q-stability conditions on Calabi–Yau-𝕏 categories

Abstract

We introduce $q$-stability conditions $(\sigma,s)$ on Calabi–Yau-$\mathbb {X}$ categories $\mathcal {D}_\mathbb {X}$, where $\sigma$ is a stability condition on $\mathcal {D}_\mathbb {X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname {QStab}_s\mathcal {D}_\mathbb {X}$ is a complex manifold of dimension $n$ for fixed $s$, where $n$ is the rank of the Grothendieck group of $\mathcal {D}_\mathbb {X}$ over $\mathbb {Z}[q^{\pm 1}]$. When $s=N$ is an integer, we show that $q$-stability conditions can be identified with the stability conditions on $\mathcal {D}_N$, provided the orbit category $\mathcal {D}_N=\mathcal {D}_\mathbb {X}/[\mathbb {X}-N]$ is well defined. To attack the questions on existence and deformation along the $s$ direction, we introduce the inducing method. Sufficient and necessary conditions are given, for a stability condition on an $\mathbb {X}$-baric heart (that is, a usual triangulated category) of $\mathcal {D}_\mathbb {X}$ to induce $q$-stability conditions on $\mathcal {D}_\mathbb {X}$. As a consequence, we show that the space $\operatorname {QStab}^\oplus \mathcal {D}_\mathbb {X}$ of (induced) open $q$-stability conditions is a complex manifold of dimension $n+1$. Our motivating examples for $\mathcal {D}_\mathbb {X}$ are coming from (Keller's) Calabi–Yau-$\mathbb {X}$ completions of dg algebras. In the case of smooth projective varieties, the $\mathbb {C}^*$-equivariant coherent sheaves on canonical bundles provide the Calabi–Yau-$\mathbb {X}$ categories. Another application is that we show perfect derived categories can be realized as cluster-$\mathbb {X}$ categories for acyclic quivers.

1. Introduction

1.1 $q$-deformation of stability conditions

The notion of a stability condition $\sigma =(Z,\mathcal {P})$ on a triangulated category $\operatorname {\mathcal {D}}$ was introduced by Bridgeland, motivated from Douglas’ work of $\Pi$-stability of D-branes in string theory. The data consist of:

• – a central charge $Z\colon K(\operatorname {\mathcal {D}})\to \mathbb {C}$, where $K(\operatorname {\mathcal {D}})$ is the Grothendieck group; and

• – a slicing $\mathcal {P}=\{\mathcal {P}(\phi )\}$, where $\mathcal {P}(\phi )$ is an additive/abelian subcategory of $\operatorname {\mathcal {D}}$, which is an $\mathbb {R}$-refinement of t-structures.

A key result established by Bridgeland [Bri07] is that all stability conditions form a complex manifold $\operatorname {Stab}\operatorname {\mathcal {D}}$, where the local coordinate is given by the central charge

\[ Z\in\operatorname{Hom}_{\mathbb{Z}}(K(\operatorname{\mathcal{D}}),\mathbb{C}). \]

In most of the cases from algebras, $K(\operatorname {\mathcal {D}})\cong \mathbb {Z}^n$ and $\dim _\mathbb {C}\operatorname {Stab}\operatorname {\mathcal {D}}=n$. Due to works of many people, the theory of stability conditions has been related/applied to various subjects in mathematics, in particular, mirror symmetry, cluster algebras, moduli spaces and Donaldson–Thomas theory (e.g. [Bri08, Bri09b, BQS20, BM11, BS15, GMN13, HKK17, Ike17, KQ20, KS08, Qiu16, QW18, Tod14]).

In this paper, we introduce $q$-stability conditions $(\sigma,s)$ on a class of triangulated categories $\operatorname {\mathcal {D}}_\mathbb {X}$, consisting of a stability condition $\sigma$ and a complex parameter $s\in \mathbb {C}$. We consider it as a $q$-deformation of a Bridgeland stability condition. Such a category $\operatorname {\mathcal {D}}_\mathbb {X}$ admits a distinguished auto-equivalence $\mathbb {X}$ (another shift) satisfying

\[ K(\mathcal{D}_{\mathbb{X}})\cong R^n,\quad R=\mathbb{Z}[q,q^{-1}]. \]

Here the $R$-module structure on $K(\operatorname {\mathcal {D}}_\mathbb {X})$ is given by the action $q[M]=[M[\mathbb {X}]]$. The compatible/ extra condition on $(\sigma,s)$ is

\[ \mathbb{X} ( \sigma)=s \cdot \sigma, \]

where the left-hand side is the $\mathbb {X}$ action and the right-hand side is the $\mathbb {C}$-action. To spell this out:

• – the central charge is $R$-linear, i.e.

  \[ Z \in \operatorname{Hom}_R(K(\operatorname{\mathcal{D}}_{\mathbb{X}}),\mathbb{C}_s), \]
  where $\mathbb {C}_s$ is still the complex plane but with the $R$-module structure through the action $q(z)=e^{{i} \pi s}\cdot z$;

• – the slicing is compatible with $Z$ under the action of $\mathbb {X}$, i.e.

  \[ \mathcal{P}(\phi + \operatorname{Re}( s)) = \mathcal{P}(\phi)[\mathbb{X}]. \]

This type of equation was considered by Toda [Tod14] to calculate the Gepner/orbit point in $\mathbb {C}\backslash \operatorname {Stab}\operatorname {\mathcal {D}}/\operatorname {Aut}$. When fixing $s$, the space $\operatorname {QStab}_s\operatorname {\mathcal {D}}$ of $q$-stability conditions forms a complex manifold of dimension $n$ that behaves as the usual spaces of stability conditions with finite dimension. In particular, when $s=N$ is an integer and the orbit category $\operatorname {\mathcal {D}}_N\colon =\operatorname {\mathcal {D}}_\mathbb {X}/[\mathbb {X}-N]$ is well-defined, $\operatorname {QStab}_s\operatorname {\mathcal {D}}$ can be embedded in $\operatorname {Stab}\operatorname {\mathcal {D}}_N$ indeed.

The main difficulty in this paper lies in finding the proper/correct definition (such as $q$-stability conditions and global dimension function) so that one can prove the corresponding results with interesting motivations/applications (to the usual Calabi–Yau-$N$ case and to $q$-deformation of quadratic differentials). The following two subsections are devoted to explain these motivation/application, cf. various further developments in [Qiu23, IQ18, FLLQ20, IQZ20, Qiu20, IOST20].

1.2 Frobenius structures

Our original motivation was to understand the link between the space of stability conditions, for a Dynkin type $Q$, and the Frobenius structure (which was called the flat structure in [Sai83]) on the unfolding space of the corresponding singularity constructed by Saito [Sai83, ST08]. This link was conjectured by Takahashi, cf. comments in [Bri09b].

Let $\mathfrak {h}$ be the Cartan subalgebra of the finite-dimensional complex simple Lie algebra $\mathfrak {g}$ corresponding to (an ADE Dynkin quiver) $Q$ and $\mathfrak {h}_{\mathrm {reg}}$ its regular part. Then for the ‘Calabi–Yau-$\infty$’ category $\operatorname {\mathcal {D}}_\infty (Q):=D^b(\mathbf {k} Q)$, we expect

(1.1)\begin{equation} \operatorname{Stab}\operatorname{\mathcal{D}}_\infty(Q)\cong\mathfrak{h}/W \end{equation}

for the Frobenius manifold $\mathfrak {h}/W$, where $W$ is the Weyl group. This has been proved for type $A$ in [HKK17] (cf. [BQS20] for the $A_2$ case). On the other hand, for the Calabi–Yau-$N$ category $\operatorname {\mathcal {D}}_N(Q):=\operatorname {\mathcal {D}}_{\rm fd}(\operatorname {\Gamma }_{N}Q)$, we expect

(1.2)\begin{equation} \operatorname{Stab}\operatorname{\mathcal{D}}_N(Q)/\operatorname{ST}_N(Q)\cong\mathfrak{h}_{\mathrm{reg}}/W, \end{equation}

where $\operatorname {ST}_N(Q)$ is the spherical twist group that can be identified with the Artin/braid group $\operatorname {Br}_Q$ [QW18]. This has been proved for type $A$ in [Ike17] (cf. [BQS20] for the $A_2$ case). In the correspondence (1.2), the central charges $Z=Z(N)$ correspond to the twisted period maps $P_\nu$ with the parameter $\nu =(N-2)/2$ [Dub04, (5.11)]. The relation between the twisted period maps with the parameter $\nu$ and the central charges of Calabi–Yau-$N$ categories was first conjectured in [Bri06] for the canonical bundle of the projective space. Note that $N$ is required to be an integer $(\ge 2)$ in the previous settings for $\operatorname {Stab}\operatorname {\mathcal {D}}_N(Q)$ where the category $\operatorname {\mathcal {D}}_N(Q)$ is well-defined. However, for the almost Frobenius structure on $\mathfrak {h}_{\mathrm {reg}}/W$, the twisted period maps can be defined for any complex parameter $\nu$. This motivates us to produce a corresponding space of stability conditions for a complex parameter $s$ with the formula $\nu =(s-2)/2$. Our construction of the $s$-fiber of the space $\operatorname {QStab}_s\operatorname {\mathcal {D}}_\mathbb {X}(Q)$ of $q$-stability conditions on $\operatorname {\mathcal {D}}_\mathbb {X}(Q):=\operatorname {\mathcal {D}}_{\rm fd}(\operatorname {\Gamma }_{\mathbb {X}}Q)$ provides one solution to this question. Here $\operatorname {\Gamma }_{\mathbb {X}}Q$ is Calabi–Yau-$\mathbb {X}$ Ginzburg differential (double) graded algebra, using Keller's construction [Kel11a]. Moreover, the distinguished auto-equivalence $\mathbb {X}$ in the Calabi–Yau-$\mathbb {X}$ categories is given by the (extra) grading shift.

Finally, we remark that the twisted period map of the almost Frobenius structure on $\mathfrak {h}_{\mathrm {reg}}/W$ can be identified with the period map associated with the primitive form of the corresponding ADE singularity in the theory of Saito [Sai83]. The complex parameter $s \in \mathbb {C}$ he introduced precisely corresponds to our $s$ for $q$-stability conditions. Thus, our construction is a first step of categorification of his period map with the parameter $s$.

1.3 Mirror symmetry

The main method used to prove (1.1) and (1.2) is to realize stability conditions as quadratic differentials. Such an observation was first made by Kontsevich and Seidel years ago and two crucial works have been carried out in [BS15, HKK17]. The idea is related to mirror symmetry. More precisely, in Kontsevich's homological mirror symmetry [Kon95], there is a conjectural derived equivalence (in general)

\[ \operatorname{\mathcal{D}}^b\operatorname{Fuk}(X) \cong \operatorname{\mathcal{D}}^b(\operatorname{Coh} X^{\vee}), \]

where $X$ is a Calabi–Yau manifold with derived Fukaya category $\operatorname {\mathcal {D}}^b\operatorname {Fuk}(X)$ on the left-hand side (A-side) and $X^{\vee }$ is the mirror partner of $X$ with bounded derived category $\operatorname {\mathcal {D}}^b(\operatorname {Coh} X^{\vee })$ on the right-hand side (B-side). In the mathematical aspects of string theory, one expects that the complex moduli space $\mathcal {M}_{\mathrm {cpx}}(X)$ for $X$ can be embedded into (a quotient of) the space $\operatorname {Stab} \operatorname {\mathcal {D}}^b(\operatorname {Coh}{X^\vee })$ of stability conditions. The key correspondence in this conjectural relation is the central charge of a stable object $S^{\vee }$ in $\operatorname {\mathcal {D}}^b(\operatorname {Coh}{X^\vee })$, is given by the integral of the complex structure via the corresponding Lagrangian $S$ in $X$:

(1.3)\begin{equation} Z(S^{\vee})=\int_{ S } \Omega, \end{equation}

where $\Omega$ is a canonical holomorphic volume form of the Calabi–Yau manifold $X$. (For details of this expectation, we refer to [ABC⁺09, § 5] and [Bri09a].)

Although the general case is hard, the surface case is attackable: the complex structure simplifies to quadratic differentials (on Riemann surfaces). In the Calabi–Yau-$3$ case, Bridgeland and Smith [BS15] prove that

(1.4)\begin{equation} \operatorname{Stab}^\circ\operatorname{\mathcal{D}}_3(\mathbf{S})/\operatorname{Aut}\cong\operatorname{Quad}_3(\mathbf{S}), \end{equation}

where $\mathbf {S}$ is a marked surface, $\operatorname {\mathcal {D}}_3(\mathbf {S})$ the associated Calabi–Yau-$3$ category and $\operatorname {Quad}_3(\mathbf {S})$ the moduli space of quadratic differentials on $\mathbf {S}$ (with predescribed singularities of GMN type, cf. [GMN13, BS15]). Here $\operatorname {\mathcal {D}}_3(\mathbf {S})$ is a subcategory of some derived Fukaya category [Smi15]. In the non-Calabi–Yau case, Haiden, Katzarkov, and Kontsevich [HKK17] prove that

(1.5)\begin{equation} \operatorname{Stab}^\circ\mathcal{D}_{\infty}(\mathbf{S})/\operatorname{Aut}\cong\operatorname{Quad}_{\infty}(\mathbf{S}), \end{equation}

where $\mathbf {S}$ is a flat surface, $\operatorname {\mathcal {D}}_\infty (\mathbf {S})=\operatorname {TFuk}(\mathbf {S})$ the associated topological Fukaya category, and $\operatorname {Quad}_\infty (\mathbf {S})$ the moduli space of quadratic differentials on $\mathbf {S}$ (with predescribed singularities of exponential type).

One key observation is that $\operatorname {\mathcal {D}}_\infty (\mathbf {S})$ should be thought as Calabi–Yau-$\infty$, which is a philosophy developed in [KQ15, Qiu15, BQS20]. When $\mathbf {S}$ is a disk, we have

\[ \operatorname{\mathcal{D}}_\infty(\mathbf{S})=\operatorname{\mathcal{D}}_\infty(A_n),\quad \operatorname{\mathcal{D}}_3(\mathbf{S})=\operatorname{\mathcal{D}}_3(A_n). \]

We focus on the surface case, relating works [BS15, HKK17] in the sequel [IQ18] by introducing the Calabi–Yau-$\mathbb {X}$ categories of quivers with superpotential.

Note that the prototype of our Calabi–Yau-$\mathbb {X}$ categories is that considered in [KS02] (cf. its mirror counterpart in [ST01]). Presumedly, the construction in § 2.5 (cf. [IQ18, § 4]) on the B-side and that in [Sei15] for the A-side should give a Calabi–Yau-$\mathbb {X}$ version of homological mirror symmetry of [LP17, Theorems A and B].

1.4 Application to cluster theory

There is a close link between stability conditions and cluster theory, cf. [KS08]. For instance, cluster theory (in particular, the cluster exchange graph) plays a key role in the proof of (1.4) in [BS15], as well as in proving simply connectedness/contractibility of spaces of stability conditions in [Qiu15, QW18, KQ20]. More precisely, the cell structure of spaces of stability conditions is encoded by Happel–Reiten–Smalø tilting, which corresponds to mutation of silting objects in the perfect derived categories (cf. [Kel11b]). On the other hand, in the categorification of cluster algebras, the Calabi–Yau-2 cluster categories $\operatorname {\mathcal {C}}_2(Q)$ are introduced by Buan et al. [BMR⁺06, Kel05]. It is generalized by Amiot, Guo, and Keller [Ami09, Kel11a, Guo10] as a Verdier quotient

(1.6)\begin{equation} \operatorname{\mathcal{C}}_{N-1}(Q)=\operatorname{per}\operatorname{\Gamma}_{N}Q/\operatorname{\mathcal{D}}_{\rm fd}(\operatorname{\Gamma}_{N}Q), \quad N\geq2(\in\mathbb{Z}) \end{equation}

of the Calabi–Yau-$N$ Ginzburg differential graded algebra $\operatorname {\Gamma }_{N}Q$. The mutation of cluster algebras corresponds to the mutation of cluster tilting objects in the cluster categories, which is also closely related to mutation of silting objects (cf., e.g., [KQ15]). One expects the perfect derived categories $\operatorname {\mathcal {D}}_\infty (Q)\cong \operatorname {per}\mathbf {k} Q$ (of an acyclic quiver $Q$) would be the Calabi–Yau-$\infty$ cluster categories $\operatorname {\mathcal {C}}_{\infty }(Q)$ as

(1.7)\begin{equation} \operatorname{\mathcal{D}}_\infty(Q) \text{`}\approx\text{'} \lim_{m\to\infty}\operatorname{\mathcal{C}}_m(Q), \end{equation}

in the sense that the fundamental domains in $\operatorname {\mathcal {D}}_\infty (Q)$ for the orbit quotient $\operatorname {\mathcal {D}}_\infty (Q)\to \operatorname {\mathcal {C}}_m(Q)$ can be chosen with inclusion relation and the limit of which is $\operatorname {\mathcal {D}}_\infty (Q)$. The corresponding statement for the spaces of stability conditions is [Qiu15, Theorem 6.2]

\[ \operatorname{Stab}\operatorname{\mathcal{D}}_\infty(Q) \cong \lim_{N\to\infty} \operatorname{Stab}\operatorname{\mathcal{D}}_N(Q)/\operatorname{Br}_Q. \]

As an application of our Calabi–Yau-$\mathbb {X}$ construction, we show that, in fact, $\operatorname {\mathcal {D}}_\infty (Q)$ can be realized as cluster-$\mathbb {X}$ category, i.e. as a Verdier quotient $\operatorname {per}\operatorname {\Gamma }_{\mathbb {X}}Q/\operatorname {\mathcal {D}}(\operatorname {\Gamma }_{\mathbb {X}}Q)$ of $\operatorname {\Gamma }_{\mathbb {X}}Q$, by relating (1.6) and (1.7) by completing a commutative diagram (1.8) of short exact sequences of triangulated categories. This unifies the theory of cluster tilting and silting, which provides a new perspective to study these categories.

1.5 Content

In § 2, we construct Calabi–Yau-$\mathbb {X}$ categories from quivers and from coherent sheaves as our motivating examples. In §§ 3–5, we introduce $q$-stability conditions (Definition 3.4) and we prove the following theorem (cf. Theorems 3.10, 4.5, and 5.11).

Theorem A Let $\operatorname {\mathcal {D}}_\mathbb {X}$ be a category satisfies Assumption 3.6, $n$ the rank of $K\operatorname {\mathcal {D}}_\mathbb {X}$ over $R=\mathbb {Z}[q^{\pm 1}]$ and $s$ a complex number. We have the following.

• – The space $\operatorname {QStab}_s\operatorname {\mathcal {D}}_\mathbb {X}$ of $q$-stability conditions is a complex manifold of dimension $n$ with local coordinate

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

• – If the orbit category $\operatorname {\mathcal {D}}_N\colon =\operatorname {\mathcal {D}}_{\mathbb {X}} \mathbin {/\mkern -6mu/} [\mathbb {X}-N]$ is well-defined (i.e. $N$-reductive), then there is a canonical injection of complex manifolds

  \[ \iota_N \colon \operatorname{QStab}_N(\operatorname{\mathcal{D}}_{\mathbb{X}}) \to \operatorname{Stab}\operatorname{\mathcal{D}}_N, \]
  whose image is open and closed.

• – If $\operatorname {\mathcal {D}}_\mathbb {X}$ is Calabi–Yau-$\mathbb {X}$, then the space $\operatorname {QStab}^{\oplus }\operatorname {\mathcal {D}}_\mathbb {X}$ of induced open $q$-stability conditions is a complex manifold of dimension $n+1$.

In § 6, we show that the perfect derived category of an acyclic quiver can be realized as a cluster-$\mathbb {X}$ category that fits into the following story (Corollary 6.8).

Theorem B Let $Q$ be an acyclic quiver and $N\ge 2$ an integer. Then there is the following commutative diagram.

(1.8)

In Appendix A, we discuss the categorification of $q$-deformed root lattices.

2. Calabi–Yau-$\mathbb {X}$ categories

All the modules and categories will be over $\mathbf {k}$, an algebraically closed field.

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 [Kel11a, 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 [Gin06, 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.

Theorem 2.4 [Kel11a, Theorem 6.3] and [Kel18, Theorem 1.1]

For a homologically smooth ddg-algebra $A$, the Calabi–Yau-$\mathbb {X}$ completion $\Pi _{\mathbb {X}}(A)$ becomes a Calabi–Yau-$\mathbb {X}$ algebra. In particular, $\operatorname {\mathcal {D}}_{\rm fd}(\Pi _{\mathbb {X}}(A))$ is Calabi–Yau-$\mathbb {X}$.

We also have the following.

Lemma 2.5 [Kel11a, 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}

Remark 2.6 All statements of this section work not only for $\mathbb {X}$ but also for any $\mathcal {N} \in \mathbb {Z} \oplus \mathbb {Z}\mathbb {X}$. However, in this paper we mainly deal with the case $\mathcal {N}=\mathbb {X}$.

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 [Gin06].

Definition 2.7 [Gin06, Kel11a]

Let $Q=(Q_0,Q_1)$ be a finite acyclic quiver with vertices $Q_0=\{1,\ldots,n\}$ and arrows $Q_1$. We introduce the Ginzburg Calabi–Yau- $\mathbb {X}$ ddg-algebra $\Gamma _{\mathbb {X}}Q :=(\mathbf {k} \bar {Q},d)$ as follows. Define a $\mathbb {Z}\oplus \mathbb {Z}\mathbb {X}$-graded quiver $\bar {Q}$ with vertices $\bar {Q}_0=\{1,\ldots,n\}$ and the following arrows:

• – an original arrow $a \colon i \to j \in Q_1$ (degree $0$);

• – an opposite arrow $a^* \colon j \to i$ for the original arrow $a \colon i \to j \in Q_1$ (degree $2-\mathbb {X}$);

• – a loop $t_i=e_i^*$ for each vertex $i \in Q_0$ (degree $1-\mathbb {X}$).

Let $\mathbf {k} \bar {Q}$ be a $\mathbb {Z} \oplus \mathbb {Z}\mathbb {X}$-graded path algebra of $\bar {Q}$, and define a differential $\operatorname {d} \colon \mathbf {k} \bar {Q} \to \mathbf {k} \bar {Q}$ of degree $1$ by:

• – $\operatorname {d} a =\operatorname {d} a^*= 0$ for $a \in Q_1$;

• – $\operatorname {d} t_i = e_i (\sum _{a \in Q_1}(aa^* -a^*a)) e_i$;

where $e_i$ is the idempotent at $i \in Q_0$. Thus, we have the ddg-algebra $\Gamma _{\mathbb {X}}Q =(\mathbf {k} \bar {Q},\operatorname {d})$. Note that the $0$th homology is given by $H^0(\Gamma _{\mathbb {X}}Q) \cong \mathbf {k} Q$.

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.

Corollary 2.8 [Kel11a, Theorem 6.3]

The Calabi–Yau-$\mathbb {X}$ completion $\Pi _{\mathbb {X}}(k Q)$ of the path algebra $k Q$ is isomorphic to the Ginzburg Calabi–Yau-$\mathbb {X}$ algebra $\Gamma _{\mathbb {X}}Q$.

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 [BB03], there is a classical generator $G \in \operatorname {\mathcal {D}}^b (\mathrm {Coh} X)$ and [Orl09] 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 [BB03, 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.

Remark 2.11 Here we show Lemma 2.9 for a smooth projective variety. However, the result also holds for a smooth quasi-projective variety. See [HK18, Proposition 3.10].

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.

Lemma 2.12

• – The functor $\Delta _* \colon \operatorname {\mathcal {D}}^b(\mathrm {Coh} X) \to \operatorname {\mathcal {D}}^b(\mathrm {Coh} (X\times X))$ is a monoidal functor.

• – The functor $F^{\rm e}\colon \operatorname {\mathcal {D}}^b (\mathrm {Coh} (X \times X)) \to \operatorname {per} A^{\rm e}$ is a monoidal functor.

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 [BM11], 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. $q$-stability conditions on $\mathbb {X}$-categories

3.1 Bridgeland stability conditions

First we recall the definition of Bridgeland stability conditions on triangulated categories from [Bri07]. 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 $n$.

Definition 3.1 Let $\operatorname {\mathcal {D}}$ be a triangulated category. A stability condition $\sigma = (Z, \mathcal {P})$ on $\operatorname {\mathcal {D}}$ consists of a group homomorphism $Z \colon K(\operatorname {\mathcal {D}}) \to \mathbb {C}$ called the central charge and a family of full additive subcategories $\mathcal {P} (\phi ) \subset \operatorname {\mathcal {D}}$ for $\phi \in \mathbb {R}$ called the slicing satisfying the following conditions:

1. (a) if $0 \neq E \in \mathcal {P}(\phi )$, then $Z(E) = m(E) \exp ({i} \pi \phi )$ for some $m(E) \in \mathbb {R}_{>0}$;

2. (b) for all $\phi \in \mathbb {R}$, $\mathcal {P}(\phi + 1) = \mathcal {P}(\phi )[1]$;

3. (c) if $\phi _1 > \phi _2$ and $A_i \in \mathcal {P}(\phi _i)\,(i =1,2)$, then $\operatorname {Hom}_{\operatorname {\mathcal {D}}}(A_1,A_2) = 0$;

4. (d) for $0 \neq E \in \operatorname {\mathcal {D}}$, there is a finite sequence of real numbers

   (3.1)\begin{equation} \phi_1 > \phi_2 > \cdots > \phi_m \end{equation}
   and a collection of exact triangles (Harder–Narasimhan (HN) filtration)
   (3.2)

   
   with HN-factors $A_i \in \mathcal {P}(\phi _i)$ for all $i$.

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 [KS08], 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 [Bri07] showed the following crucial theorem.

Theorem 3.2 [Bri07, 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 [Bri07], 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 $\operatorname {Stab}\operatorname {\mathcal {D}}$.

Definition 3.3 [Bri07, Definition 5.7]

A slicing $\mathcal {P}$ of a triangulated category $\operatorname {\mathcal {D}}$ is locally-finite if there exists a real number $\epsilon >0$ such that for all $\phi \in \mathbb {R}$ the quasi-abelian category ${\mathcal {P}(\phi -\epsilon,\phi +\epsilon)}$ is of finite length. A stability condition is locally-finite if the corresponding slicing is.

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 $\mathbb {X}(\sigma )$.

Remark 3.5 Equation (3.9) has been considered by Toda [Tod14] to study the orbit point, known as the Gepner point, of the orbitfold $\mathbb {C}\backslash \operatorname {Stab}\operatorname {\mathcal {D}}/\operatorname {Aut}$. We impose the following condition (Assumption 3.6) on our triangulated category $\operatorname {\mathcal {D}}_\mathbb {X}$, which means $\operatorname {Stab}\operatorname {\mathcal {D}}_\mathbb {X}$ is infinite dimensional. In addition, (3.9) reduces the dimension of the stability spaces. See [Qiu23, § 1.2] for further discussion.

In the rest of this paper, we assume following.

Assumption 3.6 The Grothendieck group $\operatorname {\mathrm {K}} (\operatorname {\mathcal {D}}_{\mathbb {X}})$ is free of finite rank over $R$, i.e. $\operatorname {\mathrm {K}}(\operatorname {\mathcal {D}}_{\mathbb {X}} )\cong R^{\oplus n}$ for some $n$. We will call such a category $\operatorname {\mathcal {D}}_\mathbb {X}$ an $\mathbb {X}$-category.

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).

Definition 3.7 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 the following two further conditions:

1. (e) the slicing satisfies $\mathcal {P}(\phi + \operatorname {Re}( s)) = \mathcal {P}(\phi )[\mathbb {X}]$ for all $\phi \in \mathbb {R}$;

2. (f) the central charge $Z\colon K(\operatorname {\mathcal {D}}_{\mathbb {X}}) \to \mathbb {C}_s$ is $R$-linear,

   \[ Z \in \operatorname{Hom}_R(K(\operatorname{\mathcal{D}}_{\mathbb{X}}),\mathbb{C}_s). \]

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.

Lemma 3.11 If a $q$-stability condition $(\sigma,s)$ satisfies the $q$-support property and $s\ne 0$, then it is locally-finite.

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 [Bri08, 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 [Bri07, 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 [Bri07, 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 $s$).

Lemma 3.13 The slicing $\mathcal {Q}$ satisfies condition (e) in Definition 3.7, thus $(\varsigma,s)$ is a $q$-stability condition.

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.

Lemma 3.14 The $q$-stability condition $(\varsigma,s)$ in Corollary 3.12 satisfies the $q$-support property.

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 $s\in \mathbb {C}$.

Lemma 3.15 Let $(\sigma _i,s_i)$ be $q$-stability conditions with $s_1\neq s_2$. Then they are in different connected components of $\operatorname {Stab}_s\operatorname {\mathcal {D}}_\mathbb {X}$.

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 [IQ18] (and, hence, are the most interesting as far as we are concerned).

4. Reduction

In this section, we show that under some conditions, the space of $q$-stability conditions with $s \in \mathbb {Z}$ coincides with the space of usual stability conditions (on a triangulated category with finite rank Grothendieck group). First we recall the notion of orbit categories. Let $\operatorname {\mathcal {D}}$ be a triangulated category with a functor $\Phi \colon \operatorname {\mathcal {D}} \to \operatorname {\mathcal {D}}$, the orbit category $\operatorname {\mathcal {D}} / \Phi$ is defined to be the category whose objects are the same as $\operatorname {\mathcal {D}}$ and whose morphism spaces are given by

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}} / \Phi}(E,F):=\bigoplus_{k \in \mathbb{Z}}\operatorname{Hom}_{\operatorname{\mathcal{D}}}(E,\Phi^k(F)). \]

As in the previous section, let $\operatorname {\mathcal {D}}_{\mathbb {X}}$ be a triangulated category with a distinguished auto-equivalence $\mathbb {X} \colon \operatorname {\mathcal {D}}_{\mathbb {X}} \to \operatorname {\mathcal {D}}_{\mathbb {X}}$ satisfying Assumption 3.6.

Definition 4.1 Let $N\ge 1$ be an integer. The orbit quotient

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

is defined to be the triangulated hull of the orbit category $\operatorname {\mathcal {D}}_{\mathbb {X}} / [\mathbb {X}-N]$. It is $N$-reductive if:

• – the quotient functor $\pi _N \colon \operatorname {\mathcal {D}}_{\mathbb {X}} \to \operatorname {\mathcal {D}}_N$ is exact;

• – the Grothendieck group of $\operatorname {\mathcal {D}}_N$ is free of finite rank, i.e. $K(\operatorname {\mathcal {D}}_N) \cong \mathbb {Z}^{\oplus n}$ and the induced $R$-linear map

  \[ [\pi_N] \colon K(\operatorname{\mathcal{D}}_{\mathbb{X}}) \to K(\operatorname{\mathcal{D}}_N) \]
  is a surjection given by sending $q \mapsto (-1)^N$.

In our motivating examples, $\operatorname {\mathcal {D}}_\mathbb {X}$ is constructed with dg-structures, which provides a dg enhancement of $\operatorname {\mathcal {D}}_{\mathbb {X}} \mathbin {/\mkern -6mu/} [\mathbb {X}-N]$.

Construction 4.2 Assume that $\operatorname {\mathcal {D}}_{\mathbb {X}}$ is $N$-reductive and let $(\sigma,N)$ be an $q$-stability condition on $\operatorname {\mathcal {D}}_{\mathbb {X}}$ with $\sigma =(Z,\mathcal {P})$. We define a stability condition $\sigma _N=(Z_N,\mathcal {P}_N)$ on $\operatorname {\mathcal {D}}_N$ as follows:

• – $\mathcal {P}_N(\phi ):=\mathcal {P}(\phi )$;

• – by condition (f) in Definition 3.7, $Z\colon K(\operatorname {\mathcal {D}}_{\mathbb {X}})\to \mathbb {C}$ factors through $[\pi _N]$ and thus we obtain a group homomorphism $Z_N\colon K(\operatorname {\mathcal {D}}_N)\to \mathbb {C}$ satisfying $Z=Z_N\circ [\pi _N]$.

Lemma 4.3 If $\operatorname {\mathcal {D}}_N$ is $N$-reductive, then $\sigma _N$ is a stability condition in $\operatorname {Stab}\operatorname {\mathcal {D}}_N$.

Proof. Since $(\sigma,N)$ satisfies Condition (e) in Definition 3.7, $\mathcal {P}$ on $\operatorname {\mathcal {D}}_N$ is well-defined and satisfies Condition (b) in Definition 3.1. As $\pi _N$ is exact, $\mathcal {P}$ satisfies Condition (c), (d) in Definition 3.1. and thus a slicing. Thus, the lemma follows.

For a $q$-stability condition $\sigma$, define a real number $L(\sigma )$ by

\[ L(\sigma):=\inf\bigg\{\frac{|Z(E)|}{\lVert [E] \rVert} \bigg| [E] \in \widehat{{\mathrm{ss}}}(\sigma) \bigg\}, \]

where $\lVert [E] \rVert$ is the norm in Definition 3.8. Then we note that the support property is equivalent to the condition $L(\sigma )>0$.

Proposition 4.4 Let $\{\sigma _k\}_{k \ge 1}$ is a sequence of $q$-stability conditions in $\operatorname {QStab}_s\operatorname {\mathcal {D}}_{\mathbb {X}}$ satisfying:

• – central charges $Z_k$ converge as $k \to \infty$ in $\operatorname {Hom}_{R}(K(\operatorname {\mathcal {D}}_{\mathbb {X}}),\mathbb {C}_s)$;

• – the slicing of $\sigma _k$ converges in the space $\operatorname {Slice}\operatorname {\mathcal {D}}_{\mathbb {X}}$ of slicings on $\operatorname {\mathcal {D}}_{\mathbb {X}}$, where the metric on $\operatorname {Slice}\operatorname {\mathcal {D}}_{\mathbb {X}}$ is given by

  (4.1)\begin{equation} d(\mathcal{P},\mathcal{Q}):= \sup_{0 \neq E \in \operatorname{\mathcal{D}}}\big\{ |\phi_{\mathcal{P}}^-(E) - \phi_{\mathcal{Q}}^-(E)|, |\phi_{\mathcal{P}}^+(E) - \phi_{\mathcal{Q}}^+(E)| \big\}; \end{equation}

• – there is a uniform constant $L>0$ such that

  \[ L(\sigma_k) \ge L \]
  for all $k$.

Then the sequence $\{\sigma _k \}_{k \ge 1}$ converges to some stability condition $\sigma _{\infty } \in \operatorname {QStab}_s\operatorname {\mathcal {D}}_{\mathbb {X}}$ as $k \to \infty$.

Proof. We show that for any real small number $\epsilon >0$, the limit $Z_{\infty }:=\lim _{k\to \infty }Z_k$ satisfies

\[ |Z_{\infty}(E) - Z_k(E)|<\epsilon \cdot|Z_k(E)| \]

for sufficiently large $k$ and all $[E] \in \widehat {{\mathrm {ss}}}(\sigma _k)$. Then by Theorem 3.10, there exists a unique stability condition $\sigma _{\infty }$ with the central charge $Z_{\infty }$ satisfying $d(\sigma _{\infty },\sigma _k)<\epsilon$ as required.

Consider the operator norm $\lVert \,\cdot \, \rVert$ on $\operatorname {Hom}_{R}(K(\operatorname {\mathcal {D}}_{\mathbb {X}}),\mathbb {C}_s)$ defined by

\[ \lVert W \rVert:=\sup_{0 \neq E \in \operatorname{\mathcal{D}}_{\mathbb{X}}} \frac{|W(E)|}{\lVert [E] \rVert}. \]

Then because $Z_k \to Z_{\infty }$ as $k \to \infty$, we have

\[ \lVert Z_{\infty}-Z_k \rVert<\epsilon \cdot L \]

for sufficiently large $k$. This implies

\[ |Z_{\infty}(E)-Z_k(E)|<\epsilon \cdot L\cdot \lVert [E] \rVert \]

for all $0 \neq E \in \operatorname {\mathcal {D}}_{\mathbb {X}}$. On the other hand, from the condition $L(\sigma _k )\ge L$, we have

\[ |Z_k(E)| \ge L \cdot \lVert [E] \rVert \]

for all $[E] \in \widehat {{\mathrm {ss}}}(\sigma _k)$.

Combining Lemma 4.3 and Proposition 4.4, we have the following, which is one of the motivations that we introduce $q$-stability conditions.

Theorem 4.5 If $\operatorname {\mathcal {D}}_{\mathbb {X}}$ is an $N$-reductive, then there is a canonical injection of complex manifolds

\[ \iota_N \colon \operatorname{QStab}_N(\operatorname{\mathcal{D}}_{\mathbb{X}}) \to \operatorname{Stab}\operatorname{\mathcal{D}}_N, \]

whose image is open and closed.

Proof. We show the closedness of the image of $\iota _N$ and the other part is straightforward. Take a convergent sequence $\{\sigma _n\}_{n \ge 1}$ in $\operatorname {Stab}\operatorname {\mathcal {D}}_N$ with the limit $\sigma _{\infty } \in \operatorname {Stab}\operatorname {\mathcal {D}}_N$ and assume that $\sigma _n=\iota _{N}(\tilde {\sigma }_n)$ for $\tilde {\sigma }_n \in \operatorname {QStab}_N(\operatorname {\mathcal {D}}_{\mathbb {X}})$. We show that there is some $\tilde {\sigma }_{\infty } \in \operatorname {QStab}_N(\operatorname {\mathcal {D}}_{\mathbb {X}})$ such that $\iota _N(\tilde {\sigma }_{\infty })=\sigma _{\infty }$. We check the sequence $\{\tilde {\sigma }_n\}_{n \ge 1}$ satisfies the conditions of Proposition 4.4.

The first two conditions follows by direct checking as $\sigma _n$ and $\tilde {\sigma }_n$ as their central charges and slicings are essentially the same. Next we consider the third condition. As the sequence $\{L(\tilde {\sigma }_n)\}_{n \ge 1}$ converges to some positive number $L(\tilde {\sigma }_{\infty } )>0$, there is uniform constant $L>0$ such that $L(\tilde {\sigma }_n)>L$. Then the second condition also holds because $L(\sigma _n)=L(\tilde {\sigma }_n)$ by definition.

In the next section, we introduce a special type of $q$-stability condition, the induced $q$-stability conditions, which in many cases provide the existence of $q$-stability conditions.

5. Induction

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 [AT11] (see also [FM18]).

Definition 5.1 An $\mathbb {X}$-baric heart $\operatorname {\mathcal {D}}_{\infty } \subset \operatorname {\mathcal {D}}_{\mathbb {X}}$ is a full triangulated subcategory of $\operatorname {\mathcal {D}}_{\mathbb {X}}$ satisfying the following conditions:

1. (1) if $k_1 > k_2$ and $A_i \in \operatorname {\mathcal {D}}_{\infty }[k_i\mathbb {X}]\,(i =1,2)$, then $\operatorname {Hom}_{\operatorname {\mathcal {D}}_{\mathbb {X}}}(A_1,A_2) = 0$;

2. (2) for $0 \neq E \in \operatorname {\mathcal {D}}_{\mathbb {X}}$, there is a finite sequence of integers

   \[ k_1 > k_2 > \cdots > k_m \]
   and a collection of exact triangles
   (5.1)

   
   with $A_i \in \operatorname {\mathcal {D}}_{\infty }[k_i\mathbb {X}]$ for all $i$.

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

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

Construction 5.2 Consider a triple $(\operatorname {\mathcal {D}}_\infty,\hat {\sigma },s)$ that consists of an $\mathbb {X}$-baric heart $\operatorname {\mathcal {D}}_\infty$, a (Bridgeland) stability condition $\hat {\sigma }=(\hat {Z},\hat {\mathcal {P}})$ on $\operatorname {\mathcal {D}}_\infty$ and a complex number $s$. We construct

1. (i^) the additive pre-stability condition $\sigma _{\oplus }=(Z,\mathcal {P}_{\oplus })$; and

2. (ii^) the extension pre-stability condition $\sigma _{*}=(Z,\mathcal {P}_{*})$;

where:

• – first extend $\hat {Z}$ to

  \[ Z_q\colon=\hat{Z} \otimes 1\colon K(\operatorname{\mathcal{D}}_\mathbb{X})\to\mathbb{C}[q,q^{-1}] \]
  via (5.2) and
  \[ Z=q_s\circ Z_q\colon K(\operatorname{\mathcal{D}}_\mathbb{X})\to\mathbb{C} \]
  gives a central charge function on $\operatorname {\mathcal {D}}_\mathbb {X}$;

• – the pre-slicing $\mathcal {P}_{\oplus }$ is defined as

  (5.5)\begin{equation} \mathcal{P}_{\oplus}(\phi)=\operatorname{add}^s\hat{\mathcal{P}}[\mathbb{Z}\mathbb{X}]\colon=\operatorname{add} \bigoplus_{k\in\mathbb{Z}} \hat{\mathcal{P}}(\phi-k\operatorname{Re}(s))[k\mathbb{X}]; \end{equation}

• – the pre-slicing $\mathcal {P}_{*}$ is defined as

  (5.6)\begin{equation} \mathcal{P}_{*}(\phi)=\langle \hat{\mathcal{P}}[\mathbb{Z}\mathbb{X}] \rangle^s\colon=\langle \hat{\mathcal{P}}(\phi-k\operatorname{Re}(s))[k\mathbb{X}] \rangle. \end{equation}

Note that $\sigma$ does not necessary satisfy condition (d) in Definition 3.1 and, hence, may not be a stability condition. We call such data (a central charge and collection of additive subcategories) a pre-stability condition if they satisfy conditions (a), (b), and (c) in Definition 3.1.

Remark 5.3 Clearly, $\mathcal {P}_{\oplus }(\phi )\subset \mathcal {P}_{*}(\phi )$ although their sets of simple objects coincide. Also note that $\sigma _{\oplus }$ or $\sigma _{*}$, as a pre-stability condition, may be induced from different triples.

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 [Bri07, 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 [Bri07, 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.

Theorem 5.9 Let $\operatorname {\mathcal {D}}_\mathbb {X}$ be a Calabi–Yau-$\mathbb {X}$ category satisfies Assumption 3.6. Given a stability condition $\hat {\sigma }=(\hat {Z},\hat {\mathcal {P}})$ on an $\mathbb {X}$-baric heart $\operatorname {\mathcal {D}}_\infty$ of $\operatorname {\mathcal {D}}_\mathbb {X}$, then we have the following.

1. (i^) The induced additive pre-stability condition $\sigma _{\oplus }=(Z, \mathcal {P}_{\oplus })$ is a stability condition on $\operatorname {\mathcal {D}}_\mathbb {X}$ if and only if

   (5.11)\begin{equation} \operatorname{Hom}(\hat{\mathcal{P}}(\phi_1),\hat{\mathcal{P}}(\phi_2))=0 \quad \text{for any $\phi_2-\phi_1\geq\operatorname{Re}(s)-1$.} \end{equation}

2. (ii^) The induced extension pre-stability condition $\sigma _{*}=(Z, \mathcal {P}_{*})$ is a stability condition on $\operatorname {\mathcal {D}}_\mathbb {X}$ if and only if

   (5.12)\begin{equation} \operatorname{gldim}\hat{\sigma}\le\operatorname{Re}(s)-1. \end{equation}

Clearly, both $\sigma _{\oplus }$ and $\sigma _{*}$ satisfies (3.9) and hence $(\sigma _{\oplus },s)$ and $(\sigma _{*},s)$ are both $q$-stability conditions. Finally, they satisfy $q$-support property, where $\operatorname {\mathrm {K}}(\operatorname {\mathcal {D}}_\infty )$ provides the $\mathbb {Z}^n$ lattice in condition (i$^\circ$) of Definition 3.8.

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.

Definition 5.10 An open (induced) $q$-stability condition on $\operatorname {\mathcal {D}}_{\mathbb {X}}$ is a pair $(\sigma,s)$ consisting of a stability condition $\sigma$ on $\operatorname {\mathcal {D}}_\mathbb {X}$ and a complex parameter $s$, satisfying:

• – $\sigma =\sigma _{\oplus }$ is an additive pre-stability condition induced from some triple $(\operatorname {\mathcal {D}}_\infty,\hat {\sigma },s)$ as in Construction 5.2 with

  (5.21)\begin{equation} \operatorname{gldim}\hat{\sigma}+1<\operatorname{Re}(s). \end{equation}

Denote by $\operatorname {QStab}^{\oplus }_s\operatorname {\mathcal {D}}_\mathbb {X}$ the set of all open $q$-stability conditions with the parameter $s\in \mathbb {C}$ and by $\operatorname {QStab}^{\oplus }\operatorname {\mathcal {D}}_\mathbb {X}$ the union of all $\operatorname {QStab}^{\oplus }_s\operatorname {\mathcal {D}}_\mathbb {X}$.

Similarly, a closed (induced) $q$-stability condition on $\operatorname {\mathcal {D}}_{\mathbb {X}}$ is a pair $(\sigma,s)$ consisting of a stability condition $\sigma$ on $\operatorname {\mathcal {D}}_\mathbb {X}$ and a complex parameter $s$, satisfying:

• – $\sigma =\sigma _{*}$ is an extension pre-stability condition induced from some triple $(\operatorname {\mathcal {D}}_\infty,\hat {\sigma },s)$ (note that Theorem 5.9 forces the inequality (5.12) to hold).

Denote by $\operatorname {QStab}^{*}_s\operatorname {\mathcal {D}}_\mathbb {X}$ the set of all closed $q$-stability conditions with the parameter $s\in \mathbb {C}$.

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.

Theorem 5.11 Let $\operatorname {\mathcal {D}}_\mathbb {X}$ be a Calabi–Yau-$\mathbb {X}$ category satisfying Assumption 3.6. Then $\operatorname {QStab}^{\oplus }\operatorname {\mathcal {D}}_\mathbb {X}$ is a complex manifold of dimension $n+1$.

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.

Remark 5.12 One of the key feature of the local charts we construct in Theorem 5.11 is that each point $(\sigma,s)$ admits a distinguished section $\Gamma =\{(\sigma ',s')\}$ for $s'\in (s-\epsilon,s+\epsilon )$, satisfying:

• – the sets of (semi-)stable objects are invariants along this section.

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.

Proposition 5.13 The category $\operatorname {\mathcal {D}}_\infty (Q)$ is an $\mathbb {X}$-baric heart of $\operatorname {\mathcal {D}}_\mathbb {X}(Q)$.

Proof. Regard $\Gamma _{\mathbb {X}}Q$ as a $\mathbb {Z}\mathbb {X}$-graded algebra. As $\mathbb {X}$ is the grading shift, any object admits a (HN-)filtration (5.1). The Hom vanishing property comes from the fact that $\Gamma _{\mathbb {X}}Q$ is concentrated in non-positive degrees (with respect to the $\mathbb {Z}\mathbb {X}$ grading). Therefore, $\operatorname {\mathcal {D}}_\infty (Q)$ is an $\mathbb {X}$-baric heart of $\operatorname {\mathcal {D}}_\mathbb {X}(Q)$.

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. Perfect derived categories as cluster-$\mathbb {X}$ categories

Let $Q$ be an acyclic quiver as above.

6.1 Cluster categories

The cluster categories $\operatorname {\mathcal {C}}(Q)$ were introduced in [BMR⁺06] to categorify cluster algebras associated to $Q$. Keller [Kel05] provided the construction of cluster categories as orbit categories.

Definition 6.1 [BMR⁺06, Kel05]

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-$(N-1)$.

Theorem 6.2 [Ami09, Guo10, Kel11a]

There is a natural triangle equivalence $\operatorname {\mathcal {C}}_{m}(Q)\cong \operatorname {\mathcal {C}}(\operatorname {\Gamma }_{N}Q)$ that identifies the canonical cluster tilting objects in them.

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. [ST01, Proposition 4.18], [Kel05], and [KY18, Theorem 5.1]).

Proposition 6.3 We have that $\operatorname {\mathcal {D}}_\mathbb {X}(Q) / [\mathbb {X}-N]$ is $N$-reductive, where the triangulated structure is provided by its unique triangulated hull $\operatorname {\mathcal {D}}_N(Q)$.

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.

Corollary 6.4 The quotient functor $\pi _N$ induces an injection $i_N\colon \operatorname {Aut}\operatorname {\mathcal {D}}_\mathbb {X}(Q)/[\mathbb {X}-N]\to \operatorname {Aut}\operatorname {\mathcal {D}}_N(Q)$.

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 $A$, [KS02] shows $\iota _Q^\mathbb {X}$ in (A.1) is an isomorphism. Seidel and Thomas [ST01] proved the corresponding

\[ \iota_Q^N\colon\operatorname{Br}_Q \to \operatorname{ST}_N(Q) \]

is also an isomorphism using Proposition 6.3. Recall the following result for the Dynkin case and affine type A case.

Theorem 6.5 We have that $\iota _Q^N$ is an isomorphism when:

• – $Q$ is a Dynkin quiver and $N\ge 2$ (see [QW18, Theorem B]);

• – $Q$ is affine type A and $N=2$ (see [IUU10, Corollary 37]) or $3$ (see [Qiu16, Theorem 7.3]).

Then combining with Corollary 6.4, we obtain Calabi–Yau-$\mathbb {X}$ version as a direct corollary of Corollary 6.4.

Theorem 6.6 Let $Q$ either be a Dynkin quiver or an affine type A quiver, Then $\iota _Q^\mathbb {X}:\operatorname {Br}_Q \to \operatorname {ST}_\mathbb {X}(Q)$ is an isomorphism for any $N\ge 2$.

6.3 Cluster-$\mathbb {X}$ categories

As a generalization, we defined the cluster-$\mathbb {X}$ category $\operatorname {\mathcal {C}}(\operatorname {\Gamma }_{\mathbb {X}}Q)$ as the Verdier quotient $\operatorname {per}\operatorname {\Gamma }_{\mathbb {X}}Q/\operatorname {\mathcal {D}}_\mathbb {X}(Q)$.

Theorem 6.7 The embedding $\mathbf {k} Q\to \operatorname {\Gamma }_{\mathbb {X}}Q$ induces a triangle equivalence

\[ \operatorname{\mathcal{C}}(\operatorname{\Gamma}_{\mathbb{X}}Q)\xrightarrow{\cong}\operatorname{per}\mathbf{k} Q \]

that factors through $\operatorname {per}\operatorname {\Gamma }_{\mathbb {X}}Q$.

Proof. First, the embedding $\mathbf {k} Q\to \operatorname {\Gamma }_{\mathbb {X}}Q$ induces the functor

(6.5)\begin{equation} i_*\colon \operatorname{per}\mathbf{k} Q\to \operatorname{per}\operatorname{\Gamma}_{\mathbb{X}}Q \end{equation}

sending projectives to projectives. Denote the image of $i_*$ by $\mathcal {C}$, which is generated by $\{ \operatorname {\Gamma }_{\mathbb {X}}Q[i]\}_{i\in \mathbb {Z}}$.

Recall that $\operatorname {\mathcal {D}}_\mathbb {X}(Q)$ admits the Serre functor $\mathbb {X}$ and $\operatorname {Sim}\operatorname {\Gamma }_{\mathbb {X}}Q$ is the set of simple $\operatorname {\Gamma }_{\mathbb {X}}Q$-modules. Denote by $\mathcal {S}$ the thick subcategory of $\operatorname {\mathcal {D}}_\mathbb {X}(Q)$ generated by the objects in $\operatorname {Sim}\operatorname {\Gamma }_{\mathbb {X}}Q[\mathbb {Z}]$. Consider the canonical unbounded t-structure $\operatorname {\mathcal {D}}_\mathbb {X}(Q)=\langle \mathcal {X}, \mathcal {Y} \rangle$, where $\mathcal {X}$ is generated by $\mathcal {S}[\mathbb {Z}_{\ge 0}\mathbb {X}]$ and $\mathcal {Y}$ is generated by $\mathcal {S}[\mathbb {Z}_{<0}\mathbb {X}]$.

By the Calabi–Yau-$\mathbb {X}$ duality in Lemma 2.5, we obtain

\[ \operatorname{Hom}(M,\operatorname{\Gamma}_{\mathbb{X}}Q)\cong \operatorname{Hom}(\operatorname{\Gamma}_{\mathbb{X}}Q,M[\mathbb{X}])^* \]

for any $M\in \operatorname {\mathcal {D}}_\mathbb {X}(Q)$. Noting the following calculation about $\operatorname {Hom}$ between projectives and simples:

(6.6)\begin{equation} \operatorname{Hom}^{\mathbb{Z}}_{\operatorname{per}\operatorname{\Gamma}_{\mathbb{X}}Q}(\operatorname{\Gamma}_{\mathbb{X}}Q,\mathcal{S}[j\mathbb{X}])=0 \end{equation}

for any $j\neq 0$, we have

(6.7)\begin{equation} \operatorname{Hom}_{\operatorname{per}\operatorname{\Gamma}_{\mathbb{X}}Q}(\mathcal{S}[j\mathbb{X}],\mathcal{C})=0 \end{equation}

for any $j\neq -1$ and, moreover,

\[ \operatorname{Hom}_{\operatorname{per}\operatorname{\Gamma}_{\mathbb{X}}Q}(\mathcal{S},\operatorname{\Gamma}_{\mathbb{X}}Q[\mathbb{X}]) \cong\operatorname{Hom}_{\operatorname{per}\operatorname{\Gamma}_{\mathbb{X}}Q}(\operatorname{\Gamma}_{\mathbb{X}}Q,\mathcal{S})^*\neq0, \]

which implies $\operatorname {Hom}_{\operatorname {per}\operatorname {\Gamma }_{\mathbb {X}}Q}(\mathcal {S},\mathcal {C}[\mathbb {X}])\neq 0$. Therefore, we deduce that the right perpendicular

\[ \mathcal{X}^\perp \colon=\{ Z\in\operatorname{per}\operatorname{\Gamma}_{\mathbb{X}}Q \mid \operatorname{Hom}_{\operatorname{per}\operatorname{\Gamma}_{\mathbb{X}}Q} (\mathcal{X},Z)=0 \} \]

of $\mathcal {X}$ in $\operatorname {per}\operatorname {\Gamma }_{\mathbb {X}}Q$ is generated by $\mathcal {C}[\mathbb {Z}_{\leq 0}\mathbb {X}]$ and, similarly, the left perpendicular

\[ ^\perp\mathcal{Y}\colon=\{ Z\in\operatorname{per}\operatorname{\Gamma}_{\mathbb{X}}Q \mid \operatorname{Hom}_{\operatorname{per}\operatorname{\Gamma}_{\mathbb{X}}Q} (Z,\mathcal{Y})=0 \} \]

of $\mathcal {Y}$ in $\operatorname {per}\operatorname {\Gamma }_{\mathbb {X}}Q$ is generated by $\mathcal {C}[\mathbb {Z}_{\geq 0}\mathbb {X}]$.

Next we claim that $\langle \mathcal {X},\mathcal {X}^\perp \rangle$ and $\langle ^\perp \mathcal {Y},\mathcal {Y} \rangle$ are torsion pairs in $\operatorname {per}\operatorname {\Gamma }_{X}Q$. We only need to show that they generate $\operatorname {per}\operatorname {\Gamma }_{X}Q$. Recall that the notation $\langle \mathcal {A},\mathcal {B} \rangle$ consists of object $M$ that admits a triangle

\[ A\to M\to B\to A[1] \]

with $A\in \mathcal {A}$ and $B\in \mathcal {B}$.

Consider the $\mathcal {Y}$ case first. We start by showing that $\mathcal {C}\subset \langle ^\perp \mathcal {Y}[\mathbb {X}], \mathcal {S} \rangle$. Take any projective $P_i\in \mathcal {C}$ for $i\in Q_0$. Then there is a triangle (cf. [KY11, § 2.14])

(6.8)\begin{equation} S_i[-1]\to A_1\oplus B_1 \to P_i \to S_i, \end{equation}

for

\[ A_1=\bigoplus_{ i\xrightarrow{a}j \in Q_1 } a\ P_j \quad\text{and}\quad B_1=\bigoplus_{ i\xrightarrow{b}k \in Q_1^*\cup Q_0^* } b\ P_k. \]

The arrows $b\in Q_1^*\cup Q_0^*$ are the new arrows in the double $\bar {Q}$ with $\mathbb {X}$-degree $-1$ (see Definition (2.7)). Thus, $b P_k$ are in $^\perp \mathcal {Y}[\mathbb {X}]$ and, hence, $B_1\in {^\perp }\mathcal {Y}[\mathbb {X}]$. As for $a\in Q_1$, their $\mathbb {X}$-degree is 0. Thus, the $P_j$ are still in ${^\perp }\mathcal {Y}$ and, hence, $A_1\in {^\perp }\mathcal {Y}$. Using (6.8) for $P_j\to S_j$, we obtain the following filtration of $P_i$:

for

\[ A_2= \bigoplus_{ \substack{i\xrightarrow{a}j\in Q_1 \\ j\xrightarrow{a'}j' \in Q_1}} a'a\ P_{j'} \quad\text{and}\quad B_2=B_1\oplus \bigoplus_{ \substack{i\xrightarrow{a}j\in Q_1\\ j\xrightarrow{b'}k' \in Q_1^*\cup Q_0^*}} b'a\ P_{k'} . \]

As before, we have $B_2\in {^\perp }\mathcal {Y}[\mathbb {X}]$ and $A_2\in {^\perp }\mathcal {Y}$. Again, we further decompose $P_{j'}$ using the corresponding (6.8). As $\mathbf {k} Q$ is finite-dimensional, this process will end up with a filtration

(6.9)

for $H_m$ is a direct sum whose summands are of the form $p S_{t{p}}$, where $p$ is a non-zero path of length less than $m$ in $Q_1$ with $h(p)=i$, and $B_m$ is a direct sum whose summands are of the form $b p P_{t(b)}$, where $p$ is a non-zero path of length less than $m$ in $Q_1$ and $b\in Q_1^*\cup Q_0^*$ with $h(b)=t(p),\ h(p)=i$. Here $h(-)$ and $t(-)$ are the head/tail function for arrows/paths. As above, we have $B_m\in {^\perp }\mathcal {Y}[\mathbb {X}]$. In addition, we have $H_m\in \mathcal {S}$. Therefore, (6.9) implies that $P_i\in \langle {^\perp }\mathcal {Y}[\mathbb {X}], \mathcal {S} \rangle$ via the octahedron axiom. Thus, we obtain $\mathcal {C}\subset {^\perp }\langle \mathcal {Y}[\mathbb {X}], \mathcal {S} \rangle$. Inductively, with $\mathcal {C}[-m\mathbb {X}]\subset \langle \mathcal {Y}[-m\mathbb {X}+\mathbb {X}], \mathcal {S}[-m\mathbb {X}] \rangle$ holds for any positive integer $m$, we deduce that $\mathcal {C}[-m\mathbb {X}]\subset \langle {^\perp }\mathcal {Y}, \mathcal {Y} \rangle$ holds for $m>0$ because

\[ {^\perp}\mathcal{Y}[-m\mathbb{X}-\mathbb{X}]\subset{^\perp}\mathcal{Y}\quad\text{and}\quad\mathcal{S}[-m\mathbb{X}]\subset\mathcal{Y}. \]

Hence, $\operatorname {per}\operatorname {\Gamma }_{X}Q=\langle {^\perp }\mathcal {Y},\mathcal {Y} \rangle$ as required.

For the $\mathcal {X}$ case, we only need to modify (6.8) as follows. Rewrite

\[ B_1=e_i^*\ P_i \bigoplus_{ i\xrightarrow{b}j \in Q_1^*} b\ P_k=P_i[\mathbb{X}-1]\oplus \underline{B_1} \]

and by the octahedron axiom we have the following.

Thus, we can decompose $P_i$ into a filtration with factors $S_i[-\mathbb {X}]$, $A_1\oplus \underline {B_1}$ and $P_i[1-\mathbb {X}]$, similar to the first step of (6.9). In the same way, we can prove that $\langle \mathcal {X},\mathcal {X}^\perp \rangle =\operatorname {per}\operatorname {\Gamma }_{X}Q$.

Finally, by [IY20, Theorem 1.1], we obtain the following equivalence $F$ between additive categories

(6.10)

for $\mathcal {C}=\mathcal {X}{^\perp }\cap \;{^\perp }\mathcal {Y}[1]$. What is left to show is that $F\circ i_*$ is an equivalence as triangulated categories. This follows from the fact that $i_*$ is a functor between triangulated categories and $F$ preserves shifts and triangles.

Combining the results above, we obtain the following.

Corollary 6.8 We have the following commutative diagram between short exact sequences of triangulated categories.

(6.11)

Proof. The first two vertical (exact) functors are induced from projection (6.4) as in Proposition 6.3 and the left square commutes as the corresponding horizonal functors are just inclusions. The third vertical functor is Keller's orbit quotient in Definition 6.1. The right square commutes because the projectives (generators) of $\operatorname {per}\operatorname {\Gamma }_{\mathbb {X}}Q$ are mapped to projectives of $\operatorname {per}\mathbf {k} Q$ and $\operatorname {per}\operatorname {\Gamma }_{N}Q$, respectively, and they further become the canonical $(N-1)$-cluster tilting object in $\operatorname {\mathcal {C}}_{N-1}(Q)$.

7. Example: Calabi–Yau-$\mathbb {X}$ $A_2$ quivers

In this section, we discuss the example of $\operatorname {QStab}^{\oplus }\operatorname {\mathcal {D}}_\mathbb {X}(Q)$ for $Q$ is an $A_2$ quiver. The prototype, i.e. the $N$-fiber

\[ \operatorname{QStab}^{\oplus}_N\operatorname{\mathcal{D}}_\mathbb{X}(A_2)\cong\operatorname{Stab}\operatorname{\mathcal{D}}_N(A_2) \]

for $N=2$, is calculated in [BQS20].

Let $S_1$ and $S_2$ be the simple $\Gamma _\mathbb {X} A_2$-module satisfying

\[ \operatorname{Hom}^{\mathbb{Z}^2}(S_1,S_2)=\mathbf{k}[-1],\quad \operatorname{Hom}^{\mathbb{Z}^2}(S_2,S_1)=\mathbf{k}[-\mathbb{X}+1] \]

and $\Psi _i$ be the corresponding spherical twists. The canonical $\mathbb {X}$-baric heart $\operatorname {\mathcal {D}}_\infty (A_2)$ is generated by the shifts of $S_1$ and $S_2$ and has one more indecomposable object $\Psi _1(S_2)$ (up to shift). Moreover, $\operatorname {\mathcal {D}}_\infty (A_2)$ admits a (normal) heart generated by $S_1$ and $S_2$. By Corollary 5.2 and Theorem 5.4, we have the following.

• – We have $\operatorname {ST}_\mathbb {X} A_2=\langle \Psi _1,\Psi _2 \rangle \cong \operatorname {Br}_3$.

• – The center of $\operatorname {ST}_\mathbb {X} A_2$ is generated by $(\Psi _1\circ \Psi _2)^3$.

• – Let $\tau _\mathbb {X}=\Psi _1\circ \Psi _2\circ [\mathbb {X}-2]$ which satisfies $\tau _\mathbb {X}^3=[-2]$ and

  \[ \tau_\mathbb{X}\{S_1,S_2\}=\{S_2,\Psi_1(S_2)[-1]\}. \]

• – Let $\Upsilon _\mathbb {X}=\Psi _1\circ \Psi _2\circ \Psi _1\circ [2\mathbb {X}-3]$ which satisfies $\Upsilon _\mathbb {X}^2=[X-2]$ and

  \[ \Upsilon_\mathbb{X}\{S_1,S_2\}=\{ S_2,S_1[\mathbb{X}-2] \}. \]

• – The auto-equivalence group $\operatorname {Aut}\operatorname {\mathcal {D}}_\mathbb {X}(A_2)$ is generated by $\Psi _i,[1],[\mathbb {X}]$ and sits in the short exact sequence

  \[ 1\to\operatorname{ST}_\mathbb{X}(A_2)\to\operatorname{Aut}\operatorname{\mathcal{D}}_\mathbb{X}(A_2)\to (\mathbb{Z}[1]\oplus\mathbb{Z}[\mathbb{X}])/\mathbb{Z}[3\mathbb{X}-4]\to1. \]

Applying Theorem 5.9 and the calculation of $\operatorname {gldim}$ in [Qiu23], i.e.

\[ \operatorname{gldim}\operatorname{Stab}\operatorname{\mathcal{D}}_\infty(A_2)=[1/3,\infty), \]

we have the following result.

Lemma 7.1 [BQS20, Qiu23]

• – We have that $\operatorname {QStab}^{*}_s\operatorname {\mathcal {D}}_\mathbb {X}(A_2)$ is not empty if and only if $\operatorname {Re}(s)\ge 4/3$. It is connected if and only if $\operatorname {Re}(s)\ge 2$.

• – We have that ${\operatorname {QStab}^{\oplus }}_s\operatorname {\mathcal {D}}_\mathbb {X}(A_2)$ is not empty if and only if $\operatorname {Re}(s)>4/3$. It is connected if and only if $\operatorname {Re}(s)>2$ (cf. Figures 1, 2 and 3).

• – We have that $\operatorname {QStab}^{*}_s\operatorname {\mathcal {D}}_\mathbb {X}(A_2)=\operatorname {QStab}^{\oplus }_s\operatorname {\mathcal {D}}_\mathbb {X}(A_2)$ if and only $\operatorname {Re}(s)>2$.

• – The fundamental domain for $\mathbb {C}\backslash \overline {\operatorname {QStab}^{\oplus }}_s\operatorname {\mathcal {D}}_\mathbb {X}(A_2)/\operatorname {Aut}$ is $R_s$ in Figure 1, where the coordinate $z=x+y{i}$ satisfies

  \[ e^{{i} \pi z}= Z(S_1) / Z(S_2) ,\\ \operatorname{Re}(z)=\phi(S_1)-\phi(S_2), \]
  and $l_{\pm }$ is given by the equation
  \[ l_{\pm}=\{ z=x+{i} y \mid x\in(\tfrac{1}{2},\tfrac{2}{3}], y\pi=\mp\ln(-2\cos x\pi)\}. \]
  Moreover, there are two orbitfold points on $\partial R_s$, one is $(2-s)/2$ with order 2 and the other one is $2/3$ with order 3.

• – The order 3 orbitfold point $\sigma _{G,s}$ solves the Gepner equation $\tau _\mathbb {X}(\sigma )=(-\frac {2}{3}) \cdot \sigma$ (see [Qiu23, Theorem 5.10]).

Figure 1. The region/fundamental domain $R_s$.

Figure 2. The tornado illustration of $\mathbb {C}\backslash \operatorname {QStab}^{\oplus }\operatorname {\mathcal {D}}_\mathbb {X}(A_2)$.

Figure 3. When $\operatorname {Re}(s)<2$, $\operatorname {QStab}^{\oplus }_s \operatorname {\mathcal {D}}_\mathbb {X}(A_2)$ is not connected.

Remark 7.2 In Figure 2 we present the tornado illustration of $\mathbb {C}\backslash \operatorname {QStab}^{\oplus }\operatorname {\mathcal {D}}_\mathbb {X}(A_2)$.