is the nuclear quadrupolar interaction, and
$A=-2.16\,\rm{MHz}$ is the hyperfine interaction. The experiment is performed
on an optically detected magnetic resonance setup. A magnetic field of 512 G
is applied along the NV symmetry axis ([111] crystal axis), yielding Zeeman
frequencies $\omega_{e}$ and $\omega_{n}$. The total system utilized in our
experiment is spanned by six energy levels, consisting of
$|1\rangle_{e}\\!|1\rangle_{n}$, $|1\rangle_{e}\\!|0\rangle_{n}$,
$|1\rangle_{e}\\!|\textnormal{-}1\rangle_{n}$,
$|0\rangle_{e}\\!|1\rangle_{n}$, $|0\rangle_{e}\\!|0\rangle_{n}$ and
$|0\rangle_{e}\\!|\textnormal{-}1\rangle_{n}$, relabeled by $|1\rangle$,
$|2\rangle$, $|3\rangle$, $|4\rangle$, $|5\rangle$ and $|6\rangle$ as depicted
in Fig. 2(b).

The INRM is performed by the selective microwave pulses and postselecting the
populations of the final states. The two microwave pulses, as shown in Fig.
2(c), realize the controlled gate which flips the ancilla qubit when the
system in two of the three states (see Appendix A for detailed discussion).
The three experiments correspond to the situation where the system is in three
different states to ensure that the ancilla qubit is not flipped. The
transition frequency of the microwave pulses is about 4.303 GHz, which is much
different from the nuclear spin transition frequency on the megahertz order.
These microwave pulses hardly flip the state of the nuclear spin. The non-
invasiveness of the measurement is ensured by the postselecting the
populations of the final states which indicate the case that the ancilla qubit
is not flipped. The NV center is initialized to the state $|4\rangle$ via
laser pulsesJacques_2009 . The postselection procedure is implemented by
preserving the population of $|4\rangle$, $|5\rangle$ and $|6\rangle$ in the
final states. Although the pulses in experiments will cause unwanted
transitions due to finite off-resonance, these transitions can be removed
through the postselection procedure. To verify the LG inequality, two time
correlations of the measurements, $\langle Q(t_{i})Q(t_{j})\rangle$, need to
be obtained. Regarding the initialization as the first measurement, thus
$Q(t_{1})=1$, the tested LG inequality is reduced to $\langle Q(t_{2})\rangle$
\+ $\langle Q(t_{2})Q(t_{3})\rangle$ – $\langle Q(t_{3})\rangle\leq 1$. The
value of $\langle Q(t_{2})\rangle$ can be obtained from joint probability of
measurement outcomes $P(q_{2}^{(l)},q_{3}^{(m)})$ when the controlled gates
are performed. The value $\langle Q(t_{3})\rangle$ can be obtained from
directly measurement without controlled gates intersected. The relationships
between three terms in LG function and final state populations are shown below
(see Appendix B for detailed calculations):

$\displaystyle\left\\{\begin{array}[]{lll}\langle
Q(t_{2})\rangle&=P_{4}^{1}+P_{5}^{1}+P_{6}^{1}+P_{4}^{2}+P_{5}^{2}+P_{6}^{2}\\\
&-P_{4}^{3}-P_{5}^{3}-P_{6}^{3}\\\ \langle
Q(t_{2})Q(t_{3})\rangle&=P_{4}^{1}+P_{5}^{1}-P_{6}^{1}+P_{4}^{2}+P_{5}^{2}-P_{6}^{2}.\\\
&-P_{4}^{3}-P_{5}^{3}+P_{6}^{3}\\\ \langle
Q(t_{3})\rangle&=P_{4}^{4}+P_{5}^{4}-P_{6}^{4}\end{array}\right.$ (8)

where $P_{i}^{j}$ denotes the population of ith energy level
$\left.|i\right\rangle$ of the final state $\rho_{fj}$ in Fig. 2(c), where
$j=1,2,3,4$.

Figure 3:  Dephasing time of NV center. $P_{0}$ is the population of the
electron spin state $|0\rangle_{e}$. The red line is fitted to the experiment
data (black points with error bars). The decay time, $T_{2}^{*}$, of the free
induction decay is measured to be 62(7) $\mathrm{\mu s}$. The error bars on
the data points are the standard deviation from the mean. Figure 4: (a)
Experimental results, simulation results and theoretical predictions of
$\langle Q_{2}\rangle$, $\langle Q_{2}Q_{3}\rangle$ and $\langle
Q_{3}\rangle$. (b) Experimental result (light blue bar), simulation result
(green bar) and theoretical prediction (red bar) of the LG function $K_{3}$.
The L$\mathrm{\ddot{u}}$ders bound (black line) is 1.5. The theoretical
maximum violation in a three-level system is $K_{3}=1.756$ when
$\theta=0.416\pi$. Considering dephasing noise and imperfect initialization of
ancilla qubit, the corresponding simulation result is $K_{3}^{sim}=1.632$ and
the experimental result is $K_{3}^{exp}=1.625\pm 0.022$. The error bar of the
experimental result is the standard deviation from the mean.

The evolution of the rotation on the three-level system is
$U(\theta)=e^{-i\theta S_{x}}$, where $\theta=\sqrt{2}\pi f_{rabi}t$ and
$S_{x}$ is spin operator of spin-1 system. The operation can be realized in
rotating frame by applying two radio frequency pulses simultaneously on the
nuclear spin. The frequencies of the two pulses are $\omega_{45}$ , the
resonance frequency between level $|4\rangle$ and $|5\rangle$, and
$\omega_{56}$, the resonance frequency between level $|5\rangle$ and
$|6\rangle$. $f_{rabi}$ is the Rabi frequency which is proportional to
strength of radio frequency pulses, which is set to $20\,\rm{kHz}$. In
theoryBudroni_2014 ,

$\displaystyle\left\\{\begin{array}[]{lll}\langle
Q_{2}\rangle=\frac{1}{2}+\rm{cos}(\theta)-\frac{1}{4}\rm{cos}(2\theta)\\\
\langle
Q_{2}Q_{3}\rangle=\frac{1}{2}+\rm{cos}(2\theta)-\frac{1}{4}\rm{cos}(4\theta).\\\
\langle
Q_{3}\rangle=\frac{1}{16}+\rm{cos}(\theta)-\frac{1}{16}\rm{cos}(4\theta)\end{array}\right.$
(12)

The maximal value of $K_{3}$ should be obtained when $\theta=0.416\pi$,
corresponding evolution time under the Hamiltonian is $14.71\,\mathrm{\mu s}$.
A major factor affecting the experimental results is the dephasing noise of
the electron spin. Therefore, we utilized an isotopically purified
([12C]=99.999%) sample to implement our experiments. The dephasing time of the
electron spin $T_{2}^{*}$ is measured to be 62(7) $\mathrm{\mu s}$ as shown in
Fig. 3. Under this condition, the experimentally obtained values of $\langle
Q_{2}\rangle$, $\langle Q_{2}Q_{3}\rangle$ and $\langle Q_{3}\rangle$ for
$\theta=0.416\pi$ are shown in Fig. 4(a). The corresponding values of $K_{3}$
are shown in Fig. 4(b), where the red (left) bar is the ideal prediction
according to Eq. (12), the green (middle) bar is the simulation result with
dephasing noise and initialization imperfection, and the light blue (right)
bar is the experimental result. At the point of the maximum violation, the
experimental result is $K_{3}^{exp}=1.625\pm 0.022$. The error,
$\sigma=0.022$, is the standard deviation from the mean. This result exceeds
the L$\mathrm{\ddot{u}}$ders bound of 1.5 with a $5\sigma$ level of
confidence. Considering dephasing noise and imperfect initialization of
ancilla qubit, the corresponding simulation result is $K_{3}^{sim}=1.632$ and
it is agrees with the experimental result (see Appendix C for detailed
calculations).

In summary, we have experimentally demonstrated violation of LGI in a single
spin three-level system beyond the L$\mathrm{\ddot{u}}$ders bound using the
INRM scheme. The difference in experimental result from theoretical value is
due to dephasing noise and imperfection of initialization of ancilla qubit. To
improve the experimental result, the effect of dephasing noise can be
suppressed by using the sample with long coherence time and the quantum
control which can resist noise. Some other quantities such as the quantum
witness can be measured for the further investigation of macrorealist
theoriesSchild_2015 . We hope that our results in three-level system can
extend to large scale system, paving the way to looking for macroscopic
superposition state in high-dimensional system.

This work was supported by the National Key R$\&$D Program of China (Grants
No. 2018YFA0306600 and No. 2016YFB0501603), the NNSFC (No. 81788101), the
Chinese Academy of Sciences (Grants No. QYZDY-SSW-SLH004 and No. QYZDB-SSW-
SLH005), and Anhui Initiative in Quantum Information Technologies (Grant No.
AHY050000). X. R. thanks the Youth Innovation Promotion Association of Chinese
Academy of Sciences for the support. Ya Wang thanks the Fundamental Research
Funds for the Central Universities for the support.

##  I Appendix A: Characterizing the flip probability of the electron spin
when nuclear spin is in a flip state

We implemented experiments to show the probability $p$ of the ancilla qubit
flipped when nuclear spin is in a flip state. Before applying the controlled
gate, the electron spin is initialized to the $|m_{S}=0\rangle$ state. Thus
the corresponding $|m_{S}=0\rangle$ state population $P_{0}$ is 1. When
sweeping the frequency of the microwave pulse, we observed three peaks as
shown in Fig. 5a corresponding to three different states of the nuclear spin.
Fig. 5b shows the results after implementing the controlled gate.

Compared to the simulation results, our results are quite different from the
cases with probability 0.6 and 0.8, and agree well with the case of
probability 1. In order to obtain a more precise probability $p$, we performed
an additional experiment which applied the controlled gates repeatedly. The
dynamics of the quantum state was recorded. The population of
$|m_{S}=0\rangle$ state after repeated applications of controlled gates is
shown in Fig. 6. The probability p of $0.995\pm 0.002$ can be obtained by
linear fitting.

Figure 5: (a) The optically detected magnetic resonance (ODMR) spectrum of the
electron spin before applying the controlled gate. Three peaks correspond to
three different states of the nuclear spin. (b) The ODMR spectrum after
applying the controlled gate. Black points with error bars is the experimental
results. The red solid, blue dotted and green dashed lines correspond to the
simulation results when the electron spin flip probabilities $p$ are 0.6, 0.8
and 1, respectively. Figure 6: The $|m_{S}=0\rangle$ state population $P_{0}$
after applying controlled gates (CGs) repeatedly. The fitting (red line)
agrees with the experimental data (black points with error bars). The error
bars on the data points are the standard deviations from the mean.

##  II Appendix B: LG function obtained from the populations of the final
states

At $t_{2}$, three controlled gates are applied corresponding to three nuclear
spin states. We postselect the ancilla qubit to remain in the $|0\rangle_{e}$
state, which is not flipped. Therefore, the expectation value at $t_{2}$ is

$\displaystyle\langle Q(t_{2})\rangle$
$\displaystyle=q_{2}^{(1)}(P_{4}^{1}+P_{5}^{1}+P_{6}^{1})+q_{2}^{(0)}(P_{4}^{2}+P_{5}^{2}+P_{6}^{2})$
(B1) $\displaystyle+q_{2}^{(-1)}(P_{4}^{3}+P_{5}^{3}+P_{6}^{3}).$

The correlated expectation value is

$\displaystyle\langle Q(t_{2})Q(t_{3})\rangle$
$\displaystyle=q_{2}^{(1)}q_{3}^{(1)}P_{4}^{1}+q_{2}^{(1)}q_{3}^{(0)}P_{5}^{1}+q_{2}^{(1)}q_{3}^{(-1)}P_{6}^{1}$
(B2)
$\displaystyle+q_{2}^{(0)}q_{3}^{(1)}P_{4}^{2}+q_{2}^{(0)}q_{3}^{(0)}P_{5}^{2}+q_{2}^{(0)}q_{3}^{(-1)}P_{6}^{2}$
$\displaystyle+q_{2}^{(-1)}q_{3}^{(1)}P_{4}^{3}+q_{2}^{(-1)}q_{3}^{(0)}P_{5}^{3}+q_{2}^{(-1)}q_{3}^{(-1)}P_{6}^{3}.$

In order to obtain $\langle Q(t_{3})\rangle$, there is no need to apply the
controlled gates at $t_{2}$. We still postselect the state where the ancilla
qubit does not flip. Therefore, the expectation value at $t_{3}$ is

$\langle
Q(t_{3})\rangle=q_{3}^{(1)}P_{4}^{4}+q_{3}^{(0)}P_{5}^{4}+q_{3}^{(-1)}P_{6}^{4}.$
(B3)

##  III Appendix C: Imperfections Considered in the Simulation Results

The imperfect polarization and the decoherence of the electron spin contribute
to the difference between ideal and simulation results. At the magnetic field
strength of 512 G, the spin state of the NV center is effectively polarized to
$|m_{S}=0,m_{I}=+1\rangle$. The polarization of electron spin and nuclear spin
are 95% and 98%. Our experiment was implemented on a NV center in diamond
which was isotopically purified ([12C]=99.999%). The dephasing time of the
electron spin, $T_{2}^{*}$ is measured to be 62(7) $\mathrm{\mu s}$. The
experiments suffered from the dephasing noise during pulse sequences. There
will be one more dephasing noise term, $H_{noise}=2\pi\delta_{0}S_{z}$, in the
practice Hamiltonian. The dephasing noise, $\delta_{0}$, mainly comes from the
Overhauser field (due to the interaction with the nuclear spin bath), the
magnetic field fluctuation and the instability of the microwave frequency. The
$\delta_{0}$ satisfies a Gaussian distribution
$P_{0}(\delta_{0})=\exp(-\delta_{0}^{2}/2\sigma^{2})/(\sigma\sqrt{2\pi})$ with
$\sigma=1/(\sqrt{2}\pi T_{2}^{*})$Rong_2015 . The simulation results are
obtained with the consideration of initial polarization and the dephasing
noise distribution $P_{0}(\delta_{0})$.

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