2.1 Differential double graded algebras

The aim of this section is to introduce differential double graded (ddg) algebras which are differential graded algebras graded by $\mathbb {Z} \times \mathbb {Z}$. For convenience, we identify $\mathbb {Z} \times \mathbb {Z}$ with the rank two free module $\mathbb {Z} \oplus \mathbb {Z} \mathbb {X}$ spanned by the basis $1$ and $\mathbb {X}$. Thus, a pair of integers $(m,l) \in \mathbb {Z} \times \mathbb {Z}$ is identified with $m+l \mathbb {X} \in \mathbb {Z} \oplus \mathbb {Z} \mathbb {X}$.

A ddg-algebra $A$ is a graded algebra

\[ A :=\bigoplus_{m,l \in \mathbb{Z}}A^{m+l\mathbb{X}} \]

graded by $\mathbb {Z} \oplus \mathbb {Z} \mathbb {X}$ with the differential $\operatorname {d} \colon A^{m+l\mathbb {X}} \to A^{m+1+l\mathbb {X}}$ of degree $1$. We can also define ddg-modules over $A$ similar to usual dg-modules. For ddg-module $M=\bigoplus _{m,l}M^{m+l \mathbb {X}}$ with the differential $\operatorname {d}$, the degree shift $M^{\prime }=M[k+j\mathbb {X}]$ is defined by

\[ (M^\prime)^{m+l\mathbb{X}}:=M^{m+k+(l+j)\mathbb{X}} \]

with the new differential $\operatorname {d}^{\prime }:=(-1)^k \operatorname {d}$.

Let $\operatorname {\mathcal {D}}(A)$ be the derived category of ddg-modules over $A$. The perfect derived category $\operatorname {per} A \subset \operatorname {\mathcal {D}}(A)$ is the smallest full triangulated subcategory containing $A$ and closed under taking direct summands.

2.2 Calabi–Yau algebras and Calabi–Yau categories

In this section, we recall the definition of Calabi–Yau algebras. As in the previous section, let $A$ be a ddg-algebra. The enveloping algebra of $A$ is defined by $A^{\rm e}:=A \otimes A^{{\rm op}}$, where $A^{\mathrm{op}}$ is the opposite algebra. The ddg-algebra $A$ has the $A^{\rm e}$-module structure given by the two-sided action of $A$.

A ddg-algebra $A$ is called homologically smooth if

\[ A \in \operatorname{per} A^{\rm e}. \]

Let $A$ be a homologically smooth ddg-algebra. The inverse dualizing complex $\Theta _A$ is defined by the cofibrant replacement of

\[ \operatorname{RHom}_{A^{\rm e}}(A,A^{\rm e}) \]

considered as an object of $\operatorname {\mathcal {D}}(A^{\rm e})$,

We treat $\Theta _A$ as an object in $\operatorname {per} A^{\rm e}$. Denote by $\operatorname {\mathcal {D}}_{\rm fd}(A)$ the full subcategory of $\operatorname {\mathcal {D}}(A)$ consisting of ddg-modules with finite-dimensional total cohomology, i.e.

\[ \operatorname{\mathcal{D}}_{\rm fd}(A):=\bigg\{M \in \operatorname{\mathcal{D}}(A) \biggm| \sum_k \dim H^k(M)<\infty \bigg\}. \]

By using $\Theta _A$, the Serre duality is described as follows.

Lemma 2.1 [Reference KellerKel11a, Lemma 3.4]

Let $A$ be a homologically smooth ddg-algebra and $\Theta _A$ be its inverse dualizing complex. Then, for any ddg-modules $M,N \in \operatorname {\mathcal {D}}_{\rm fd}(A)$, there is a canonical isomorphism

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}(A)}(M \otimes_A \Theta_A, N) \xrightarrow{\sim} \mathrm{D}\operatorname{Hom}_{\operatorname{\mathcal{D}}(A)}(N,M). \]

The following Calabi–Yau property of a ddg-algebra is due to Ginzburg and Kontsevich [Reference GinzburgGin06, Definition 3.2.3].

Definition 2.2 Let $\mathcal {N} \in \mathbb {Z}\oplus \mathbb {Z}\mathbb {X}$. A ddg-algebra $A$ is called a Calabi–Yau- $\mathcal {N}$ algebra if $A$ is homologically smooth and

\[ \Theta_A \xrightarrow{\sim} A[-\mathcal{N}] \]

in $\operatorname {per} A$.

A triangulated category $\operatorname {\mathcal {D}}$ is called Calabi–Yau- $\mathcal {N}$ (CY-$\mathcal {N}$) if, for any objects $X,Y$ in $\mathcal {D}$ we have a natural isomorphism

(2.1)\begin{equation} \mathfrak{S}:\operatorname{Hom} (X,Y) \xrightarrow{\sim} \mathrm{D}\operatorname{Hom} (Y,X[\mathcal{N}]). \end{equation}

If $A$ is a Calabi–Yau-$\mathcal {N}$ algebra, then $\operatorname {\mathcal {D}}_{\rm fd}(A)$ becomes a Calabi–Yau-$\mathcal {N}$ category.

2.3 Calabi–Yau-$\mathbb {X}$ completions for ddg-algebras, following Keller

As in the previous section, we consider ddg-algebras indexed by $\mathbb {Z} \oplus \mathbb {Z}\mathbb {X}$.

Definition 2.3 Let $A$ be a homologically smooth ddg-algebra and $\Theta _A$ be its inverse dualizing complex. The Calabi–Yau- $\mathbb {X}$ completion of $A$ is defined by

\[ \Pi_{\mathbb{X}}(A):=\mathrm{T} _A(\theta)= A \oplus \theta \oplus (\theta \otimes_A \theta) \oplus \cdots, \]

where $\theta :=\Theta _A[\mathbb {X}-1]$ and $\mathrm {T} _A(\theta )$ is the tensor algebra of $\theta$ over $A$.

Keller's result can be adopted in this setting.

We also have the following.

Lemma 2.5 [Reference KellerKel11a, Lemma 4.4]

The canonical projection (on the first component) $\Pi _{\mathbb {X}}(A)\to A$ induces an Lagrangian-$\mathbb {X}$ immersion

\[ \mathcal{L}\colon \operatorname{\mathcal{D}}_{\rm fd}(A)\to\operatorname{\mathcal{D}}_{\rm fd}( \Pi_{\mathbb{X}}(A) ), \]

in the sense that for $L,M\in \operatorname {\mathcal {D}}_{\rm fd}(\mathcal {A})$, we have

(2.2)\begin{equation} \operatorname{RHom}_\Pi(\mathcal{L}(L),\mathcal{L}(M))=\operatorname{RHom}_{\mathcal{A}}(L,M) \oplus D\operatorname{RHom}_{\mathcal{A}}(M,L)[-\mathbb{X}]. \end{equation}

2.4 Acyclic quiver case

In the case when $A$ is a path algebra of an acyclic quiver, the Calabi–Yau-$\mathbb {X}$ completion $\Pi _{\mathbb {X}}(A)$ has the following explicit description, known as the Ginzburg Calabi–Yau algebra [Reference GinzburgGin06].

Denote by $\operatorname {\mathcal {D}}_\infty (Q)\colon =\operatorname {\mathcal {D}}^b(\mathbf {k} Q)$ the bounded derived category of $\mathbf {k} Q$ and

\[ \operatorname{\mathcal{D}}_\mathbb{X}(Q)\colon=\operatorname{\mathcal{D}}_{\rm fd}(\Gamma_{\mathbb{X}}Q) \]

the finite-dimensional derived category of $\Gamma _{\mathbb {X}}Q$. Then we have the following Calabi–Yau-$\mathbb {X}$ version of Keller's Calabi–Yau completion.

In particular, $\operatorname {\mathcal {D}}_\mathbb {X}(Q)$ is Calabi–Yau-$\mathbb {X}$. There is an Lagrangian immersion

(2.3)\begin{equation} \mathcal{L}_Q\colon\operatorname{\mathcal{D}}_\infty(Q)\to\operatorname{\mathcal{D}}_\mathbb{X}(Q). \end{equation}

By abuse of notation, we may not distinguish $\operatorname {\mathcal {D}}_\infty (Q)$ and $\mathcal {L}_Q(\operatorname {\mathcal {D}}_\infty (Q))$.

2.5 $\mathbb {C}^*$-equivariant coherent sheaves on canonical bundles

Let $X$ be a smooth projective variety over $\mathbb {C}$ and $\operatorname {\mathcal {D}}^b (\mathrm {Coh} X)$ the bounded derived category of coherent sheaves on $X$. Due to [Reference Bondal and van den BerghBB03], there is a classical generator $G \in \operatorname {\mathcal {D}}^b (\mathrm {Coh} X)$ and [Reference OrlovOrl09] gives the direct description $G=\bigoplus _{i=1}^{d+1}L^i$ where $L$ is the ample line bundle on $X$ and $d=\dim X$. Consider the endomorphism dg-algebra $A:=\operatorname {RHom}(G,G)$. Then we have the equivalence of derived categories

(2.4)\begin{equation} F:=\operatorname{RHom}(G,-) \colon \operatorname{\mathcal{D}}(\mathrm{Qcoh} X)\to \operatorname{\mathcal{D}}(A). \end{equation}

As $G$ is a classical generator of $\operatorname {\mathcal {D}}^b (\mathrm {Coh} X)$, the restriction of $F$ gives the equivalence

\[ F \colon \operatorname{\mathcal{D}}^b (\mathrm{Coh} X) \xrightarrow{\cong} \operatorname{per} A. \]

By [Reference Bondal and van den BerghBB03, Lemma 3.4.1], the object $G \boxtimes G^*$ is a generator of $\operatorname {\mathcal {D}}^b (\mathrm {Coh} (X \times X))$. Thus, we also have the equivalence

(2.5)\begin{equation} F^{\rm e}:=\operatorname{RHom}(G \boxtimes G^*,-) \colon \operatorname{\mathcal{D}}^b (\mathrm{Coh} (X \times X)) \xrightarrow{\cong} \operatorname{per} A^{\rm e} \end{equation}

because $\operatorname {RHom}(G \boxtimes G^*,G \boxtimes G^*)\cong A \otimes A^{{\rm op}}=A^{\rm e}$.

We note that $\operatorname {per} X \times X \cong D^b(\mathrm {Coh} (X \times X))$ because $X$ is smooth. For a perfect object $\mathcal {E} \in \operatorname {per} X \times X$, define the derived dual by

\[ \mathcal{E}^{\vee}:={\mathcal{H}}om_{\mathcal{O}_{X \times X}}(\mathcal{E},\mathcal{O}_{X \times X}) \in \operatorname{per} X \times X. \]

Similarly for a perfect object $M \in \operatorname {per} A^{\rm e}$, let $M^{\vee }:=\operatorname {Hom}_{A^{\rm e}}(M,A^{\rm e})$ be its derived dual. By definition, $M^{\vee }$ has a natural left $A^{\rm e}$-module structure (so the right $(A^{\rm e})^{{\rm op}}$-module structure). We note $(A^{\rm e})^{{\rm op}}\cong A^{{\rm op}}\otimes A$. Through the algebra homomorphism

\[ \begin{array}{ccc} \tau \colon A\otimes A^{{\rm op}} & \to & A^{{\rm op}}\otimes A,\\ \quad a \times b & \mapsto & (-1)^{|a||b|}b \otimes a \end{array} \]

exchanging two components, we can define the right $A^{\rm e}$-module structure on $M^{\vee }$. Thus, $M^{\vee }$ can be regarded as an object in $\operatorname {per} A^{\rm e}$.

Lemma 2.9 Let $\Delta \colon X \to X \times X$ be the diagonal embedding. Then

\[ F^{\rm e}(\Delta_*\mathcal{K}_X^{-1})=\Theta_{A}[d]. \]

Proof. First we check that $F^{\rm e}({\Delta _* \mathcal {O}_X})=A$. By definition, $G \boxtimes G^*=p_1^*G \otimes p_2^* G^*$ where $p_i$ are projections from $X \times X$ to the $i$th component. Then

\begin{align*} F^{\rm e}(\Delta_* \mathcal{O}_X) &=\operatorname{RHom}(p_1^*G \otimes p_2^* G^*, \Delta_* \mathcal{O}_X) \\ &=\operatorname{RHom}(p_1^*G ,p_2^* G \otimes {\Delta_* \mathcal{O}_X}) \\ &=\operatorname{RHom}(p_1^*G ,\Delta_*( G \otimes \mathcal{O}_X)) \\ &=\operatorname{RHom}(\Delta^* p_1^*G , G )=A. \end{align*}

By definition, the derived dual $A^{\vee }$ in $\operatorname {per}(A^{\rm e})$ is $\Theta _A$. On the other hand, the derived dual $(\Delta _* \mathcal {O}_X)^{\vee }$ in $\operatorname {per} X\times X$ can be computed as follows. By using the Grothendieck–Verdier duality, we have

\[ (\Delta_*\mathcal{O}_X)^{\vee}={\mathcal{H}}om(\Delta_* \mathcal{O}_X, \mathcal{O}_{X \times X}) \cong \Delta_* {\mathcal{H}}om (\mathcal{O}_X, \Delta^* \mathcal{O}_{X\times X} \otimes \mathcal{K}_{\Delta}[-d]), \]

where $d=\dim X$ is the relative dimension of $\Delta$, and

\[ \mathcal{K}_{\Delta}:=\mathcal{K}_X \otimes \Delta^* \mathcal{K}_{X \times X}^{-1} \]

is the relative dualizing bundle of $\Delta$. Note that $\Delta ^* \mathcal {O}_{X \times X} \cong \mathcal {O}_X$. Thus, we have

\[ \Delta_* {\mathcal{H}}om_{\mathcal{O}_X} (\mathcal{O}_X, \Delta^* \mathcal{O}_{X \times X} \otimes \mathcal{K}_{\Delta}[-d])\cong \Delta_*(\mathcal{K}_X \otimes \Delta^* \mathcal{K}_{X \times X}^{-1}[-d]). \]

Finally, the adjunction formula for $\Delta \colon X \to X \times X$ implies

\[ \mathcal{K}_X \cong \Delta^* \mathcal{K}_{X \times X} \otimes \bigwedge^{\operatorname{rank} \mathcal{N}}\mathcal{N}, \]

where $\mathcal {N}$ is the normal bundle and in this case, $\mathcal {N} \cong \mathcal {T}_X$. Thus, $\Delta ^* \mathcal {K}_{X \times X}\cong \mathcal {K}_X^{\otimes 2}$ and, hence, we have $(\Delta _* \mathcal {O}_X)^{\vee } \cong \mathcal {K}_X^{-1}[-d]$. As $F^{\rm e}$ commutes with taking the derived dual by next lemma, the result follows.

We show $F^{\rm e}$ commutes with taking the derived dual.

Lemma 2.10 There is an isomorphism of right $A^{\rm e}$-modules

\[ F^{\rm e}(\mathcal{E}^{\vee})\cong F^{\rm e}(\mathcal{E})^{\vee} \]

for $\mathcal {E} \in \operatorname {per} (X\times X)$.

Proof. The left-hand side is

\[ F^{\rm e}(\mathcal{E}^{\vee})=\operatorname{Hom}(G \boxtimes G^{\vee}, \mathcal{E}^{\vee}). \]

The right-hand side is

\begin{align*} F^{\rm e}(\mathcal{E})^{\vee}&=\operatorname{Hom}_{A^{\rm e}}(\operatorname{Hom}_{X \times X}(G \boxtimes G^{\vee},\mathcal{E}),A^{\rm e})\\ &\cong \operatorname{Hom}_{A^{\rm e}}(\operatorname{Hom}_{X \times X}(G \boxtimes G^{\vee},\mathcal{E}), \operatorname{Hom}_{X \times X}(G \boxtimes G^{\vee},G \boxtimes G^{\vee})) \\ &\cong \operatorname{Hom}_{X \times X}(\mathcal{E}, G \boxtimes G^{\vee}) \\ &\cong \operatorname{Hom}_{X \times X}( (G \boxtimes G^{\vee})^{\vee},\mathcal{E}^{\vee})\\ &\cong \operatorname{Hom}_{X \times X}( G^{\vee} \boxtimes G,\mathcal{E}^{\vee}). \end{align*}

Clearly, $F^{\rm e}(\mathcal {E}^{\vee })$ is isomorphic to $F^{\rm e}(\mathcal {E})^{\vee }$ as complexes of vector spaces. As $A \otimes A^{{\rm op}}\cong \operatorname {Hom}_X(G,G)\otimes \operatorname {Hom}_X(G^{\vee },G^{\vee })$ acts on this through the exchange

\[ \tau \colon \operatorname{Hom}_X(G,G)\otimes \operatorname{Hom}_X(G^{\vee},G^{\vee})\to \operatorname{Hom}_X(G^{\vee},G^{\vee}) \otimes \operatorname{Hom}_X(G,G), \]

$A^{\rm e}$-module structures coincide.

Let

\[ p_{ij} \colon X \times X \times X \to X \times X \]

be projections on the $i$th and $j$th component for $1 \le i< j \le 3$.

Define the convolution product

\[ * \colon \operatorname{\mathcal{D}}^b(\mathrm{Coh} (X\times X))\times \operatorname{\mathcal{D}}^b(\mathrm{Coh} (X\times X)) \to \operatorname{\mathcal{D}}^b(\mathrm{Coh} (X\times X)) \]

by

\[ (\mathcal{E},\mathcal{F})\mapsto \mathcal{E}*\mathcal{F}:= p_{13 *}(p_{12}^* \mathcal{E} \otimes p^*_{23}\mathcal{F}). \]

Then $\operatorname {\mathcal {D}}^b(\mathrm {Coh} (X\times X))$ has the monoidal structure by the convolution product with the unit object $\Delta _* \mathcal {O}_X$. On the other hand, the category $\operatorname {per} A^{\rm e}$ has the monoidal structure by the tensor product $M \otimes _A N$ for $M,N \in \operatorname {per} A^{\rm e}$ with the unit object $A$. It is easy to check the following.

Let $Y:=\mathrm {V}(\mathcal {K}_X)$ be the total space of the canonical bundle of $X$. We note that as a scheme, $Y$ is given by

\[ Y=\mathrm{Spec}_X \mathrm{T} \mathcal{K}_X^{-1}, \]

where $\mathrm {T} \mathcal {K}_X^{-1}$ is the tensor algebra of the inverse of the canonical sheaf. (As $\mathcal {K}_X^{-1}$ is rank one, the tensor algebra $\mathrm {T} \mathcal {K}_X^{-1}$ coincides with the symmetric algebra $\mathrm {S} \mathcal {K}_X^{-1}$.) We note that the bounded derived category of coherent sheaves on $Y$ can be identified with the bounded derived category of finite rank $\mathrm {T} \mathcal {K}_X^{-1}$-modules on $X$:

\[ \operatorname{\mathcal{D}}^b (\mathrm{Coh} Y) \cong \operatorname{\mathcal{D}}^b(\mathrm{mod-} \mathrm{T} \mathcal{K}_X^{-1} ). \]

As Lemmas 2.9 and 2.12 imply

\[ F^{\rm e}(\Delta_* \mathrm{T} \mathcal{K}_X^{-1})=\mathrm{T}(\Theta_A[d])=\Pi_{d+1}(A), \]

we have an equivalence

\[ \operatorname{\mathcal{D}}^b(\mathrm{Coh} Y) \cong \operatorname{per} \Pi_{d+1}(A). \]

Denote by $\operatorname {\mathcal {D}}^b_c(Y)$ the derived category of coherent sheaves on $Y$ with compact support cohomology. Then we also have an equivalence

\[ \operatorname{\mathcal{D}}^b_c(\mathrm{Coh} Y) \cong \operatorname{\mathcal{D}}_{\rm fd}(\Pi_{d+1}(A)). \]

Next we consider the fiber scaling action of $\mathbb {C}^*$ on $Y$ and $\mathbb {C}^*$-equivariant coherent sheaves on $Y$. For a $\mathbb {C}^*$-module $M$, we have the weight decomposition

\[ M:=\bigoplus_{m \in \mathbb{Z}}M^m \]

where $t \in \mathbb {C}^*$ acts on $M^d$ by $t^d$. We denote by $M\{1\}$ the weight-one shift of $M$, namely $M\{1\}^m:=M^{m+1}$. The $\mathbb {C}^*$-action on $Y$ induces the weight decomposition on $\mathrm {T} \mathcal {K}_X^{-1}$:

\[ (\mathrm{T} \mathcal{K}_X^{-1})_{\mathrm{gr}}:=\mathrm{T}(\mathcal{K}_X^{-1}\{1\})= \bigoplus_{m \ge 0}\mathcal{K}_X^{-m}\{m\}, \]

where $t\in \mathbb {C}$ acts on $\mathcal {K}_X^{-m}\{m\}$ as $t^{-m}$. We regard $(\mathrm {T} \mathcal {K}_X^{-1})_{\mathrm {gr}}$ as a $\mathbb {Z}$-graded algebra through the above weight decomposition. Then a $\mathbb {C}^*$-equivariant coherent sheaf on $Y$ can be identified with a finite rank $\mathbb {Z}$-graded $(\mathrm {T} \mathcal {K}_X^{-1})_{\mathrm {gr}}$-module over $X$. Again there is an equivalence

\[ \operatorname{\mathcal{D}}^b_{\mathbb{C}^*}(Y) \cong \operatorname{\mathcal{D}}^b(\mathrm{grmod-}(\mathrm{T} \mathcal{K}_X^{-1})_{\mathrm{gr}} ), \]

where the left-hand side is the bounded derived category of $\mathbb {C}^*$-equivariant coherent sheaves on $Y$ and the right-hand side is the bounded derived category of finite rank $\mathbb {Z}$-graded $(\mathrm {T} \mathcal {K}_X^{-1})_{\mathrm {gr}}$-modules over $X$.

Through the identification of $\mathcal {K}_X^{-1}$ with $\Theta _A[d]$, one can define the extra $\mathbb {Z}$-grading structure on $\mathrm {T}(\Theta _A[d])$ by

\[ \mathrm{T}(\Theta_A[d])_{\mathrm{gr}}:=\bigoplus_{m \ge 0}(\Theta_A[d]\{1\})^{\otimes m}. \]

If we set $[\mathbb {X}]:=[d+1]\{1\}$, then, by definition, we regard $\mathrm {T}(\Theta _A[d])_{\mathrm {gr}}$ as the Calabi–Yau-$\mathbb {X}$ completion $\Pi _{\mathbb {X}}(A)$. Thus, we obtain the following equivalences.

Proposition 2.13 There is an equivalence of derived categories

(2.6)\begin{equation} \operatorname{\mathcal{D}}^b_{\mathbb{C}^*}(Y) \cong \operatorname{per} \Pi_{\mathbb{X}}(A) \end{equation}

and the shift $[d+1]\{1\}$ in the left-hand side coincides with the shift $[\mathbb {X}]$ in the right-hand side. By restricting objects in the left-hand side to those with compact support cohomology, we have an equivalence

(2.7)\begin{equation} \operatorname{\mathcal{D}}^b_{c,\mathbb{C}^*}(Y) \cong \operatorname{\mathcal{D}}_{\rm fd}(\Pi_{\mathbb{X}}(A) ). \end{equation}

In particular, $\operatorname {\mathcal {D}}^b_{c,\mathbb {C}^*}(Y)$ is Calabi–Yau-$[d+1]\{1\}$.

Example 2.14 Let $X$ be the projective line $\mathbb {P}^1$ and $Q$ be the Kronecker quiver (i.e. type $\widetilde {A_{1,1}}$). Then there is a triangulated equivalence

(2.8)\begin{equation} \operatorname{\mathcal{D}}^b(\operatorname{Coh} X) \cong \operatorname{\mathcal{D}}_\infty(Q). \end{equation}

Then the proposition above implies the Calabi–Yau-$\mathbb {X}$ version of this equivalence

(2.9)\begin{equation} \operatorname{\mathcal{D}}^b_{c,\mathbb{C}^*}(Y) \cong \operatorname{\mathcal{D}}_\mathbb{X}(Q), \end{equation}

where $Y=T^* \mathbb {P}^1$ and $\mathbb {X}=[2]\{1\}$. In this case, we can describe the above category as the derived category of the graded preprojective algebra of $Q$ as follows. Consider the graded double quiver $\tilde {Q}_{\mathrm {gr}}$

with the grading $\operatorname {deg} a_i=0$ and $\operatorname {deg} a^*_i=-1$ for $i=1,2$. Let

\[ \Pi(Q)_{\mathrm{gr}}:=\mathbb{C} \tilde{Q}_{\mathrm{gr}} / (a_1 a_1^*-a_1^* a_1,a_2 a_2^*-a_2^* a_2) \]

be the graded preprojective algebra of $Q$ and $\operatorname {\mathcal {D}}^b(\mathrm {grmod-}\Pi (Q)_{\mathrm {gr}})$ be the bounded derived category of finite-dimensional graded modules over $\Pi (Q)_{\mathrm {gr}}$. Then this grading fits into the weight of $\mathbb {C}^*$-action on $Y$ and we have an equivalence

\[ \operatorname{\mathcal{D}}^b_{c,\mathbb{C}^*}(Y) \cong \operatorname{\mathcal{D}}^b(\mathrm{grmod-}\Pi(Q)_{\mathrm{gr}}). \]

Example 2.15 When $X$ is the projective plane $\mathbb {P}^2$, then there is a corresponding version of (2.8), where $Q$ is the following quiver

with some commutative relations

\[ a_1 b_2=b_1 a_2,\quad a,b\in\{x,y,z\}. \]

We can also upgraded that to a Calabi–Yau-$\mathbb {X}$ version (2.9). Again we can describe the above category as follows. Consider the following graded quiver with graded potential $(\tilde {Q}_{\mathrm {gr}},W_{\mathrm {gr}})$

where $\operatorname {deg} x_3, y_3, z_3=-1$ and gradings of other arrows are zero. Then we have an associated graded Jacobi algebra $\mathrm {J}(\tilde {Q}_{\mathrm {gr}},W_{\mathrm {gr}})$. Thus, there is an equivalence

\[ \operatorname{\mathcal{D}}^b_{c,\mathbb{C}^*}(Y) \cong \operatorname{\mathcal{D}}^b(\mathrm{grmod-}\mathrm{J}(\tilde{Q}_{\mathrm{gr}},W_{\mathrm{gr}})). \]

Finally, we note that its $\mathbb {X}=3$-reduction implies to forget the grading from $(\tilde {Q}_{\mathrm {gr}},W_{\mathrm {gr}})$ because $\mathbb {X}=[3]\{1\}$. Therefore, the $\mathbb {X}=3$-reduction gives the equivalence

(2.10)\begin{equation} \operatorname{\mathcal{D}}^b_{\mathbb{P}^2}(Y) \cong \operatorname{\mathcal{D}}^b(\mathrm{mod-} \mathrm{J}(\tilde{Q},W)), \end{equation}

between the derived category of coherent sheaves on the local $\mathbb {P}^2$, considered in [Reference Bayer and MacrBM11], and the derived category of finite-dimensional modules over the Jacobi algebra of $(\tilde {Q},W)$ which is obtained by forgetting the grading of $(\tilde {Q}_{\mathrm {gr}},W_{\mathrm {gr}})$. We explore this direction/application in the future studies.

3.1 Bridgeland stability conditions

First we recall the definition of Bridgeland stability conditions on triangulated categories from [Reference BridgelandBri07]. Throughout this section, we assume that for a triangulated category $\operatorname {\mathcal {D}}$, its Grothendieck group $K(\operatorname {\mathcal {D}})$ is free of finite rank, i.e. $K(\operatorname {\mathcal {D}}) \cong \mathbb {Z}^{\oplus n}$ for some $n$.

For each non-zero object $0 \neq E \in \operatorname {\mathcal {D}}$, we define two real numbers by $\phi _{\sigma }^+(E):=\phi _1$ and $\phi _{\sigma }^-(E):=\phi _m$ where $\phi _1$ and $\phi _m$ are determined by the axiom (d). Non-zero objects in $\mathcal {P}(\phi )$ are called semistable of phase $\phi$ and simple objects in $\mathcal {P}(\phi )$ are called stable of phase $\phi$. For a non-zero object $E \in \operatorname {\mathcal {D}}$ with extension factors $A_1,\ldots,A_m$ given by axiom (d), define the mass of $E$ by

\[ m_{\sigma}(E):=\sum_{i=1}^m |Z(A_i)|. \]

Following [Reference Kontsevich and SoibelmanKS08], we assume an additional condition, called the support property. For a stability condition $\sigma = (Z,\mathcal {P})$, we introduce the set of semistable classes ${\mathrm {ss}}(\sigma ) \subset K(\operatorname {\mathcal {D}})$ by

(3.3)\begin{equation} {\mathrm{ss}}(\sigma) :=\{\alpha \in K(\operatorname{\mathcal{D}})\,\vert\,\text{there is a semistable object } E \in \operatorname{\mathcal{D}} \text{ such that } [E] = \alpha\}. \end{equation}

Let $\lVert \,\cdot \, \rVert$ be some norm on $K(\operatorname {\mathcal {D}}) \otimes \mathbb {R}$. A stability condition $\sigma =(Z,\mathcal {P})$ satisfies the support property if there is a some constant $C=C_\sigma >0$ such that

(3.4)\begin{equation} C \cdot |{Z(\alpha)}| > \lVert \alpha \rVert \end{equation}

for all $\alpha \in {\mathrm {ss}}(\sigma )$. Let $\operatorname {Stab}\operatorname {\mathcal {D}}$ be the set of all stability conditions on $\operatorname {\mathcal {D}}$ satisfying the support property. We define the distance

(3.5)\begin{equation} d \colon \operatorname{Stab}\operatorname{\mathcal{D}}\times \operatorname{Stab}\operatorname{\mathcal{D}}\to [0,\infty] \end{equation}

by

(3.6)\begin{equation} d(\sigma,\varsigma):= \sup_{0 \neq E \in \operatorname{\mathcal{D}}}\bigg\{ |\phi_{\sigma}^-(E) - \phi_{\varsigma}^-(E)|, |\phi_{\sigma}^+(E) - \phi_{\varsigma}^+(E)|, \bigg|\log \frac{m_{\sigma}(E)}{m_{\varsigma}(E)} \bigg| \bigg\}. \end{equation}

Under the topology induced by the distance $d$ on $\operatorname {Stab}\operatorname {\mathcal {D}}$, Bridgeland [Reference BridgelandBri07] showed the following crucial theorem.

Theorem 3.2 [Reference BridgelandBri07, Theorem 1.2]

The projection map of taking central charges

\[ \begin{array}{ccc} \mathcal{Z}\colon \operatorname{Stab}\operatorname{\mathcal{D}} & \longrightarrow & \operatorname{Hom}(K(\operatorname{\mathcal{D}}),\mathbb{C}),\\ \quad(Z,\mathcal{P}) & \mapsto & Z \end{array} \]

is a local homeomorphism of topological spaces. In particular, $\mathcal {Z}$ induces a complex structure on $\operatorname {Stab}\operatorname {\mathcal {D}}$.

Here we consider the case when the Grothendieck group $K(\operatorname {\mathcal {D}})$ is of finite rank over $\mathbb {Z}$. However in [Reference BridgelandBri07], he also deals with the case when $K(\operatorname {\mathcal {D}})$ is of infinite rank. In that case, one should consider the set of all locally-finite stability conditions $\operatorname {Stab}\operatorname {\mathcal {D}}$.

On the space of all stability conditions $\operatorname {Stab}\operatorname {\mathcal {D}}$, we can define two group actions commuting with each other. The first one is the natural $\mathbb {C}$ action

\[ s \cdot (Z,\mathcal{P})=(Z \cdot e^{-{i} \pi s},\mathcal{P}_{-\operatorname{Re}(s)}), \]

where $\mathcal {P}_x(\phi )=\mathcal {P}(\phi +x)$. There is also a natural action on $\operatorname {Stab}\operatorname {\mathcal {D}}$ induced by $\operatorname {Aut}\operatorname {\mathcal {D}}$, namely

\[ \Phi (Z,\mathcal{P})=\big(Z \circ \Phi^{-1}, \Phi (\mathcal{P}) \big). \]

3.2 $q$-stability conditions

Convention Let $\operatorname {\mathcal {D}}_{\mathbb {X}}$ be a triangulated category with a distinguished auto-equivalence

\[ \mathbb{X} \colon \operatorname{\mathcal{D}}_{\mathbb{X}} \to \operatorname{\mathcal{D}}_{\mathbb{X}}. \]

Here $\mathbb {X}$ is not necessarily the Serre functor. We write $E[l \mathbb {X}]$ instead of $\mathbb {X}^l(E)$ for $l \in \mathbb {Z}$ and $E \in \operatorname {\mathcal {D}}_{\mathbb {X}}$. Set

(3.7)\begin{equation} R=\mathbb{Z}[q^{\pm 1}] \end{equation}

and define the $R$-action on $K(\operatorname {\mathcal {D}}_{\mathbb {X}})$ by

(3.8)\begin{equation} q^l \cdot [E] := [E[l \mathbb{X}]]. \end{equation}

Then $K(\operatorname {\mathcal {D}}_{\mathbb {X}})$ has an $R$-module structure. Moreover, when we consider such a category $\operatorname {\mathcal {D}}_\mathbb {X}$, the auto-equivalence group $\operatorname {Aut}\operatorname {\mathcal {D}}_\mathbb {X}$ will only consists of those that commute with $\mathbb {X}$.

Definition 3.4 A $q$-stability condition $(\sigma,s)$ consists of a (Bridgeland) stability condition $\sigma =(Z,\mathcal {P})$ on $\operatorname {\mathcal {D}}_\mathbb {X}$ and a complex number $s \in \mathbb {C}$ satisfying

(3.9)\begin{equation} \mathbb{X} ( \sigma)=s \cdot \sigma. \end{equation}

We may write $\sigma [\mathbb {X}]$ for $\mathbb {X}(\sigma )$.

In the rest of this paper, we assume following.

For a fixed complex number $s \in \mathbb {C}$, consider the specialization

\[ q_s\colon\mathbb{C}[q,q^{-1}]\to\mathbb{C},\quad q\mapsto e^{{i} \pi s}. \]

Denote by $\mathbb {C}_s$ the complex numbers with the $R$-module structure through the specialization $q_s$. To spell out the conditions for (3.9), we have the following equivalent conditions of (3.9).

Similar to the usual stability conditions, we consider the support property as follows.

Definition 3.8 A $q$-stability condition $(\sigma,s)$ satisfies the $q$-support property if there is some lattice $\Gamma :=\mathbb {Z}^n \subset K(\operatorname {\mathcal {D}}_{\mathbb {X}})$ satisfying $\Gamma \otimes _{\mathbb {Z}}R \cong K(\operatorname {\mathcal {D}}_{\mathbb {X}})$ and a subset

(3.10)\begin{equation} \widehat{{\mathrm{ss}}}(\sigma) \subset \bigg\{ \alpha \in K(\operatorname{\mathcal{D}}_{\mathbb{X}}) \biggm| \alpha= \sum_{j=0}^l q^j \alpha_j \, (\alpha_j \in \Gamma) \bigg\} \end{equation}

such that:

1. (i) the set of semistable classes ${\mathrm {ss}}(\sigma )$ is given by

   \[ {\mathrm{ss}}(\sigma)=\bigcup_{k \in \mathbb{Z}}q^k \cdot \widehat{{\mathrm{ss}}}(\sigma); \]

2. (ii) for some norm $\lVert \,\cdot \, \rVert$ on a finite-dimensional vector space $\Gamma \otimes _{\mathbb {Z}}\mathbb {R}$, there is some constant $C>0$ such that (3.4) holds for all $\alpha \in \widehat {{\mathrm {ss}}}(\sigma )$ (and hence all $\alpha \in {\mathrm {ss}}(\sigma )$), where

   \[ \bigg\| \sum_{j=0}^l q^j \alpha_j \bigg\| \colon= \sum_{j=0}^l | e^{{i} \pi j s}|\cdot \lVert \alpha_j \rVert; \]

3. (iii) ($\mathbb {X}$-$\operatorname {Hom}$-bounded) for any semistable object $E$ with $[E]\in \Gamma$, there exists $N_0$ such that for any stable object $F$ with $[F]\in \Gamma$,

   (3.11)\begin{equation} \operatorname{Hom}(E,F[k\mathbb{X}])=0=\operatorname{Hom}(F,E[k\mathbb{X}]) \end{equation}
   when $|k|>N_0$.

Remark 3.9 (Change of basis)

Given any other lattice $\Gamma '\subset K(\operatorname {\mathcal {D}}_{\mathbb {X}})$ satisfying $\Gamma \otimes _{\mathbb {Z}}R \cong K(\operatorname {\mathcal {D}}_{\mathbb {X}})$, then there is another subset

\[ \widehat{{\mathrm{ss}}'}(\sigma) \subset \bigg\{ \alpha \in K(\operatorname{\mathcal{D}}_{\mathbb{X}})\biggm| \alpha= \sum_{j=0}^l q^j \alpha_j (\alpha_j \in \Gamma') \bigg\} \]

such that conditions (ii$^\circ$) and (iii$^\circ$) hold for $\widehat {{\mathrm {ss}}'}(\sigma )$ and $\Gamma '$, respectively. We make this more precise.

• – The second condition holds for all $\alpha \in {\mathrm {ss}}(\sigma )$ and will not be effect either by the choices of $\Gamma '$ or by $\widehat {ss}'(\sigma )$.

• – The $\mathbb {X}$-$\operatorname {Hom}$-boundedness holds because the Hom-vanishing property is preserved under (iterated) extension.

In other words, the $q$-support property is intrinsic (that does not depend on the choice of the lattice $\Gamma$).

Denote by $\operatorname {QStab}_s\operatorname {\mathcal {D}}_\mathbb {X}$ the set of all $q$-stability conditions satisfying the $q$-support property and with fixed $s$. As the space $\operatorname {QStab}_s\operatorname {\mathcal {D}}_\mathbb {X}$ is a subset of the space of usual stability conditions $\operatorname {Stab}(\operatorname {\mathcal {D}}_{\mathbb {X}})$ on $\operatorname {\mathcal {D}}_{\mathbb {X}}$, the distance $d$ (3.6) on $\operatorname {Stab}\operatorname {\mathcal {D}}$ induces a topology on $\operatorname {QStab}_s\operatorname {\mathcal {D}}_\mathbb {X}$. We proceed to show the analogous result of Theorem 3.2.

Theorem 3.10 The projection map of taking central charges

(3.12)\begin{equation} \begin{array}{ccc} \mathcal{Z}_s \colon \operatorname{QStab}_s\operatorname{\mathcal{D}}_{\mathbb{X}} & \longrightarrow & \operatorname{Hom}_R(K(\operatorname{\mathcal{D}}_{\mathbb{X}}),\mathbb{C}_s),\\ \quad((Z,\mathcal{P}),s) & \mapsto & Z \end{array} \end{equation}

is a local homeomorphism of topological spaces. In particular, $\mathcal {Z}_s$ induces a complex structure on $\operatorname {QStab}_s\operatorname {\mathcal {D}}_{\mathbb {X}}$.

The next subsection is devoted to the proof of this theorem.

3.3 Proof of the deformation theorem

We divide the proof into five steps, recall that we fix the complex number $s$. The outline is that we first prove the locally finiteness and adapt Bridgeland's deformation strategy to obtain stability conditions, which we need to show that it is a $q$-stability condition that satisfies/preserves the corresponding properties.

Step I. The $q$-support property implies locally-finiteness.

Proof. Suppose not, then without loss of generality we can assume that there is an infinite chain of subobjects $\cdots \subset E_m\subset \cdots \subset E_1=E$ in the quasi-abelian category $\mathcal {P}(\phi -\epsilon,\phi +\epsilon )$ for some $\phi$ and $\epsilon <1/2$. As in the proof of [Reference BridgelandBri08, Lemma 4.4], the norms $|Z(E_m)|$ of central charges are bounded, say by $K>0$. Thus, the set

\begin{align*} \{\alpha \in {\mathrm{ss}}(\sigma)\mid\, & |Z(\alpha)| < K, \;\exists M \in \mathcal{P}(\phi-\epsilon,\phi+\epsilon) \text{ s.t.}\\ & [M]=\alpha,\;\operatorname{Hom}(M,E)\ne0 \}, \end{align*}

denoted by $\Lambda _{\sigma }$, is infinite. Taking their HN-factors if necessary, we may assume that both $E$ and $M$ are semisimple when considering $\Lambda _{\sigma }$.

By condition (i$^\circ$) of $q$-support property, any $\alpha \in \Lambda _{\sigma }$ equals $q^k\cdot \hat {\alpha }$ for some $\hat {\alpha }\in \widehat {{\mathrm {ss}}}$. Thus, $\Lambda _{\sigma }=\bigcup _{k} \Lambda _k$, where $\Lambda _k$ is

\begin{align*} \{ \hat{\alpha} \in \widehat{{\mathrm{ss}}}(\sigma)\mid\, &|e^{ki\pi s}\cdot Z(\hat{\alpha})| < K, \;\exists M \in \mathcal{P}(\phi-\epsilon,\phi+\epsilon)\text{ s.t.}\\ & [M]=q^k\hat{\alpha},\;\operatorname{Hom}(M,E)\ne0 \}. \end{align*}

Note that $[E[l\mathbb {X}]]\in \widehat {{\mathrm {ss}}}(\sigma )$ for some $l\in \mathbb {Z}$. Then by condition (iii$^\circ$) (i.e. $\mathbb {X}\operatorname {Hom}$-bounded) of the $q$-support property, only finitely many $\Lambda _k$ are not empty. Therefore, there exists $k_0\in \mathbb {Z}$ such that $\Lambda _{k_0}$ is infinite.

On the other hand, by support property (3.4),

\[ \{ \hat{\alpha} \in \widehat{{\mathrm{ss}}}(\sigma)\mid |Z(\hat{\alpha})| < K' \} \]

is finite for any $K'>0$. Taking $K'=|e^{-ki\pi s}| \cdot K$, then $\Lambda _k$ is finite for any $k$, which is a contradiction.

Step II. Adopting the proof of Bridgeland [Reference BridgelandBri07, Theorem 1.2], we obtain the following deformation statement as follows.

Corollary 3.12 Let $(\sigma,s)$ be a $q$-stability condition satisfying the $q$-support property, where $\sigma =(Z,\mathcal {P})$. Then there exists $0<\epsilon <1/8$, such that for any $W\in \operatorname {Hom}_R(K(\operatorname {\mathcal {D}}_\mathbb {X}),\mathbb {C}_s)$ satisfying that

(3.13)\begin{equation} | W(E)-Z(E) | < \sin(\epsilon\pi)|Z(E)| \end{equation}

holds for any $\sigma$-stable object $E$, there exists a slicing $\mathcal {Q}$ so that $\varsigma =(W,\mathcal {Q})$ forms a stability conditions on $\operatorname {\mathcal {D}}_\mathbb {X}$ with $d(\sigma,\varsigma )<\epsilon$. Here the distance $d$ is defined as in (3.5).

For the later proof, let us sketch the construction of the slicing $\mathcal {Q}$ (from [Reference BridgelandBri07, Theorem 1.2]), which is the key to the proof of the corollary. Fix $\epsilon$ so that any (quasi-abelian) subcategory $\mathcal {P}(t-\eta,t+\eta )$ is of finite length (i.e. both artinian and noetherian) for any $\eta \le \epsilon$. Then $\mathcal {Q}(\psi )$ is the full additive subcategories of $\operatorname {\mathcal {D}}_\mathbb {X}$ consisting of the zero objects together with those object $E$, which is $W$-semistable with phase $\psi$ in the subcategory

(3.14)\begin{equation} \mathcal{P}( \psi-\epsilon,\psi+\epsilon ). \end{equation}

Note that $E$ is, in fact, $W$-semistable in any subcategory $\mathcal {P}(I)$ if the interval $I$ contains ${( \psi -\epsilon,\psi +\epsilon )}$.

Step III. Now we prove that the deformed stability condition $\varsigma$ in Corollary 3.12 is a $q$-stability condition (together with $s$).

Proof. We only need to prove that, if $E$ is in $W(\psi )$, then $E[\mathbb {X}]$ is $W(\psi +\operatorname {Re}(s))$. By construction, $E$ is $W$-semistable with phase $\psi$ in $\mathcal {P}( \psi -\epsilon,\psi +\epsilon )$. Now suppose that $E[\mathbb {X}]$ is not $W$-semistable in $\mathcal {P}( \psi +\operatorname {Re}(s)-\epsilon,\psi +\operatorname {Re}(s)+\epsilon )$. Then there is a short exact sequence

\[ 0\to A\to E[\mathbb{X}]\to B\to 0 \]

in $\mathcal {P}( \psi +\operatorname {Re}(s)-\epsilon,\psi +\operatorname {Re}(s)+\epsilon )$ such that $\phi _W(A)>\phi _W(E[\mathbb {X}])>\phi _W(B)$. As $\mathcal {P}$ satisfies condition (e) of Definition 3.7, here is a short exact sequence

\[ 0\to A[-\mathbb{X}]\to E\to B[-\mathbb{X}]\to 0 \]

in $\mathcal {P}( \psi -\epsilon,\psi +\epsilon )$ such that $\phi _W(A[-\mathbb {X}])>\phi _W(E)>\phi _W(B[-\mathbb {X}])$. This contradicts the fact that $E$ is $W$-semistable in $\mathcal {P}( \psi -\epsilon,\psi +\epsilon )$, that finishes the proof.

Step IV. Next we check the $q$-support property is preserved under deformation.

Proof. Let $\widehat {{\mathrm {ss}}}(\varsigma )$ be any set in the form of (3.10) satisfying the first condition in (3.8), with respect to the lattice $\Gamma$. This can be done by choosing one representative $q^{k_0}\alpha$ in each subset $\{ q^k\alpha \mid k\in \mathbb {Z} \}\subset {\mathrm {ss}}(\varsigma )$, where $k_0$ is bigger enough.

Now let $\alpha =[E]\in \widehat {{\mathrm {ss}}}(\varsigma )$ with $\varsigma$-semistable $E$. Consider the HN filtration (3.2) of $E$ with respect to $\sigma$. Thus, $[A_i]\in {\mathrm {ss}}(\sigma )$ and

\[ \alpha=[E]=\sum_{i}^m [A_i]. \]

By (3.13), we have

\[ |W(\alpha)|>(1-\sin(\epsilon\pi))|W(\alpha)|. \]

Moreover, because $E$ is in $\mathcal {P}( \psi -\epsilon,\psi +\epsilon )$ in (3.14), the phases of $A_i$ and $E$, with respect to $\sigma$ (or $\mathcal {P}$), is within an open interval of length $2\epsilon$. Thus,

\[ |Z(E)|=\bigg| \sum_{i}^m Z(A_i) \bigg|> \sum_{i}^m \cos(2\epsilon) | Z(A_i) |. \]

Combining the calculations above and the fact that $[A_i]$ satisfies (3.4) (as $\sigma$ satisfies support property), we have

\begin{align*} |W(\alpha)|&>(1-\sin(\epsilon\pi))|Z(\alpha)|\\ &>(1-\sin(\epsilon\pi)) \cdot \cos(2\epsilon) \displaystyle\sum_{i}^m| Z(A_i) | \\ &>(1-\sin(\epsilon\pi)) \cdot \cos(2\epsilon) \cdot C_\sigma^{-1}\displaystyle\sum_{i}^m \lVert [A_i] \rVert\\ &>(1-\sin(\epsilon\pi)) \cdot \cos(2\epsilon) \cdot C_\sigma^{-1} \bigg\|\displaystyle\sum_{i}^m [A_i]\bigg\|\\ &>(1-\sin(\epsilon\pi)) \cdot \cos(2\epsilon) \cdot C_\sigma^{-1} \lVert \alpha \rVert. \end{align*}

Thus, we can take $C_\varsigma =(1-\sin (\epsilon \pi ))^{-1} \cdot \cos (2\epsilon )^{-1} \cdot C_\sigma$ so that $\varsigma$ satisfies the second condition in Definition 3.8.

Finally, the third condition holds as explained in Remark 3.9.

This completes the proof.

3.4 Gluing $q$-stability conditions

By Assumption 3.6, $\operatorname {QStab}_s\operatorname {\mathcal {D}}_\mathbb {X}$ is of (complex) dimension $n$.

With respect to the original topology of Bridgeland, $\operatorname {QStab}_s\operatorname {\mathcal {D}}_\mathbb {X}$ are in different connected components of $\operatorname {Stab}\operatorname {\mathcal {D}}_\mathbb {X}$ for different $s\in \mathbb {C}$.

Proof. Recall the distance $d$ on $\operatorname {Stab}\operatorname {\mathcal {D}}_\mathbb {X}$ is defined in (3.6). We have

\[ \phi_{\sigma_1}^\pm(E[k\mathbb{X}]) - \phi_{\sigma_2}^\pm(E[k\mathbb{X}])= \phi_{\sigma_1}^\pm(E) - \phi_{\sigma_2}^\pm(E)+k(\operatorname{Re}(s_1)-\operatorname{Re}(s_2)) \]

for any $k\in \mathbb {Z}$ by condition (e) of Definition 3.7. Thus, if $\operatorname {Re}(s_1)\ne \operatorname {Re}(s_2)$, then we have

\[ d(\sigma_1,\sigma_2)\ge \sup_{k\in\mathbb{Z}} \big\{ \big| \phi_{\sigma_1}^\pm(E) - \phi_{\sigma_2}^\pm(E)+k(\operatorname{Re}(s_1)-\operatorname{Re}(s_2)) \big|\big\}=\infty. \]

Similarly, we have

\[ \log \frac{m_{\sigma_1}(E[k\mathbb{X}])}{m_{\sigma_2}(E[k\mathbb{X}])} =\log \frac{m_{\sigma_1}(E)}{m_{\sigma_2}(E)}+k(\operatorname{Im}(s_1)-\operatorname{Im}(s_2)) \]

for any $k\in \mathbb {Z}$ by condition (f) of Definition 3.7. Thus, if $\operatorname {Im}(s_1)\ne \operatorname {Im}(s_2)$, then we have

\[ d(\sigma_1,\sigma_2)\ge \sup_{k\in\mathbb{Z}} \bigg\{\bigg| \log \frac{m_{\sigma_1}(E)}{m_{\sigma_2}(E)}+k(\operatorname{Im}(s_1)-\operatorname{Im}(s_2)) \bigg|\bigg\}=\infty. \]

Thus, the question now is whether/how we can glue different connected components $\operatorname {QStab}_s\operatorname {\mathcal {D}}_\mathbb {X}$ in $\operatorname {Stab}\operatorname {\mathcal {D}}_\mathbb {X}$ together. We hope that one can deform along the $s$ direction to reveal the relations between these complex manifolds $\operatorname {QStab}_s\operatorname {\mathcal {D}}_\mathbb {X}$. Set

\[ \operatorname{QStab}\operatorname{\mathcal{D}}_\mathbb{X}:=\bigcup_{s\in\mathbb{C}}\operatorname{QStab}_s\operatorname{\mathcal{D}}_\mathbb{X}. \]

Conjecture 3.16 We conjecture that $\operatorname {QStab}\operatorname {\mathcal {D}}_\mathbb {X}$ admits the structure of a complex manifold of dimension $n+1$ and the projection map

\[ \pi \colon \operatorname{QStab}\operatorname{\mathcal{D}}_\mathbb{X} \to \mathbb{C},\quad (\sigma,s) \mapsto s \]

is holomorphic.

A partial answer to this conjecture is provided in Theorem 5.11, that an open subspace of $\operatorname {QStab}\operatorname {\mathcal {D}}_\mathbb {X}$ consisting of ‘induced’ $q$-stability conditions does glue together. Moreover, such a subspace for the type $A_2$ quiver is calculated in § 7. These induced $q$-stability conditions are identified with multi-valued quadratic differentials in the surface case in the sequel [Reference Ikeda and QiuIQ18] (and, hence, are the most interesting as far as we are concerned).

5.1 $\mathbb {X}$-baric hearts and induced pre $q$-stability conditions

In this section, we introduce $\mathbb {X}$-baric heart (a triangulated category) in an $\mathbb {X}$-category $\operatorname {\mathcal {D}}_\mathbb {X}$ as the $\mathbb {X}$-analogue of the usual heart (an abelian category) of a triangulated category. Note that this is a special case of the baric structure studied by Achar and Treumann [Reference Achar and TreumannAT11] (see also [Reference Fiorenza and MarchettiFM18]).

Note that, by definition, classes of objects in $\operatorname {\mathcal {D}}_{\infty }$ span $K(\operatorname {\mathcal {D}}_{\mathbb {X}})$ over $R$ and we have a canonical isomorphism

(5.2)\begin{equation} K(\operatorname{\mathcal{D}}_{\infty}) \otimes_{\mathbb{Z}} R \cong K(\operatorname{\mathcal{D}}_{\mathbb{X}}). \end{equation}

The triangulated category $\operatorname {\mathcal {D}}_\mathbb {X}$ is Calabi–Yau-$\mathbb {X}$ if $\mathbb {X}$ is the Serre functor, i.e. there is a natural isomorphism

(5.3)\begin{equation} \mathbb{X}:\operatorname{Hom} (X,Y) \xrightarrow{\sim}\operatorname{Hom} (Y,X[\mathbb{X}])^\vee, \end{equation}

where $V^{\vee }$ is the (graded, if $V$ is) dual space of $\mathbf {k}$-vector space $V$. For an $\mathbb {X}$-baric heart $\operatorname {\mathcal {D}}_{\infty }$ in a Calabi–Yau-$\mathbb {X}$ category $\operatorname {\mathcal {D}}_\mathbb {X}$, condition $(1)$ can be refined as

(5.4)\begin{equation} \operatorname{Hom}_{\operatorname{\mathcal{D}}_{\mathbb{X}}}(A_1,A_2) = 0, \end{equation}

for $A_i\in \operatorname {\mathcal {D}}_\infty [k_i\mathbb {X}]$ and $k_1-k_2\notin \{0,1\}$.

Recall that we have the specialization

\[ q_s\colon\mathbb{C}[q,q^{-1}]\to\mathbb{C},\quad q\mapsto e^{{i} \pi s}. \]

By construction, for any object $E\in \hat {\mathcal {P}}(\phi ), k\in \mathbb {Z}$,

\[ Z_q(E[k\mathbb{X}])=q^k\cdot m(E)\cdot e^{{i} \pi \phi}, \]

where $m(E)\in \mathbb {R}_{>0}$.

5.2 On global dimensions of stability conditions

Definition 5.4 (Global dimension)

Given a slicing $\mathcal {P}$ on a triangulated category $\operatorname {\mathcal {D}}$, define the global dimension of $\mathcal {P}$ by

(5.7)\begin{equation} \operatorname{gldim}\mathcal{P}=\sup\{ \phi_2-\phi_1 \mid \operatorname{Hom}(\mathcal{P}(\phi_1),\mathcal{P}(\phi_2))\neq0\}. \end{equation}

For a stability condition $\sigma =(Z,\mathcal {P})$ on $\operatorname {\mathcal {D}}$, its global dimension $\operatorname {gldim}\sigma$ is defined to be $\operatorname {gldim}\mathcal {P}$.

Remark 5.5 Given a heart $\mathcal {H}$ in $\operatorname {\mathcal {D}}$, let $\mathcal {P}$ be the associated slicing with $\mathcal {P}(\phi )=\mathcal {H}[\phi ]$ for $\phi \in \mathbb {Z}$ and $\mathcal {P}(\phi )=\emptyset$ otherwise. Then we have

\[ \operatorname{gldim}\mathcal{P}=\operatorname{gldim} \mathcal{H}. \]

Lemma 5.6 Let $\mathcal {P}$ be a slicing on $\operatorname {\mathcal {D}}$ with heart $\mathcal {H}_\phi =\mathcal {P}[\phi,\phi +1)$. Then

\[ \vert \operatorname{gldim}\mathcal{P}-\operatorname{gldim}\mathcal{H}_\phi \vert\leq 1. \]

Recall that $\operatorname {Slice}\operatorname {\mathcal {D}}$ the space of (locally-finite) slicings on $\operatorname {\mathcal {D}}$ with the generalized metric [Reference BridgelandBri07, Lemma 6.1]

(5.8)\begin{align} d(\mathcal{P},\mathcal{Q})&=\inf\{ \epsilon\in\mathbb{R}_{\geq0}\mid \mathcal{Q}(\phi)\subset\mathcal{P}[\phi-\epsilon,\phi+\epsilon],\ \forall \,\phi\in\mathbb{R} \}\nonumber\\ &=\sup\big\{ |\phi^+_{\mathcal{Q}}(E)-\phi^+_{\mathcal{P}}(E)|, |\phi^-_{\mathcal{Q}}(E)-\phi^-_{\mathcal{P}}(E)| \mid 0\neq E\in\operatorname{\mathcal{D}}\!\big\}. \end{align}

Moreover, the generalized metric on $\operatorname {Stab}\operatorname {\mathcal {D}}$ can be defined as [Reference BridgelandBri07, Proposition 8.1]

\[ d(\sigma_1,\sigma_2)=\sup\bigg\{ |\phi^+_{\sigma_2}(E)-\phi^+_{\sigma_1}(E)|, |\phi^-_{\sigma_2}(E)-\phi^-_{\sigma_1}(E)| , \bigg|\log\frac{m_{\sigma_2}(E)}{m_{\sigma_1}(E)}\bigg| \,\bigg|\, 0\neq E\in\operatorname{\mathcal{D}}\!\bigg\}. \]

Therefore, we have the following.

Lemma 5.7 The function $\operatorname {gldim} \colon \operatorname {Slice}\operatorname {\mathcal {D}} \to \mathbb {R}_{\ge 0}$ is continuous and, hence, induces a continuous function

(5.9)\begin{equation} \operatorname{gldim}\colon\operatorname{Stab}\operatorname{\mathcal{D}}\to\mathbb{R}_{\ge 0} \end{equation}

on $\operatorname {Stab}\operatorname {\mathcal {D}}$.

Proof. We use the first line of (5.8) as definition for the topology of $\operatorname {Slice}\operatorname {\mathcal {D}}$. For any $\epsilon >0$, we claim that if $d(\mathcal {P},\mathcal {Q})<\epsilon /2$, then

\[ \vert\operatorname{gldim}\mathcal{P}-\operatorname{gldim}\mathcal{Q}\,\vert<\epsilon. \]

For any $\delta >0$, there exists $\phi _1,\phi _2\in \mathbb {R}$ such that $\phi _2-\phi _1\in (\operatorname {gldim}\mathcal {Q}-\delta,\operatorname {gldim}\mathcal {Q}]$ and

\[ \operatorname{Hom}\big( \mathcal{Q}(\phi_1),\mathcal{Q}(\phi_2) \big)\neq0. \]

As $\mathcal {Q}(\phi _i)\subset \mathcal {P}(\phi _i-\epsilon /2,\phi _i+\epsilon /2)$, we have

\[ \operatorname{Hom}\big( \mathcal{P}(\phi_1-\epsilon/2,\phi_1+\epsilon/2), \mathcal{P}(\phi_2-\epsilon/2,\phi_2+\epsilon/2) \big)\neq0. \]

Thus,

\[ \operatorname{gldim}\mathcal{P}\geq (\phi_2-\epsilon/2 )- (\phi_1+\epsilon/2)>\operatorname{gldim}\mathcal{Q}-\delta-\epsilon, \]

which implies that $\operatorname {gldim}\mathcal {P}>\operatorname {gldim}\mathcal {Q}-\epsilon$. Similarly, we have the inequality in the other direction, which completes the claim (and, hence, the proof).

5.3 The inducing theorem

Lemma 5.8 Suppose $\mathcal {P}$ is either $\mathcal {P}_{\oplus }$ in (5.5) or $\mathcal {P}_{*}$ in (5.6), where $\hat {\mathcal {P}}$ is the slicing of a stability condition $\operatorname {\mathcal {D}}_\infty$ of an $\mathbb {X}$-baric heart of $\hat {\sigma }$ of some Calabi–Yau-$\mathbb {X}$ category $\operatorname {\mathcal {D}}_\mathbb {X}$. If $\operatorname {Re}(s)\geq \operatorname {gldim}\hat {\sigma }$, then

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}_\mathbb{X}}(\mathcal{P}(\phi_1),\mathcal{P}(\phi_2))=0 \]

for $\phi _1>\phi _2$.

Proof. We need to show that for any $\phi _1>\phi _2, k_1,k_2\in \mathbb {Z}$,

(5.10)\begin{equation} \operatorname{Hom}_{\operatorname{\mathcal{D}}_\mathbb{X}}(\hat{\mathcal{P}}(\phi_1-k_1\operatorname{Re}(s))[k_1\mathbb{X}],\hat{\mathcal{P}}(\phi_2-k_2\operatorname{Re}(s))[k_2\mathbb{X}])=0. \end{equation}

There are three cases.

• – If $k_2-k_1<0$ or $k_2-k_1>1$, (5.10) follows from (5.4).

• – If $k_1=k_2$, the left-hand side of (5.10) equals

  \[ \operatorname{Hom}_{\operatorname{\mathcal{D}}_\infty}(\hat{\mathcal{P}}(\phi_1-k_1\operatorname{Re}(s)),\hat{\mathcal{P}}(\phi_2-k_1\operatorname{Re}(s))), \]
  which is zero because $\phi _1-k_1\operatorname {Re}(s)>\phi _2-k_1\operatorname {Re}(s)$.

• – If $k_2-k_1=1$, then apply Calabi–Yau-$\mathbb {X}$ duality, the left-hand side of (5.10) equals

  \[ D\operatorname{Hom}_{\operatorname{\mathcal{D}}_\infty}(\hat{\mathcal{P}}(\phi_2-k_2\operatorname{Re}(s)),\hat{\mathcal{P}}(\phi_1-k_1\operatorname{Re}(s))). \]
  As
  \begin{align*} \big( \phi_1-k_1\operatorname{Re}(s) \big) - \big( \phi_2-k_2\operatorname{Re}(s) \big)&=\phi_1-\phi_2+\operatorname{Re}(s)\\ &>\operatorname{gldim}\hat{\mathcal{P}}, \end{align*}
  the $\operatorname {Hom}$ vanishes.

Proof. We only prove the result for additive case where the extension case is just a slight variation. Note that $q$-support property can be checked directly once we have shown $(\sigma _{\oplus },s)$ and $(\sigma _{*},s)$ are both $q$-stability conditions.

First, we prove the ‘if’ part. We need to show that $\sigma =(q_s\circ Z_q, \mathcal {P})$ is a stability condition on $\operatorname {\mathcal {D}}_\mathbb {X}$, where $\mathcal {P}$ is defined as $\mathcal {P}=\operatorname {add}^s\hat {\mathcal {P}}[\mathbb {Z}\mathbb {X}]$. Clearly, $Z=q_s\circ Z_q$ is a group homomorphism that is compatible with $\mathcal {P}$ in the sense that

\[ Z(E)=m(E) e^{{i} \pi \phi} \]

for some $m(E)\in \mathbb {R}_{>0}$ if $E\in \mathcal {P}(\phi )$. By Lemma 5.8, $\mathcal {P}$ satisfies the $\operatorname {Hom}$-vanishing properties. Thus, what is left to show is any object exists (and, hence, unique) a HN-filtration (with respect to $\mathcal {P})$.

Any object $M$ in $\operatorname {\mathcal {D}}_\mathbb {X}$ admits a filtration, with respect to the $\mathbb {X}$-baric heart $\operatorname {\mathcal {D}}_\infty$,

(5.13)\begin{equation} \operatorname{filt}_{\mathbb{X}}(M)=\{E_1[k_1\mathbb{X}],\ldots,E_m[k_m\mathbb{X}]\mid E_i\in\operatorname{\mathcal{D}}_\infty, k_1>\cdots>k_m (\in\mathbb{Z}) \}. \end{equation}

Moreover, each $E_i\in \operatorname {\mathcal {D}}_\infty$ admits a filtration with respect to the slicing $\hat {\mathcal {P}}$ and, hence, (5.13) can be refined as

(5.14) \begin{align} \operatorname{filt}_0(M)&=\bigl\{E_{1,1}[k_1\mathbb{X}],\ldots,E_{1,l_1}[k_1\mathbb{X}],\nonumber\\ &\qquad E_{2,1}[k_2\mathbb{X}],\ldots,\nonumber\\ &\qquad\qquad\qquad\ldots\ldots E_{m,l_m}[k_m\mathbb{X}]\mid E_{i,r}\in\hat{\mathcal{P}}(\phi_{i,r})\nonumber\\ &\qquad k_1>\cdots>k_m,\, l_1,\ldots,l_m\in\mathbb{Z} \bigr\} \end{align}

or simply as

(5.15)\begin{equation} \operatorname{filt}_0(M)=\{ F_1,\ldots,F_t\mid F_j\in\mathcal{P}(\phi_j)\}. \end{equation}

Now we claim that we can exchange the position of the factors in $\operatorname {filt}_0(M)$ inductively so that it becomes a filtration for $\mathcal {P}$

\[ \operatorname{filt}(M)=\{ F_1',\ldots,F_t'\mid F_j'\in\mathcal{P}(\phi_j'), \phi_1'>\cdots>\phi_t'\}. \]

This is equivalent to showing that (and then use induction):

• – for $j$ in (5.15) with $\phi _j<\phi _{j+1}$, we have

  (5.16)\begin{equation} \operatorname{Hom}_{\operatorname{\mathcal{D}}_\mathbb{X}}(F_{j+1},F_j[1])=0, \end{equation}
  which implies, by the octahedral axiom, that the exchange of $F_j$ and $F_{j+1}$ is admissible.

Note that $F_j=M[a\mathbb {X}], F_{j+1}=L[b\mathbb {X}]$ for some $M,L\in \operatorname {\mathcal {D}}_\infty, a\geq b\in \mathbb {Z}$. There are three cases.

• – If $a=b$, then $F_j,F_{j+1}$ are factors of the same $E_i$ (with respect to the slicing $\hat {\mathcal {P}}$) and, hence, $\phi _j>\phi _{j+1}$ which contradicts to $\phi _j<\phi _{j+1}$.

• – If $a>b+1$, then (5.16) follows from (5.4).

• – If $a=b+1$, then $\phi _j<\phi _{j+1}$ is equivalent to

  \[ \varphi_{\hat{\mathcal{P}}} (M)+\operatorname{Re}(s) < \varphi_{\hat{\mathcal{P}}} (L) , \]
  where $\varphi _{\hat {\mathcal {P}}}$ denotes the phase of objects in $\operatorname {\mathcal {D}}_\infty$ with respect to $\hat {\mathcal {P}}$. Hence,
  \[ \operatorname{gldim}\hat{\mathcal{P}}<\operatorname{Re}(s)-1 = \varphi_{\hat{\mathcal{P}}} (L)-\varphi_{\hat{\mathcal{P}}} (M[1]), \]
  which implies $\operatorname {Hom}_{\operatorname {\mathcal {D}}_\infty }(M[1],L)=0$. Therefore, we have
  \[ \operatorname{Hom}_{\operatorname{\mathcal{D}}_\mathbb{X}}(F_{j+1},F_j[1])=\operatorname{Hom}_{\operatorname{\mathcal{D}}_\mathbb{X}}(L,M[1+\mathbb{X}]) =D\operatorname{Hom}_{\operatorname{\mathcal{D}}_\mathbb{X}}(M[1],L)=0. \]

In all, $\mathcal {P}$ is a slicing, $(q_s\circ Z_q, \hat {\mathcal {P}})$ is in $\operatorname {Stab}\operatorname {\mathcal {D}}_\mathbb {X}$ and $\sigma =\mathcal {L}_*^s(\hat {\sigma })=(Z_q,\hat {\mathcal {P}},s)$ is in $\operatorname {QStab}^{\oplus }_s\operatorname {\mathcal {D}}_\mathbb {X}$.

Next, let us prove the ‘only if’ part. Suppose that (5.11) does not hold, then there exists $M,L$ in $\operatorname {\mathcal {D}}_\infty$ such that

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}_\infty}(M[1],L)\neq0\quad \text{and}\quad \varphi_{\hat{\mathcal{P}}}(L)-\varphi_{\hat{\mathcal{P}}}(M[1])\geq\operatorname{Re}(s)-1, \]

where $\varphi _{\hat {\mathcal {P}}}$ is the phase with respect to the slicing $\hat {\mathcal {P}}$.

Now consider $M,L$ in $\operatorname {\mathcal {D}}_\mathbb {X}$. We have

(5.17)\begin{equation} 0\neq D\operatorname{Hom}_{\operatorname{\mathcal{D}}_\mathbb{X}}(M[1],L)=\operatorname{Hom}_{\operatorname{\mathcal{D}}_\mathbb{X}}(L,M[1+\mathbb{X}]) \end{equation}

and (here $\varphi _{\hat {\mathcal {P}}}$ denote the phase with respect to $\hat {\mathcal {P}}$)

(5.18)\begin{equation} \varphi_{\hat{\mathcal{P}}}(L)-\varphi_{\hat{\mathcal{P}}}(M[\mathbb{X}])= \varphi_{\hat{\mathcal{P}}}(L)-\varphi_{\hat{\mathcal{P}}}(M)-\operatorname{Re}(s)\geq0. \end{equation}

By (5.17), there exists an object $E$ that sits in the non-trivial triangle

\[ M[\mathbb{X}]\to E\to L\to M[1+\mathbb{X}]. \]

Consider the filtration of $E$ with respect to the $\mathbb {X}$-baric heart $\operatorname {\mathcal {D}}_\infty$, which must be the triangle above, i.e.

(5.19)\begin{equation} \operatorname{filt}_{\mathbb{X}}(E)=\{M[\mathbb{X}],L\}. \end{equation}

Consider the filtration of $E$ with respect to the slicing $\mathcal {P}=\operatorname {add}^s\hat {\mathcal {P}}[\mathbb {Z}\mathbb {X}]$

(5.20)\begin{equation} \operatorname{filt}(E)=\{ F_1,\ldots,F_t\mid F_j\in\mathcal{P}(\phi_j), \phi_1>\cdots>\phi_t\}. \end{equation}

Again, using the octahedral axiom (and vanishing $\operatorname {Hom}$) we can rearrange the order of the factors so that it becomes

\[ \operatorname{filt}(E)=\{ F_1',\ldots,F_t'\mid F_j'=E_i[k_i\mathbb{X}], E_i\in\mathcal{R}_0, k_1\geq\cdots\geq k_t\}. \]

Comparing with the filtration (5.19), we deduce that $1=k_1=\cdots =k_j,k_{j+1}=\cdots =k_t=0$ and there are filtrations

\[ \operatorname{filt}(M)=\{ E_1,\ldots,E_j \},\quad \operatorname{filt}(L)=\{ E_{j+1},\ldots,E_t \}. \]

As $M$ and $L$ are (semi)stable objects, then the phase of $E_i$ equal phase of $M$ (with respect to $\mathcal {P}$) for $1\leq i\leq j$ and the phase of $E_i$ equal phase of $L$ (with respect to $\mathcal {P}$) for $j+1\leq i\leq t$. As $\{E_i[k_i]\}$ are rearrangement of $F_i\in \mathcal {P}(\phi _j)$, we deduce that $j=1$ and $t=2$, i.e. the filtration (5.19) coincide with the filtration (5.20). However, then (5.18) implies $\phi _1=\varphi _{\hat {\mathcal {P}}}(M[\mathbb {X}])\leq \varphi _{\hat {\mathcal {P}}}(L)=\phi _2$ that contradicts to $\phi _1>\phi _2$, which completes the proof.

By comparing inequalities (5.21) and (5.12) one deduces that $\operatorname {QStab}^{\oplus }_s\operatorname {\mathcal {D}}_\mathbb {X}\subset \operatorname {QStab}^{*}_s\operatorname {\mathcal {D}}_\mathbb {X}$. We prefer to consider open $q$-stability conditions as we can glue them together. In fact, in most of the cases we are interested in (with $\operatorname {Re}(s)\ge 2$), we expect they coincide.

Proof. Given an open $q$-stability condition $(\sigma,s)$, suppose that $\sigma$ is induced from a triple $(\operatorname {\mathcal {D}}_\infty,\hat {\sigma },s)$. Then by Theorem 5.9, we have (5.11). As $\operatorname {gldim}$ is continuous by Lemma 5.7, there is a neighborhood $U(\hat {\sigma })$ of $\hat {\sigma }$ in $\operatorname {Stab}\operatorname {\mathcal {D}}_\infty$ satisfying

\[ \operatorname{Re}(s)>\operatorname{gldim}\hat{\sigma}'+1+\epsilon \]

condition for any $\hat {\sigma }'\in U(\hat {\sigma })$ and some positive real number $\epsilon$. Hence, for any $\hat {\sigma }'$ in $U(\hat {\sigma })$ and $s'\in (s-\epsilon,s+\epsilon )$, the triple $(\operatorname {\mathcal {D}}_\infty,\hat {\sigma }',s')$ induces an open $q$-stability. This gives a local chart for $\operatorname {QStab}^{\oplus }\operatorname {\mathcal {D}}_\mathbb {X}$, that isomorphic to

\[ U(\hat{\sigma})\times(s-\epsilon,s+\epsilon). \]

This type of chart provides the required complex manifold structure.

By Theorem 5.9, the minimal value of the global function on the $\mathbb {X}$-baric heart is important concerning the question that if $\operatorname {QStab}_s\operatorname {\mathcal {D}}_\mathbb {X}$ is empty or not.

5.4 Example of acyclic quivers

Here is a class of examples of $\mathbb {X}$-baric hearts.

Thus, we can apply our general results, namely Theorems 5.9 and 5.11. In particular, we can construct $q$-stability conditions on $\operatorname {\mathcal {D}}_\mathbb {X}(Q)$ for any $s$ with $\operatorname {Re}(s)\ge 2$.

6.1 Cluster categories

The cluster categories $\operatorname {\mathcal {C}}(Q)$ were introduced in [Reference Buan, Marsh, Reineke, Reiten and TodorovBMR⁺06] to categorify cluster algebras associated to $Q$. Keller [Reference KellerKel05] provided the construction of cluster categories as orbit categories.

Definition 6.1 [Reference Buan, Marsh, Reineke, Reiten and TodorovBMR⁺06, Reference KellerKel05]

For any integer $m\geq 2$, the $m$-cluster shift is the auto-equivalence of $\operatorname {per}\mathbf {k} Q\cong \operatorname {\mathcal {D}}_\infty (Q)$ given by $\operatorname {\Sigma }_{m}=\tau ^{-1}\circ [m-1]$. The $m$-cluster category $\operatorname {\mathcal {C}}_m(Q)$ is the orbit category

(6.1)\begin{equation} \operatorname{\mathcal{C}}_m(Q)\colon=\operatorname{\mathcal{D}}_\infty(Q)/\operatorname{\Sigma}_{m}. \end{equation}

Note that $\operatorname {\mathcal {C}}_m(Q)$ is Calabi–Yau-$m$ and the classical case (corresponding to cluster algebras) is when $m=2$.

Another way to realize cluster categories is via the Verdier quotient. Set

\[ N=m+1 \]

and let

(6.2)\begin{equation} \operatorname{\mathcal{C}}(\operatorname{\Gamma}_{N}Q)\colon=\operatorname{per}\operatorname{\Gamma}_{N}Q/\operatorname{\mathcal{D}}_N(Q) \end{equation}

be the generalized cluster category associated to $\operatorname {\Gamma }_{N}Q$, i.e. it sits in the short exact sequence of triangulated categories:

(6.3)

Note that $\operatorname {\mathcal {C}}(\operatorname {\Gamma }_{N}Q)$ is Calabi–Yau-$(N-1)$.

Here, a cluster tilting object in a Calabi–Yau-$m$ cluster category is the direct sum of a maximal collection of non-isomorphic indecomposables $\{M_i\}$ such that $\operatorname {Ext}^k(M_i, M_j)=0$, for all $1\leq k\leq m-1$.

6.2 $N$-reduction

In Definition 2.7, replacing $\mathbb {X}$ with an integer $N\ge 2$, we obtain the usual Ginzburg dg-algebra $\operatorname {\Gamma }_{N}Q$ and the corresponding Calabi–Yau-$N$ category $\operatorname {\mathcal {D}}_N(Q)$. On the level of differential (double) graded algebras, there is a projection

(6.4)\begin{equation} \pi_N\colon\operatorname{\Gamma}_{\mathbb{X}}Q\to\operatorname{\Gamma}_{N}Q \end{equation}

collapsing the double degree $(a,b)\in \mathbb {Z}\oplus \mathbb {Z}\mathbb {X}$ into $a+bN\in \mathbb {Z}$, that induces a functor

\[ \pi_N\colon\operatorname{\mathcal{D}}_\mathbb{X}(Q)\to\operatorname{\mathcal{D}}_N(Q). \]

Thus, we obtain the following (cf. [Reference Seidel and ThomasST01, Proposition 4.18], [Reference KellerKel05], and [Reference Kalck and YangKY18, Theorem 5.1]).

Thus, in this case we have

\[ \operatorname{\mathcal{D}}_N(Q)=\operatorname{\mathcal{D}}_\mathbb{X}(Q) \mathbin{/\mkern-6mu/} [\mathbb{X}-N]. \]

Similarly, there is an $N$-reduction $\pi _N\colon \operatorname {per}\operatorname {\Gamma }_{\mathbb {X}}Q\to \operatorname {per}\operatorname {\Gamma }_{N}Q$. A direct corollary is the following.

For $\operatorname {\mathcal {D}}_N(Q)$, we also have the corresponding spherical twist group $\operatorname {ST}_N(Q)$, generated by spherical twists along the $N$-spherical simple $\operatorname {\Gamma }_{N}Q$-modules. When $Q$ is of type $A$, [Reference Khovanov and SeidelKS02] shows $\iota _Q^\mathbb {X}$ in (A.1) is an isomorphism. Seidel and Thomas [Reference Seidel and ThomasST01] proved the corresponding