Quiver algebras as Fukaya categories

Let $(\mathbb{S},\mathbb{M})$ be a marked bordered surface. An ideal triangulation $T$ of $\mathbb{S}$ with vertices at $\mathbb{M}$ has an associated quiver with potential $(Q(T),W(T))$, which can be defined over any algebraically closed field $k$. The construction is indicated schematically in the case where $T$ is non-degenerate in Figure 1 and Equation (2.2), and defined more generally in [32]. There is a triangulated CY₃ category ${\mathcal{D}}(T)$ of finite type (i.e. cohomologically finite) over $k$ defined by the Ginzburg algebra construction [19]; this has a distinguished heart, equivalent to the category of finite-dimensional modules for the complete Jacobi algebra of the quiver with potential $(Q(T),W(T))$. Results of Keller-Yang and Labardini-Fragoso imply that ${\mathcal{D}}(T)$ depends up to quasi-isomorphism only on the underlying marked bordered surface $(\mathbb{S},\mathbb{M})$. We write ${\mathcal{D}}(\mathbb{S},\mathbb{M})$ for any category in this equivalence class.

A meromorphic quadratic differential $\phi$ on a Riemann surface $S$ has an associated marked bordered surface $(\mathbb{S},\mathbb{M})$. The surface $\mathbb{S}$ is obtained as the real blow-up of $S$ at poles of $\phi$ of order $\geqslant 3$; the distinguished tangent directions of the horizontal foliation of $\phi$ define boundary marked points in $\mathbb{M}$, and poles of order $\leqslant 2$ define the punctures (interior marked points) $\operatorname{\mathbb{P}}\subset\mathbb{M}$. We will write $\mathrm{Pol}(\phi)$ for the set of poles, and $\mathrm{Pol}_{\geqslant i}(\phi)$, $\mathrm{Pol}_{=2}(\phi)$ for poles of constrained or specified orders. For $q\in\mathrm{Pol}(\phi)$ let $ord(q)$ denote the order of the corresponding pole.

All quadratic differentials considered in this paper have simple zeroes.

Let $\operatorname{Quad}(\mathbb{S},\mathbb{M})$ denote the complex orbifold parametrizing equivalence classes of pairs comprising a Riemann surface $S$ and a meromorphic quadratic differential $\phi$ with simple zeroes whose associated marked bordered surface is diffeomorphic to $(\mathbb{S},\mathbb{M})$. This has an open dense subset $\operatorname{Quad}(\mathbb{S},\mathbb{M})_{0}$ of pairs where the differential $\phi$ has poles of order exactly $2$ at $\operatorname{\mathbb{P}}$, or equivalently for which the flat metric defined by $\phi$ is complete. There is an unramified $2^{|\operatorname{\mathbb{P}}|}:1$ cover $\operatorname{Quad}^{\pm}(\mathbb{S},\mathbb{M})_{0}\rightarrow\operatorname{Quad}(\mathbb{S},\mathbb{M})_{0}$ whose points are signed meromorphic differentials, meaning that we fix a choice of sign of the residue of the differential $\phi$ at each double pole; the cover extends as a ramified cover $\operatorname{Quad}^{\pm}(\mathbb{S},\mathbb{M})\rightarrow\operatorname{Quad}(\mathbb{S},\mathbb{M})$.

Fix a signed complete differential $(S,\phi)\in\operatorname{Quad}^{\pm}(\mathbb{S},\mathbb{M})_{0}$. Denote by $M$ the divisor $\sum_{p\in\mathrm{Pol}(\phi)}\lceil ord(p)/2\rceil p$ on $S$, where $\lceil\alpha\rceil$ denotes the smallest integer greater than or equal to $\alpha$, and write

corresponding to the obvious decomposition $\mathrm{Pol}(\phi)=\mathrm{Pol}_{=2}\amalg\mathrm{Pol}_{\geqslant 3}$. Take a rank two holomorphic vector bundle $V$ on $S$ with $\det(V)=K_{S}(M)$. We perform an elementary modification of the vector bundle $\mathrm{Sym}^{2}(V)$ at each double pole $p$ of $\phi$, to obtain a rank three bundle $W$ fitting into a short exact sequence

We then remove from $S$ all poles of $\phi$ of order $\geqslant 3$.

The determinant map $\det\colon\mathrm{Sym}^{2}(V)\longrightarrow K_{S}(M)^{\otimes 2}$ restricts to a quadratic map $\det_{W}$ on the bundle $W$ which is rank one at points of $M_{=2}\subset S$. The threefold

is an affine conic fibration over $S-\mathrm{Pol}_{\geqslant 3}(\phi)$, with nodal fibres over the zeroes of $\phi$, fibres singular at infinity over the double poles, and empty fibres over higher order poles. The topology of the fibre over a double pole $p$ depends on the choice of elementary modification. A choice of line in the fibre of $V$ at $p$ determines a distinguished elementary modification, with the property that the resulting fibre of $Y_{\phi}$ at $p$ is isomorphic to the disjoint union of two planes $\mathbb{C}^{2}_{p,+}\amalg\mathbb{C}^{2}_{p,-}$. We will always consider elementary modifications with this property.

We fix a linear Kähler form on $\mathbb{P}(W\oplus\mathcal{O})$. This induces a Kähler form $\omega$ on $Y_{\phi}$. A Moser-type argument, see Lemma 3.17, shows that (having fixed the parameters determining the cohomology class of the Kähler form appropriately) the symplectic manifold underlying $Y_{\phi}$ depends up to isomorphism only on the pair $(\mathbb{S},\mathbb{M})$. $Y_{\phi}$ has vanishing first Chern class, and $\phi$ determines a distinguished homotopy class $\eta(\phi)$ of trivialisation of the canonical bundle of $Y_{\phi}$.

When $\partial\mathbb{S}\neq\emptyset$, the Kähler form $\omega$ is exact, and $Y_{\phi}$ has a well-defined exact Fukaya category $\mathcal{F}(Y_{\phi})$, which may be constructed over any field $k$, see [47]. More generally, we will consider Lagrangian submanifolds $L\subset Y_{\phi}$ with the following property: there is an almost complex structure $J_{L}$ on $Y_{\phi}$, taming the symplectic form and co-inciding with the given integrable structure at infinity, for which $L$ bounds no $J_{L}$-holomorphic disk and does not meet any $J_{L}$-holomorphic sphere. For ease of notation, we will refer to such $(L,J_{L})$ as strictly unobstructed. When $\mathbb{S}$ is closed, $Y_{\phi}$ has a strictly unobstructed Fukaya category (which we again denote by) $\mathcal{F}(Y_{\phi})$, now defined over the Novikov field $\Lambda_{\mathbb{C}}$. This version of the Fukaya category appears, for instance, in [2] and [49]. The strictly unobstructed hypothesis rules out bubbling of holomorphic disks, which simplifies the technical construction of $\mathcal{F}(Y_{\phi})$, see [49, Sections 3b, 3c] for a detailed discussion under slightly weaker hypotheses.

Let $Y$ be a symplectic manifold with well-defined Fukaya category $\mathcal{F}(Y)$. For each $b\in H^{2}(Y;\mathbb{Z}_{2})$, there is a category $\mathcal{F}(Y;b)$, the $b$-twisted strictly unobstructed Fukaya category, which for $b=0$ recovers the category considered previously. Objects of $\mathcal{F}(Y;b)$ are closed oriented graded strictly unobstructed Lagrangians $L$, which are equipped with a relative spin structure¹¹1In other words, $b|_{L}=w_{2}(TL)$ and we fix a trivialisation of $\xi_{B}\oplus TL$ over the 2-skeleton of $L$, where $\xi_{b}\rightarrow Y$ is the unique real 2-plane bundle with $w_{1}(\xi_{b})=0$ and $w_{2}(\xi_{b})=b$., relative to the background class $b$. The choice of background class $b\in H^{2}(Y;\mathbb{Z}_{2})$ serves to change the signs with which holomorphic polygons contribute to the $A_{\infty}$-operations $\mu^{d}_{\mathcal{F}}$, and can be seen as fixing a particular coherent orientation scheme for the Fukaya category; compare to [47, Section 11 & Remark 12.1].

Each category $\mathcal{F}(Y_{\phi};b)$ is a $\mathbb{Z}$-graded $A_{\infty}$-category, linear over the appropriate field $k$. Let ${\mathcal{D}}\,\mathcal{C}$ denote the cohomological category $H^{0}(\mathrm{Tw}\,\mathcal{C})$ of the category of twisted complexes over an $A_{\infty}$-category $\mathcal{C}$.

Now consider the threefold $Y_{\phi}\rightarrow S$. For each $p\in M\subset S$, the fibre $\pi^{-1}(p)$ is reducible. Let $\mathbb{C}^{2}_{+,p}$ denote one component of this fibre. We fix the background class $b_{0}\in H^{2}(Y_{\phi};\mathbb{Z}_{2})$ represented by the locally finite cycle

given by (either) one of the components of the reducible fibre at each point of $\mathrm{Pol}_{=2}(\phi)$, or equivalently each point of $\operatorname{\mathbb{P}}$. The class is non-trivial by Lemma 3.11. Different choices of cycle representative for $b_{0}$ are related by monodromy by Lemma 3.12.

The untwisted Fukaya category $\mathcal{F}(Y_{\phi})$, which differs from $\mathcal{F}(Y_{\phi};b_{0})$ by certain signs, is different, and is discussed in Section 5.4.

One drawback of Theorem 1.1 is that it does not give a symplectic characterisation of the image of the embedding ${\mathcal{D}}(\mathbb{S},\mathbb{M})\hookrightarrow{\mathcal{D}}\mathcal{F}(Y_{\phi};b_{0})$. There is an obvious candidate for such a characterisation, which we now explain.

Any embedded path $\gamma:[0,1]\rightarrow S$, with end-points distinct zeroes of $\phi$ and otherwise disjoint from the zeroes and poles of $\phi$, defines a Lagrangian 3-sphere $L_{\gamma}\subset Y_{\phi}$, fibred over the arc $\gamma\subset\Sigma$ via Donaldson’s “matching cycle” construction [47, III, Section 16g]. The matching spheres $L_{\gamma}$ are exact if $\partial\mathbb{S}\neq\emptyset,$ and strictly unobstructed (with $J_{L}$ the canonical complex structure on $Y_{\phi}$) when $\mathbb{S}$ is closed and $g(\mathbb{S})>0$. A Lagrangian sphere is relatively spin for any choice of background class $b$, hence equipped with a grading defines a Lagrangian brane in $\mathcal{F}(Y_{\phi};b)$. A non-degenerate ideal triangulation $T$ of $S$ defines a full subcategory $\mathcal{A}(T;b)\subset\mathcal{F}(Y_{\phi};b)$, generated by the matching spheres associated to the edges of the cellulation dual to $T$. Theorem 1.1 is proved by showing that ${\mathcal{D}}\mathcal{A}(T;b_{0})\simeq{\mathcal{D}}(T)$ for particularly well-behaved triangulations $T$. Since the category ${\mathcal{D}}(T)$ does not depend on $T$, it follows that ${\mathcal{D}}\mathcal{A}(T;b_{0})$ also depends only on the pair $(\mathbb{S},\mathbb{M})$, up to triangulated equivalence.

Let $\mathcal{K}(Y_{\phi};b_{0})\subset\mathcal{F}(Y_{\phi};b_{0})$ be the full $A_{\infty}$-subcategory generated by Lagrangian matching spheres. This manifestly depends only on the pair $(\mathbb{S},\mathbb{M})$. It seems likely that ${\mathcal{D}}(\mathbb{S},\mathbb{M})\simeq{\mathcal{D}}\mathcal{K}(Y_{\phi};b_{0})$. We outline one tentative approach to proving that, which amounts to proving that $\mathcal{A}(T;b_{0})$ generates $\mathcal{K}(Y_{\phi};b_{0})$, in Section 4.9, but elaborating the details of that sketch would be a substantial task. Going further, it seems likely that all Lagrangian spheres in $Y_{\phi}$ are quasi-isomorphic to matching spheres (the result is proved in special cases in [2, 50]), in which case the category ${\mathcal{D}}\mathcal{K}(Y_{\phi};b_{0})$ would be a symplectic invariant of $Y_{\phi}$, carrying an action of the subgroup of $\pi_{0}\mathrm{Symp}(Y_{\phi})$ preserving the class $b_{0}\in H^{2}(Y_{\phi};\mathbb{Z}_{2})$. The question of whether the embedding $\mathcal{K}(Y_{\phi};b_{0})\hookrightarrow\mathcal{F}(Y_{\phi};b_{0})$ itself split-generates is also open, though here there seems to be less evidence either way. We hope to return to these questions elsewhere.

The imposed condition $\mathbb{M}\neq\emptyset$ gives a substantial simplication since holomorphic polygons are constrained for grading considerations that do not pertain when $\phi$ is globally holomorphic, cf. Remarks 3.22 and 4.15. The constraint on the number of punctures $|\mathbb{M}|\geqslant 3$ when $\mathbb{S}$ is closed arises from a similar constraint in work of Geiss, Labardini-Fragoso and Schröer [18], who study the action of right equivalences on potentials on the quivers $Q(T)$, cf. Theorem 2.1. We conjecture that, for a closed surface $\mathbb{S}$ of genus $g(\mathbb{S})>0$, Theorem 1.1 holds under the weaker hypothesis $|\mathbb{M}|>1$. (Once-punctured surfaces are special: not every pair of signed ideal triangulations are related by pops and flips, and when $g=1$ the analogue of Theorem 2.1 is actually false, cf. [18].)

One can relax the strict unobstructedness hypothesis at the cost of invoking the deep obstruction theory of [14] in the construction of $\mathcal{F}(Y_{\phi};b)$. If $\mathbb{S}$ is closed and $g(\mathbb{S})=0$ then $Y_{\phi}$ may contain rational curves, so only the more complicated construction is available. This is the reason for the genus constraint in the first part of Theorem 1.1.

In many cases, the paper [5] computes the space of stability conditions on the category ${\mathcal{D}}(\mathbb{S},\mathbb{M})$ in terms of moduli spaces of quadratic differentials. If either $\mathbb{S}$ is closed with $g(\mathbb{S})>0$ and with at least two punctures, or $\partial\mathbb{S}\neq\emptyset$ and $\mathbb{S}$ is not a sphere with fewer than six punctures, there is a connected component $\operatorname{Stab}_{\Delta}(\mathbb{S},\mathbb{M})$ of $\operatorname{Stab}({\mathcal{D}}(\mathbb{S},\mathbb{M}))$ and a subgroup $\mathrm{Aut}_{\Delta}$ of autoequivalences which preserve this component modulo those which act trivially upon it, with

where $\operatorname{Quad}_{\heartsuit}(\mathbb{S},\mathbb{M})$ has the same coarse moduli space as $\operatorname{Quad}(\mathbb{S},\mathbb{M})$ but additional orbifolding along the incomplete locus, see [5] for details. This gives a non-trivial computation of the space of stability conditions on (a subcategory of) the Fukaya category of a symplectic six-manifold, and enables one to understand the Donaldson-Thomas invariants of these categories.

One can construct a complex (3,0)-form $\Omega$ on $Y_{\phi}$ with the following property: if the path $\gamma\subset\Sigma$ is a saddle connection for $\phi$, the associated matching sphere $L_{\gamma}$ is special, i.e. $\Omega$ has constant phase on $L_{\gamma}$. (We are not working with the Ricci-flat metric, so these are not strictly special in the traditional sense.) There are similarly $\Omega$-special Lagrangian submanifolds $S^{1}\times S^{2}$ of $Y_{\phi}$ associated to homotopically non-trivial closed geodesics for the flat metric defined by $\phi$. Theorem 1.4 of [5] implies that the Donaldson-Thomas invariants of $Y_{\phi}$, defined with respect to the stability condition associated to $\phi$, count such special Lagrangian submanifolds, the existence and numerics of which are therefore governed by the Joyce-Song and Kontsevich-Soibelman wall-crossing formulae [24, 30]. This confirms, in this special case, a long-standing expectation of Joyce and others.

It is natural to conjecture that moduli spaces of holomorphic quadratic differentials with simple zeroes arise as spaces of stability conditions (modulo autoequivalences) on the Fukaya categories of the local threefolds of [8] (see Equation 3.1). However, these categories do not appear to admit descriptions in terms of quivers, and different techniques would be required to analyse them and the corresponding spaces of stability conditions.

The threefolds $Y_{\phi}$ are associated to meromorphic maps of Riemann surfaces into the versal deformation space of the $A_{1}$-surface singularity $\mathbb{C}^{2}/\mathbb{Z}_{2}$. There are also local threefolds associated to meromorphic maps to the versal deformation spaces of other ADE singularities: in the case of a holomorphic map, the relevant threefolds are studied in [8, 55]. Already for the $A_{2}$-surface, however, the geometry is substantially more complicated, and the relationship to stability conditions rather less clear.

The natural data required to write down a quasi-projective Calabi-Yau threefold fibred over a Riemann surface $S$ in the $A_{n}$-case is a tuple comprising (perhaps meromorphic) sections of $K_{S}^{\otimes i}$ for $2\leqslant i\leqslant n$. For instance, when $n=2$ and supposing one is working with holomorphic rather than meromorphic differentials, one takes a vector bundle $V=L_{1}\oplus L_{2}$ with $det(V)\cong K_{S}$, and considers the hypersurface

where $\phi\in H^{0}(K_{S}^{\otimes 2})$ and $\psi\in H^{0}(K_{S}^{\otimes 3})$. For generic choices of $(\phi,\psi)$ this is a smooth Calabi-Yau, the total space of a Lefschetz fibration over $S$, with generic fibre the Milnor fibre of $\mathbb{C}^{2}/\mathbb{Z}_{3}$ (symplectically, the plumbing of two copies of $T^{*}S^{2}$).

However, the moduli space of such data – a complex structure on $\mathbb{S}$ and a tuple of differentials – has dimension smaller than the dimension of the space of stability conditions on the corresponding category, or more mundanely smaller than the rank of $H_{3}$ of the associated threefold (compare to Remark 3.15). This is a familiar problem: whilst one expects spaces of complex structures on a symplectic Calabi-Yau to embed into the space of stability conditions on the Fukaya category, there is no reason to expect that embedding to be onto an open subset. The fortunate accident in the $A_{1}$-case is that Teichmüller space has the same dimension as the space of quadratic differentials. At least for $A_{n}$-fibred threefolds, it seems natural following [33] to expect the “higher Teichmüller space”, i.e. Hitchin’s contractible component [22] of the variety of flat $\mathbb{P}SL(n,\mathbb{R})$-connections, to play a role in resolving this discrepancy.

Finally, we note that computations in [17] indicate that the CY₃ -categories arising in higher rank (even when one restricts to meromorphic differentials) are appreciably more complicated; whereas the Donaldson-Thomas invariants of the categories studied in this paper are always $+1$ or $-2$, cf. [5], and non-vanishing only for primitive classes, in the higher rank case one expects there are classes $[\gamma]$ for which the DT-invariants (conjecturally related to counts of special Lagrangian submanifolds in the corresponding threefold) in class $k[\gamma]$ grow exponentially with $k$. The symplectic topology of (1.4) is the subject of work in progress by the author.

The arguments for the two cases of Theorem 1.1 are rather similar. For definiteness, for the rest of the paper we consider the (more complicated) case when $\mathbb{S}$ is closed, leaving the required modifications for the case with non-empty boundary to the interested reader.

In the closed case all the marked points $\mathbb{M}=\operatorname{\mathbb{P}}\subset\mathbb{S}$ are punctures. To avoid transversality issues arising from rational curves and their multiple covers, we also exclude the (interesting) case of threefolds fibring over the 2-sphere. Therefore the discrete topological data will henceforth be indexed by a pair $(\mathbb{S},\mathbb{M})$ with $g(\mathbb{S})>0$ and $\#\mathbb{M}>0$.

Acknowledgements. Tom Bridgeland was originally to be a co-author; his influence is pervasive. I am indebted to Daniel Labardini-Fragoso for explanations of his joint work with Geiss and Schröer [18], on which we rely essentially. Thanks also to Mohammed Abouzaid, Bernhard Keller, Yankl Lekili, Andy Neitzke, Oscar Randal-Williams, Tony Scholl, Paul Seidel, Balazs Szendroi and Richard Thomas for helpful conversations; Abouzaid and Seidel provided assistance with a crucial sign computation. Finally, I am grateful to the anonymous referee for suggesting many improvements to the exposition.

Let $Q$ be a quiver, specified by sets of vertices and arrows $Q_{0}$, $Q_{1}$, and source and target maps $s,t\colon Q_{1}\to Q_{0}$. We write $kQ$ for the path algebra of $Q$ over the field $k$, and $\widehat{kQ}$ for the completion of $kQ$ with respect to path length. A potential on $Q$ is an element $W\in\widehat{kQ}$ of the closure of the subspace of $\widehat{kQ}$ spanned by all cyclic paths in $Q$ of length $\geqslant 2$. A potential is called reduced if it lies in the closure of the subspace spanned by cycles of length $\geqslant 3$.

We say that two potentials $W$ and $W^{\prime}$ are cyclically equivalent if $W-W^{\prime}$ lies in the closure of the subspace generated by differences $a_{1}\ldots a_{s}-a_{2}\ldots a_{s}a_{1}$, where $a_{1}\ldots a_{s}$ is a cycle in the path algebra. $W$ and $W^{\prime}$ are right-equivalent if there is an automorphism $\phi:\widehat{kQ}\rightarrow\widehat{kQ}$ of the completed path algebra which fixes the zero-length paths and such that $\phi(W)$ and $W^{\prime}$ are cyclically equivalent. Following [18], we say $W$ and $W^{\prime}$ are weakly right equivalent if $W$ and $tW^{\prime}$ are right-equivalent, for some invertible scalar $t\in k^{*}$.

Consider minimal $A_{\infty}$ categories $\mathcal{C}$ whose objects $S_{i}$ are indexed by the vertices of $Q$, and such that, as a graded vector space

where $V_{ij}$ is the space with basis consisting of arrows in $Q$ connecting vertex $i$ to vertex $j$. There is an obvious non-degenerate pairing

Thus, if we define

an $A_{\infty}$ product of degree $2-(n-1)$,

is equivalently described by a linear map of degree $-n$

Let us insist that $\mathcal{C}$ is cyclic as an $A_{\infty}$ category, meaning that the tensors $c_{n}$ are cyclically invariant in the graded sense. If we further insist that the $A_{\infty}$ structure on $\mathcal{C}$ is strictly unital then the whole structure is determined by the elements $c_{n}(f_{n},\cdots,f_{1})$ when all the $f_{i}$ have degree 1. For background on this construction, see [43].

Let $W$ be a reduced potential on $Q$. Decomposing the potential into homogeneous pieces, and cyclically symmetrising, we obtain linear maps

Setting $c_{n}=W_{n}$ gives a well-defined $A_{\infty}$ category $\mathcal{C}(Q,W)$. Define ${\mathcal{D}}(Q,W)$ to be the homotopy category of the category of twisted complexes over $\mathcal{C}(Q,W)$

The associated graded category of ${\mathcal{D}}(Q,W)$ contains $\mathcal{C}$ as a full subcategory.

The same category ${\mathcal{D}}(Q,W)$ admits an alternative (Koszul dual) description in terms of the derived category of a dg algebra $A(Q,W)$ called the complete Ginzburg algebra (see for instance [30, Theorem 9] or [26, Section 5]). To define it, first double $Q$, adding a dual edge $a^{*}$ for each $a\in Q$, and then add loops $t_{i}$ based at each vertex $i$ of $Q$. The resulting quiver $Q^{*}$ has a grading given by

Let $A(Q,W)$ be the completion of the path algebra $\widehat{kQ^{*}}$ as a graded algebra, with respect to the ideal generated by the arrows of $Q^{*}$. There is a unique continuous differential $d$ satisfying

Thus $A(Q,W)$ is a dg algebra. The category ${\mathcal{D}}(Q,W)$ can then be equivalently described as the full subcategory of the derived category of the dg algebra $A(Q,W)$ consisting of finite-dimensional modules. By a general result of Keller and Van den Bergh [27], this description shows that ${\mathcal{D}}(Q,W)$ is a CY₃ triangulated category.

Keller and Yang [28, Lemma 2.9] showed that if $W$ and $W^{\prime}$ are right-equivalent potentials, they have isomorphic complete Ginzburg algebras, and hence yield equivalent categories ${\mathcal{D}}(Q,W)\simeq{\mathcal{D}}(Q,W^{\prime})$. Ladkani [34, Proposition 2.7], see also [18, Lemma 8.5], showed that the same conclusion holds when $W$ and $W^{\prime}$ are only weakly right-equivalent. Indeed, there is a natural $k^{*}$-action on the set of minimal $A_{\infty}$-structures on the category $\mathcal{C}$, where $\lambda\in k^{*}$ acts by rescaling the operation $m_{n}$ by $\lambda^{n-2}$. $A_{\infty}$-structures related by the $k^{*}$-action are $A_{\infty}$-equivalent even though not gauge-equivalent in the usual sense (the required equivalence does not act by the identity on cohomology but by a multiple of the Euler vector field). The potentials $W$ and $tW$ on $Q$ give rise to $A_{\infty}$-categories $\mathcal{C}(Q,W)$ and $\mathcal{C}(Q,tW)$ which are related by the $k^{*}$-action. Since $A_{\infty}$-equivalences induce equivalences on categories of twisted complexes by [47, Lemma 3.20], the category ${\mathcal{D}}(Q,W)$ depends only on the weak right equivalence class of $W$.

The fact that $A(Q,W)$ is concentrated in non-positive degrees implies [28, Lemma 5.2] that ${\mathcal{D}}(Q,W)$ is equipped with a canonical bounded t-structure, whose heart $\mathcal{A}(Q,W)$ is equivalent to the category of nilpotent representations of the completed Jacobian algebra

In particular, $\mathcal{A}(Q,W)\subset{\mathcal{D}}(Q,W)$ is a finite-length heart. Since it admits a bounded t-structure, the category ${\mathcal{D}}(Q,W)$ is split-closed, i.e. agrees with its own idempotent completion, see [35].

Suppose again that $\mathbb{S}$ is a closed oriented surface of genus $g(\mathbb{S})>0$, now equipped with a non-empty set of $d\geqslant 2$ marked points $\mathbb{M}\subset\mathbb{S}$.

The quiver $Q(T)$ has vertices at the midpoints of the edges of $T$, and is obtained by inscribing a small clockwise 3-cycle inside each face of $T$, as in Figure 1. There are two obvious systems of cycles in $Q(T)$, namely a clockwise 3-cycle $T(f)$ in each face $f$, and an anticlockwise cycle $C(p)$ of length at least 3 encircling each point $p\in\mathbb{M}$. Define a potential

When $\epsilon\equiv 1$ we will sometimes omit it from the notation.

Consider the derived category of the completed Ginzburg algebra of the quiver with potential $(Q(T),W(T,\epsilon))$ over $k$, and let ${\mathcal{D}}(T,\epsilon)$ be the full subcategory consisting of modules with finite-dimensional cohomology. As a special case of the discussion of Section 2.1, this is a CY₃ triangulated category of finite type over $k$, and comes equipped with a canonical t-structure, whose heart $\mathcal{A}(T,\epsilon)\subset{\mathcal{D}}(T,\epsilon)$ is equivalent to the category of finite-dimensional modules for the completed Jacobi algebra of $(Q(T),W(T,\epsilon))$.

Suppose two non-degenerate ideal triangulations $T_{i}$ are related by a flip, in which the diagonal of a quadilateral is replaced by its opposite diagonal. The resulting quivers with potential $(Q(T_{i}),W(T_{i},\epsilon))$ (in which the signing is unchanged) are related by a mutation at the vertex corresponding to the edge being flipped. It follows from general results of Keller and Yang [28] that there exist distinguished $k$-linear triangulated equivalences $\Phi_{\pm}\colon{\mathcal{D}}(T_{1},\epsilon)\cong{\mathcal{D}}({T}_{2},\epsilon)$.

Labardini-Fragoso [32] extended the above definitions so as to encompass a larger class of signed ideal triangulations (ones containing self-folded triangles, in which two of the three edges co-incide; in this case there may be punctures of valency one, and the mutation operation can change the signing). He moreover proved that flips also induce right-equivalences in this more general context. There is another operation on signed ideal triangulations, which involves changing the signing at a given puncture without changing the triangulation, and a corresponding “pop” equivalence which relates the associated categories. Under our hypothesis on $(\mathbb{S},\mathbb{M})$ that $d=|\mathbb{M}|\geqslant 2$, any two of these more general signed ideal triangulations are related by a finite chain of flips and pops. It follows that up to $k$-linear triangulated equivalence, the category ${\mathcal{D}}(T,\epsilon)$ is independent of the chosen triangulation and of the choice of signing; see [5, Sections 8 & 9] for a more detailed discussion. We denote by ${\mathcal{D}}(\mathbb{S},\mathbb{M})$ any triangulated category in this quasi-equivalence class.

Given the quiver $Q(T)$, one can define a CY₃ category by taking any potential on $Q(T)$, not necessarily the potential $W(T,\epsilon)$ described above. We will say that two potentials $W_{1}$ and $W_{2}$ are disjoint if no cycle occuring in $W_{1}$ is cyclically equivalent to a cycle appearing in $W_{2}$. The following result is due to Geiss, Labardini-Fragoso and Schröer [18].

Geiss, Labardini-Fragoso and Schröer furthermore prove that every pair $(\mathbb{S},\mathbb{M})$ with $g(\mathbb{S})>0$ and with $|\mathbb{M}|\geqslant 3$ admits some triangulation $T$ which satisfies the hypotheses, i.e. which contains no self-folded triangle or loop, in which every vertex has valency at least 4, and for which the associated quiver has no double arrow. (In the case when $\mathbb{S}$ has non-empty boundary, the same result holds without further hypotheses on the number of punctures.) The proof of Theorem 2.1 involves a delicate, iterative construction of a suitable right-equivalence by hand, obtained as an infinite composition of equivalences which, to first approximation, increase the minimal length of any cycle appearing in the remainder term $W^{\prime}$; the actual proof is more complicated than this suggests.

For our purposes, these results yield a finite-determinacy theorem for $A_{\infty}$-structures on the total endomorphism algebra of the category $\mathcal{C}$ of Section 2.1 in the special case $(Q,W)=(Q(T),W(T,\epsilon))$. Considering the description of the category ${\mathcal{D}}(Q,W)$ as a category of twisted complexes over an $A_{\infty}$-algebra given in Section 2.1, Theorem 2.1 implies in particular that different choices of scalars $\{\lambda_{p}\}_{p\in\operatorname{\mathbb{P}}}$ for the potential (2.3) yield equivalent categories ${\mathcal{D}}(Q,W)$, whilst $A_{\infty}$-products encoded by the “remainder” term $W^{\prime}$ can be gauged away.

Let $(S,\phi)$ denote a pair comprising a Riemann surface $S$ and meromorphic quadratic differential $\phi$ with poles of order precisely 2 at the points of a divisor $M\subset S$ comprising $d$ reduced points, and with simple zeroes. Thus, the marked bordered surface associated to $(S,\phi)$ is diffeomorphic to $(\mathbb{S},\mathbb{M})$. Let $\mathrm{Zer}_{\phi}\subset S$ denote the set of zeroes, so

At a point of $S\backslash\{M\cup\mathrm{Zer}_{\phi}\}$ there is a distinguished local co-ordinate $z$ with respect to which

This local co-ordinate is uniquely defined up to changes $z\mapsto\pm z+constant$. At simple zeroes, respectively double poles, there is a canonical co-ordinate with respect to which

We refer to the value $m$ in the second case as the residue at the double pole.

The surface $S\backslash M$ inherits a flat metric $|\phi|^{2}$ with singularities; at each $p\in\mathrm{Zer}_{\phi}$, the metric has a cone angle of $3\pi$. The length element of the metric is defined by

in an arbitrary local parameter $w$, so the length of a curve $\gamma\subset S\backslash M$ is given by

This is well-defined for curves passing through zeroes of $\phi$, but diverges to infinity for curves through double poles. The area of the flat surface

is infinite, since a neighbourhood of each point of $M$ is isometric to a semi-infinite flat cylinder of circumference $2\pi|m|$, with $m$ as in Equation 2.4.

The differential $\phi\in H^{0}(K_{S}(M)^{\otimes 2})$ defines a horizontal foliation of $S\backslash M$, given by the lines along which $arg(\phi)=0$. In the natural local co-ordinate, the horizontal foliation is given by lines $\operatorname{Im}(z)=constant$. The local trajectory structure at a zero shows the horizontal foliation is not transversely orientable. The natural $S^{1}$-action by rotation, $\phi\mapsto e^{i\theta}\phi$, does not change the underlying flat surface, but changes which in the circle of foliations defined by $\arg\,\phi(z)=constant$ is regarded as horizontal.

A saddle connection is a finite length maximal horizontal trajectory. Any such has both end-points at (not necessarily distinct) zeroes of $\phi$.

Suppose the quadratic differential $\phi$ is complete and saddle-free, meaning that it has no finite-length maximal horizontal trajectory. It then defines a canonical isotopy class of triangulation of $S$ with vertices at $M$, called the WKB-triangulation, see [5, Section 10]. There is a dual “Lagrangian cellulation”, with trivalent vertices the zeroes of $\phi$ and which has exactly one face for each point of $M$.

In general, the WKB-triangulation may contain self-folded triangles. Given a quadratic differential $\psi$ whose associated WKB-triangulation contains a self-folded triangle, there is an edge in the Lagrangian cellulation which goes from a zero to itself. The quiver prescription of Labardini-Fragoso differs in this case [32]. For simplicity we will restrict attention to the non-degenerate case:

For brevity, we will call a triangulation as provided by Lemma 2.2 a non-degenerate WKB triangulation.

Let $S$ be a closed Riemann surface of genus $g\geqslant 1$, equipped with a reduced divisor $M$ comprising $d=|M|$ points. Fix a rank two holomorphic vector bundle

The determinant defines a fibrewise quadratic map

Consider an elementary modification of the symmetric square $S^{2}(V)$ along $M$, fitting into a short exact sequence of sheaves

Elementary modifications along $M$ are not unique, but depend on the choice of $\alpha$ in (3.3). It will be important for us to choose the elementary modification compatibly with the quadratic map (3.2).

Note that a choice of complex line $\ell$ in the fibre $V_{p}$ of $V$ at $p$, equivalently of a parabolic structure on $V$ at $p$, induces an elementary modification as in Lemma 3.2, where the subspace $W_{p}\subset S^{2}V_{p}$ is identified with the quadratic forms on $V_{p}^{*}$ vanishing on the annihilator of $\ell\subset V_{p}$.

Let $\phi\in H^{0}(K_{S}(M)^{\otimes 2})$ be a meromorphic quadratic differential on $S$ with simple zeroes and a pole of order exactly $2$ at each $x\in M$. Define the hypersurface

inside the total space of the vector bundle $W$. We shall write

for the fibrewise projective completion of $Y_{\phi}$. Being fibred in quadrics, this is the zero-locus of a section of $\pi^{*}K_{S}(M)^{\otimes 2}(2)\rightarrow\mathbb{P}(W\oplus\mathcal{O})$, where $\pi:\mathbb{P}(W\oplus\mathcal{O})\to S$ denotes projection.

The previous description of the determinant map $\det_{W}$ shows that the natural map $X_{\phi}\rightarrow S$ is a fibration by projective quadrics, with generic fibre $\mathbb{P}^{1}\times\mathbb{P}^{1}$, nodal fibres over zeroes of $\phi$, and reducible fibre $\mathbb{P}^{2}\cup_{l}\mathbb{P}^{2}$ at each point of $M$, i.e. the union of two planes joined along a line $l$. The locus of reducible quadrics has codimension $>1$ in the space of all quadric hypersurfaces in $\mathbb{P}^{3}$, so a generic one-parameter family of quadrics would have no such singular fibres. The singular fibre $\mathbb{P}^{2}\cup_{l}\mathbb{P}^{2}$ is not locally smoothable (i.e. not the $0$-fibre of a smooth three-fold $\mathcal{X}\rightarrow D$); an infinitesimal smoothing is determined by a section of the tensor product $\nu_{l/\mathbb{P}^{2}}^{*}\otimes\nu_{l/\mathbb{P}^{2}}^{*}\cong\mathcal{O}(-2)$ of the normal bundles to the normal crossing locus $l$ in the two components of the singular fibre.

The quasi-projective variety $Y_{\phi}$ comes with a natural projection map $\pi:Y_{\phi}\rightarrow S$.

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  The generic fibre of $\pi$ is a smooth affine quadric $\{ab-c^{2}=t\}$ with $t\neq 0$, abstractly diffeomorphic to the cotangent bundle $T^{*}S^{2}$;

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  At a zero $p$ of $\phi$, $\pi^{-1}(p)$ is defined by the quadratic $\{ab-c^{2}=0\}\subset\mathbb{C}^{3}$, which has an isolated nodal singularity;

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  At a point $x\in M$, recalling that by hypothesis $\phi(x)=t^{\prime}\neq 0$, the fibre $\pi^{-1}(x)$ is given by an affine quadric $\{c^{2}=t^{\prime}\}\subset\mathbb{C}^{3}$, a disjoint union of two planes.