Global dimension function on stability conditions and Gepner equations

Abstract

We study the global dimension function ${\text {gldim}}:{\text {Aut}}\backslash {\text {Stab}}{\mathcal {D}}/\mathbb {C}\rightarrow \mathbb {R}_{\ge 0}$ on the quotient of the space of Bridgeland stability conditions on a triangulated category ${\mathcal {D}}$ as well as Toda’s Gepner equation $\Phi (\sigma )=s\cdot \sigma$ for some $\sigma \in {\text {Stab}}{\mathcal {D}}$ and $(\Phi ,s)\in {\text {Aut}}{\mathcal {D}}\times \mathbb {C}$. For the bounded derived category ${\mathcal {D}}^b(\textbf{k}Q)$ of a Dynkin quiver Q, we show that there is a unique minimal point $\sigma _G$ of ${\text {gldim}}$ (up to the $\mathbb {C}$-action), with value $1-2/h$. which is the solution of the Gepner equation $\tau (\sigma )=(-2/h)\cdot \sigma$. Here $\tau$ is the Auslander–Reiten functor and h is the Coxeter number. This solution $\sigma _G$ was constructed by Kajiura–Saito–Takahashi. We also show that for an acyclic non-Dynkin quiver Q, the minimal value of ${\text {gldim}}$ is 1. Our philosophy is that the infimum of ${\text {gldim}}$ on ${\text {Stab}}{\mathcal {D}}$ is the global dimension for the triangulated category ${\mathcal {D}}$. We explain how this notion could shed light on the contractibility conjecture of the space of stability conditions.

Appendix A. q-stability conditions on Calabi–Yau-\(\mathbb {X}\) categories

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

$$\begin{aligned} \mathbb {X}:{\mathcal {D}}_{\mathbb {X}} \rightarrow {\mathcal {D}}_{\mathbb {X}}. \end{aligned}$$

We will write $E[l \mathbb {X}]$ instead of $\mathbb {X}^l(E)$ for $l \in \mathbb {Z}$ and $E \in {\mathcal {D}}_{\mathbb {X}}$. Set

$$\begin{aligned} R=\mathbb {Z}[q^{\pm 1}] \end{aligned}$$

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

$$\begin{aligned} q^n \cdot [E] := [E[n \mathbb {X}]]. \end{aligned}$$

Then $K({\mathcal {D}}_{\mathbb {X}})$ has an R-module structure. Let ${\text {Aut}}{\mathcal {D}}_\mathbb {X}$ be the group of auto-equivalences of ${\mathcal {D}}_\mathbb {X}$ that commute with $\mathbb {X}$ and ${\text {Hom}}^{\mathbb {Z}^2}(M,N):=\bigoplus _{k,l\in \mathbb {Z}}{\text {Hom}}(M,N[k+l\mathbb {Z}])$.

1.1 Calabi–Yau-$\mathbb {X}$ categories from quivers

For an acyclic quiver Q, denote by ${\Gamma }_{\mathbb {X}}Q$ the Calabi–Yau-$\mathbb {X}$ Ginzburg differential $\mathbb {Z}\oplus \mathbb {Z}[\mathbb {X}]$ graded algebra of Q, that is constructed as follows (cf. [12] and [16, Sect. 7.2]):

• Let ${\overline{Q}}$ be the graded quiver whose vertex set is $Q_0$ and whose arrows are: the arrows in Q with degree 0; an arrow $a^*:j\rightarrow i$ with degree $2-\mathbb {X}$ for each arrow $a:i\rightarrow j$ in Q; a loop $e^*:i\rightarrow i$ with degree $1-\mathbb {X}$ for each vertex e in Q.

• The underlying graded algebra of ${\Gamma }_{\mathbb {X}}Q$ is the completion of the graded path algebra $\textbf{k}{\overline{Q}}$ in the category of graded vector spaces with respect to the ideal generated by the arrows of ${\overline{Q}}$.

• The differential of ${\Gamma }_{\mathbb {X}}Q$ is the unique continuous linear endomorphism homogeneous of degree 1 which satisfies the Leibniz rule and takes the following (non-zero) values on the arrows of ${\overline{Q}}$

  $$\begin{aligned} {\text {d}}\sum _{e\in Q_0} e^*=\sum _{a\in Q_1} \, [a,a^*] . \end{aligned}$$

Write ${\mathcal {D}}_\mathbb {X}(Q)$ for ${\mathcal {D}}_{fd}(\mod {\Gamma }_{\mathbb {X}}Q)$, the finite dimensional derived category of ${\Gamma }_{\mathbb {X}}Q$, which admits the Serre functor $\mathbb {X}$ that corresponds to grading shift of (0, 1).

Note that there is an embedding

$$\begin{aligned} \mathcal {L}:{\mathcal {D}}_\infty (Q)\rightarrow {\mathcal {D}}_\mathbb {X}(Q), \end{aligned}$$

(A.1)

induced from the projection ${\Gamma }_{\mathbb {X}}Q\rightarrow \textbf{k}Q$, which is a Lagrangian immersion, in the sense that

$$\begin{aligned} {\text {RHom}}_{{\mathcal {D}}_\mathbb {X}(Q)}(\mathcal {L}(L),\mathcal {L}(M))={\text {RHom}}_{{\mathcal {D}}_\infty (Q)}(L,M) \oplus D{\text {RHom}}_{{\mathcal {D}}_\infty (Q)}(M,L)[-\mathbb {X}], \end{aligned}$$

(A.2)

Moreover, $\mathcal {L}$ also induces an R-isomorphism

$$\begin{aligned} \mathcal {L}_*:K({\mathcal {D}}_\mathbb {X}(Q))\cong _R K({\mathcal {D}}_\infty (Q)) \otimes R=R^n, \end{aligned}$$

(A.3)

where the simple $\textbf{k}Q$-modules provide a $\mathbb {Z}$-basis for $K({\mathcal {D}}_\infty (Q))\cong \mathbb {Z}^n$ and the simple ${\Gamma }_{\mathbb {X}}Q$-modules provide an R-basis for $K({\mathcal {D}}_\mathbb {X}(Q))$.

By abuse of notation, we will not distinguish objects in ${\mathcal {D}}_\infty (Q)$ and their images in ${\mathcal {D}}_\mathbb {X}(Q)$ (under the fixed canonical Lagrangian immersion $\mathcal {L}$).

1.2 q-Stability conditions

We recall q-stability conditions from [12].

Definition A.1

[12, Def. 3.4] Suppose that ${\mathcal {D}}$ is a triangulated category with Grothendieck group $K({\mathcal {D}}_\mathbb {X})\cong _R R^n$. An q-stability condition consists of a (Bridgeland) stability condition $\sigma =(Z,\mathcal {P})$ on ${\mathcal {D}}_\mathbb {X}$ and a complex number $s\in \mathbb {C}$, satisfying

$$\begin{aligned} \mathbb {X}(\sigma )=s \cdot \sigma . \end{aligned}$$

(A.4)

We may write $\sigma [\mathbb {X}]$ for $\mathbb {X}(\sigma )$. Denote by ${\text {QStab}}_s{\mathcal {D}}_{\mathbb {X}}$, the space of q-stability conditions consisting of $(\sigma , s)$ with q-support property.

We have the following results.

Proposition A.2

[12, Thm. 3.10] The projection map defined by taking central charges

$$\begin{aligned} \pi _Z :{\text {QStab}}_s{\mathcal {D}}_\mathbb {X}\longrightarrow {\text {Hom}}_{R}(K({\mathcal {D}}_{\mathbb {X}}),\mathbb {C}_s), \quad (Z,\mathcal {P}) \mapsto Z \end{aligned}$$

(A.5)

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

Theorem A.3

[12, Thm. 5.9] Let $\sigma =(Z,\mathcal {P})$ be a stability condition on ${\text {Stab}}{\mathcal {D}}_\infty (Q)$ and let $s\in \mathbb {C}$ satisfies

$$\begin{aligned} \Re (s)\ge {\text {gldim}}\sigma +1. \end{aligned}$$

Then $\mathcal {L}$ induces a q-stability condition $(\sigma ^{\mathcal {L}}_s,s)$ such that $\sigma ^{\mathcal {L}}_s=(Z^{\mathcal {L}}_s,\mathcal {P}^{\mathcal {L}}_s)$ is defined as

• $Z^{\mathcal {L}}_s=q_s\circ \big ( Z \otimes 1 \big ):K({\mathcal {D}}_\mathbb {X}(Q))\rightarrow \mathbb {C}$ (recall that we have (A.3)), where $q_s$ is the specialization

  $$\begin{aligned} q_s:\mathbb {C}[q,q^{-1}]\rightarrow \mathbb {C},\quad q\mapsto e^{\textbf{i} \pi s}. \end{aligned}$$

• $\mathcal {P}^{\mathcal {L}}_s(\phi )=\langle \mathcal {P}(\phi +k\Re (s)) [k\mathbb {X}] \mid k\in \mathbb {Z}\rangle .$

As we have calculate the image of ${\text {gldim}}$ on ${\text {Stab}}{\mathcal {D}}_\infty (Q)$ in Theorem 4.8, a direct corollary is the following.

Corollary A.4

Let Q be a Dynkin quiver and $\Re (s)\ge {\text {gd}}_Q+1$. Then ${\text {QStab}}_s{\mathcal {D}}_\mathbb {X}(Q)$ is non-empty.

In particular, for any $\Re (s)\ge {\text {gd}}_Q+1$, denote by $\sigma _{G,s}^{\mathcal {L}}$ the induced q-stability condition in ${\text {QStab}}_s{\mathcal {D}}_\mathbb {X}(Q)$ from the Gepner point $\sigma _G\in {\text {Stab}}{\mathcal {D}}_\infty (Q)$ with ${\text {gldim}}\sigma _G={\text {gd}}_Q$. In the rest of this section, we shall prove that $\sigma _{G,s}^{\mathcal {L}}$ is a Gepner point of ${\text {QStab}}_s{\mathcal {D}}_\mathbb {X}$, in the sense that it satisfies one more Gepner equation (2.2) other than (A.4).

1.3 Center of the braid group

Definition A.5

[6] Denote by ${\text {Br}}(Q)$ the braid group (a.k.a Artin group) associated to a Dynkin quiver Q, which admits the following presentation

$$\begin{aligned}\begin{array}{rll} &{}{}{\text{ Br }}(Q)=\langle b_i, i\in Q_0 \mid {\text{ Co }}(b_i,b_{j}), \text{ no } \text{ arrows } \text{ between } i \text{ and } j;\\ {} &{}{} {\text{ Br }}(b_j,b_k), \exists ! \text{ arrow } \text{ between } i \hbox { and }j \rangle . \end{array}\end{aligned}$$

where ${\text {Co}}(a,b)$ means the commutation relation $ab=ba$ and ${\text {Br}}(a,b)$ means the braid relation $aba=bab$.

Provided the labeling of vertices as in (4.9), define

$$\begin{aligned}&\zeta _Q=b_n \circ ... \circ b_1, \end{aligned}$$

(A.6)

$$\begin{aligned}&\delta _Q={\left\{ \begin{array}{ll} 1, \quad &{}~\hbox {if}~Q~\hbox {is of type}~A_n, D_{2l+1}, E_6;\\ 1/2, \quad &{}~\hbox {if}~Q~\hbox {is of type}~D_{2l}, E_7, E_8. \end{array}\right. } \end{aligned}$$

(A.7)

Then it is well-known that $z_Q=\zeta _Q^{\delta _Q h_Q}$ generates of the center of the braid group ${\text {Br}}(Q)$.

Definition A.6

An object S in a CY-$\mathbb {X}$ category ${\mathcal {D}}$ is ($\mathbb {X}$-)spherical if

$$\begin{aligned} {\text {Hom}}(S, S[k+l\mathbb {X}]){\left\{ \begin{array}{ll} \textbf{k},&{} \text {if }k=0\text { and }l\in \{0,1\};\\ 0,&{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

and induces a twist functor $\Psi _S\in {\text {Aut}}{\mathcal {D}}$, such that

$$\begin{aligned} \Psi _S(X)={\text {Cone}}\left( S\otimes {\text {Hom}}^{\mathbb {Z}^2}(S,X)\rightarrow X\right) \end{aligned}$$

with inverse

$$\begin{aligned} \Psi _S^{-1}(X)={\text {Cone}}\left( X\rightarrow S\otimes {\text {Hom}}^{\mathbb {Z}^2}(X,S)^\vee \right) [-1]. \end{aligned}$$

In the case when ${\mathcal {D}}_\mathbb {X}(Q)={\mathcal {D}}_{fd}({\Gamma }_{\mathbb {X}}Q)$, any simple $S_i$ for $i\in Q_0$ in the canonical heart

$$\begin{aligned} \mathcal {H}_Q^\mathbb {X}:=\mod {\Gamma }_{\mathbb {X}}Q \end{aligned}$$

is spherical and the spherical twist group is

$$\begin{aligned} {\text {ST}}_\mathbb {X}(Q):=\langle \Psi _{S_i} \mid i\in Q_0 \rangle \subset {\text {Aut}}{\mathcal {D}}_\mathbb {X}(Q). \end{aligned}$$

We have the following.

Theorem A.7

[12, Thm. 6.6] There is a canonical isomorphism ${\text {Br}}(Q)\cong {\text {ST}}_\mathbb {X}(Q)$, sending the standard generators to the standard ones.

Thus the generator $\zeta _Q$ of $Z({\text {Br}}(Q))$ becomes

$$\begin{aligned} \zeta _Q^{\mathcal {L}}=\Psi _{S_n}\circ \cdots \circ \Psi _{S_1}. \end{aligned}$$

1.4 $\mathbb {X}$-Auslander–Reiten functor

First, we prove the following lemma.

Lemma A.8

$\mathcal {L}$ induces an injection $\mathcal {L}_*:{\text {Aut}}{\mathcal {D}}_\infty (Q)\rightarrow {\text {Aut}}{\mathcal {D}}_\mathbb {X}(Q)$ that fits into the commutative diagram

for any $\Phi \in {\text {Aut}}{\mathcal {D}}_\infty (Q)$.

Proof

By Kozsul duality,

$$\begin{aligned} {\mathcal {D}}_\infty (Q)={\mathcal {D}}_{fd}(\textbf{k}Q)\cong {\text {per}}{\mathcal {E}}_Q, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {E}}_Q={\text {RHom}}_{{\mathcal {D}}_\infty (Q)}(S_Q,S_Q) \end{aligned}$$

is the dg endomorphism algebra for $S_Q=\bigoplus S_i$. Similarly, we have

$$\begin{aligned} {\mathcal {D}}_\mathbb {X}(Q)={\mathcal {D}}_{fd}({\Gamma }_{\mathbb {X}}Q)\cong {\text {per}}{\mathcal {E}}_Q^\mathbb {X}, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {E}}_Q^\mathbb {X}={\text {RHom}}^{\mathbb {Z}^2}_{{\mathcal {D}}_\mathbb {X}(Q)}(S_Q, S_Q) \end{aligned}$$

is the differential $\mathbb {Z}^2$-graded endomorphism algebra. Then any auto-equivalence $\Phi$ in ${\text {Aut}}{\mathcal {D}}_\infty (Q)$ maps ${\mathcal {E}}_Q$ to another dg endomorphism algebra

$$\begin{aligned} \Phi ({\mathcal {E}}_Q)={\text {RHom}}^{\mathbb {Z}^2}_{{\mathcal {D}}_\mathbb {X}(Q)}(\Phi (S_Q), \Phi (S_Q)) \end{aligned}$$

that $\Phi$ can be realized as ${\text {per}}{\mathcal {E}}_Q\rightarrow {\text {per}}\Phi ({\mathcal {E}}_Q)$. After applying $\mathcal {L}$ that passes to ${\mathcal {D}}_\mathbb {X}(Q)$, we obtain an auto-equivalence

$$\begin{aligned} \mathcal {L}_*(\Phi ):{\text {per}}{\mathcal {E}}_Q^\mathbb {X}\rightarrow {\text {per}}\mathcal {L}_*\left( \Phi ({\mathcal {E}}_Q^\mathbb {X})\right) . \end{aligned}$$

Finally, if $\mathcal {L}_*(\Phi )={\text {id}}$ preserves $S_i$ in ${\mathcal {D}}_\mathbb {X}(Q)$ (and the ${\text {Hom}}$s between them), then $\Phi$ preserves them in ${\mathcal {D}}_\infty (Q)$, which must be identity. $\square$

Now, let $\tau _\mathbb {X}^{\mathcal {L}}=\mathcal {L}_*(\tau )\in {\text {Aut}}{\mathcal {D}}_\mathbb {X}(Q)$. We have the following.

Proposition A.9

Let Q be a Dynkin quiver. Then

$$\begin{aligned} \tau _\mathbb {X}^{\mathcal {L}}=[\mathbb {X}-2]\circ \zeta _Q^{\mathcal {L}}\end{aligned}$$

(A.8)

satisfies $(\tau _\mathbb {X}^{\mathcal {L}})^h=[-2]$.

Proof

The calculation is exactly the same as the Calabi–Yau-N case in the proof of [24, Prop. 6.4.1]. Note that the assumption there, i.e. the isomorphism ${\text {Br}}(Q)\cong {\text {ST}}_N(Q)$, has been proved in Theorem A.7 (cf. [27]). $\square$

Recall we have a Gepner point $\sigma _G=(Z_G, \mathcal {P}_G)$ on ${\text {Stab}}{\mathcal {D}}_\infty (Q)$ and it induces a q-stability condition $(\sigma _{G,s}^{\mathcal {L}},s)$ for $\Re (s)\ge {\text {gd}}_Q+1$, where $\sigma _{G,s}^{\mathcal {L}}=(Z^{\mathcal {L}}_s,\mathcal {P}^{\mathcal {L}}_s)$ is constructed in Theorem A.3.

Theorem A.10

$\sigma _{G,s}^{\mathcal {L}}\in {\text {QStab}}_s{\mathcal {D}}_\mathbb {X}(Q)$ satisfies the Gepner equation

$$\begin{aligned} \tau _\mathbb {X}^{\mathcal {L}}(\sigma )=\left( -\frac{2}{h}\right) \cdot \sigma \end{aligned}$$

(A.9)

for $\Re (s)\ge {\text {gd}}_Q+1$.

Proof

As $\tau _\mathbb {X}^{\mathcal {L}}$ is induced from $\tau$ via $\mathcal {L}$, we have $\tau _\mathbb {X}^{\mathcal {L}}= \tau \otimes R$ on the Grotendieck groups. Thus, for the central charge we have

$$\begin{aligned}\begin{array}{rcl} Z_{G,s}^\mathcal {L}\circ \tau _\mathbb {X}^{\mathcal {L}}&{}=&{} \left( q_s\circ (Z_G\otimes R) \right) \circ \tau _\mathbb {X}^{\mathcal {L}}\\ &{}=&{}q_s\circ \left( (Z_G\otimes R)\circ \tau _\mathbb {X}^{\mathcal {L}}\right) \\ &{}=&{}q_s\circ \left( (Z_G\circ \tau )\otimes R\right) \\ &{}=&{}q_s\circ \left( (e^{2\pi \textbf{i}/h}\cdot Z_G)\otimes R\right) \\ &{}=&{} e^{2\pi \textbf{i}/h}\cdot \big ( q_s\circ (Z_G\otimes R) \big ) \\ &{}=&{} e^{2\pi \textbf{i}/h}\cdot Z_{G,s}^\mathcal {L}, \end{array}\end{aligned}$$

where we use the Gepner property of $\sigma _G$ that $Z_G\circ \tau = e^{2\pi \textbf{i}/h}\cdot Z_G$. For the slicing, we have $\tau _\mathbb {X}^{\mathcal {L}}(\mathcal {P})=\tau (\mathcal {P})$ (recall that we identify $\mathcal {P}(\phi )\subset {\mathcal {D}}_\infty (Q)$ with its image in ${\mathcal {D}}_\mathbb {X}(Q)$ under $\mathcal {L}$) and hence

$$\begin{aligned}\begin{array}{rcl} \tau _\mathbb {X}^{\mathcal {L}}\big ( \mathcal {P}_{G,s}^\mathcal {L}(\phi ) \big ) &{}=&{} \big \langle \tau \big ( \mathcal {P}(\phi -k\Re (s)) \big ) [k\mathbb {X}] \mid k\in \mathbb {Z}\big \rangle \\ &{}=&{} \big \langle \mathcal {P}_{(-2/h)}(\phi -k\Re (s) ) [k\mathbb {X}] \mid k\in \mathbb {Z}\big \rangle \\ &{}=&{} ( \mathcal {P}_{G,s}^\mathcal {L})_{(-2/h)} (\phi ) \end{array} \end{aligned}$$

where we use the Gepner property of $\sigma _G$ that $\tau (\mathcal {P})=\mathcal {P}_{(-2/h)}$. Thus $\sigma _{G,s}^{\mathcal {L}}$ satisfies (A.9). $\square$

Remark A.11

A reachable Lagrangian immersion $\mathcal {L}'$, by definition, is of the form $\mathcal {L}'=\Upsilon \circ \mathcal {L}$ for $\Upsilon \in {\text {Aut}}{\mathcal {D}}_\mathbb {X}(Q)$, where $\mathcal {L}$ is the fixed initial Lagrangian immersion in (A.1). By the construction in Theorem A.3, we have

$$\begin{aligned} \sigma _{G,s}^{\mathcal {L}'}=\Upsilon \circ \sigma _{G,s}^{\mathcal {L}}, \end{aligned}$$

which solves the equation

$$\begin{aligned} \tau _\mathbb {X}^{\mathcal {L}'}(\sigma )=\left( -\frac{2}{h}\right) \cdot \sigma , \end{aligned}$$

for

$$\begin{aligned} \tau _\mathbb {X}^{\mathcal {L}'}=(\mathcal {L}')_*(\tau )= \Upsilon \circ \tau _\mathbb {X}^{\mathcal {L}}\circ \Upsilon ^{-1}. \end{aligned}$$

Therefore, all such Gepner points $\sigma _{G,s}^{\mathcal {L}'}$ correspond to the same point $\sigma _G$ in ${\text {Stab}}{\mathcal {D}}_\mathbb {X}(Q)/\mathbb {C}$.