Ginzburg algebras of triangulated surfaces and perverse schobers

1.1 Relative Ginzburg algebras of triangulated surfaces

A quiver Q consists of a finite set of vertices, denoted $Q_{0}$ , and a finite set of arrows, denoted $Q_{1}$ , together with source and target maps $s,t:Q_{1}\rightarrow Q_{0}$ . A quiver is called graded if each arrow carries an integer labelling. Given a graded quiver Q, we denote by $kQ$ the graded path algebra over a commutative ring k. A potential W for a quiver Q is an element of the cyclic path algebra $kQ^{\operatorname {cyc}}$ , meaning the algebra of k-linear sums of cyclic paths.

For the definition of the Ginzburg algebra, due to [Reference GinzburgGin06], we follow [Reference KellerKel11]. Consider a quiver with potential $(Q,W)$ . We denote by $Q^{\prime }$ the graded quiver with the same set of vertices as Q and graded arrows of the following three kinds:

• • an arrow $a:i\rightarrow j$ in degree $0$ for each $a:i\rightarrow j\in Q_{1}$ ,

• • an arrow $a^{*}:j\rightarrow i$ in degree $1$ for each $a:i\rightarrow j\in Q_{1}$ and

• • an arrow $l_{i}:i\rightarrow i$ in degree $2$ for each $i\in Q_{0}$ .

The cyclic derivative $\partial _{a}:kQ_{\operatorname {cyc}}\rightarrow kQ$ with respect to $a\in Q_{1}$ is the k-linear map taking a cycle c to $\partial _{a}c=\sum _{c=uav}uv$ , where $u,v\in kQ$ are allowed to be lazy paths. We denote the lazy path at a vertex $i\in Q_{0}$ by $p_{i}$ . We define the Ginzburg algebra $\mathscr {G}(Q,W)$ to be the dg-algebra whose underlying graded algebra is given by the graded path algebra $kQ^{\prime }$ and whose differential d is determined by the following action on the generators:

$$ \begin{align*} a & \mapsto 0,\\ a^{*} & \mapsto \partial_{a} W,\\ l_{i} & \mapsto \sum_{a\in Q_{1}}p_{i}[a,a^{*}]p_{i}. \end{align*} $$

Note that $\mathscr {G}(Q,W)$ is not the completed Ginzburg algebra, as considered, for example, in [Reference Keller and YangKY11, Reference SmithSmi15]. We will not consider completed Ginzburg algebras in this paper. In terms of the associated derived $\infty $ -categories of these dg-algebras, this does not mean much of a loss, because the derived $\infty $ -category of the completed Ginzburg algebra can be realised as a full subcategory of the derived $\infty $ -category of the noncompleted Ginzburg algebra. This perspective, however, neglects the additional topological structure of the completed Ginzburg algebra; see, for example, [Reference Keller and YangKY11 , Appendix].

We now introduce a relative version of the Ginzburg algebra $\mathscr {G}\left (Q_{\mathcal {T}},W^{\prime }_{\mathcal {T}}\right )$ associated to an ideal triangulation $\mathcal {T}$ of an oriented marked surface $\mathbf {S}$ . We define a quiver $Q_{\mathcal {T}}$ by adapting the definition of the quiver $Q^{\circ }_{\mathcal {T}}$ to include the boundary of $\mathbf {S}$ . We let $Q_{\mathcal {T}}$ be the quiver with a vertex for each edge of $\mathcal {T}$ (including boundary edges) and an arrow $a:i\rightarrow j$ for each ideal triangle containing the edges $i,j$ , where the edge j follows the edge i in the clockwise order. If $\mathcal {T}$ contains self-folded triangles, we additionally include an arrow $a:i\rightarrow i$ for each self-folded edge i of $\mathcal {T}$ . The quiver $Q_{\mathcal {T}}$ contains a clockwise $3$ -cycle $T(f)$ for each ideal triangle f of $\mathcal {T}$ . We define the potential

$$ \begin{align*} \overline{W}^{\prime}_{\mathcal{T}}=\sum_{f} T(f)\in kQ_{\mathcal{T}}^{\operatorname{cyc}}. \end{align*} $$

We denote by $\tilde {Q}_{\mathcal {T}}$ the graded quiver with the same set of vertices as $Q_{\mathcal {T}}$ and graded arrows of the following three kinds:

• • an arrow $a:i\rightarrow j$ in degree $0$ for each $a:i\rightarrow j\in (Q_{\mathcal {T}})_{1}$ ,

• • an arrow $a^{*}:j\rightarrow i$ in degree $1$ for each $a:i\rightarrow j\in (Q_{\mathcal {T}})_{1}$ and

• • an arrow $l_{i}:i\rightarrow i$ in degree $2$ for each vertex $i\in (Q_{\mathcal {T}})_{0}$ given by an internal edge of $\mathcal {T}$ .

We define the relative Ginzburg algebra $\mathscr {G}_{\mathcal {T}}$ to be the dg-algebra whose underlying graded algebra is given by the graded path algebra $k\tilde {Q}_{\mathcal {T}}$ and whose differential is determined by the following action on the generators:

$$ \begin{align*} a & \mapsto 0,\\ a^{*} & \mapsto \partial_{a} \overline{W}^{\prime}_{\mathcal{T}},\\ l_{i} & \mapsto \sum_{a\in \left(Q_{\mathcal{T}}\right)_{1}}p_{i}[a,a^{*}]p_{i}. \end{align*} $$

The relative Ginzburg algebra $\mathscr {G}_{\mathcal {T}}$ is an example of the more general relative Ginzburg algebras associated to ice quivers with potential [Reference WuWu21]. An ice quiver is a quiver equipped with the further datum of a subquiver, whose vertices and arrows are called frozen. The ice quiver underlying $\mathscr {G}_{\mathcal {T}}~$ is given by $Q_{\mathcal {T}}$ , with frozen vertices given by the boundary edges of $\mathcal {T}$ and no frozen arrows. The potential is $\overline {W}_{\mathcal {T}}^{\prime }$ .

The quiver $Q_{\mathcal {T}}^{\circ }$ is the full subquiver of $Q_{\mathcal {T}}$ spanned by the vertices corresponding to internal edges. The potential $W_{\mathcal {T}}^{\prime }=\sum _{f}T(f)\in \left (kQ^{\circ }_{\mathcal {T}}\right )^{\operatorname {cyc}}$ consists of all $3$ -cycles inscribed into internal ideal triangles of $\mathcal {T}$ . Note that if the boundary of $\mathbf {S}$ is empty, then $\left (Q_{\mathcal {T}},\overline {W}_{\mathcal {T}}^{\prime }\right )=\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ and the relative Ginzburg algebra $\mathscr {G}_{\mathcal {T}}~$ is equivalent to $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ . As an example, let $\mathbf {S}$ be the $3$ -gon and $\mathcal {T}$ a triangle. The relative Ginzburg algebra $\mathscr {G}_{\mathcal {T}}$ is then given by the graded path algebra of the graded quiver

(1)

with differential d mapping each arrow of degree $1$ to the composite of the two opposite arrows of degree $0$ . The Ginzburg algebra $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ of the triangle $\mathcal {T}$ is, however, zero.

Theorem 1 extends to relative Ginzburg algebras in the following way:

Theorem 2. Let $\mathcal {T}$ be an ideal triangulation of an oriented marked surface $\mathbf {S}$ with dual ribbon graph $\Gamma $ . The $\infty $ -category of global sections of the parametrised perverse schober $\mathcal {F}_{\mathcal {T}}$ satisfies

$$ \begin{align*} \mathcal{H}(\Gamma,\mathcal{F}_{\mathcal{T}})\simeq \mathcal{D}(\mathscr{G}_{\mathcal{T}}). \end{align*} $$

In [Reference KellerKel11, Section 7.6] it shown that mutation of quivers with potential induces derived equivalences between the respective Ginzburg algebras. In [Reference Labardini-FragosoLF09] it is shown that if two ideal triangulations $\mathcal {T},\mathcal {T}^{\;\prime }$ are related by a flip of an edge, the associated quivers with potentials $\left (Q^{\circ }_{\mathcal {T}},W_{\mathcal {T}}\right )$ and $\left (Q^{\circ }_{\mathcal {T}^{\;\prime }},W_{\mathcal {T}^{\;\prime }}\right )$ are related by quiver mutation. In combination, these two results show that flips of ideal triangulations induce derived equivalences of the associated Ginzburg algebras. We extend the derived equivalences to the relative Ginzburg algebras.

Theorem 3. Let $\mathbf {S}$ be an oriented marked surface with two ideal triangulations $\mathcal {T},\mathcal {T}^{\;\prime }$ related by a flip of an edge e of $\mathcal {T}$ . Then there exists an equivalence of $\infty $ -categories

$$ \begin{align*} \mu_{e}: \mathcal{D}(\mathscr{G}_{\mathcal{T}})\simeq \mathcal{D}\left(\mathscr{G}_{\mathcal{T}^{\;\prime}}\right). \end{align*} $$

We will prove Theorem 3 in Section 6.4 using an intrinsic feature of the theory of parametrised perverse schobers, namely equivalences of global sections induced from contractions of the underlying ribbon graphs.

We thank Bernhard Keller for informing us about an alternative approach to Theorem 3. A result of Yilin Wu [Reference WuWu21] extends the argument from [Reference KellerKel12, Section 7.6] to relative Ginzburg algebras, showing that the mutations of ice quivers with potential of [Reference PresslandPre20] induce derived equivalences between the associated relative Ginzburg algebras. Theorem 3 may then be recovered by additionally extending the results of [Reference Labardini-FragosoLF09] relating flips of the ideal triangulation and mutations of quivers with potentials to ice quivers.

1.2 Perverse schobers and Fukaya categories

Perverse schobers are a conjectured categorification of the notion of perverse sheaves [Reference Kapranov and SchechtmanKS14]. An approach to the categorification of a perverse sheaf on a disc was suggested in [Reference Kapranov and SchechtmanKS14]. The datum of a perverse sheaf on a disc with a single singularity in the centre is equivalent to the datum of a certain quiver diagram; the proposed ad hoc categorification of the quiver description is a spherical adjunction. In this paper, we extend this ad hoc categorification to perverse schobers on oriented marked surfaces. We combinatorially describe perverse schobers using ribbon graphs. Such a ribbon graph arises as the dual to an ideal triangulation of the marked surface. Given a ribbon graph $\Gamma $ , we define a poset $\operatorname {Exit}(\Gamma )$ with

• • objects the vertices and edges of $\Gamma $ and

• • morphisms of the form $v\rightarrow e$ , with v a vertex and e an incident edge.

For each n-valent vertex v of $\Gamma $ , there exists a subposet $\operatorname {Exit}(\Gamma )_{v/}\subset \operatorname {Exit}(\Gamma )$ consisting of the vertex v and the n incident edges. We define a perverse schober $\mathcal {F}$ parametrised by $\Gamma $ to be a functor $\mathcal {F}:\operatorname {Exit}(\Gamma )\rightarrow \operatorname {St}$ into the $\infty $ -category of stable $\infty $ -categories such that the restriction to $\operatorname {Exit}(\Gamma )_{v/}$ is for every vertex v equivalent to a particular diagram obtained from a spherical adjunction. The exact definition is based on the categorified Dold–Kan correspondence of [Reference DyckerhoffDyc21] and categorifies the ‘fractional spin’ description of perverse sheaves on a disc of [Reference Kapranov and SchechtmannKS16a]. The definition of a parametrised perverse schober captures the idea that a perverse schober on a surface is a collection of suitably glued-together spherical adjunctions, categorifying the description of perverse sheaves on surfaces given in [Reference Kapranov and SchechtmannKS16a]. The $\infty $ -category of global sections $\mathcal {H}(\Gamma ,\mathcal {F})$ of a parametrised perverse schober $\mathcal {F}$ is defined as the limit of $\mathcal {F}$ in $\operatorname {St}$ . Under mild technical assumptions, the global sections of $\mathcal {F}$ are equivalent to a suitable colimit of the dual to $\mathcal {F}$ (left adjoint diagram), which describes a constructible cosheaf, see Section 4.3.

Given an ideal triangulation $\mathcal {T}$ without self-folded triangles of an oriented marked surface $\mathbf {S}$ , Smith [Reference SmithSmi15] defines a Calabi–Yau $3$ -fold Y with an affine conic fibration $\pi :Y\rightarrow \mathbf {S}$ . The relation to Ginzburg algebras is as follows:

• • The derived category of finite modules over $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ arises as a full subcategory of the derived Fukaya category $\operatorname {Fuk}(Y)$ of Y, where $W_{\mathcal {T}}^{\prime }$ is the potential of $Q_{\mathcal {T}}^{\circ }$ consisting of clockwise $3$ -cycles.

• • The derived category of finite modules over $\mathscr {G}\left (Q^{\circ }_{\mathcal {T}},W_{\mathcal {T}}\right )$ arises as a full subcategory of the derived Fukaya category $\operatorname {Fuk}(Y,b)$ of Y with a twisting background class $b \in H^{2}(Y,\mathbb {Z}_{2})$ . Here, $W_{\mathcal {T}}=W_{\mathcal {T}}^{\prime }+W_{\mathcal {T}}^{\prime \prime }$ is the potential consisting of clockwise $3$ -cycles and anticlockwise cycles.

The geometry of $\pi $ becomes clear when considering its fibres, which are given as follows:

• • The generic fibre of $\pi $ is diffeomorphic to $T^{*}S^{2}$ .

• • In the interior of each ideal triangle of $\mathcal {T}$ , there exists exactly one singular value with singular fibre given by the $2$ -dimensional $A_{1}$ -singularity.

• • The fibres of the interior marked points in $\mathbf {S}$ are given by $\mathbb {C}^{2}\amalg \mathbb {C}^{2}$ .

We denote by $\Sigma := \mathbf {S}\backslash (M\cap \mathbf {S}^{\circ })$ , with $\mathbf {S}^{\circ }$ the interior of $\mathbf {S}$ , the surface without the interior marked points and $Y^{\circ } := \pi ^{-1}(\Sigma )$ . Note that the restriction $\pi \rvert _{Y^{\circ }}:Y^{\circ }\rightarrow \Sigma $ of $\pi $ is a Lefschetz fibration.

The twist by the background class $b\in H^{2}(Y,\mathbb {Z}_{2})$ changes signs in the signed count of pseudoholomorphic curves passing through the fibres of the interior marked points. Without the background class, the signed count of such pseudoholomorphic curves always vanishes, so that the derived Fukaya category of $Y^{\circ }$ is equivalent to the derived Fukaya category of Y. The change in the $A_{\infty }$ -structure of the derived Fukaya category of Y induced by the background class b accounts exactly for the difference between the potentials $W^{\prime }_{\mathcal {T}}$ and $W_{\mathcal {T}}$ .

We expect the $\infty $ -category of global sections of the parametrised perverse schober $\mathcal {F}_{\mathcal {T}}$ of Theorem 1 to describe (the $\operatorname {Ind}$ -completion of) a partially wrapped Fukaya category of $Y^{\circ }$ . We further expect the global sections with support on $\Gamma ^{\circ }$ (the graph obtained from $\Gamma $ by removing boundary edges) to then correspond to (the $\operatorname {Ind}$ -completion of) the wrapped Fukaya category of $Y^{\circ }$ . In the case of the unpunctured n-gon, where $Y^{\circ }=Y$ is the $3$ -dimensional $A_{n-3}$ -singularity and $Q_{\mathcal {T}}^{\circ }$ the $A_{n-3}$ -quiver, it is shown in [Reference Lekili and UedaLU21] that $\mathcal {W}(Y^{\circ })\simeq \mathcal {D}\left (\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )\right )^{\operatorname {perf}}$ , meaning that $\mathcal {H}_{\Gamma ^{\circ }}(\Gamma ,\mathcal {F}_{\mathcal {T}})$ is equivalent to the $\operatorname {Ind}$ -completion of the wrapped Fukaya category of $Y^{\circ }$ .

We describe in Section 1.3 how the geometry of the Lefschetz fibration manifests itself in the definition of $\mathcal {F}_{\mathcal {T}}$ . We expect that the twisting by the background class b can be described as a deformation of the wrapped Fukaya category. It would be interesting to study the relation between such a deformation and the description in terms of parametrised perverse schobers.

1.3 The gluing construction of Ginzburg algebras

We now describe the construction of the perverse schober $\mathcal {F}_{\mathcal {T}}$ appearing in Theorem 1 and Theorem 2. We assume for simplicity that all ideal triangles of $\mathcal {T}$ are not self-folded. The ribbon graph $\Gamma $ dual to $\mathcal {T}$ parametrising $\mathcal {F}_{\mathcal {T}}$ consists of a vertex for each ideal triangle and an edge for each edge of $\mathcal {T}$ . Boundary edges of $\mathcal {T}$ correspond to external edges of the ribbon graph. Parametrised perverse schobers can, as can sheaves, be glued. To define $\mathcal {F}_{\mathcal {T}}$ , it thus suffices to define $\mathcal {F}_{\mathcal {T}}$ locally at each vertex of $\Gamma $ . The local datum at each vertex is a spherical adjunction, which we choose to be

(2) $$ \begin{align} f^{*}:\mathcal{D}(k)\longleftrightarrow \operatorname{Fun}\left(S^{2},\mathcal{D}(k)\right):f_{*}, \end{align} $$

where $\operatorname {Fun}\left (S^{2},\mathcal {D}(k)\right )$ is the $\infty $ -category of local systems on the $2$ -sphere with values in $\mathcal {D}(k)$ and $f^{*}$ is the pullback functor along $S^{2}\rightarrow \ast $ . This adjunction is shown in [Reference ChristChr20] to be spherical.

The $\infty $ -category $\operatorname {Fun}\left (S^{2},\mathcal {D}(k)\right )$ is equivalent to the derived $\infty $ -category of the polynomial algebra $k[t_{1}]$ with generator $t_{1}$ in degree $1$ (see Proposition 5.5). This derived $\infty $ -category is, by a result of [Reference AbouzaidAbo11], equivalent to the Ind-completion of the wrapped Fukaya of the cotangent bundle $T^{*}S^{2}$ , which is the generic fibre of the Lefschetz fibration $\pi \rvert _{Y^{\circ }}$ . Under these equivalences, the image $f^{*}(k)$ corresponds to the Lagrangian zero section of $T^{*}S^{2}$ . The fibration $\pi \rvert _{Y^{\circ }}$ has exactly one singular value in each ideal triangle of $\mathcal {T}$ , so that up to homotopy of $\Gamma $ , the vertices of $\Gamma $ lie at the singular values of $\pi \rvert _{Y^{\circ }}$ . The singular fibres are given by the $A_{1}$ -singularity. The relation between the geometry of $\pi \rvert _{Y^{\circ }}$ and the definition of $\mathcal {F}_{\mathcal {T}}$ can thus be summarised as follows:

• • The wrapped Fukaya category of the generic fibre $T^{*}S^{2}$ of $\pi \rvert _{Y^{\circ }}$ gives rise to the $\infty $ -category on the right of formula (2). This $\infty $ -category describes the generic stalk of $\mathcal {F}_{\mathcal {T}}$ .

• • Each vertex of $\Gamma $ corresponds to a singular value of $\pi \rvert _{Y^{\circ }}$ . The $\infty $ -category on the left of formula (2) describes the categorification of the vector space of vanishing cycles at that singularity of $\pi \rvert _{Y^{\circ }}$ . Since the $A_{1}$ -singularity has a unique vanishing cycle, this $\infty $ -category is given by $\mathcal {D}(k)$ .

• • The spherical adjunction $f^{*}\dashv f_{*}$ arises from a spherical object, the Lagrangian zero section, in the wrapped Fukaya category of $T^{*}S^{2}$ describing the vanishing cycle.

We further note that the perverse schober only models the Lefschetz fibration $\pi \rvert _{Y^{\circ }}$ , not the full fibration $\pi $ . The fibres $\mathbb {C}^{2}\amalg \mathbb {C}^{2}$ of $\pi $ over the interior marked points of $\mathbf {S}$ are not encoded in $\mathcal {F}_{\mathcal {T}}$ .

The parametrised perverse schober $\mathcal {F}_{\mathcal {T}}$ in total corresponds to the datum of a diagram

$$ \begin{align*} \mathcal{F}_{\mathcal{T}}: \operatorname{Exit}(\Gamma)\rightarrow \operatorname{St}\\[-15pt] \end{align*} $$

in the $\infty $ -category $\operatorname {St}$ of stable $\infty $ -categories indexed by the poset $\operatorname {Exit}(\Gamma )$ (see Section 1.2). The computations in Section 5 show that the parametrised perverse schober $\mathcal {F}_{\mathcal {T}}$ assigns the following:

• • To each vertex of $\Gamma _{\mathcal {T}}$ , a stable $\infty $ -category equivalent to the derived $\infty $ -category of the relative Ginzburg algebra of the $3$ -gon, depicted in diagram (1); this uses the fact that each vertex of $\Gamma _{\mathcal {T}}$ is trivalent.

• • To each edge of $\Gamma _{\mathcal {T}}$ , a stable $\infty $ -category equivalent to the derived $\infty $ -category of the polynomial algebra $k[t_{1}]$ with generator $t_{1}$ in degree $1$ . Note that $k[t_{1}]$ is equivalent to the $2$ -Calabi–Yau completion of k in the sense of [Reference KellerKel11] – that is, a $2$ -dimensional Ginzburg algebra.

The equivalence $\mathcal {H}(\Gamma ,\mathcal {F}_{\mathcal {T}})\simeq \mathcal {D}(\mathscr {G}_{\mathcal {T}})$ of Theorem 2 thus expresses that the derived $\infty $ -category of the relative Ginzburg algebra $\mathscr {G}_{\mathcal {T}}$ is glued from relative Ginzburg algebras of $3$ -gons along $2$ -dimensional Ginzburg algebras. We further illustrate the gluing construction of $\mathscr {G}_{\mathcal {T}}$ in two examples in Section 6.2.

Notation and conventions

We follows the notation and conventions of [Reference LurieLur09] and [Reference LurieLur17]. In particular, we always use the homological grading.