Derived equivalences of gentle algebras via Fukaya categories

Following the approach of Haiden-Katzarkov-Kontsevich arXiv:1409.8611, to any homologically smooth graded gentle algebra $A$ we associate a triple $(\Sigma_A, \Lambda_A; \eta_A)$, where $\Sigma_A$ is an oriented smooth surface with non-empty boundary, $\Lambda_A$ is a set of stops on $\partial \Sigma_A$ and $\eta_A$ is a line field on $\Sigma_A$, such that the derived category of perfect dg-modules of $A$ is equivalent to the partially wrapped Fukaya category of $(\Sigma_A, \Lambda_A ;\eta_A)$. Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of $\Sigma_A$ on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella-Alaminos-Geiss in math/0607348, as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in arXiv:1705.06023.


Introduction
Given a Liouville manifold (M, ω = dλ), a rigorous definition of the compact Fukaya category, F (M), appears in the monograph [20]. This is a triangulated A ∞ -category linear over some base ring K. Roughly speaking, the objects of F (M) are compact, exact, oriented Lagrangian submanifolds in M, equipped with spin structures (if charK = 2). The orientations on each Lagrangian determine a Z 2 -grading on F (M), and the spin structures enter in orienting the moduli spaces of holomorphic polygons that enter into the definition of structure constants of the A ∞ operations. It is often convenient to upgrade the Z 2grading on F (M) to a Z-grading, which can be done under the additional assumption that 2c 1 (M) = 0 (see [15], [19]). Under this assumption, one defines a notion of a grading structure on M, and correspondingly considers only graded Lagrangians as objects of F (M), which now becomes a Z-graded category. We refer to [19] for these general notions. In this paper, we focus our attention to the case where M = Σ is punctured (real) 2-dimensional surface, equipped with an area form. A grading structure on Σ can be concretely described as a homotopy class of a section η of the projectivized tangent bundle of P(T Σ). Note that there is an effective H 1 (Σ)'s worth of choices (see Sec. 1). A Lagrangian can be graded if the winding number of η along L vanishes, and in such a situation a grading is a choice of a homotopy from the tangent lift L → T L ⊂ T Σ to η |L along L. These gradings extend in a straightforward manner to the wrapped Fukaya category W(Σ) which contains F (Σ) as a full subcategory, but also allows non-compact Lagrangians in Σ and more generally, partially wrapped category W(Σ, Λ), as studied in [10,Sec. 2.1], where Σ is a surface with boundary and Λ is a collection of stops (i.e., marked points) on ∂Σ.
Given two graded surfaces with stops, (Σ i , Λ i ; η i ) for i = 1, 2, a homeomorphism φ : Σ 1 → Σ 2 , which restricts to a bijection Λ 1 → Λ 2 , and a homotopy between φ * (η 1 ) and η 2 , one gets an equivalence between the partially wrapped Fukaya categories W(Σ 1 , Λ 1 ; η 1 ) and W(Σ 2 , Λ 2 ; η 2 ). Thus, it is important to have a set of explicit computable invariants of a line field η on a surface with boundary that determine the orbit of η under the action of the mapping class group of Σ. Our first result (see Theorem 1.2.5) gives such invariants in terms of winding numbers of η. In the most interesting case when genus is ≥ 2, the invariants consist of the winding numbers along all the boundary components, plus two more invariants, each taking values 0 and 1. The first of them is a Z 2 valued invariant which decides whether the line field η is induced by a non-vanishing vector field, while the second is the Arf-invariant of a certain quadratic form over Z 2 . Note that from the numerical invariants of Theorem 1.2.5 one can also recover the genus of the surface and the numbers of stops on the boundary components, so if these invariants match then then the corresponding partially wrapped Fukaya categories are equivalent.
Next, we use this result to construct derived equivalences between gentle algebras, introduced by Assem and Skowrónski in [3]. This is a remarkable class algebras with monomial quadratic relations of special kind with a well understood structure of indecomposable modules. Furthermore, their derived categories of modules also enjoy many nice properties (see [7] and references therein). Avella-Alaminos and Geiss [5] gave a combinatorial definition of derived invariants of finite-dimensional gentle algebras, which form a collection of pairs of non-negative integers (m, n) with multiplicities. We refer to these as AAG-invariants. It is known that these invariants do not completely determine the derived Morita class of a gentle algebra in general (for example, see [1]).
We consider Z-graded gentle algebras and their perfect derived categories (the classical case corresponds to algebras concentrated in degree 0). For such an algebra A, we denote by D(A) the perfect derived category of dg-modules over A viewed as a dg-algebra with zero differential. The category D(A) has a natural dg-enhancement which we take into account when talking about equivalences involving D(A).
The connection between graded gentle algebras and Fukaya categories was established by Haiden, Katzarkov and Kontsevich in [10] (cf. [6]): they constructed collections of formal generators in some partially wrapped Fukaya categories whose endomorphism algebras are graded gentle algebras. In Theorem 3.2.2 we give an inverse construction 1 : starting from a homologically smooth graded gentle algebra A we construct a graded surface with stops (Σ A , Λ A ; η A ) together with a set formal generators whose endomorphism algebra is isomorphic to A. This leads to an equivalence of the partially wrapped Fukaya category W(Σ, Λ) with the derived category D(A). In addition, we generalize the combinatorial definition of AAG-invariants to possibly infinite-dimensional graded gentle algebras and show that they can be recovered from the winding numbers of η A along all boundary components. Now recalling our numerical invariants of graded surfaces with stops from Theorem 1.2.5 we obtain a sufficient criterion for derived equivalence between homologically smooth graded gentle algebras. Namely, if we start with two such algebras A and A ′ and find that the corresponding invariants from Theorem 1.2.5, determined by winding numbers of η A and η A ′ , coincide then we get a derived equivalence between A and A ′ . Note that this involves checking that A and A ′ have the same AAG-invariants, and in addition that two more invariants with values in {0, 1} match.
As an application, using the above approach we obtain a sufficient criterion for derived equivalence of homologically smooth graded gentle algebras given purely in terms of AAGinvariants (see Corollary 3.2.5). Using Koszul duality, we also get a sufficient criterion for derived equivalence of finite-dimensional gentle algebras with grading in degree 0 (see Corollary 3.2.6).
In a different direction, we construct derived equivalences between stacky nodal curves studied in [16], Namely, these are either chains or rings of weighted projective lines glued to form stacky nodes, locally modelled by quotients (xy = 0)/(x, y) ∼ (ζ k x, ζy), where ζ r = 1 and k ∈ (Z/r) * . In [16,Thm. B] we constructed an equivalence of the derived category of coherent sheaves on such a stacky curve with the partially wrapped Fukaya category of some graded surface with stops (this can be viewed as an instance of homological mirror symmetry). Thus, using Theorem 1.2.5 we get many nontrivial derived equivalences between our stacky curves. In the case of balanced nodes (those with k = −1) we recover the equivalences between tcnc curves from [21].
Acknowledgments. Y.L. is partially supported by the Royal Society (URF) and the NSF grant DMS-1509141, and would like to thank Martin Kalck for pointing out the reference [5]. A.P. is supported in part by the NSF grant DMS-1700642 and by the Russian Academic Excellence Project '5-100'. While working on this project, A.P. was visiting King's College London, Institut des Hautes Etudes Scientifiques, and Korea Institute for Advanced Study. He would like to thank these institutions for hospitality and excellent working conditions. 1. Line fields on surfaces 1.1. Basics on line fields. Let Σ be an oriented smooth surface of genus g(Σ) with nonempty boundary with connected components ∂Σ = b i=1 ∂ i Σ. The mapping class group of Σ is M(Σ) = π 0 (Homeo + (Σ, ∂Σ)), where Homeo + (Σ, ∂Σ) is the space of orientation preserving homeomorphism of Σ which are the identity pointwise on ∂Σ. Definition 1.1.1. An (unoriented) line field η on Σ is a section of the projectivized tangent bundle P(T Σ). We denote by G(Σ) = π 0 (Γ(Σ, P(T Σ))) the set of homotopy classes of unoriented line fields.
A non-vanishing vector field i.e. a section of the unit tangent bundle SΣ induces a line field via the bundle map SΣ → P(T Σ) which is a fibrewise double covering. However, not all line fields come from non-vanishing vector fields: a section of P(T Σ) may not lift to a section of SΣ.
The trivial circle fibration S 1 ι − → P(T Σ) p − → Σ induces an exact sequence A line field η determines a class [η] ∈ H 1 (P(T Σ)) such that ι * [η]([S 1 ]) = 1 by taking the Poincaré-Lefschetz dual of the class of the image [η(Σ)] ⊂ H 2 (P(T Σ), ∂P(T Σ)). Via this construction, we get an identification where ζ ∈ H 1 (S 1 ) is the generator which integrates to 1 along S 1 . Thus, the set G(Σ) is a torsor over H 1 (Σ). We denote the corresponding action of c ∈ H 1 (Σ) on G(Σ) by The mapping class group M(Σ) acts on G(Σ) on the right. Our goal in this section is to understand the orbit decomposition of G(Σ) with respect to this action.
The winding number w η (γ) with respect to η only depends on the homotopy class of η and the regular homotopy class of γ. From the definition we immediately get the following compatibility with the action of H 1 (Σ): Throughout, ∂Σ is oriented with respect to the natural orientation as the boundary of Σ. In particular, w η (∂D 2 ) = 2 for the unique homotopy class of line fields on D 2 . For a boundary component B ⊂ ∂Σ with the opposite orientation, we write −B. Then, we have w η (−B) = −w η (B).

1.2.
Invariants under the action of the mapping class group. The winding numbers along boundary components of Σ gives the first set of invariants of elements of G(Σ). To go further, we need to study the winding numbers along non-separating curves on Σ. As is well-known, the winding number invariants do not descend to a map from H 1 (Σ). Indeed, if S ⊂ Σ is a compact subsurface with boundary ∂S = n i=1 ∂ i S, by Poincaré-Hopf index theorem (see [11,Ch. 3]), we have: However, considering the reduction modulo 2 we still get a well-defined homomorphism: i.e an element H 1 (Σ; Z 2 ).
We have a natural inclusion induced map The cokernel of i is isomorphic to Z 2g 2 and comes equipped with a non-degenerate intersection pairing.
Note that the numbers r i (η) mod 2 are precisely the values of [w η ] (2) on the boundary cycles. Thus, if r i (η) is odd for at least one i then σ(η) = 1. If all r i (η) are even then we can check whether σ(η) = 0 by looking at the winding numbers of a collection of cycles projecting to a basis of the cokernel of i. given by where α i are simple closed curves. It satisfies where a, b ∈ H 1 (Σ; Z 4 ), and a · b denotes the intersection pairing on H 1 (Σ; Z 4 ).
Proof. In the case when η comes from a vector field v, we have w η (a) = 2w v (a), where w v (·) is the winding number of the vector field. Hence, the assertion in this case follows from [12, Thm 1A, Thm 1B]. In general we have [η] = η 0 + c, for some c ∈ H 1 (Σ). Thus, the function q η (a) := q η 0 (a) + c, a has the claimed properties. Proof.
Thus, the study of the M(Σ)-orbits on G(Σ) reduces to the study of M(Σ)-orbits on the set of functions q : Let us denote by Quad 4 = Quad 4 (Σ) the set of all such functions (it is an H 1 (Σ, Z 4 )-torsor).
Recall that given a symplectic vector space V, (− · −) over Z 2 , one can consider the set Quad(V ) of quadratic forms q : V → Z 2 satisfying For every q ∈ Quad(V ), the Arf-invariant ( [2], [8]) is the element of Z 2 given by where (a i , b i ) is a symplectic basis of V . The Arf invariant is the value that q attains on the majority of vectors in V . In the case when r i (η) = w η (∂ i Σ) + 2 ∈ 4Z for every i = 1, . . . , d, and the quadratic function q = q η takes values in 2Z 4 , we can associate to q a certain quadratic form on a Z 2 -vector space, and its Arf-invariant will give us an additional invariant of η modulo the mapping class group action.
Let us set H := H 1 (Σ, Z 4 ), K = im(i * : H 1 (∂Σ, Z 4 ) → H 1 (Σ, Z 4 )), H = H/2H, K = K/2K. Since K lies in the kernel of the intersection pairing, for any q ∈ Quad 4 the restriction q| K is a homomorphism K → Z 4 . Note that for q = q η the value of this homomorphism on [∂ i Σ] is r i (η) mod 4. Now let q ∈ Quad 4 be such that q| K is zero. Then it is easy to see that q descends to a well defined function q H/K on H/K. Assume in addition that σ(η) = 0, i.e., q takes values in 2Z 4 . In this case we have q H/K = 2q, where q is a function H/K → Z 2 satisfying (1.3). It is easy to see that q(x + 2y) = q(x), so q can be viewed as a Z 2 -valued quadratic form on H/K ≃ Z 2g 2 . Thus, q is an element of Quad(H/K) and we define A(η) as the Arf-invariant of q.
Here α, β are simple curves such that [α] and [β] project to a basis of Then two line fields η and θ are in the same M(Σ) orbit if and only if the following conditions are satisfied: Proof. (i) This follows immediately from the fact that G(Σ) is an H 1 (Σ)-torsor and the boundary curves ∂ i Σ generate the group H 1 (Σ). (ii) This is proved in the same way as Theorem 2.8 in [13]. (iii) We need to prove that if the invariants match then η and θ are in the same M(Σ)orbit. Note that σ(η) is determined by whether the quadratic function q η is trivial modulo 2 or not. By Lemma 1.2.4, it is enough to prove that the quadratic functions q η and q θ are in the same M(Σ)-orbit.
First, let us analyze the result of the action of a transvection T a (x) = x + (a · x)a on quadratic functions in Quad 4 . We have Assume now that q ∈ Quad 4 is such that q| K is surjective, i.e., the reduction of q| K modulo 2 is nonzero. Then we claim that any q ′ ∈ Quad 4 with q ′ | K = q| K lies in the M(Σ)-orbit of q. Indeed, we have q ′ − q = (a·?) for some a ∈ H. By surjectivity of q| K we can find k ∈ K such that q(k) = −1 − q(a), i.e., q(a + k) = −1. Then from (1.4) we get Next, let us consider q ∈ Quad 4 such that q| K takes values in 2Z 4 . Assume also that q mod 2 = 0. We claim that in this case the M(Σ)-orbit of q is determined by q| K . Note that q mod 2 is a homomorphism H → Z 2 trivial on K, so it is an element of Hom(H/K, Z 2 ). Since M(Σ) acts transitively on nonzero elements in Hom(H/K, Z 2 ), it is enough to prove that if q ′ ≡ q mod 2 and q ′ | K = q| K then q ′ and q are in the same M(Σ)-orbit. As before we deduce that q ′ − q = 2(a·?) for some a ∈ H. If q(a) ≡ 1 mod 2 then this immediately gives q ′ = qT 2 a . On the other hand, if q ′ (a) ≡ q(a) ≡ 0 mod 2 then for any element b with q(b) ≡ 1 mod 2 we have (1.5) 2. In the case g(Σ) = 1, let α, β be the standard non-separating curves in Σ. Then, it can be shown as in [13, Lemma 2.6] that We also note that in the case d = 1, w η (∂Σ) = −2, hence this invariant reduces to gcd(w η (α), w η (β)) considered in [1].
3. In the case σ(η) = 0, the line field η is induced by a non-vanishing vector field v. This induces a spin structure on the surface Σ (by considering its mod 2 reduction). The condition that w η (∂ i Σ) ∈ 2 + 4Z means that this spin structure extends to a spin structure on the compact surface obtained from Σ by capping off the boundaries with a disk. Now, it is a theorem of Atiyah [4] (see also [12]) that the action of the mapping class group on the spin structures on a compact Riemann surface has exactly 2 orbits distinguished by the Arf invariant.

Partially wrapped Fukaya categories
The partially wrapped Fukaya category W(Σ, Λ; η) (with coefficients in a field K) is associated to a graded surface (Σ, Λ; η), where Σ is a connected compact surface with non-empty boundary ∂Σ, Λ ⊂ ∂Σ is a collection of marked points called stops, and η is a line field on Σ. There is a combinatorial description of W(Σ, Λ; η) provided in [10]. A set of pairwise disjoint and non-isotopic Lagrangians {L i } in Σ\Λ generates the partially wrapped Fukaya category W(Σ, Λ; η) as a triangulated category if the complement of the Lagrangians is a union of disks D f each of which has exactly one stop on its boundary. Figure 1 illustrates how each D f may look like, where the blue arcs are in i L i while the black arcs lie in ∂Σ.
Furthermore, in this case, the associative K-algebra is formal, and it can be described by a graded gentle algebra (see Def. 3.1.1). The generators of this quiver can easily be described following the flow lines corresponding to rotation around the boundary components of Σ connecting the Lagrangians. Note that each boundary component of Σ is an oriented circle (where the boundary orientation is induced by the area form on Σ). Specifically, a flowline that goes from L j to L i gives a generator for hom(L i , L j ) (note the reversal of indices). The data of Λ enters by disallowing flows that pass through a marked point. The algebra structure is given by concatenation of flow lines. Given α i ∈ hom(L i , L i+1 ) for i = 1, . . . , n, we write for their product, read from right-to-left, and if non-zero, this expression corresponds to a flow from L n to L 1 . Finally, the line field η is used to grade the morphism spaces. A convenient way to determine the line field η is by describing its restrictions along each of the disks D f . Each such disk is as in Figure 1. Different disks are glued along the curves L i (the blue parts in their boundary). Changing a line field by homotopy, we can arrange that it is tangent to L i (as L i are contractible). Thus, every line field on Σ (up to homotopy) can be glued out of such line fields on the disks D f .
Note that if we have an embedded segment c ⊂ Σ and a line field η, which is transversal to c at the ends p 1 , p 2 of c, then we can define the winding number w η (c) (first, one can trivialize T Σ along c in such a way that the tangent line to c is constant, then count the number of times (with sign) η coincides with the tangent line to c along c. An equivalent definition is given in [10,Sec. 3.2]). Now a line field on a disk D f , tangent to {L i }, is determined (up to homotopy) by the integers θ i , for i = 1, . . . , m, given by its winding numbers along the boundary parts on ∂Σ (the black parts in Figure 1). By definition, these numbers are the degrees of the corresponding morphisms in the wrapped Fukaya category.
The numbers θ i can be chosen arbitrarily subject to the constraint This last constraint is the topological condition that needs to be satisfied in order for the line field to extend to the interior of the disk. (Note that the stops do not play a role in this discussion.) The gentle algebra A L• is always homologically smooth since so is W(Σ, Λ; η). In what follows, it will be convenient to consider A op L• as a quiver algebra KQ/I, so that flow lines from L i to L j correspond to arrows from the i th vertex to j th vertex. Note that the collection {L i } generates the partially wrapped Fukaya category W(Σ, Λ; η). Therefore, we have an equivalence , where the category on the left denotes the bounded derived category of perfect (left) dgmodules over A op L• .

Gentle algebras and Fukaya categories
3.1. Graded gentle algebras and AAG-invariants. A quiver is a tuple Q = (Q 0 , Q 1 , s, t) where Q 0 is the set of vertices, Q 1 is the set of arrows, s, t : Q 1 → Q 0 is the functions that determine the source and target of the arrows. We always assume Q to be finite. A path in Q is a sequence of arrows α n . . . α 2 α 1 such that s(α i+1 ) = t(α i ) for i = 1, . . . , (n − 1). A cycle in Q is a path of length ≥ 1 in which the beginning and the end vertices coincide but otherwise the vertices are distinct. For K a field, let KQ be the path algebra, with paths in Q as a basis and multiplication induced by concatenation. Note that the source s and target t maps have obvious extensions to paths in Q.
Definition 3.1.1. A gentle algebra 2 A = KQ/I is given by a quiver Q with relations I such that (1) Each vertex has at most two incoming and at most two outgoing edges.
(2) The ideal I is generated by composable paths of length 2.
(3) For each arrow α, there is at most one arrow β such that αβ ∈ I and there is at most one arrow β such that βα ∈ I. (4) For each arrow α, there is at most one arrow β such that αβ / ∈ I and there is at most one arrow β such that βα / ∈ I. In addition, we always assume Q to be connected.
We will consider Z-graded gentle algebras, i.e., every arrow in Q should have a degree assigned to it. For a Z-graded gentle algebra A we denote by D(A) the derived category of perfect dg-modules over A, where A is viewed as a dg-algebra with its natural grading and zero differential.   . But the latter space can be computed using the standard Koszul complex, and the presence of forbidden cycles would mean that for some S the space Ext * A (S, S) is infinite-dimensional.
We will use the following notions from [5]. such that all (α i ) are distinct and for all i = 1, . . . , (n − 2), α i+1 α i ∈ I. It is a forbidden thread if for all β ∈ Q 1 neither βα n . . . α 2 α 1 nor α n . . . α 2 α 1 β is a forbidden path. In addition, if v ∈ Q 0 with #{α ∈ Q 1 |s(α) = v} ≤ 1, #{α ∈ Q 1 |t(α) = v} ≤ 1, then we consider the idempotent e v as a (trivial) forbidden thread in the following cases: • either there are no α with s(α) = v or there are no α with t(α) = v; • we have β, γ ∈ Q 1 with s(γ) = v = t(β) and γβ ∈ I. The grading of a forbidden thread is defined by such that all (α i ) are distinct and for all i = 1, . . . , (n − 1), α i+1 α i / ∈ I, and it is a permitted thread if for all β ∈ Q 1 neither βα n . . . α 2 α 1 nor α n . . . α 2 α 1 β is a permitted path. In addition, if v ∈ Q 0 with #{α ∈ Q 1 |s(α) = v} ≤ 1, #{α ∈ Q 1 |t(α) = v} ≤ 1, then we consider the idempotent e v as a (trivial) permitted thread in the following cases: • either there are no α with s(α) = v or there are no α with t(α) = v; • we have β, γ ∈ Q 1 with s(γ) = v = t(β) and γβ / ∈ I. The grading of a permitted thread is defined by Remark 3.1.6. Inclusion of the idempotents as forbidden and permitted threads ensures that every vertex appears in exactly two forbidden threads/cycles and exactly two permitted threads/cycles. such that s(f i ) = s(p i ) for i ∈ Z/n, and t(p i ) = t(f i+1 ) for i ∈ Z/n with the following condition: The winding number associated to a combinatorial boundary component b of type I is defined to be We also denote the number n of forbidden threads in b as n(b).
A combinatorial boundary component of type II (that can appear only if A is not proper) is simply a permitted cycle pc = α m . . . α 1 . The winding number associated to such a cycle is A combinatorial boundary component of type II' (that can appear only if A is not homologically smooth) is simply a forbidden cycle The winding number associated to such a cycle is For combinatorial boundary components of types II and II' we set n(b) = 0. Proof. This follows directly from the description of the AAG-invariants in [5,Sec. 3]. Note that the pair (0, m) in Step (3)

3.2.
Relation to Fukaya categories. The definition of the combinatorial boundary component for a gentle algebra is motivated by the following proposition: Proof. Figure 2 shows an example of the way the surface Σ looks around a boundary component B. Assume first that there is at least one stop on B. Let q 1 (1), . . . , q 1 (k 1 ), q 2 (1), . . . , q 2 (k 2 ), . . . , q n (1), . . . , q n (k n ) be the endpoints of the Lagrangians ending on B, ordered compatibly with the orientation of B. Here we assume that there are no stops between q i (j) and q i (j + 1) and there is exactly one stop s i between q i (k i ) and q i+1 (1), for i ∈ Z/n. Then for every i ∈ Z/n we have a permitted thread p i = β i (k i − 1) . . . β i (1), where β i (j) is the generator of A corresponding to the flow on B from q i (j) to q i (j + 1). On the other hand, each stop s i lies on a unique disk D, and by looking at the pieces of ∂D formed by other boundary components of Σ, we obtain a forbidden thread f i = α m i . . . α 1 starting at the Lagrangian corresponding to q i (1) and ending at the one corresponding to q i−1 (k i−1 ). Thus, we get a combinatorial boundary component of type I, b = p n f n . . . p 1 f 1 .
The winding number of η along the arc passing through the stop, oriented in the opposite direction to the boundary direction, is determined using the constraint (2.1) to be On the other hand, the winding number of η along the arc corresponding to the permitted thread p is simply |p|. Thus, we get the equality w η (B) = w(b).
In the case of a boundary component B ⊂ ∂Σ with no stops, the sequence of flows between the corresponding ends of Lagrangians on B gives a permitted cycle, i.e., a combinatorial boundary component of type II. Again, the winding numbers match.
It is easy to see that in this way we get a bijection between the boundary components B and the combinatorial boundary components of A. Figure 2. The boundary component is given by the cyclic sequence p 2 f 2 p 1 f 1 where f 1 = α 3 α 2 α 1 , p 1 = β 2 β 1 , f 2 = γ 2 γ 1 and p 2 = δ 1 . Note that if instead of f 1 , we considered the forbidden threadf 1 =β 2 β 1 , the condition (⋆) is violated.
Theorem 3.2.2. (i) Given a homologically smooth graded gentle algebra A over a field K (with |Q 1 | > 0), there exists a graded (connected) surface with stops (Σ A , Λ A , η A ), with non-empty boundary and a derived equivalence Furthermore, the AAG-invariants of A are given by the collection of pairs Proof. (i) We define a ribbon graph R A whose vertices are in bijection with the collection of forbidden threads in Q, and whose edges are in bijection with vertices of Q.
Recall that there are precisely two forbidden threads that pass through a vertex of Q. The corresponding edge on R A is defined to connect the two forbidden threads. Furthermore, we can equip the set of edges in R A incident to a vertex f with a total ordering. Namely, the set of edges incident to a vertex f of R is in bijection with the set of vertices of Q which appear in the forbidden thread f . Hence, we can use the order in which these vertices appear in the forbidden thread. This linear order of edges incident to vertices of R A induces a ribbon structure on R A , i.e., a cyclic order of edges incident to each vertex. Therefore, we can consider the associated thickened surface Σ A such that R A is embedded as a deformation retract of Σ A . (A graph with the additional data of a linear ordering on the edges incident to a vertex is called a ciliated fat graph [9].) Thus, to construct Σ A we replace each vertex of R A with a 2-disk D 2 and each edge with a strip, a thin oriented rectangle [−ǫ, ǫ] × [0, 1], where the rectangles are attached to the boundary of the disks according to the given cyclic orders at the vertices. On the boundary of each disk associated to the vertex of R A we also mark a point, called a stop as follows. If the linear order on edges incident to this vertex is given by e 1 < e 2 < . . . < e k , the stop e 0 appears in the circular order such that e k < e 0 < e 1 . We define Λ A by taking the union of all such points. In particular, the cardinality of Λ A , is equal to the number of forbidden threads in A.
We claim that the ribbon graph R A and hence the associated surface Σ A is connected. Indeed, for every vertex v of Q let e(v) be the corresponding edge in R A , viewed as a subgraph in R A . Since Q is connected, it is enough to check that if v and v ′ are connected by an edge α in Q then e(v) and e(v ′ ) intersect in R A . Indeed, let f be a forbidden thread containing α (it always exists). Then f is a vertex of both e(v) and e(v ′ ). This proves our claim that R A is connected.
Dual to the edges of R A we obtain a disjoint collection of non-compact arcs L v indexed by vertices of Q. Thus, Σ A is a surface with non-empty oriented boundary, Λ A is a set of marked points in its boundary, and {L v : v ∈ Q 0 } is a collection pair-wise disjoint and non-isotopic Lagrangian arcs in Σ A \ Λ A . Furthermore, the complement is a union of disks D f indexed by forbidden threads f in Q, with exactly one marked point on its boundary (see Examples 3.2.7, 3.2.8 below). In particular, the collection {L v } gives a generating set.
By construction, there is a bijection between arrows in the quiver Q and the generators of the endomorphism algebra A L := v,w hom(L v , L w ) since each edge α in Q is in exactly one forbidden thread f , and the corresponding D f has a flow associated to α. Furthermore, two flows α 1 : L v 2 → L v 1 and α 2 : L v 3 → L v 2 can be composed in A L if and only if α i is in a forbidden thread f i , for i = 1, 2, such that the disks D f 1 and D f 2 are glued along the edge corresponding to v 2 . But this means that the corresponding elements of A satisfy α 2 α 1 / ∈ I, as otherwise condition (3) of Definition 3.1.1 would be violated. This imples that A is naturally identified with A op L as an ungraded algebra. We define the line field η A on Σ A as follows. We require that the line field is tangent to each L v . Then it suffices to describe its restrictions to the disks D f . Each D f is a 2m-gon as in Figure 1. The homotopy class of a line field on D f is determined by the winding numbers θ i along the boundary arcs of D f , α i , for i = 1, . . . , (m − 1), avoiding the unique stop (black in Figure 1) between the Lagrangians (blue in Figure 1). Indeed, the remaining winding number θ m along the boundary arc that passes through the stop is determined by the condition m i=1 θ i = m − 2, and we can define η A | D f as the unique line field with these winding numbers. Now we set θ i , for i = 1, . . . , m − 1, to be the degree of the generator of A corresponding to α i .
With this definition A and A op L are identified as graded algebras. Since we also know that the collection Finally, the last statement follows from Proposition 3. Combining this with the previous formula we get v(R A ) = 2|Q 0 | − |Q 1 |, so we deduce that χ(R A ) = χ(Q).
Using formula (1.5) we derive the following property of the AAG-invariants.
..,d be the AAG-invariants of a homologically smooth graded gentle algebra A. Then where g ≥ 0 is the genus of the corresponding surface Σ A . As a particular case of the last Corollary, we can describe some cases when already looking at the AAG-invariants gives the derived equivalence. We can use Koszul duality to convert our results about homologically smooth graded gentle algebras into those about finite-dimensional gentle algebras. Namely let A be a finite-dimensional gentle algebra with grading in degree 0. Let A ! be the Koszul dual gentle algebra (with respect to the generators given by the edges). We equip A ! with the grading for which all edges have degree 1 (i.e., path-length grading). Then the result of Keller in [14, Sec. 10.5] ("exterior" case) gives an equivalence where D f (A) is the bounded derived category of finite-dimensional A-modules (and D(A ! ) is the perfect derived category of A ! viewed as a dg-algebra, as before).
Furthermore, it is easy to check that the AAG-invariants of A and A ! are the same. Thus, Corollary 3.2.5 leads to the following result.    The associated ribbon graph is given in Figure 4, where the cyclic order at vertices are given by counter-clockwise rotation. Figure 5 depicts the corresponding surface, together with the dual arcs L 1 , L 2 , L 3 , L 4 .  The corresponding surface is given in Figure 7.
Remark 3.2.9. An optimist's conjecture would be that conversely if A and B are homologically smooth graded gentle algebras which are derived equivalent, then there exists a homeomorphism φ : Σ A → Σ B inducing a bijection Λ A → Λ B and such that φ * (η A ) is homotopic to η B . Note that to prove this, one needs to show that the topological type of (Σ A , Λ A ; η A ) is a derived invariant of A. This is encoded by the numerical invariants of η A introduced in Theorem 1.2.5 (from which one can recover the topological type of the surface), together with the numbers of marked points on each boundary component.
Remark 3.2.10. In Theorem 3.2.2, it is possible to drop the assumption that A is smooth. Assume for simplicity that A is proper. In this case, the surface Σ would be glued together from the disks D f associated to forbidden threads as before, and also disks D c with an interior hole, associated with forbidden cycles. In other words, D c is an annulus whose inner boundary component has no marked points and is not glued to anything, while its outer boundary component is connected by strips, corresponding to the vertices in c, to other disks (this boundary component of D c still has no stops). In the presence of unmarked boundary components, there is a dual construction to the construction of partially wrapped Fukaya categories, W(Σ, Λ; η), namely, the infinitesimal wrapped Fukaya categories F (Σ, Λ; η), studied in [16]. Its objects are graded Lagrangians which do not end on the unmarked components of the boundary. Thus, for non-smooth proper gentle algebras, a version of Theorem 3.2.2 should state the equivalence However, we have not checked that the collection of Lagrangians {L v } given by the construction in Theorem 3.2.2 (and modified as above) generates F (Σ A , Λ A ; η A ).

Derived equivalences between stacky curves
4.1. Chains. Recall that in [16] we considered stacky curves C(r 0 , . . . , r n ; k 1 , . . . , k n−1 ) obtained by gluing weighted projective lines B(r 0 , r 1 ), B(r 1 , r 2 ), . . . , B(r n−1 , r n ) into a chain, where k i ∈ (Z/r i ) * are used to determine the stacky structure of the nodes in this chain. We showed in [16,Thm. B] that the bounded derived category of coherent sheaves on such a stacky curve is equivalent to the partially wrapped Fukaya category of a surface obtained by a certain linear gluing of the annuli. Namely, let A(r, r ′ ) denote the annulus with ordered boundary components that has r marked points p − 1 , . . . , p − r on the first component and r ′ marked points p + 1 , . . . , p + r ′ on the second boundary component (the points are ordered cyclically compatibly with the orientation of the boundary). Given a collection of permutations σ i ∈ S r i , i = 1, . . . , n − 1, we consider the surface Σ lin (r 0 , . . . , r n ; σ 1 , . . . , σ n−1 ) obtained by gluing the annuli A(r 0 , r 1 ), A(r 1 , r 2 ), . . . , A(r n−1 , r n ) in the following way. For each i = 1, . . . , n − 1, j = 1, . . . , r i , we glue a small segment of the boundary around the marked point p + j in A(r i−1 , r i ) with a small segment of the boundary around the point p − σ i (j) in A(r i , r i+1 ) by attaching a strip, as in Figure 8. Note the resulting surface has two special boundary components equipped with r 0 and r n marked points, respectively (there are no other marked points on the other boundary components). The boundary components that arise because of the gluing are from n − 1 groups, so that components in the ith group are in bijection with cycles in the cycle decomposition of the commutator [σ i , τ ] ∈ S r i , where τ is the cyclic permutation j → j −1.
We equip each annulus with the standard line field that has zero winding numbers on both boundary components. These line fields glue into a line field η on the surface Σ lin (r 0 , . . . , r n ; σ 1 , . . . , σ n−1 ). More precisely, we take η that corresponds to the horizontal direction in Figure 8.
It is easy to see that the boundary invariants of η are given as follows. For the two special boundary components the winding numbers are equal to zero, so the corresponding invariant is 2. For a boundary component corresponding to a k-cycle in the cycle decomposition of [σ i , τ ] the winding number is −2k, so the invariant is 2 − 2k.
Note that for each i the commutator [σ i , τ ] is given by x → x + k i + 1 mod(r i ), so its cycle decomposition has p i = gcd(k i + 1, r i ) cycles of length r i /p i . Thus, the corresponding boundary invariants are 2, 2 (for the special boundary components) and for each i = 1, . . . , n − 1, the number 2 − 2r i /p i repeated p i times.

4.1.2.
Trade-off for balanced nodes. More generally, let I ⊂ [1, n − 1] be the subset of indices i such that k i = −1, and let r I = i∈I r i . Then, we have a homeomorphism Σ lin (r 0 , . . . , r n ; k 1 , . . . , k n−1 ) ≃ Σ lin (r 0 , r I , (r i ) i ∈I , r n ; −1, (k i ) i ∈I ) preserving the line fields. This can be either derived from Theorem 1.2.5 as above or constructed directly. As before, this leads to a derived equivalence of the corresponding stacky curves. 4.1.4. Genus ≥ 2. Because of the two special components with the boundary invariant 2, the Arf-invariant never appears. Thus, two surfaces Σ lin (r • ; k • ) and Σ lin (r ′ • ; k ′ • ) of genus g ≥ 2 are homeomorphic as surfaces with a line field, whenever we have r 0 = r ′ 0 , r n = r ′ n and the sequence ((r 1 /p 1 ) p 1 , . . . , (r n−1 /p n−1 ) p n−1 ) differs from the corresponding sequence for (r ′ • , k ′ • ) by a permutation (here (r i /p i ) p i means the number r i /p i repeated p i times). For example, we can specialize to the case n = 2, r 0 = r 2 = 0, r 1 = r. Note that the corresponding stacky curve C(0, r, 0; k) is the global quotient of the affine coordinate cross xy = 0 by the µ r -action ζ · (x, y) = (ζ k x, ζy). We obtain that for k, k ′ ∈ (Z/r) * , such that gcd(k + 1, r) = gcd(k ′ + 1, r), there exists an equivalence D b Coh(C(0, r, 0; k)) ≃ D b Coh(C(0, r, 0; k ′ )).

4.2.
Rings. Now let us consider another class of stacky curves considered in [16], denoted by R(r 1 , . . . , r n ; k 1 , . . . , k n ). They are defined by gluing the weighted projective lines B(r 1 , r 2 ), B(r 2 , r 3 ), . . . , B(r n , r 1 ) into a ring, where as before k i ∈ (Z/r i ) * are used to determine the stacky structure of the nodes.
On the symplectic side we can modify our definition of the surfaces Σ lin (r 0 , . . . , r n ; σ 1 , . . . , σ n−1 ) as follows. Starting with the annuli A(r 1 , r 2 ), A(r 2 , r 3 ), . . . , A(r n , r 1 ) we can glue them circularly using permutations σ 1 , . . . , σ n . Thus, the corresponding surface could be represented similarly to Figure 8 but with the right and left ends identified (so that the corresponding boundary components disappear). We denote the resulting surface by Σ cir (r 1 , . . . , r n ; σ 1 , . . . , σ n ). Similarly to the case of a linear gluing it is equipped with a natural line field η that corresponds to the horizontal direction when the surface is depicted as on Figure 8.

4.3.
Case of irreducible stacky curve. This is the case n = 1. Let r = r 1 . Let us consider the case of k ∈ Z r such that gcd(k + 1, r) = 1 (note that this is possible only when r is odd). Then the surface Σ cir (r; k) has genus g = (r + 1)/2 and one boundary component with the winding number −2r, i.e., the invariant 2 − 2r. Note that 2 − 2r is divisible by 4, so to determine the orbit of the line field under the mapping class group we have to calculate the corresponding Arf-invariant. This invariant will depend on k.
First, let us consider the case k = 1. Let us look at the simple curves α i , i = 1, . . . , r − 1, depicted on Figure 9. In addition, we have two simple curves α and β corresponding to a vertical and horizontal line on Figure 9.
Hence, the Arf-invariant is given in this case by 1 + (r−1)/2 Next, assume in addition that r is not divisible by 3 and consider the case k = 2 (then gcd(k + 1, r) = 1). Then we claim that the classes [α i ], together with (α, β) still project to a basis of H 1 (Σ, Z 2 ), however, their intersection numbers are now given by where i < j (we still have α i · α = α i · β = 0 and α · β = 1. It is easy to see that by renumbering the classes (α i ) as follows: we get the quadratic form of Example 4.5.2. Hence, the Arf invariant is given by 1 + (r − 1)/2 mod 2. Thus, we deduce the following derived equivalence. Proposition 4.3.1. Assume that r ≥ 7 is not divisible by 3 and r ≡ ±1 mod (8). Then the stacky curves C ring (r; 1) and C ring (r; 2) are derived equivalent.

4.4.
Merging two nodes into one. Let us fix an odd r. Then the surfaces Σ cir (r, r; 1, 1) and Σ cir (2r; 1) are homeomorphic: they both have genus r and 2 boundary components. One can ask whether they are homeomorphic as surfaces with line fields. The boundary invariant on each component is equal to 2 − 2r, so we need to look at the Arf-invariant.
Proposition 4.4.1. The Arf-invariant of the form associated to the line field on Σ cir (r, r; 1, 1) is equal to 1. The Arf-invariant of the form associated to the line field on Σ cir (2r; 1) is equal to (r + 1)/2 mod 2. Hence, if r ≡ 1 mod(4) then the stacky curves C ring (r, r; 1, 1) and C ring (2r; 1) are derived equivalent.
Proof. In the case of the surface Σ cir (r, r; 1, 1) we have two collections of simple curves (α 1 , . . . , α r−1 ), (α ′ 1 , . . . , α ′ r−1 ) associated with each of the two segments where the gluing happens. In addition we have two standard curves α and β as before. So the corresponding quadratic space will be a direct sum of two copies of V r−1 together with the 2-dimensional space spanned by (α, β). Thus, the Arf-invariant is equal to 1.

Computation of Arf-invariants.
Example 4.5.1. Let V n be a Z 2 -vector space with the basis α 1 , . . . , α n , and the even pairing given by α i · α j = 1 for i = j. Let q be the unique quadratic form in Quad(V n ) such that q(α i ) = 0 for all i. First, assume that n is even. Then we claim that this pairing is nondegenerate and the Arf-invariant of q is given by A(q) = n/2 2 mod 2.
Indeed, it is enough to prove that the Gauss sum is equal to ±2 n/2 . Then the sign will determine the Arf-invariant. It is easy to see that q(x) = (−1) ( k 2 ) , where k is the number of nonzero coordinates of x. Thus, we have G(q) = n k=0 n k (−1) ( k 2 ) .
Now, let us assume that n is odd. Then the vector v 0 = n k=1 α k lies in the kernel of the pairing and q(v 0 ) = n 2 mod 2. Thus, if we assume in addition that n ≡ 1 mod 4 then we have q(v 0 ) = 0 and so the form q descends to a well-defined quadratic form q on V n−1 = V n / v 0 . We claim that its Arf-invariant is A(q) = n − 1 4 mod 2.
We have Example 4.5.2. Now let V be a Z 2 -vector space with the basis α 1 , . . . , α n , where n ≥ 4 is even, the even pairing given by the rule α i · α j = 1, i < j < i + n/2, 0, j ≥ i + n/2, and the quadratic form q in Quad(V ) such that q(α i ) = 0 for all i. Assume also that n ≡ 2 mod(3). Then we claim that the pairing is nondegenerate and A(q) = n/2 mod 2.
We will prove this by relating (V, q) with another quadratic form. For every k ≥ 0, such that k ≡ 2 mod(3), let us consider a Z 2 -vector space W k with the basis β 1 , γ 1 , . . . , β k , γ k , the even pairing given by the rule β i · β j = 1 for i = j; γ i · γ j = 1 for i = j; β i · γ j = 1 for i ≤ j; β i · γ j = 0 for i > j, and the quadratic form q k in Quad(W k ) such that q k (β i ) = q(γ i ) = 1 for all i.
First, we will prove that A(q) = A(q n/2−2 ) and then we will prove that A(q k ) = k mod 2 (4.1) To relate (V, q) with (W n/2−2 , q n/2−2 ) let us consider the 2-dimensional isotropic subspace I ⊂ V spanned by α 1 and α n . Then we have q| I ≡ 0, so the Arf-invariant of q is equal to that of the induced quadratic form on I ⊥ /I. Now setting γ i = α 2 + α 2+i , β i = α n/2+1 + α n/2+1+i , for i = 1, . . . , n/2 − 2, we get an identification of I ⊥ /I with W n/2−2 , compatible with the quadratic forms.
To prove (4.1) we use induction on k. It is easy to check that A(q 1 ) = 1 (and A(q 0 ) = 0 for trivial reasons), so it is enough to establish the formula A(q k ) = A(q k−3 ) + 1.
To this end we consider the 2-dimensional isotropic subspace J ⊂ W k spanned by β k + γ 1 and β 1 + β k + γ k . We have q k | J = 0, and our formula follows from the identification where the standard basis of W k−3 corresponds to the elements (β 2 + β 2+i mod J, γ 2 + γ 2+i mod J) 1≤i≤k−3 while a copy of W 1 spanned by β k mod J and γ k mod J.

King's College London
University of Oregon, National Research University Higher School of Economics, and Korea Institute for Advanced Study