Stability conditions for affine type A

We construct maximal green sequences of maximal length for any affine quiver of type $A$. We determine which sets of modules (equivalently $c$-vectors) can occur in such sequences and, among these, which are given by a linear stability condition (also called a central charge). There is always at least one such maximal set which is linear. The proofs use representation theory and three kinds of diagrams shown in Figure 1. Background material is reviewed with details presented in two separate papers [arxiv:1706.06986] and [arXiv:1706.06503].


Introduction
This paper addresses the question of linearity of maximal green sequences of maximal length. This question originates in a conjecture by Reineke [R] in which he asks for a linear stability condition on a Dynkin quiver which makes all indecomposable modules stable. Isomorphism classes of such modules are in bijection with the positive roots of the underlying root system. Reineke showed that the sequence of stable positive roots corresponding to the stable modules of a linear stability condition gives a quantum dilogarithm identity and he wanted that identity to have one term for every positive root. Yu Qiu [Q] has shown that, for every Dynkin quiver, there exists an orientation of the quiver and a linear stability condition given by a central charge which makes all indecomposable modules stable. This had already been done in type A n with straight orientation by Reineke [R]. So, [Q] dealt with quivers of other Dynkin types.
It is very easy to see that there are nonlinear stability conditions (called "maximal green sequences") which make all positive roots stable, namely take all indecomposable modules going, depending on sign convention, either from left to right (Reineke's sign convention) or from right to left (our sign convention) in the Auslander-Reiten quiver of the path algebra. More precisely, we order the indecomposable modules in such a way that, for every irreducible map A → B, B comes before A. Thus the question is mainly about the linearity of the stability condition.
In this paper we prove Reineke's original conjecture in type A n with any orientation and we give a complete resolution to the extension of this question to quivers of type A n−1 . Since there are infinitely many positive roots in that case, the corresponding problem is to find maximal green sequences of maximal finite length and to determine which are linear. Our results are the following.
Recall that A a,b , for positive integers a, b, denotes a cyclic quiver with a arrow going clockwise and b arrows going counterclockwise. For example, there are, up to isomorphism, two quivers of type A 3,2 which we denote: (See (3.2) for the sign notation.) • • ✿ ✿ ✿ ✿ y y r r r r r r r rÃ For b = 1 this is already known [14]. Theorem M1 is shown in two steps: In Theorem 6.10 we construct maximal green sequences of this length and in Theorem 5.11 we show that there are no maximal green sequences of greater length. Although maximal green sequences were originally defined combinatorially [6], we use the language of representation theory which we review in 1.1. We take a fixed field K and, for any acyclic quiver Q, we take the path algebra Λ = KQ of Q. We use the "wall-crossing" definition of a maximal green sequence from [7], [4] which we review in Section 3. A maximal green sequence for a finite dimensional algebra Λ is a finite sequence of indecomposable modules M 1 , · · · , M m for which there exists a "green path" γ going through the walls D(M 1 ), · · · , D(M m ) in that order and no other walls. The set of modules M i in this sequence are called the stable modules of the sequence.
Maximal green sequences for Q are in bijection with those for KQ given by the wall crossing definition in Section 3 and the dimension vectors of the stable module M i are the c-vectors of the corresponding combinatorially defined maximal green sequence. (See [7]).
For quivers of Dynkin type, the longest green path passes through all of the walls and all indecomposable modules are stable. For quivers of type A a,b there are infinitely many indecomposable modules and each maximal green sequence makes only finitely many of them stable. In this paper we determine all possible sets of stable modules of the maximum size given in Theorem M1 above. This is summarized by the following two theorems.
Theorem M2. For every quiver of type A a,b with (a, b) = (2, 2) there are exactly ab possible sets of stable modules for the maximal green sequences of length a+b 2 + ab.
These sets are denoted S kℓ where ε k = +, ε ℓ = − and 0 < k < ℓ < k + n ≤ 2n. See Returning to the linearity question, we prove first that Reineke's conjecture holds for quivers of type A n with any orientation.
Theorem L1 (Corollary 1.9). For a quiver of type A n with any orientation, there exists a standard linear stability condition making all indecomposable modules stable.
For every linear stability condition Z (also called a central charge 1.2) there is a linear green path γ Z : R → R n given by γ Z (t) = tb − a. If there are only finitely many stable modules, these modules form a maximal green sequence and we call the stability condition a finite linear stability condition.
In the affine case A a,b , we show that the following analogue of Reineke's conjecture hold.
Theorem L2. For any quiver of type A a,b there exists a finite linear stability condition for which the number of stable modules is a+b 2 + ab.
Theorem 6.6 describes which of the sets S kℓ are linear, i.e., realized by linear stability conditions. For example, S kℓ is linear when |k − ℓ| ≤ 2. Theorem 6.6 implies the following.
Theorem L3. For the quiver A ε a,b , if either a or b is ≤ 2, every possible set of stable modules of maximal size is linear. Otherwise (when a, b ≥ 3) there is at least one orientation of the quiver (choice of ε) for which one of the sets of stable modules is not linear.
The smallest nonlinear example is A +++−−− 3,3 where 1 of the ab = 9 sets of stable modules is nonlinear. And, in fact, this example is the cause of all nonlinearity. For every nonlinear example, + + + − −− will be a subsequence of the sign pattern ε up to cyclic order.
By the Deletion Lemma 5.15, maximal green sequences for A ε a,b restrict to maximal green sequences for A ε ′ a ′ ,b ′ for many subsequence ε ′ of ε (in particular, a ′ ≤ a, b ′ ≤ b). By Remark 5.17, this includes the exceptional case where b = 0. Although " A n,0 " has infinite representation type (being an oriented cycle), only modules of length ≤ n occur in a maximal green sequence since longer modules are not "bricks" (See Proposition 3.4). In this paper we also exclude modules of length n. Equivalently, we mod out rad n−1 , and we have one of the well-known cluster-tilted algebras of type D n for n ≥ 4 considered in [8]. In Theorem 5.18 and Corollary 5.19 we show the following.
Corollary L4. For Λ n = KQ n /rad n−1 the path algebra of the oriented n-cycle Q n modulo the relation rad n−1 = 0, the longest maximal green sequence has length n 2 + n − 1, there are n different sets of stable modules of this size and all of them are linear. Figure 1. Three diagrams for the same linear stability condition on a quiver of type A 3 with stable modules S 3 , P 3 , S 1 , P 2 , S 2 . Each diagrams show I 2 to be unstable: (1) γ does not pass through the wall D(I 2 ) (in red), (2) the chord I 2 is outside the polygon and (3) the vertex I 2 has a red line above it.
In order to prove these theorem we use the following three types of diagrams where (2) and (3) are always planar. (Figure 1 gives an example.) (1) Wall crossing diagrams. M is stable when a green path γ : R → R n passes through the interior of the wall D(M ). The stability condition given by γ is linear when γ is a straight line. (2) Chords in the "stability polygon". Certain chords represent stable modules in a linear stability condition. (3) Wire diagrams. Stable modules are indicated by certain crossings of wires in the plane. When the wires are straight lines, this is a linear stability condition. In Section 3 we use wall crossing diagrams to prove basic theorems about maximal green sequences, linear and nonlinear. Wire diagrams, introduced in Sections 2, 4, are used in Section 5 to show that a+b 2 + ab is an upper bound for the maximum length of a maximal green sequence on any quiver of typeÃ a,b . Finally, in Section 6, chord diagrams, introduced in Section 1, are used to realize this upper bound.
At the end of the paper (Sec 7) we give a summary of notation, definitions and constructions and a statement of the precise correspondence between there three kinds of diagrams.

Chord diagrams for A n
In this section we will review the representation theory of quivers, give the precise statement of Reineke's conjecture and a proof of the conjecture in type A n using chord diagrams.
1.1. Representations of quivers. Suppose that Q is a quiver, i.e., a finite oriented graph, with vertex set Q 0 = {1, 2, · · · , n}, arrow set Q 1 and no oriented cycles. An important case is when Q is a linear quiver, in which case we say it has type A n . For example, There are 2 n−1 possible orientations for the arrows in a quiver of type A n . We specify the orientation with a sign function which we define to be any mapping ε : [0, n] = {0, 1, 2, · · · , n} → {−, 0, +}, written ε(i) = ε i , so that ε i = 0 iff i = 0 or n. In the corresponding linear quiver, denoted A ε n , the ith arrow points left i ← i + 1 when ε i = + and right i → i + 1 when ε i = −. Thus, the example above is A −−+ 4 where we drop the values v 0 = v n = 0 from the notation. A representation M of a quiver Q over a field K is defined to be a sequence of finite dimensional vector spaces M i , i = 1, · · · , n and linear maps M a : M i → M j for every arrow The dimension of M is the dot product Recall that the positive roots of the Dynkin diagram A n are the integer vectors β ij := e i+1 + e i+2 + · · · + e j = (0, · · · , 0, 1, 1, · · · , 1, 0, · · · , 0) with 1s in positions i + 1, i + 2, · · · , j for any 0 ≤ i < j ≤ n. These are the dimension vectors of the indecomposable representations of A ε n for any ε. A representation of an acyclic quiver Q is equivalent to a finitely generated modules over the path algebra Λ = KQ. See, e.g., [1]. A representation is indecomposable if it is indecomposable as a Λ-module. We say N is a subrepresentation or submodule of M if, considered as Λ-modules, N is a submodule of the module M . Equivalently, The following proposition is an easy exercise. Proposition 1.1. As representations of the quiver A ε n , M ij is a subrepresentation of M pq if and only if the following three conditions are satisfied. ( Linear stability conditions. We consider the dimension vectors of Λ-modules to be elements of K 0 Λ = Z n . A central charge on Λ = KQ is defined to be an additive mapping For a Λ-module M , the slope of M is then defined to be In the standard case, b · dim M = dim K M . The slope is undefined for M = 0.
Because of this definition we often refer to Z as a linear stability condition. The problem is to determine the maximum finite number of stable modules given by a linear stability condition. It is an easy exercise to show that any Z-stable module is indecomposable.
Reineke's original conjecture states: For Q a Dynkin quiver, there exist a standard linear stability condition making all indecomposable modules stable.
Lutz Hille has claimed, privately to the second author, that this conjecture is not true in type E 6 . Yu Qiu has shown [15] that this conjecture holds for at least one orientation of each Dynkin diagram if we drop the restriction that the linear stability condition should be standard. We will prove the original conjecture for A ε n for any sign function ε using chord diagrams.
1.3. Chord diagrams for type A n . For a central charge Z : K 0 (KA ε n ) → C given by Z(x) = a·x+ib·x, we will construct a "stability polygon" C(Z) ⊆ R 2 which will visually display which roots β ij are stable. This "polygon" might be degenerate, i.e., one-dimensional.
The vertices of the stability polygon (also called dual vertices of the quiver) are given by p i = (x i , y i ) := (b 1 + · · · + b i , a 1 + · · · + a i ) for i = 0, · · · , n. In particular p 0 = (0, 0). For standard Z, x i = i for all i. The sign of p i is ε i and we sometimes write p + k (or p − ℓ ) to mean p k which has sign ε k = + (or: p ℓ with ε ℓ = −). We say that the vertex p k is positive, resp negative, if ε k = +, resp ε k = −. Nonnegative means either positive of equal to p 0 or p n which have sign 0. Nonpositive is similarly defined.
Lemma 1.4. The slope of the line segment V ij := p i p j is equal to the slope of M ij : The line segments V ij will be called chords of the stability polygon C(Z) defined below.
In the sequel we will use the words "above" and "below " to refer to relative position in the plane. Thus, a point (x 0 , y 0 ) is above, resp. below, a subset S ⊆ R 2 if S contains a point (x 0 , z) with y 0 > z, resp. y 0 < z. We use "higher " and "lower " when referring only to the difference in the y-coordinates. (1) For all i < k < j with ε k = +, the point p + k lies on or above the chord V ij .
(2) For all i < ℓ < j with ε ℓ = −, the point p − ℓ lies on or below the chord V ij . M ij is Z-stable if and only if it is Z-semistable and p i , p j are the only vertices on V ij .
Proof. We discuss only the stable case. The semistable case is similar.
(⇒) Suppose M ij is Z-stable. Since M ik and M ℓj are submodules of M ij for ε k = +, ε ℓ = − the slopes of the chords V ik , V ℓj must be greater than the slope of V ij . This holds if and only if p k is above V ij and p ℓ is below V ij . Thus (1) and (2) hold.
(⇐) Given (1) and (2), we have seen that submodules of M ij of the form M ik , M ℓj will have slope greater than the slope of M ij . The remaining indecomposable submodules are M ℓk . But V ℓk has slope greater than that of V ij since it starts at the point p − ℓ below V ij and ends at p Figure 2. The chords V ik , V ℓk and V ℓj have slope greater than that of V ij if p + k is above and p − ℓ is below the chord V ij .
We will reformulate this theorem in terms of a polygon C(Z) whose vertices are the points p i , i = 0, · · · , n. The main property of C(Z) will be that M ij is Z-semistable if and only if V ij ⊆ C(Z). We define the stability polygon C(Z) to be the intersection C(Z) = C + (Z)∩C − (Z) where C + (Z), C − (Z) are defined below. Figure 3 gives an example.
Let P + = {p k : ε k ≥ 0}, be the set of all nonnegative vertices. Number the elements of P + as p k i where 0 = k 0 < k 1 < · · · < k m = n. In Figure 3, these are k i = 0, 1, 3, 8. For every pair of consecutive elements p k i−1 , p k i , Let C + i (Z) be union of the convex hull of the points p j for all k i−1 ≤ j ≤ k i with the set of all points below this convex hull. Let C + (Z) = C + i (Z) be the union of these sets. In the example in Figure 3, C + (Z) is the union of three sets: C + 1 (Z) is the chord V 01 and points below, C + 2 (Z) is the chord V 13 and points below and C + 3 (Z) is the union of the two chords V 34 and V 48 and the points below these two chords, i.e., C + (Z) is the blue curve and everything below the blue curve. Lemma 1.6. For any 0 ≤ i < j ≤ n, the chord V ij lies in C + (Z) if and only if p + k lies on or above V ij for all positive vertices p + k between p i and p j , i.e., so that i < k < j. Proof. (⇒) C + (Z) does not contain any of the points above a positive vertex p + k . Thus, if p + k is below V ij then V ij cannot be contained in C + (Z).
(⇐) Suppose that each p + k for i < k < j lies on or above V ij . Let L be the piecewise linear curve going from p i to p j , which goes through all positive vertices p + k between p i and p j and which bends only at these positive vertices. We see that L, and thus all points below L, is contained in C + (Z). Since each vertex on L lies on or above V ij , the entire curve L lies on or above V ij . So, V ij is contained in C + (Z) as claimed.
We define C − (Z) analogously to C + (Z): Let P − be the set of nonpositive vertices p ℓ . For any pair of consecutive vertices p ℓ i−1 , p ℓ i in P − , let C − i (Z) be the union of the convex hull of all vertices p j for ℓ i−1 ≤ j ≤ ℓ i and all points in R 2 above this convex hull. Let C − (Z) = C − i (Z). Then we have the following Lemma analogous to Lemma 1.6 above. Lemma 1.7. For any 0 ≤ i < j ≤ n, the chord V ij lies in C − (Z) if and only if p − ℓ lies on or below V ij for all negative vertices p − ℓ so that i < ℓ < j. Proof. This follows immediately from Theorem 1.5 and Lemmas 1.6, 1.7 above.
1.4. Reineke's conjecture. We can now prove Reineke's conjecture for A n with any orientation. Corollary 1.9 (Reineke's conjective for A n ). For a quiver A ε n with any sign function ε, there is a standard linear stability condition making all indecomposable modules stable.
Proof. By Theorem 1.8 above it suffices to find a standard central charge Z (with b = (1, 1, · · · , 1)) so that the stability polygon C(Z) is convex and so that no three vertices are collinear. Such a central charge is given by inscribing the stability polygon C(Z) in the circle of radius n/2 centered at (n/2, 0) and letting the positive vertices p + k lie on the upper semi-circle and the negative p − ℓ lie on the lower semi-circle. And p 0 , p n on the x-axis. More precisely: p 0 = (0, 0), p n = (n, 0), Figure 4. Figure 4. Proof of Reineke's conjecture: The stability polygon C(Z) is convex with no three vertices collinear. So, all chords V ij are stable.

Linear wire diagrams for A n
We briefly discuss the wire diagrams of type A n . Given a central charge Z(x) = a·x+ib·x for the quiver A ε n , the wires L i ⊆ R 2 , i = 0, · · · , n are defined to be the graphs of the functions f i : R → R given by: . This is: We have the following easy theorem.
Proof. This follows from Theorem 1.5 since (1) is equivalent to the condition that p k lies above the chord V ij and (2) is equivalent to the condition that p ℓ lies below V ij . Proof of equivalence for (1): Since f k − f i has negative slope and becomes 0 at t ik , f k (t ij ) > f i (t ij ) if and only if the slope of V ik which is t ik is greater than t ij , the slope of V ij . But this condition is equivalent to p k being above V ij . The proof for (2) is similar. Figure 5 shows the use of colors to determine stability. M ij is stable if L i ∩ L j is below all positive wires (in blue) of intermediate slope (between those of L i , L j ) and above all negative wires (in red) of intermediate slope. For the wire diagram in Figure 1 in the introduction we see that, in that example, all modules are stable except M 03 = I 2 . Figure 5. M ij is stable when all positive lines L k for i < k < j (in blue) are above and all negative lines L ℓ for i < ℓ < j (in red) are below L i ∩ L j .

Wall crossing and maximal green sequences
We recall the wall crossing version of stability from Bridgeland [2], [3], Derksen-Weyman [5], [10] which define a maximal green sequence. See [7], [4] for details of this particular formulation given by a green path passing through a finite sequence of "walls" D(M ). In this section, Λ = KQ is the path algebra of an arbitrary acyclic quiver Q.
The stability set of M , denoted intD(M ), is defined to be the subset of The sets D(M ) are also called walls since they divide R n into "chambers" as we explain below.

Note that D(M ) is a convex subset of the hyperplane
Remark 3.1 translates into the following condition on the linear path Therefore λ Z (t 0 ) ∈ D(M ). The same calculation proves the converse. This proves the first statement. The second statement follow from this and the definition of intD(M ).
We need the following important observation from [7]. Recall that a module M is called a brick if every nonzero endomorphism of M is an automorphism. The simplest nontrivial example is the three dimensional algebra KA − 2 with sign function ε = (0, −, 0). This is the quiver 1 → 2. There are three indecomposable right KA − 2 -modules S 1 , S 2 , P 1 . The semistability sets D(S 1 ), D(S 2 ), D(P 1 ) ⊂ R 2 are shown in Figure 6.
Definition 3.5. By a green path for Λ we mean a smooth (C 1 ) path γ : R → R n having the following properties.
We say that γ passes through the wall D(M ) in the green direction if (3) holds.
Three examples of green paths are drawn in Figure 6 with γ 3 being linear. The vector (1, 1) is always green. So, we see that each γ i passes through the walls in the green direction.

3.2.
Properties of green paths. One of the fundamental properties which holds for any Λ is the following.
This implies that t 0 is uniquely determined. We denote it t M . The proof of Lemma 3.7 follows the proof of Theorem 3.9 below.
Definition 3.8. A green path γ will be called finite if there are only finitely many modules M 1 , · · · , M m (up to isomorphism) for which γ passes through D(M i ) and if, furthermore, γ meets the interior of each D(M i ) at distinct times t i and t 1 < t 2 < · · · < t m . By a maximal green sequence (MGS) for Λ we mean a finite ordered sequence M 1 , · · · , M m given by some finite green path γ. The MGS will be called linear, resp. standard linear, if it is given by a linear green path, resp. standard linear path.
For any green path we have: Hom Λ (M i , M j ) = 0 whenever i < j. This is Corollary 3.10 which follows easily from the following theorem that we will need later in this paper. (See [7], [8], [4] for more details and other equivalent definitions of a maximal green sequence.) Theorem 3.9. Suppose that γ is a green path for Λ and (M, t 0 ) is γ-stable. Then γ crosses the hyperplanes of proper submodules M ′ M after t 0 and it crosses the hyperplanes of proper quotient modules M ′′ before t 0 . More precisely: In other words, Since there are only finitely many dimension vectors of the form dim M ′ , µ is the minimum of a finite collection of linear functions. So, µ is continuous.
For every x ∈ H(M ) we have the following.
If γ is a green path, it must cross the graph of the function x → h x (µ(x)) at some point. By (3), this graph is a union of semistability sets D(M ′ ). So, the "green" condition (3.5 (3)) means it passes from below to above. Thus, γ can only cross this graph once from below to above. By definition of µ, every hyperplane H(M ′ ) for every M ′ M lies on or above this graph. So, if γ(t ′ ) ∈ H(M ′ ) for any such M ′ , it must be after γ crosses this graph. Given that (M, t 0 ) is γ-stable, γ(t 0 ) must be in the interior of D(M ) which, by (5), is below the graph of the function h x (µ(x)). So, γ will hit the graph of the function after time t 0 . This implies t ′ > t 0 . So, (a) holds.
The proof of (b) is similar using ν(x), the largest real number so that h x (ν(x)) ∈ H(M ′′ ) for some proper quotient M ′′ of M .
Proof of Lemma 3.7. The statement is that, for (M, t 0 ) γ-stable, t 0 is unique. When γ crosses D(M ), it must already have crossed the graph of h x (ν(x)) and has not yet crossed the graph of h x (µ(x)). So, it can never cross H(M ) − D(M ) which is below the first graph and above the second. Also, γ can cross D(M ) only once in the green direction.
The path γ must pass through the hyperplane H(X) at least once since it starts from its negative side and ends on its positive side. Let t x be one of these times. Then, by (b) in Theorem Similarly, by (a) and Lemma 3.7, t 2 ≤ t x since X ⊆ M 2 . This contradicts the assumption that t 1 < t 2 .
Remark 3.11. It is show in [7] for Λ hereditary and in [8] for Λ cluster-tilted of finite type (which includes all examples in this paper) that maximal green sequences are characterized by this hom-orthogonality condition. More precisely, a sequence of bricks M 1 , · · · , M m is a MGS if and only if Hom Λ (M i , M j ) = 0 for i < j and the sequence (M i ) is maximal with this property. Lemma 3.7 and Theorem 3.9 give necessary conditions for (M, t 0 ) to be γ-stable. However, we need necessary and sufficient conditions such as the following.
Theorem 3.12. Let γ be a green path and λ(t) = a + bt any linear green path for Λ so that  3.3. Affine quivers of type A. In this paper we are interested in quivers of type A a,b . These have n = a + b vertices labeled with integers modulo n and n arrows arranged in a circle with a arrows going clockwise and b going counterclockwise. There are 2 n possible orientations of such a quiver which we indicate with a sign function ε : [1, n] → {+, −} or, equivalently, an n-periodic sign function ε : Z → {+, −}. The sign ε i is positive or negative depending on whether there is an arrow i ← i + 1 or i → i + 1, respectively.
For example we have: The exponent indicates the sign function. All quivers of type A a,b , a, b ≥ 1, are of infinite type. They have infinitely many indecomposable representations up to isomorphism. However, in a maximal green sequence it is well-known that only "string modules" occur. These are the modules M ij , i < j which we now describe.
We write the finite cyclic quiver as an infinite n-periodic linear quiver: with vertices and arrows labeled with integers so that the direction of α i is the same as that of α i+kn for any integer k. The direction of these arrows is given by the n-periodic version of the sign function ε = (· · · , −, −, −, +, +, −, −, −, +, +, · · · ) which we denote ε = (ε 1 , · · · , ε n ) which is (−, +, +, −, −) in this case. This periodic linear quiver is the universal covering of the quiver of type A −++−− 2,3 from (3.2). The representation M ij of A ε a,b is "pushed-down" from the indecomposable representation M ij of the infinite quiver with dimension vector e i+1 + · · · + e j , just as in the finite case. The push-down M of a representation M of the infinite quiver is given by M i = k M i+kn with negative arrows i → i + 1 inducing k M α i+kn : M i+kn → M i+kn+1 and similarly for positive arrows.

Wire diagrams for A a,b
Any green path can be written as where a, b : R → R n are C 1 functions with velocity vectors a ′ (t), b ′ (t) equal to zero for |t| large (giving four vectors a(∞), a(−∞), b(∞), b(−∞)) and so that b i (t) > 0 for all t and i. For example, we could let b(t) be the constant vector b = (1, 1, · · · , 1) and let a(t) = tb − γ(t). Although the decomposition of γ into the two parts tb(t) and −a(t) is not unique, it gives a very useful interpretation of γ for quivers of type A n and A a,b . Also, in Theorem 3.12 we can take the linear green path λ to be λ t 0 given by since, clearly, λ t 0 (t 0 ) = γ(t 0 ).

4.1.
Slope. Given a fixed decomposition γ(t) = tb(t) − a(t), we can define the slope of any nonzero Λ-module M at time t by We have the following easy calculation.
The following characterization of stable pairs follows from Theorem 3.12 using λ = λ t 0 . ( Note that, since the numerator of (2) is equivalent to the following condition: Proof. By (4.2), (1) is equivalent to the condition γ(t 0 ) ∈ H(M ). Now apply Theorem 3.12 using λ = λ t 0 defined in (4.1). For any M ′ M , the formula for σ t 0 shows that t ′ = σ t 0 (M ′ ) is the unique real number so that λ t 0 (t ′ ) ∈ H(M ′ ). Therefore, (2) is equivalent to the condition t ′ > t 0 in Theorem 3.12 which is equivalent to (M, t 0 ) being γ-stable.
Next, we use the well-known fact that, for Λ hereditary, the modules M in any maximal green sequence are exceptional which means that End Λ (M ) = K and Ext 1 Λ (M, M ) = 0. We also use the well-known fact that Λ of type A a,b is a "gentle algebra" in which all exceptional modules are "string modules" of the form M ij for i < j with j − i not divisible by n with dimension vector where e k is the p-th unit vector for p ∈ [1, n] congruent to k modulo n. The indices i, j satisfy one more condition: when ε i = ε j , we must have |i − j| < n, otherwise M ij is not exceptional. This condition also appears later in Proposition 5.7.

4.2.
The functions f i (t). For Λ of type A a,b , the "wires" in our "wire diagram" are the graphs of functions f i : R → R, i ∈ Z, given by where c i = a 1 + a 2 + · · · + a i , taking indices modulo n, and Lemma 4.2. For i < j and t 0 ∈ R, the following are equivalent. ( Proof. Since f i (t) − f j (t) = (b i+1 + · · · + b j )t − (a i+1 + · · · + a j ) the values of t 0 so that f i (t 0 ) = f j (t 0 ) are given by So, (1) is equivalent to t 0 = σ t 0 (M ij ) which, by (4.2), is equivalent to (2).
Let T ij = {t ij } denote the set of all real numbers so that f i (t ij ) = f j (t ij ). Thus T ij = T ji = γ −1 H(M ij ) by Lemma 4.2 above. Since f i (t) < f j (t) for t << 0 and f i (t) > f j (t) for t >> 0 (when i < j), T ij is always nonempty. The smallest and largest elements of T ij will be denoted t 0 ij and t 1 ij respectively. The order theorems, from Section 3.2, which hold for any finite dimensional algebra imply the following for A a,b . Lemma 4.3. Let M ij be γ-stable for a green path γ. Then t ij ∈ T ij is unique.
Proof. By Lemma 3.7, γ −1 H(M ij ) = T ij has only one element. Proof. We again use Theorem 3.12 with λ(t) = tb(t ij ) − a(t ij ). Let L λ i , L λ j , etc. denote the linear wires for λ. Let t λ ij be the x-coordinate of L λ i ∩ L λ j . If L λ k lies above L λ i ∩ L λ j and L λ ℓ lies below L λ i ∩ L λ j then the t coordinates of their intersection points are arranged: Also, t λ ij < t λ ℓk when ℓ < k. Since ε k = + and ε ℓ = −, M ik M ij , M ℓj M ij and M ℓk M ij when ℓ < k. These are all of the proper indecomposable submodules of M ij . So, by Theorem 3.12, M ij is γ-stable.
Conversely, if M ij is γ-stable then, by Theorem 3.12, t ij must be less than t λ ik and t λ ℓj for i < k, ℓ < j, ε k = +, ε ℓ = −. This implies that the straight line L λ k goes above L λ i ∩ L λ j and as claimed. When γ is linear, we can reinterpret Theorem 4.4 in terms of chords V ij as follows.
Corollary 4.6. Let γ(t) = bt − a be the linear green path with a, b ∈ R n constant with b i > 0 for every i, i.e., γ = λ Z for the central charge Z given by Z(x) = a · x + ib · x. Let p i ∈ R 2 be given by p i = (b 1 + b 2 + · · · + b i , a 1 + a 2 + · · · + a i ) Let V ij be the line segment from p i to p j . Then M ij is γ-stable if and only if the following two conditions are satisfied: (1) For every i < k < j with ε k = + the point p k lies above the line segment V ij .
Proof. The slope of V ij , which is equal to σ Z (M ij ) = t ij , is less than t ik and t ℓj , the slopes of V ik and V ℓj , if and only if Conditions (1) and (2) hold.
Theorem 4.7. Let M ij be γ-stable for a green path γ. Then t ij is unique and we have the following.
(1) Suppose i < k < j and ε k = +. Then Similarly, for i < ℓ < j, and ε ℓ = − we have t 1 iℓ < t ij < t 0 ℓj . Conversely, when (1), (2) both hold and t ij is unique we have: These three statements are illustrated in Figure 7. Figure 7. The wires L i , L j cross only once, at t ij , since M ij is stable. The statement t ij < t 0 ik (the first crossing of L i , L k ) is equivalent to saying that L k does not cross L i to the left of t ij . Similarly t 1 kj < t ij says L k does not cross L j to the right of t ij . (All three values of t kj are to the left of t ij .) Proof. As in the case of finite A n quivers, the positivity of k implies that the kth arrow points to the left: M ik ← · · · . Therefore, M ik M ij . By Theorem 3.9, t ij < t ′ for any t ′ ∈ γ −1 H(M ik ). We are using the notation t ′ = t ik . So, t ij < t ik . The other three cases are similar. The converse statement follows by examination of Figure 7. E.g., L k cannot go under L i ∩ L j without crossing L i to the left of t ij .
Definition 4.8. If i < k < j with ε k = + and either t 0 ik < t ij or t ij < t 1 kj we say that k is a witness to the instability of M ij . Similarly, ℓ is a witness to the instability of M ij if i < ℓ < j with ε ℓ = − satisfies either t 0 ℓj < t ij or t ij < t 1 iℓ . Theorem 4.9. Let M ij be γ-stable for a green path γ. Then t ij is unique and we have the following.

4.3.
Example: a nonlinear maximal green sequence. The wire diagram in Figure 9 shows an example of a nonlinear maximal green sequence for A −+−+ (1) This is a wire diagram since each curve is the graph of a continuous function R → R, all crossing are transverse and the wires L 0 , · · · , L 5 are in ascending order on the left and descending order on the right. (2) The diagram gives a maximal green sequence since all stable crossings are green (with lower numbered wire having higher slope). (3) All modules M ij are stable since L i , L j cross only once and the intermediate blue, resp. red, wires pass above, resp. below, the crossing point. E.g., M 05 is stable since L 2 , L 4 go over the crossing point L 0 ∩ L 5 and L 1 , L 3 go under that point.
(4) The diagram is nonlinear since, if it were linear, the wires L 0 , L 5 could be moved to the left to make the crossings t 01 , t 45 , resp. t 02 , t 35 , line up with t 23 , resp. t 14 . But this would move the crossing t 05 to the left contradicting Pappus' Theorem. This proves the following.  Figure 9 gives an example of a maximal green sequence of maximal length, 15, which is not given by any linear stability condition.

Critical line and upper bound
In this section we introduce the critical line and use the order of the points on the critical line to determine which wire crossings can be stable. We then use this to find the upper bound on the length of a maximal green sequence by assuming that all of these crossings are indeed stable. In Section 6 we will show that this upper bound is actually attained.
Throughout this section we assume that γ is a generic green path for a quiver of type A ε a,b with associated functions f i and wires L i which are the graphs of these functions. Note that, since γ is generic, no three wires meet at any point. So, M ij is (γ)-stable if and only if it is (γ)-semistable. We will also say that the intersection t ij is stable in that case.

The critical line. Due to the periodicity of the
. . , or more generally that t 0 i,i+n = t 0 j,j+n for all i, j. This leads to the following definition. The critical line is so called because we can use it to determine several conditions for stability of intersections, to be described shortly. Note that the signs of the lines in the wire diagram determine a sequence on the critical line, such as the one in Figure 10. Figure 10. Four points on the critical line with ε i = ε k = + and ε ℓ = −. The figure indicates i ⊳ k ⊳ ℓ ⊳ j but does not preclude the presence of other points on the critical line above, below or between these points.

5.2.
Lemmas. For each i, j, we will determine all p so that M i,j+pn can be stable. More precisely we show that M i,j+pn is unstable (not semistable) for all but two values of p. To show that M ij is unstable, we find t ij ∈ T ij so that γ(t ij ) / ∈ D(M ij ). In that case we say that the intersection of the wires L i and L j at t ij is unstable. This will depend on the sign of i, j and/or the position of other curves L k , L ℓ on the critical line.
Proof. In both cases there are two subcases: (a) i ⊳ j and (b) j ⊳ i. We may assume t ij is unique since, otherwise, M ij is not stable. In Subcase (a), t ij > t 0 and, in Subcase (b), t ij < t 0 . In all four subcases, k is a witness to the instability of M ij (Definition 4.8). For example, in Case (1a), f i (t) = f k (t) for some t ≤ t 0 . So, t 0 ik ≤ t 0 < t ij making k a witness. In Case (1b), f j (t), f k (t) switch order at some t ≥ t 0 . So t 1 kj ≥ t 0 > t ij making k a witness. Cases (2a), (2b) are similar. Thus M ij is not stable. j · k − i · or j · k + i · I.e., k ⊳ i ⊳ j with ε k = + in the first case and i ⊳ j ⊳ k with ε k = − in the second case. Then there is at most one stable intersection of the form t i,j ′ where k < i < k + n and j = j + pn for some p, namely the one given by k < i, j ′ < k + n.
Proof. Case 1: If j ′ < k then j ′ < k < i making M j ′ i unstable by Lemma 5.2 (1). Similarly, if k + n < j ′ , then i < k + n < j ′ making M ij ′ unstable. So, there is only one possible value of j ′ = j + pn for which M ij ′ could be stable, namely the one with k < j ′ < k + n.

Lemma 5.4 (Two-Point Lemma).
Suppose there are two points on the critical line in one of the following two arrangements: j · j − i + or i · I.e., i ⊳ j and either ε i = + or ε j = −. Then there are at most two stable intersections of the form t i,j ′ with j ′ = j + pn given by taking the two values of j ′ so that i − n < j ′ < i + n.
Proof. Take the first case. If j ′ < i − n then apply Lemma 5.2 with k = i − n to see that M ij ′ is unstable. If j ′ > i + n we let k = i + n. Then, again, M ij ′ is unstable by Lemma 5.2. So, the only possible stable M ij ′ are the two with i − n < j ′ < i + n.
The second case is similar.
Lemma 5.5 (Four-Point Lemma). Suppose there are four points on the critical line in the following arrangement: j · ℓ − k + i · I.e, i ⊳ k ⊳ ℓ ⊳ j, ε k = + and ε ℓ = −. Then there are at most two stable intersections of the form t i,j+pn . Choosing i, j, k, ℓ so that k < j, ℓ < k + n and ℓ < i < ℓ + n, the possible stable intersections are t ij and t i,j+n .
For any j ′ = j + pn < k we have t i,j ′ < c 0 < t 1 kℓ . So, M i,j ′ is unstable by Theorem 4.9. Similarly, if j ′ > k + 2n we have t 0 ℓ+n,k+2n < c 0 < t i,j ′ making M i,j ′ unstable. Thus the only possible values of j ′ = j + pn making stable M i,j ′ are the two with k < j ′ < k + 2n. Lemma 5.6 (Finiteness Lemma). Suppose that, on the critical line t = c 0 , all positive points are above all negative points. I.e., k ⊲ ℓ whenever ε k = + and ε ℓ = −. Then there are an infinite number of stable modules.
Proof. For any positive integer p we consider the (finite) set of all pairs (i, j) with −pn ≤ i < j ≤ pn so that ε i = +, ε j = −. Since f i (c 0 ) > f j (c 0 ) the curves L i , L j must cross at some point t ij < c 0 . Choose (i, j) so that t ij < c 0 is maximal and, if there is a tie, choose whichever has smaller value of j − i. Then we claim that this intersection is stable. To see this consider any i < k < j with ε k = +. If f k (t ij ) ≤ f j (t ij ) then L k , L j cross at some point t ij ≤ t kj < c 0 . Since j − k < j − i this contradicts the choice of (i, j).
Now increase p by 1. Then −(p + 1)n ≤ i − n < j + n ≤ (p + 1)n. Since t ij < c 0 we have: So, the wires L i−n and L j+n cross at some point t ij < t i−n,j+n < c 0 . So, there exists another stable M i ′ ,j ′ with t ij < t i ′ j ′ < c 0 . Proceeding in this way, we get an infinite sequence of nonisomorphic stable modules.
Proposition 5.7. Let M ij be a stable module in a maximal green sequence for the quiver A ε a,b . Suppose ε i = ε j . Then j − i < n. Proof. We exclude first the case when i < j are congruent modulo n. In that case, f i (c 0 ) = f j (c 0 ). By the Finiteness Lemma, there must be either a negative point ℓ ⊳ i or a positive point k ⊲ i. By adding multiples of n we can assume i < k, ℓ < j. Then M ij is unstable by Theorem 4.4. In the remaining cases when i < j are not congruent module n, by the 2-point Lemma 5.4 we must have j − i < n.

5.3.
The upper bound. From the above lemmas, we can determine an upper bound for the number of stable intersections when finite: By the Finiteness Lemma there must be at least one positive point below a negative point. We will show that the following pattern of points on the critical line is the one which gives the largest finite number of stable intersections.  In every maximal green sequence of this length, the pattern of points on the critical line must be as given in (5.1) above.
Proof. Considering the points on the critical line, let f k (c 0 ) be minimal among all points with ε k = + and let f ℓ (c 0 ) be maximal for ε ℓ = −. By the Finiteness Lemma 5.6, k ⊳ ℓ. By adding a multiple of n to ℓ if necessary, we may assume k < ℓ < k + n. Let A denote the set of all j with k < j < k + n so that k ⊳ j. Let A ′ be the subset of A consisting of those j with ℓ j. Then A ′ contains ℓ and the points above ℓ which are all positive. Since k is positive and below ℓ, a ′ = |A ′ | ≤ a.
Similarly, let B denote the set of all i with ℓ − n < i < ℓ so that i ⊳ ℓ and let B ′ be the subset of B consisting of those i with i k. Then b ′ = |B ′ | ≤ b. On the critical line t = c 0 , the set of points corresponding to A, resp A ′ , is complementary to the set corresponding to Consider all pairs i, j (up to translation by n) so that i ⊳ j. Since f p (c 0 ) takes n values (given by p = 1, 2, · · · , n) there are n 2 such pairs i, j. Claim (a): If i ∈ B ′ and j ∈ A ′ there are at most 2 values of p so that M i,j+pn is stable, namely p = 0, −1 (i.e., only M ij and M i,j−n might be stable.) Claim (b): If i / ∈ B ′ + nZ or j / ∈ A ′ + nZ, there is at most one value of p so that M i,j+pn is stable.
In other words, the pair (i, j) gives at most one stable module unless i ∈ B ′ and j ∈ A ′ in which case there might be two stable modules. So, the Claim implies that there are at most n 2 + a ′ b ′ stable modules. This is maximal when a ′ = a and b ′ = b which happens exactly when the pattern of points on the critical line is given by (5.1). Thus, these claims imply the Theorem.
Proof of Claim (b): Suppose i / ∈ B ′ + nZ. Then i, j ∈ A (up to translation by a multiple of n). By the 3-point Lemma (5.3), only M ij might be stable for i, j ∈ A. Similarly, if j / ∈ A ′ + nZ then we may assume i, j ∈ B in which case, again, only M ij might be stable. Proof of Claim (a): Given i ∈ B ′ , j ∈ A ′ we apply the 4-point Lemma 5.5 to ℓ < i + n < ℓ + n and k < j < k + n to give the statement of Claim (a) in the case when i = k and j = ℓ. In the special cases when either i = k or j = ℓ, the statement of Claim (a) is give by the 2-point Lemma 5.4.
This proves Claims (a) and (b). The Theorem follows.
Claims (a) and (b) in the above proof also imply the following converse of the Finiteness Lemma. Given a green path γ with corresponding functions f i we say that (k, ℓ) is an essential pair for γ if ε k = +, ε ℓ = − and k ⊳ ℓ (f k (c 0 ) < f ℓ (c 0 )). Proof. This condition is necessary by the Finiteness Lemma 5.6. The converse follows from the proof of Theorem 5.8 above since Claims (a) and (b) in the proof imply that there are at most n 2 + a ′ b ′ stable modules where a ′ = |A ′ |, b ′ = |B ′ | are the sizes of the two finite sets in the proof of Theorem 5.8 which exist when there is an essential pair.

Sets of stable modules.
Examining the details of the proof of Theorem 5.8, we obtain the list of stable modules of the maximum length: Definition 5.10. For any k < ℓ < k + n with ε k = +, ε ℓ = −, let A kℓ , B kℓ denote the following sets of integers.
A kℓ := {ℓ} ∪ {j : k < j < k + n, ε j = +} B kℓ := {k} ∪ {i : ℓ − n < i < ℓ, ε i = −} Note that |A kℓ | = a, |B kℓ | = b. Let S kℓ denote the set of modules in the following list. ( Theorem 5.11. For any maximal green sequence of length a+b 2 + ab, the set of stable modules is one of the sets S kℓ defined above. Proof. The sets A ′ , B ′ in the proof of Theorem 5.8 are equal to A kℓ , B kℓ in the case when the pattern of points on the critical line is as given in (5.1). The possible stable modules of Claim (b) in the proof are listed in (1) and (2)  Proof. We will show that, if a = 1 or a ≥ 3, the sets S kℓ are distinct, i.e., the set S = S kℓ determines k and ℓ (modulo n). The case b = 2 is analogous.
Case 1: a = 1. Then k is unique modulo n. Given k, the set S = S kℓ determines ℓ since ℓ is the only integer between k and k + n for which M kℓ and M ℓ−n,k are both in the set S. So, the b sets S kℓ are distinct.
Case 2: a ≥ 3. Then we claim that the set S = S kℓ uniquely determines ℓ modulo n. The reason is that ℓ is the unique integer modulo n with ε ℓ = − which is "doubly paired" with only one integer k modulo n with sign ε k = + where, by "doubly paired", we mean there are two values of p for which M ℓ,k+pn are in S. (The negative points not equal to ℓ are paired with the a − 1 ≥ 2 positive points not equal to k.) Furthermore, ℓ determines k modulo n since k and ℓ are doubly paired.
In the next section we will show (Theorem 6.10) that each of the sets S kℓ occur as the stable set of modules of some maximal green sequence for A ε a,b for any sign function ε. Together with the above lemma, this implies the following , the formula for S kℓ gives: Thus S 14 = S 23 with neither equal to S 13 = S 24 . So, there are three distinct sets S kℓ . 5.5. Deletion Lemma. One immediate consequence of Theorem 4.4 and Corollary 4.6, two theorems we had earlier, is the observation that, when a curve L k is deleted, stable intersections of other curves remain stable. However, previously unstable intersections, possibly an infinite number of them, might become stable. Using Theorem 5.9 we can prevent this from happening.
We need some notation. For Q a quiver of type A a,b and X a subset of the arrow set of Q having at most a + b − 2 elements, we say that Q ′ is obtained by collapsing the arrows in X if Q ′ is given by deleting each arrow α i (from i to i + 1 or i + 1 to i) in X and identifying the source and target of α i . This induces an epimorphism π : Q 0 = {1, 2, · · · , n} → Q ′ 0 = {1, 2, · · · , n − |X|} which is the reduction modulo n−|X| of a map π : Z ։ Z which is periodic in the sense that π(i + n) = π(i) + n − |X| and so that π(1) = 1, π(i) ≤ π(j) when i ≤ j and π(i) = π(i + 1) when α i ∈ X. Given a module M ij for KQ, let π(M ij ) be the KQ ′ module given by Proposition 5.15 (Deletion Lemma). Let M 1 , · · · , M m be a maximal green sequence for Λ = KQ for Q a quiver of type A a,b . Let γ be a corresponding green path. Let Q ′ be a quiver obtained from Q by collapsing a set of arrows X in Q where X is disjoint from at least one essential pair for γ. Then there is a maximal green sequence for KQ ′ containing as a subsequence the nonzero elements of the sequence π(M 1 ), · · · , π(M m ). Furthermore, if the first sequence is linear, so is the second.
Remark 5.16. It is an easy exercise to show that collapsing arrows in X sends S kℓ to S kℓ (a different set with the same name) as long as X is disjoint from the essential pair (k, ℓ).

5.6.
Cluster-tilted D n . The analysis above also applies to the case when b = 0. Then the quiver is a single oriented cycle of length n = a. Call this quiver Q n . We impose the condition rad n−1 = 0, i.e., the composition of any n − 1 arrows is zero. The path algebra KQ n modulo rad n−1 is the "Jacobian algebra" of a quiver with potential Λ n = J(Q n , W ). This is "cluster-tilted" of type D n . For more details, please see the paper [8] which was written to explain cluster-tilted algebras of Dynkin type and the problem of finding a maximal green sequence for these algebras.
The algebra Λ n has, up to isomorphism, n(n − 1) indecomposable modules M ij where 1 ≤ i ≤ n and i < j < i + n. Given any green path γ(t) = tb(t) − a(t) for Λ n we have associated functions f i : R → R n given, just as in the case of A a,b , by Equation (4.3). The critical line t = c 0 is defined as before. The Finiteness Lemma does not hold since there are only finitely many stable modules. However Proposition 5.7 holds by definition and the 2-, 3-and 4-point lemmas all hold with the same proofs as before.
Maximal green sequences of maximal length for Λ n will arise from the collapsing process described in the Deletion Lemma 5.15 using the following observation.
Remark 5.17. Let Λ = KQ be the algebra of type A ε n,1 with sign function ε i = − if and only if i is a multiple of n + 1. Then Q n is obtained from Q by collapsing the unique edge in Q with negative sign. The collapsing map π from the Deletion Lemma sends the set of Λ-modules S k,n+1 to the set of Λ n -modules S k := {M ij : k ≤ i < j ≤ k + n, j − i < n} which has n 2 + n − 1 elements. We observe that the n sets S k are distinct. Theorem 5.18. Maximal green sequences for Λ n = KQ n /rad n−1 have length at most n 2 + n − 1 and each such maximal green sequence has stable set of modules S k for some k.
Proof We give a central charge Z : K 0 Λ n → C which realizes each maximal stability set S k . Together with Theorem 5.18 above, this will prove Corollary L4 from the introduction. (See Figure 14 for the corresponding chord diagram.) Corollary 5.19. For every k there is a standard linear stability condition on Λ n with stable set of modules S k .

Chord diagrams for A a,b
In this section we use (periodic) chord diagrams to construct linear stability conditions having the maximum number of stable modules. We also construct piecewise linear green paths having any given maximal set S kℓ of stable modules as given in Definition 5.10 completing the proof of Theorem M2.
6.1. Periodic stablility polygon C(Z). The first step is to show that, without loss of generality, we may assume that m = 0 and c, the critical slope of Z, is also zero. We have already noted that the value of m is irrelevant. Setting m = 0 makes: We translate all wires L i to the left by c units by replacing f i with . This corresponds to a new central charge Z c given by Z c (x) = (a − cb) · x + ib · x. Then Z c has the same stable modules as does Z with the slopes of all modules decreased by c. Thus, without loss of generality, we may replace Z with Z c and assume that c = 0. Equivalently, a 1 + · · · + a n = 0.
We say that Z is normalized if it has these two properties (m = c = 0).
As in the finite A n case we define the periodic dual vertices p i ∈ R 2 for i ∈ N by Since c = 0, p i+n = p i + (B, 0) for all i ∈ N where B = b 1 + · · · + b n . For i < 0 we can then define p i = p i+kn − k(B, 0) for sufficiently large k. For all integers i < j the chord V ij is the line segment with endpoints p i , p j . As in the finite case (Theorem 1.5) we have the following. For example, M i,i+2n is never Z-stable since p i+n is in the interior of V i,i+2n . In the following corollary it is essential that Z be normalized. Then p i , p i+n have the same height y i = y i+n for all i ∈ Z. So, at most n distinct heights are attained. Corollary 6.2. If the height y k of every positive dual vertex p k is greater than the height y ℓ of every negative dual vertex p ℓ then there are infinitely many nonisomorphic Z-stable modules M ij .
Proof. Let p k be a positive dual vertex with minimum height y k . Let p ℓ be a negative dual vertex of maximum height y ℓ . Then y k > y ℓ = y ℓ+sn for all integers s. Then any interior point in any chord of the form V k,ℓ+sn has y-coordinate greater than y ℓ and less than y k . By Theorem 6.1 the corresponding modules M k,ℓ+sn are Z-stable for all integers s.
In analogy with the finite case, the periodic stability polygon C(Z) is defined to be the intersection where C + (Z), C − (Z) are defined as follows. Let p k i be the positive dual vertices with fixed indexing k i , i ∈ Z. For every pair of consecutive elements in this set, p k i−1 , p k i , let C + i (Z) be the union of the convex hull of the points p j for k i−1 ≤ j ≤ k i with the set of all points below this convex hull, i.e., points with x-coordinate equal and y-coordinate less than a point in this convex hull. Define C + (Z) to be the union of these sets C + i (Z). Similarly, C − (Z) is the union of sets C − j (Z) which are defined to be the convex hull of all dual vertices p i between and including two consecutive negative vertices union the set of all points above this convex hull. Theorem 6.1 is equivalent to the following affine analogue of Theorem 1.8. Theorem 6.3. For a normalized central charge Z on K A ε ab , the string module M ij is Zsemistable if and only if V ij ⊂ C(Z). M ij is Z-stable if, in addition, the chord V ij contains no dual vertex p k in its interior.
6.2. Linearity of maximal stable sets. Theorem 6.4. Suppose k < ℓ < k + n. Then, a sufficient condition for the stable set S kℓ to be linear is if, in the sequence of signs ε j for k < j < ℓ, all negative signs come before all positive signs. Another sufficient condition is if, in the sequence of signs ε j for ℓ < j < k + n, all positive signs come before all negative signs. Remark 6.5. Theorem 6.6 below shows that if neither of these conditions holds, the stable set S kℓ is nonlinear. Thus, these conditions, taken together, are necessary and sufficient.
To draw more general cases, we should insert vertices along the chords V 14 and V 36 along a curve (in green in Figure 11) which is slightly concave up along V 14 and slightly concave down along V 36 . As long as all the positive points between ℓ and k + n come before all the negative points, such a figure will be accurate and make all chords in S kℓ stable. Theorem 6.6. The stable set of modules S kℓ is nonlinear if and only if there exist k < k ′ < ℓ ′ < ℓ < ℓ ′′ < k ′′ < k + n so that ε k ′ , ε k ′′ are positive and ε ℓ ′ , ε ℓ ′′ are negative.
Proof. If k ′ , k ′′ , ℓ ′ , ℓ ′′ do not all exist then the stable set S kℓ is linear by Theorem 6.4. Therefore, it suffices to show that, when they do exist, S kℓ is nonlinear.
By the Deletion Lemma 5.15, we may delete (collapse) all the other edges of the quiver and assume that the quiver is A +++−−− 33 with k = 2, ℓ = 5. Consider the periodic stability polygon C(Z) with vertices p i . If we write p i = (x i , y i ) then y i = y i+6m = a 1 + · · · + a i = f i (0). By construction of the maximal stability set S kℓ = S 25 we therefore have: (6.1) y 0 , y 4 < y 2 < y 5 < y 1 , y 3 But S 25 contains M 03 , M 14 , M 36 , M 47 . Since these are stable, the dual vertex p 2 must be above the chords V 03 , V 14 . So, y 2 must be greater than the y-coordinate of (s 1 , s 2 ) = V 03 ∩V 14 .
Similarly, the y-coordinate of (t 1 , t 2 ) = V 36 ∩ V 47 must be greater than y 5 . So, However, examination of Figure 12 shows that t 2 < s 2 which gives a contradiction. The proof is: move p 1 (and p 7 ) left until it is right above p 0 (and p 7 above p 6 ). Then move p 3 = (x 3 , y 3 ) right until x 3 = x 4 . Both moves decreases s 2 and increase t 2 . After the moves we have s 2 = t 2 . So, the original values must be related by s 2 > t 2 . This is a contradiction showing that S 25 is not the stability set corresponding to the height pattern (6.1). But the analysis of the critical line shows that this height pattern is the only one which can give this stability set. Therefore S 25 is not the stability set of any central charge.
Corollary 6.7. If either a or b is ≤ 2, all maximal stable sets of modules are linear.
6.3. Nonlinear maximal stable sets. Let S kℓ be one of the maximal stability sets identified as being nonlinear in Theorem 6.6. We will construct a piecewise linear green path γ which crosses the walls D(β) for β ∈ S kℓ and no other walls. This green path will be formed Figure 12. When M 03 , M 14 are stable, p 2 is higher than (s 1 , s 2 ) = V 03 ∩V 14 .
When M 36 , M 47 are stable, p 5 is lower than (t 1 , t 2 ) = V 36 ∩ V 47 . But t 2 < s 2 , contradicting the assumption that p 2 is lower than p 5 .
from two linear green paths λ, λ ′ with the same value at t = 0: λ(0) = λ ′ (0) by We say that γ is given by splicing together λ and λ ′ at t = 0. Since linear green paths are given by λ(t) = tb − a, two linear green paths take the same value at t = 0 if and only if they have the same vector a ∈ R n . Lemma 6.8. Let γ be given by splicing together linear green paths λ Z , λ Z ′ for stability functions Z, Z ′ with the same vector a. Suppose that Z, Z ′ have no semi-stable modules of slope 0. Then the set of γ-stable, resp. γ-semistable, modules is the union of the following two sets.
(1) The set of Z-stable, resp. Z-semistable, modules M with negative slope.
(2) The set of Z ′ -stable, resp. Z ′ -semistable, modules N with positive slope. Remark 6.9. Having the same vector a means that, in the periodic chord diagrams C(Z), C(Z ′ ), the corresponding dual vertices p i , p ′ i have the same y-coordinates. The stable/semistable chords of γ are the stable/semistable chords of C(Z) of negative slope and the stable/semistable chords of C(Z ′ ) of positive slope. Theorem 6.10. Given Λ = K A ε ab and k < ℓ < k + n with ε k = +, ε ℓ = − there exists a piecewise linear green path γ, given by splicing together two linear green paths as described above, so that all γ-semistable modules are stable and the set of γ-stable modules is S kℓ .
Proof. We will construct the periodic stability polygons C(Z), C(Z ′ ) having the required properties. The functions Z, Z ′ are determined by the coordinates of the dual vertices p i , p ′ i of C(Z), C(Z ′ ), resp. As required, p k , p ℓ will have y-coodinates y k < y ℓ . Then all modules of length ≥ 2n will be unstable. So, the set of stable objects is finite and cannot be it greater than S kℓ . We assume that p k−1 , p k+1 are positive and p ℓ−1 , p ℓ+1 are negative. There is no loss of generality since, by the Deletion Lemma 5.15, we can delete these dual vertices and the remaining objects in S kℓ will remain stable and give the set S kℓ for the smaller quiver.
For the cluster-tilted algebra Λ n = J(Q n , W ) = KQ n /rad n−1 , the linear maximal green sequence of Corollary 5.19 is illustrated in Figure 14 in the case n = 5, k = 1.  Figure 14. The n 2 + n − 1 = 14 chords V ij for 1 ≤ i < j ≤ 6 are stable.

Summary of definitions
We summarize the definitions and constructions for linear stability conditions. We summarize the changes in the nonlinear case at the end.

C(Z):
The stability polygon of Z has vertices p i . V ij : The chords are V ij = p i p j .
f i : R → R are the linear functions (2) given by f i (t) = t(m − b 1 − · · · − b i ) + a 1 + · · · + a i where m is any convenient real number. L i : The wires L i ⊆ R 2 are the graphs of the linear functions f i . t ij : This is the x-coordinate of the intersection point L i ∩ L j . Also, This is independent of the choice of m. For Λ = A ε ab (3.3) we have: A ε ab : The quiver with vertices 1, 2, · · · , n, taken to be modulo n, with a arrows k ← k + 1 for ε(k) = + and b arrows ℓ → ℓ + 1 for ε(ℓ) = −. Equivalently, this is an n-periodic quiver with vertex set Z. Q n : = " A n0 " is a single oriented cycle of length n modulo rad n−1 . I.e., the composition of any n − 1 arrows is zero. This is cluster-tilted of type D n for n ≥ 4. ε: The periodic sign function ε : Z → {+, −} so that ε i = ε i+n and ε(k) = + for a values of k in [1, n] and ε(ℓ) = − for b values of ℓ in [1, n]. η: The null root η = (1, 1, · · · , 1). c: = t 0n , the critical slope (5.1), is the slope of the null root c = σ Z (η) = a 1 + · · · + a n b 1 + · · · + b n = slope of V 0n = x-coordinate of L 0 ∩ L n p i , V ij , f i , t ij : Periodic version of these are defined for all i < j ∈ Z with the same formulas as in the finite A n case. C(Z): The periodic stability polygon (6.1) has p i the periodic dual vertices as vertices. In the nonlinear case we make the following modifications: a: R → R n , b :R → (0, ∞) n are C 1 curves with velocity a ′ (t) = b ′ (t) = 0 for |t| large. f i : R → R are the smooth functions given by f i (t) = t(m − b 1 (t) − · · · − b i (t)) + a 1 (t) + · · · + a i (t) where m is any convenient fixed real number. L i : ⊆ R 2 are the graphs (now smooth curves) of the functions f i : R → R. T ij : The set of x-coordinates t ij of the elements of L i ∩ L j ⊂ R 2 . t 0 ij , t 1 ij : The minimum and maximum values of t ij . c 0 = t 0 0n : The (smallest) critical slope. S kℓ : The nonlinear stability sets of maximum size (5.10) for any k < ℓ < k + n with ε k = +, ε ℓ = − consisting of: (1) M ij for all i, j in the set A of size a consisting of ℓ and all k < t < k + n with sign ε t = + (2) M ij for all i, j in the set B of size b consisting of k and all ℓ − n < s < ℓ with sign ε s = − (3) M ij and M j,i+n for all i ∈ B, j ∈ A. The main result of this paper is to show that the sets S kℓ are the only stability sets of the maximum size a 2 + b 2 + 2ab and to determine which are given by linear stability conditions. For (a, b) = (2, 2) the ab sets S kℓ are distinct.
The three kinds of diagrams used in this paper are based on the following theorem proved in the text. (2) chord diagram (1,6): For all i < k, ℓ < j with ε k = +, ε ℓ = −, the point p k is above or on the chord V ij and the point p ℓ is below or on V ij . (3) wire diagram (2,4): For all i < k, ℓ < j with ε k = +, ε ℓ = −, i.e., the intersection point L i ∩ L j is on or below L k and on or above L ℓ . Furthermore, t in (1) is equal to σ Z (M ij ) = t ij in (3) is equal to the slope of V ij in (2).
The relation between chord diagrams and mixed cobinary trees is explained in [11,Sec 8]. This will be extended to periodic trees in a revised version of [12]. We also expect that m-noncrossing trees [9] and periodic versions of such trees can be used to find the maximal lengths of m-maximal green sequences using [13].