Cluster Configuration Spaces of Finite Type

For each Dynkin diagram $D$, we define a ''cluster configuration space'' ${\mathcal{M}}_D$ and a partial compactification ${\widetilde {\mathcal{M}}}_D$. For $D = A_{n-3}$, we have ${\mathcal{M}}_{A_{n-3}} = {\mathcal{M}}_{0,n}$, the configuration space of $n$ points on ${\mathbb P}^1$, and the partial compactification ${\widetilde {\mathcal{M}}}_{A_{n-3}}$ was studied in this case by Brown. The space ${\widetilde {\mathcal{M}}}_D$ is a smooth affine algebraic variety with a stratification in bijection with the faces of the Chapoton-Fomin-Zelevinsky generalized associahedron. The regular functions on ${\widetilde {\mathcal{M}}}_D$ are generated by coordinates $u_\gamma$, in bijection with the cluster variables of type $D$, and the relations are described completely in terms of the compatibility degree function of the cluster algebra. As an application, we define and study cluster algebra analogues of tree-level open string amplitudes.

1 Introduction 1.1. The configuration space M 0,n of n distinct points on P 1 is a smooth affine algebraic The closure (M 0,n ) ≥0 of (M 0,n ) >0 in M 0,n (R) is a stratified space that is homeomorphic to the face stratification of the associahedron polytope.
Let W denote the union of those boundary divisors in M 0,n whose intersection with (M 0,n ) ≥0 is empty, and let M 0,n := M 0,n \ W . The divisors that do intersect (M 0,n ) ≥0 correspond to ways to divide {1, 2, . . . , n} into two cyclic intervals, each of size greater than or equal to two. For example, M 0,5 has ten boundary divisors, and M 0,5 includes five of them, corresponding to the five sides of the pentagon (the associahedron of dimension two). Somewhat surprisingly, the partial compactification M 0,n is an affine algebraic variety, and its ring of regular functions has the following description. Let u ij be variables labeled by the diagonals (i, j) (not including sides) of a n-gon P n . Then C M 0,n is isomorphic to the polynomial ring C[u ij ] modulo the relations R ij := u ij + (k, ) crossing (i,j) u k − 1, (i, j) varying over all diagonals, (1.1) and M 0,n ⊂ M 0,n is the locus where u ij = 0. The u ij are called dihedral coordinates. Brown [11] describes the same space using a presentation with more relations (see Section 10.1); the extra relations are implied by our smaller set. The u ij are cross-ratios (see (7.1)) on M 0,n and appeared in the study of scattering amplitudes in string theory and for the bi-adjoint φ 3 -theory [1].

1.2.
In this paper, we construct in an analogous manner two affine algebraic varieties M D ⊂ M D for each Dynkin diagram D of finite type by considering the relations Here, γ and ω denote mutable cluster variables of a cluster algebra A of type D [18], and (ω||γ) denotes the compatibility degree. We call M D the cluster configuration space of type D. In the case D = A n−3 , we have M A n−3 = M 0,n and M A n−3 = M 0,n . Amongst many remarkable properties of these relations, let us immediately note that u γ = 0 forces u ω = 1 for all ω such that (γ||ω) = 0 (or equivalently, (ω||γ) = 0). Thus, factorization is manifest in (1.2). Some of the results of this work were reported in [4], and M D is an example of the notion of "binary geometry" discussed therein. Whereas M 0,n has a stratification indexed by the faces of the associahedron, the space M D has a stratification (Proposition 3.5) indexed by the faces of the Chapoton-Fomin-Zelevinsky generalized associahedron for D ∨ [12,17]. We show (Theorem 3.  [2]. the locus where all cluster variables are non-vanishing. We show (Theorem 4.2) that M D is isomorphic to the (free) quotient ofX B by the cluster automorphism group T B , generalizing the construction of M 0,n from Gr (2, n). The functions u γ are particular T B -invariant rational functions on X B . The u γ are related to some of the "cluster X-coordinates" in the sense of Fock and Goncharov [16] by the equation u = X/(1 + X). The cluster X-coordinates appearing here are exactly those encountered in the Auslander-Reiten walk through cluster variables, beginning from an acyclic quiver and mutating only on sources. It is important to note that while the u γ are simply related to the cluster X-variables in this way, they are actually in bijection with the cluster A-variables, while in general there are more cluster X-variables than cluster A-variables.
We do not have a good understanding of the relationship between M D and cluster X -varieties; for example, M D does not contain a collection of (cluster) torus charts.
Our approach depends crucially on the flexibility in the choice ofB. WhenB =B univ is the extended exchange matrix for the universal coefficient cluster algebra [19,27], the relation (1.2) is obtained from the primitive exchange relations of A B univ by setting all mutable cluster variables to 1, and sending the universal frozen variables z γ to u γ . The non-primitive exchange relations give rise to other relations of the form U + U = 1, where U and U are monomials in the u γ -s.
WhenB =B prin is the extended exchange matrix for the principal coefficient cluster algebra, the functions u γ become identified with certain ratios of the F -polynomials F γ (y). Bazier-Matte, Douville, Mousavand, Thomas, and Yildrim have shown [8] in the case that D is simply-laced that the Newton polytope of F γ (y) has normal fan a coarsening of the g-vector fan N D ∨ of D ∨ , and this result was extended to skew-symmetric cluster algebras by Fei [15]. We extend via folding this description to the case that D is multiply-laced finite type Dynkin diagram. The identification of M D with an open subset of the toric variety X N (D ∨ ) depends crucially on this analysis.
As an application of our results on quotients of cluster varieties and on F -polynomials, we identify (Theorem 9.2) the positive tropicalization Trop >0 M D of the cluster configuration space with the cluster fan N D ∨ . In particular, we resolve a conjecture of Speyer and Williams [29,Conjecture 8.1] on positive tropicalizations of cluster varieties of finite type; see also [22]. 1.4. Inspired by similar questions for M 0,n , we proceed with studying the topology of M D (C) and M D (R). We identify M Bn with the complement to the Shi-hyperplane arrangement and thereby compute point counts over finite fields, and the Euler characteristics of M Bn (R) and M Bn (C). We give a configuration space style description of M Cn (Proposition 7.5) but were not able to determine whether M Cn is a hyperplane arrangement complement. Nevertheless, we were able to compute the point count for M Cn (F q ), and the number of connected components of M Cn (R). We found numerically the point counts for types D 4 , D 5 and G 2 , and obtained numerically that the point count of M D 4 (F q ) over a finite field F q is not a polynomial in q but a quasi-polynomial.
1.5. One of the main motivations for us are scattering amplitudes in string theory. In [3], we introduced integral functions, called stringy canonical forms, where p j (x) is a positive Laurent polynomial. We showed in [3] that the leading order lim α →0 (α ) n I is a rational function that for fixed c j -s coincides with the canonical rational function [2] of the Minkowski sum of the Newton polytopes of p j (x). Tree-level n-point open superstring amplitudes are integrals on M 0,n . It turns out that for a suitable parametrization of M 0,n , these amplitudes can be written as an integral I A n−3 in the form (1.3), where the p j (x) are the F -polynomials for the type A n−3 cluster algebra. The importance of the u ij -variables appears in the rewriting (see [3,Section 9] or [ The poles of I D are made manifest by rewriting in terms of the u γ -s, and the leading order of I D is controlled by the combinatorics of the generalized associahedron of D ∨ .

Background on cluster algebras and generalized associahedra
In this section we review basic facts concerning cluster algebras. The most important cluster algebra references for us are [8,31]. For cluster varieties, our conventions follow [23].

2.1.
Let D be a finite Dynkin diagram with vertex set I, and let A = (a ij ) denote the n × n Cartan matrix of D, where n = |I|. Let B be a skew-symmetrizable exchange matrix, i.e., there exists a matrix Z with positive diagonal entries such that ZB is skew-symmetric. We say that B = (B ij ) has type D if In standard cluster algebra language, B corresponds to an acyclic initial seed of a cluster algebra of finite type D. Given D, the possible exchange matrices B of type D are in bijection with orientations of the underlying tree of D: writing i → j for the directed edges of this orientation, we have For an (n+m)×n extended exchange matrixB extending B, we let A B denote the corresponding cluster algebra of geometric type [18]. By convention, A B is the C-algebra generated by all mutable cluster variables, all frozen variables, and the inverses of all frozen variables. We let X B = Spec A B denote the cluster variety [23]. This is a complex affine algebraic variety, and in general it differs from the union of cluster tori, which is sometimes called a cluster manifold.

2.2.
We say thatB (or A or X) has full rank ifB has rank n. We say thatB (or A or X) has really full rank if the rows ofB span Z n . IfB has full rank, then X B is a smooth affine algebraic variety [24, Theorem 7.7].
2.3. Let Π = Π(B) be the indexing set for cluster variables, which depends only on B. Set r := |Π|. For γ ∈ Π, we let x γ ∈ A B denote the corresponding cluster variable. (Abusing terminology, sometimes we will refer to elements of Π as cluster variables.) We give Π the structure of a simplicial complex, called the cluster complex, by declaring the maximal faces to be the clusters {γ 1 , . . . , γ n }. The set Π can be identified with the following set of pairs of integers: where r i , i ∈ I are some positive integers. The initial cluster is {(0, i) | i ∈ I}. We let Π + ⊂ Π denote the subset of non-initial cluster variables, i.e., those γ = (t, j) with t = 0.
Remark 2.1. In [31], the set Π is identified with the set of weights {c s ω i | 0 ≤ s ≤ r i }. We have chosen to index using the pairs of integers (s, i) instead. The choice c of a Coxeter element in [31] corresponds to our choice of an orientation of D in determining the exchange matrix B.
Remark 2.2. Starting from the initial cluster {x (0,i) | i ∈ I}, the cluster {x (1,i) | i ∈ I} is obtained by mutating each vertex of I once, always mutating at sources. This process is repeated to obtain all the cluster variables. In particular, the cluster variable x (t,j) is obtained by mutation from x (t−1,j) ; see Proposition 2.5 for the exchange relation. We refer the reader to [8] for an explanation of this Auslander-Reiten walk, the relation to quiver representations, and many examples.

2.4.
There is an involution * : I → I sending i to i * induced by the longest element of the Weyl group of the root system of D. This involution is the identity in all types except for A n , D 2n+1 , E 6 , and in these types * : I → I is the non-trivial automorphism of D (as a graph). We shall use the notation (−1, i) := (r i * , i * ); see [ We have (ω||γ) = 0 if and only if (γ||ω) = 0 and in this case we say that ω and γ are compatible. Otherwise, we call ω and γ incompatible. If (ω||γ) = (γ||ω) = 1, we say that ω and γ are exchangeable. The faces of the cluster complex consist of sets of cluster variables that are pairwise compatible.

2.7.
Let A prin = A B prin denote the cluster algebra with principal coefficients [19]. Thus B prin is a 2n × n matrix whose top half is equal to B and bottom half is equal to the identity matrix. In this case, the initial mutable variables are denoted x 1 , x 2 , . . . , x n and the principal frozen variables are denoted y 1 , y 2 , . . . , y n . We have a Z n -grading on the principal coefficient cluster algebra A prin given by Each mutable cluster variable is homogeneous with respect to this grading, and we define the g-vector by g γ := deg(x γ ) for γ ∈ Π.
2.8. For γ ∈ Π + , define the F -polynomial F γ (y) by setting the initial cluster variables to 1 in the Laurent expansion of the cluster variable x prin γ in the cluster algebra A prin with principal coefficients: By convention, we have F γ (y) = 1 if γ is initial. Computations of g-vectors and F -polynomials are given in Examples 6.7 and 6.8. Further examples can be found in [8,19].
2.9. The cluster fan N (B) is the collection of cones spanned by {g γ 1 , . . . , g γs } as {γ 1 , . . . , γ s } varies over collections of cluster variables that belong to the same cluster, called compatible cluster variables. Recall that a cone C is called simplicial if dim(C) is equal to the number of extremal rays of C, and a fan is called simplicial if all its cones are. A fan N in R n is called smooth if it is simplicial and for each maximal cone C ∈ N the primitive integer vectors v 1 , . . . , v n spanning C form an integral basis for Z n . 12,17,21]). The collection of cones N (B) is a smooth, complete polyhedral fan.
A generalized associahedron of type B is any polytope whose normal fan is equal to N (B). Often, we will say "generalized associahedron of type D", with the choice of B of type D understood.

2.10.
For γ ∈ Π, let P γ denote the Newton polytope of the F -polynomial F γ (y). By convention, if γ is initial, we have set F γ (y) = 1 and P γ = {0}. Let F (y) = γ∈Π F γ (y). Then the Newton polytope P of F (y) is the Minkowski sum γ∈Π P γ . The following result is established in [8] when D is simply-laced (and extended to not necessarily acyclic initial seeds in [15]), and in Theorem 6.1 we extend the result to multiply-laced finite type D with acyclic initial seed.
Theorem 2.4. The (outer) normal fan of the Minkowski sum γ∈Π P γ is equal to N (B ∨ ).
An exchange relation for A B is called primitive if it is of the form x γ x ω = M + M , where one of the two monomials M , M does not contain any mutable cluster variables. The primitive exchange relations are exactly the ones of the form x τ γ x γ = M + M .

2.12.
Let A univ = A B univ denote the cluster algebra with universal coefficients, from [19,Theorem 12.4], [31,Section 5], and [27,Theorem 10.12 and Remark 10.13]. ThusB univ is a (n + r) × n matrix whose top part is equal to B and whose bottom part has rows given by the g-vectors of the cluster algebra with exchange matrix B T , see [27]. The bottom r rows ofB univ are again indexed by Π, and we denote the corresponding frozen variables by z γ , for γ ∈ Π.
Proposition 2.5 ([31, Proposition 5.6]). The primitive exchange relations of A B univ are given by for j ∈ I and 0 ≤ t ≤ r j .
2.13. For γ ∈ Π + , define the universal F -polynomial F univ γ (z) by setting the initial cluster variables to 1 in the Laurent expansion of x univ By convention, we have F univ γ (z) = 1 if γ is initial.

The cluster configuration space M D
In this section, we define the cluster configuration space M D and its partial compactification M D , and we state some geometric properties of these spaces. We also give examples of the cluster compatibility degree appearing in the defining relations.
for γ ∈ Π, and (ω||γ) denotes the compatibility degree. The following result will be proved in Section 5.4.

3.2.
Let us give the relations R γ explicitly in types A, B, C, D, G. In the following discussion, we use models for Π involving diagonals of a polygon; see [18] for further details. The precise correspondence with (2.1) depends on the choice of initial cluster (for example, a choice of triangulation of the polygon in type A), or equivalently the choice of B, or equivalently the choice of orientation of D. 3.2.2. Type C n−1 . Let P 2n be the 2n-gon with vertices cyclically labeled 1, 2, . . . , n,1,2, . . . ,n.
The set Π is identified with the union of the long diagonals, and pairs of centrally symmetric diagonals in P 2n . In total we have |Π| = n 2 − n.
The compatibility degree (γ||ω) is equal to the number of crossings of one of the diagonals representing ω with the diagonals representing γ. Thus for example [1,1] [11] + u [22] u [33] u 2 [23] , 1 = u [23] + u [11] u [13] u [12] (3.3) and the three cyclic rotations of each. This case is obtained from A 2n−3 by folding. There is an action of the two-element group Γ on P 2n mapping i ↔ī. This induces the natural map ν :Π → Π sending diagonals of P 2n to Γorbits on the diagonals of P 2n . We may verify Proposition 2.6(2): for example [2,3] The automorphism τ is inherited from the rotation of the 2n-gon P 2n .
3.2.3. Type D n . Let P P n denote an n-gon P n with vertices 1, 2, . . . , n (in clockwise order) and an additional marked point 0 in the middle. The set Π consists of certain arcs in P P n connecting vertices and 0: (a) for 1 ≤ i = j ≤ n and i = j + 1 mod n we have an arc (i, j) connecting i to j going counterclockwise around 0, and (b) for each 1 ≤ i ≤ n we have two arcs [i] and [ĩ] connecting i to 0. We denote the corresponding u-variables by u ij , and u i and uĩ. See Figure 1.
(We caution the reader that the notationĩ here is unrelated to the notationγ used for foldings.) In this case, the automorphism τ is the composition of the rotation of P n with "changing the tagging at 0" (i.e., switching from [i] to [ĩ] if the arc is incident to 0).   [11] u [13] u [12] and the three cyclic rotations of each. Note that M B 2 is isomorphic to M C 2 under a non-trivial re-indexing of the u-variables. However M Bn and M Cn are not isomorphic for n > 2. Type B n−1 can be obtained from type D n by folding. Let Γ be the two-element group acting onΠ = Π(D n ) by sending [ [23] u [24] u [14] u [13] , 1 = u [13] + u 2 [44] u [14] u [24] u [24] u [34] , 1 = u [11] + u [22] u [33] u [44] u [23] u [34] u [24] . (3.5) The automorphism τ is inherited from the rotation of the n-gon in type D n .
3.2.5. Type G 2 . In type G 2 , we have |Π| = 8 and we denote u-variables by a i , b i for i = 1, 2, 3, 4. The u-equations for type G 2 are and the cyclic rotations under the group Z/4Z. Type G 2 can be obtained from D 4 by folding.
(Though at present we only consider foldings of simply-laced diagrams, G 2 can also be obtained from B 3 by folding.)

Let
we find that u ω = 1 for all ω incompatible with γ. Let Π(γ) ⊂ Π be the subset of κ ∈ Π that are compatible with γ. Setting u ω = 1 in R κ for κ ∈ Π(γ), we get It follows that the coordinate ring of M D (F ) has the following presentation Proposition 3.6. Suppose γ = (t, j) and removing j from D disconnects D into connected components D 1 , . . . , D s . Then we have Proof . Let γ ∈ Π. Since γ defines a facet of the generalized associahedron for B, it follows that the collection of clusters containing γ are connected by mutation, without mutating γ. It follows that Π(γ) is the cluster complex of a cluster algebra of finite type associated to the disjoint union of D 1 , D 2 , . . . , D s .
Note that removing a vertex from a finite type Dynkin diagram produces at most three components, so in Proposition 3.6 we have s ≤ 3. By applying Proposition 3.6 repeatedly, we have the following result.  3.6. Let γ ∈ Π. Then as in Proposition 3.6, we can uniquely associate Dynkin diagrams Proposition 3.8. Suppose that γ ∈ Π and s(γ) = 1. Then we have a natural morphism The proof of Proposition 3.8 is delayed to Section 4.7. The map of Proposition 3.8 corresponds to "forgetting a marked point" in the case of M A n−3 = M 0,n . We expect (3.7) to be a fibration, similar to the M 0,n case.

3.7.
Let D be a folding ofD and let Γ and ν :Π → Π be as in Section 2.14.
Proposition 3.9. The quotient of C[u]/ID by the ideal generated by the equations uγ = u g·γ forγ ∈Π and g ∈ Γ is canonically isomorphic to C[u]/I D .

M D as a quotient of a cluster variety
In this section we show that M D can be obtained as a quotient of an open subspaceX of the cluster variety X B by the action of the cluster automorphism torus T considered in [23]. An important role is played by principal and universal coefficients, whereB =B prin orB =B univ . In particular, the defining relations of M D are obtained from the primitive exchange relations of X B univ .

4.1.
LetB be a full rank extended exchange matrix. Let T = T B be the cluster automorphism group [23] of A B : this is the group of algebra automorphisms φ : A B → A B such that for each (mutable or frozen) cluster variable x, we have φ(x) = ζ(x)x for ζ(x) ∈ C * . Thus, T acts on any cluster torus of the cluster variety X B by scaling the coordinates. By [23, Proposition 5.1], we have SinceB has full rank, the group T is a (possibly disconnected) abelian algebraic group of dimension m. The character group of T is the lattice Z n+m /BZ n . By definition, the torus T acts on each cluster variable by a character, and we denote the weight of the cluster variable x ∈ A B by wt(x) ∈ Z n+m /BZ n .
Lemma 4.1. LetB have full rank. ThenB has really full rank if and only if Z n+m /BZ n has no torsion, or equivalently, the group T is connected, and thus a torus of dimension m.
Proof . The rows ofB span Z n if and only ifBQ n ∩ Z n+m =BZ n if and only if Z n+m /BZ n has no torsion.
Let X = X B be the cluster variety, which is a smooth affine algebraic variety. LetX ⊂ X be the locus where all mutable cluster variables are non-vanishing. In terms of rings, we have ThusX is a smooth affine subvariety of the initial (or any) cluster subtorus of X, and it follows immediately from the definitions that the action of T preservesX, and furthermore the action of T is free onX. The geometric invariant theory quotient X / / T := Spec C X T is again a smooth affine algebraic variety, and furthermore, there is a bijection between closed points ofX / / T and T -orbits onX. We thus simply denoteX / / T byX/T . Explicitly, the ring C X T consists of all weight zero Laurent polynomials in cluster variables.

Let
where M only involves frozen variables. For each primitive exchange relation, we define the rational function By definition, f γ ∈ C X , and it is easy to see that Theorem 4.2. Suppose thatB is a full rank extended exchange matrix, acyclic and of finite type. Let X = X B . Then the quotientX/T is a smooth affine variety isomorphic to M D , Theorem 4.3. Suppose thatB is a full rank extended exchange matrix, acyclic and of finite type. Let X = X prin B have principal coefficients. Then M D ∼ =X prin /T prin is isomorphic to the locus X prin (1) ⊂ X prin , where all initial mutable cluster variables have been set to 1.
The coordinate ring C[M D ] is isomorphic to the subring of C(y 1 , y 2 , . . . , y n ) generated by F ±1 γ (y) and y ±1 i .
Theorem 4.4. Suppose thatB is a full rank extended exchange matrix, acyclic and of finite type. Let X = X univ B have universal coefficients. Then M D ∼ =X univ /T univ is isomorphic to the locus X univ (1) ⊂ X univ , where all mutable cluster variables have been set to 1. The isomorphism 4.3. The relations in the following corollary will be discussed in further detail in Section 10.1.
Corollary 4.5. The ideal I D has a natural set of generators of the form U + U − 1, given by the images of all exchange relations of X univ .
The ideal I D also contains the |Π|−n distinguished elements which are images of 1−F univ γ (z).

Proof of Theorem 4.4.
Recall that the mutable cluster variables of A univ are denoted x γ and the frozen variables are denoted z γ , where γ ∈ Π. Let X univ (1) ⊂X univ ⊂ X univ be the locus {x γ = 1}, where all mutable cluster variables have been set to 1. By Proposition 2.5, the primitive exchange relations are of the form where S is a monomial in the mutable cluster variables. So, and thus on X univ (1) we have (f γ )| X univ (1) = z γ and the relation We will now show that the multiplication map gives an isomorphism or equivalently, every T univ -orbit onX univ intersects X univ (1) in exactly one point. The character group of T univ is naturally isomorphic to Z n+r /BZ n , which is a free abelian group of rank r = |Π|. Thus each cluster variable x γ has a weight (or degree) wt(x γ ) ∈ Z n+r /BZ n (see (6.2) for the weight of initial and frozen variables). By Proposition 6.5, the set {wt(x γ ) | γ ∈ Π} form a basis of the lattice Z n+r /BZ n . Thus we have a projectionX univ → T univ given by sending x ∈X univ to the coordinates (x γ ) γ∈Π , and the fiber of this projection is X univ (1). This is an inverse to the multiplication map T univ × X univ (1) →X univ , and we deduce that T univ × X univ (1) ∼ =X univ .
We conclude that C X univ T univ ∼ = C[X univ (1)]. Now, any T univ -invariant function in C X univ is a linear combination of T univ -invariant Laurent monomials in mutable and frozen variables. Each such Laurent monomial restricts to a Laurent monomial in the z γ -s on C[X univ (1)]. It follows that the functions f γ and their inverses generate C X univ T univ , and by (4.2) satisfy the same relations that u γ ∈ C[M D ] satisfy. Finally, we check that the generators f γ do not satisfy any further relations. Suppose we have a polynomial identity p(f γ ) = 0 inside C X univ T univ . The equality p(f γ ) = 0 is equivalent to an equality q(x γ , z γ ) = 0 inside C X univ , where q(x γ , z γ ) is a Laurent polynomial. We claim that the primitive exchange relations allow us to eliminate all the non-initial cluster variables, i.e., q(x γ , z γ ) = r(x 1 , x 2 , . . . , x n , z γ ) mod ideal generated by primitive exchange relations, where r(x 1 , x 2 , . . . , x n , z γ ) is a Laurent polynomial and the ideal is taken inside C X univ T univ . To see this, first note that deg(x 1 ), . . . , deg(x n ) and deg(z γ ), γ ∈ Π together span Z n+r , and thus we can always multiply q(x γ , z γ ) by a T univ -invariant monomial so that the denominator involves only initial x i and the z γ . Next, we have where R = M +M xτγ xγ −1 is a primitive exchange relation (divided by x τ γ x γ ). This allows us (modulo the ideal) to replace x γ by an expression involving x τ γ and M + M . If γ = (t, j), the mutable cluster variables that appear in M + M are either of the form (t − 1, i) or of the form (t, i), where i → j (see Proposition 2.5). It follows that x γ will not appear again when this process is repeated. This proves our claim.
But r(x 1 , x 2 , . . . , x n , z γ ) = 0 as an element of C X univ T univ ⊂ C X univ ⊂ C(X univ ) only if the polynomial r is 0, since x 1 , x 2 , . . . , x n , z γ are algebraically independent. We conclude that p(f γ ) lies in the ideal generated by primitive exchange relations. Thus the ideal of relations satisfied by the f γ is generated by (4.2). We thus have an isomorphism of rings

4.5.
Proof of Theorem 4.2. By the defining property of universal coefficients, we have a homomorphism of rings φ : is a Laurent monomial in the frozen variables x n+1 , . . . , x n+m of A. The homomorphism φ may not be surjective, for example this would be the case ifB has rows equal to 0, or rows that are repeated. However, the image A := φ(A univ ) is itself a cluster algebra: it is generated by φ(x univ γ ) and the monomials φ(z γ ). The monomials φ(z γ ) and their inverses generate a Laurent polynomial Since A has full rank, the quotient X/T = X /T has dimension n, and A also has full rank.
Replacing A by A , we now assume that φ : A univ → A is surjective, and thus we have a closed immersion ψ :X →X univ . The monomials φ(z γ ), γ ∈ Π together define a surjective linear map C : Z r → Z m . Extending by the identity in the first n coordinates, we get a linear map Z n+r → Z n+m , represented by a matrix C satisfying CB univ =B. Suppose that t ∈ T = Hom Z n+m /BZ n , C * . Then composing t with C, we get an element t ∈ T univ = Hom Z n+r /B univ Z n , C * . Since C is surjective, the induced map ψ : T → T univ is injective and thus the inclusion of a subgroup.
We need to show that φ : C X univ T univ → C X T is an isomorphism. For surjectivity, suppose that f ∈ C X T . Then we may assume that f is a Laurent monomial in mutable and frozen variables. Let g ∈ C X univ be such that φ(g) = f . It is immediate that g is invariant under T , i.e., the weight wt(g) ∈ Z n+r /B univ Z n of g satisfies C wt(g) = 0 in Z n+m /BZ n . Thus, there exists u ∈B univ Z n such that C(wt(g) + u) = 0 ∈ Z n+m . The matrix C is the identity in the first n-coordinates, so the first n coordinates of wt(g) + u is 0. Let M = γ∈Π z −aγ γ , where (a γ ) are the last m coordinates of wt(g). Then by construction we have wt(gM ) + u = 0, i.e., wt(gM ) = 0 ∈ Z n+r /B univ Z n . Furthermore, φ(M ) = 1 and thus gM ∈ C X univ T univ satisfies φ(gM ) = f , proving surjectivity.
For injectivity, suppose that φ(g) = 0, where g ∈ C X univ T univ is nonzero. We have already shown that X univ /T univ is an irreducible affine variety in Section 4.4. The affine variety Spec C X T is thus identified with a subvariety of X univ /T univ of lower dimension. But this is impossible, since dim(X/T ) = n = dim(X univ /T univ ).
The isomorphism C[M D ] ∼ = C X T given by u γ → f γ now follows from Section 4.4.

4.6.
Proof of Theorem 4.3. The group Z 2n /B prin Z n can be naturally identified with the subgroup Z n = Z [1,n] ⊂ Z [1,2n] = Z 2n consisting of vectors which vanish in the last n-coordinates. Under this identification, the torus T prin has character lattice Z n , and the grading on A prin is given by (2.2). By Theorem 4.2, we have M D ∼ =X prin /T prin . It follows from wt(x i ) = e i that X prin /T prin is identified with the subvarietyX prin (1) ⊂X prin , where all initial cluster variables are set to 1. The function f prin γ onX prin (1) restricts to the rational function in y 1 , . . . , y n given by (see [31,Theorem 1.5]) Proposition 4.6. The rational functions {f γ (y) | γ ∈ Π} and {y 1 , . . . , y n } ∪ {F γ (y) | γ ∈ Π + } are related by an invertible monomial transformation.
The proof of Proposition 4.6 is delayed until Section 6.4.

4.7.
Proof of Proposition 3.8. Using τ , let us assume that γ = (0, j) so that x γ = x j is an initial mutable cluster variable. LetB be full rank of type D and let A B j denote the cluster algebra of type D 1 that is obtained by freezing the variable x j in A B . The extended exchange matrixB j is obtained fromB by removing the j-th row and we have The action of the cluster automorphism group T B extends to an action on A B j and we can identify

M D as an affine open in a projective toric variety
In this section, we show that the partial compactification M D is an affine open subspace of the projective toric variety X N (B ∨ ) associated to the cluster fan of B ∨ . The stratification (Proposition 3.5) of M D is inherited from the natural stratification of X N (B ∨ ) by torus orbits.
Our approach follows that of [3].

5.1.
Let C(y) = C(y 1 , . . . , y n ) denote the field of rational functions. Recall that for γ ∈ Π, Some examples of f γ (y) are computed in Examples 6.7 and 6.8.
Proof . There is a surjective ring homomorphism ϕ : C[u] → R B given by u γ → f γ (y). We already know that the kernel K of ϕ contains the ideal I D ⊂ C[u]. We need to show that the homomorphism ϕ :

5.2.
We give another description of R B ⊂ C(y). Let R(y) = P (y)/Q(y) ∈ C(y) be a rational function such that P (y), Q(y) ∈ Z[y] have positive integer coefficients. Then Trop(R(y)) is the piecewise-linear function on R n given by the formal substitution For example, Trop 3y 2 . Note that the coefficients are unimportant since, for example, The domains of linearity of the piecewise-linear function L(Y) = Trop(R(y)) define the structure of a complete fan on Proposition 5.2. The ring R B is equal to the subring of C(y) generated by rational functions R(y) satisfying Proof . Let R(y) be a Laurent monomial in {y i , F γ (y)}. By Theorem 2.4, the domains of linearity of the function L(Y) = Trop(R(y)) is a coarsening of the negative of the cluster fan −N (B ∨ ). Thus L(Y) is uniquely determined by b γ = L(−g γ ) as γ varies over Π, and g γ denotes a g-vector. As in the proof of Proposition 4.6, we have Thus the subring of rational functions R(y) satisfying (1) and (2) is exactly the subring R D .

The Laurent polynomial ring
n is the coordinate ring of an n-dimensional torus T y . Recall that F (y) = γ F γ (y). The following result is an application of [3, Section 10]. Proof . For any g ∈ R n , the quantity Trop(F (y))(g) is equal to the minimum value that the linear function Y → Y · g takes on the Newton polytope P of F (y). Thus by Theorem 2.4, the outer normal fan of P is equal to N B ∨ . Recall that a lattice polytope Q is called very ample if for sufficiently large integers r > 0, every lattice point in rQ is a sum of r (not necessarily distinct) lattice points in Q. For any lattice polytope Q, it is known that some integer dilation cQ is very ample. So let c ∈ Z >0 be such that cP is very ample and let {v 1 , . . . , v k } = cP ∩ Z n be the set of all lattice points in cP . For v ∈ Z n , let y v be the monomial with exponent vector v. Then X N (B ∨ ) can be explicitly realized as the closure of the set of points inside the projective space P k−1 . The polynomial F (y) c can be identified with a hyperplane section of X N (B ∨ ) in this projective embedding, and the affine open V := {F (y) = 0} is the complement of this hyperplane section. The coordinate ring C[V ] is generated by the functions [3,Section 10]), and we have C[V ] = R B as subrings of C(y). The theorem now follows from Theorem 5.1.
Question 5.4. Is P , the Newton polytope of F (y) = γ F γ (y), very ample? Is P normal? Question 5.5. Is the polynomial F (y) saturated? Question 5.6. Is every lattice point in P a sum of lattice points in P γ ? Fei [14] has shown that F γ (y) is saturated in the simply-laced case (and in more general situations). Thus Questions 5.5 and 5.6 are equivalent in that case.

5.4.
Proof of Theorem 3.3. By Theorem 2.3, the fan N B ∨ is a smooth, simplicial, polytopal, complete fan. Thus X N (B ∨ ) is a smooth projective toric variety and the torus-orbit closure stratification of X N (B ∨ ) is simple normal-crossing.

Properties of F -polynomials
We establish some technical properties of F γ (y) and F univ γ (z), following the approach of [8].
The statements are first established in the case of simply-laced D; the multiply-laced case follows from folding. Another closely related approach is that of [26], which would presumably avoid folding.
A key technical result is Theorem 6.6 which gives the values of the tropicalization Trop(f γ (y)) on a (negated) g-vector.
In this section, we will assume that D is a finite type Dynkin diagram whose underlying tree has been given an orientation, and we let B denote the corresponding exchange matrix. Recall that D ∨ denotes the dual Dynkin diagram, and we let B ∨ denote the exchange matrix of type D ∨ , satisfying the condition: 6.1. Let B be the exchange matrix corresponding to the oriented Dynkin diagram D. Let R Π be the vector space with basis indexed by Π, and write (p γ ) γ∈Π for a typical vector in R Π . Define Π + := {(s, i) | 1 ≤ s ≤ r i } ⊂ Π and let c = (c γ ) γ∈Π + denote a typical vector in R Π + . Following [8], we consider the c-deformed mesh relations where (t, j) ∈ Π + . (Compare with (2.3), and note that if i → j then B ij > 0, but if j → i then B ij < 0.) If c = 0, we call (6.1) the 0-mesh relations. For c = (c γ ) ∈ R Π + , we let E c ⊂ R Π denote the solutions to (6.1), and let U c := E c ∩ R Π ≥0 denote the intersection of E c with the positive orthant. Let π : R Π → R n denote the projection onto the coordinates p γ , where γ varies over {(r i , i) | i = 1, 2, . . . , n}. (Up to the action of τ −1 , this is the same as projection onto the initial cluster variables.) We use the notation U(D) c and E(D) c resp. U D ∨ c and E D ∨ c to denote these objects for B or D resp. B ∨ or D ∨ . In the following, e γ denotes the unit basis vector in R Π + >0 .
Proof of Theorem 2.4. By Theorem 6.1(3) the Newton polytope P γ of F γ (y) is π U D ∨ eγ . The Newton polytope P of γ F γ (y) is the Minkowski sum of the P γ , and by Theorem 6.1(1), we conclude that P is a generalized associahedron.
We let g ∨ γ denote the g-vector for B ∨ indexed by the element of Π(B ∨ ) corresponding to γ under the bijection of Section 2.5.

6.2.
Proof of Theorem 6.1. For D simply-laced, we have D = D ∨ and Theorem 6.1 is proven in [8]. We now prove it for multiply-laced D via folding.
Note that all monomials in F univ γ have the same weight moduloB univ Z n + span(e 1 , . . . , e n ).
Proposition 6.5. The sets are bases of Z n+r / B univ Z n + span(e 1 , . . . , e n ) and Z n+r /B univ Z n respectively.
Proof . The first statement implies the second. By Theorem 6.1(2), for γ ∈ Π + , we have wt F univ γ ∈ E(D ∨ ) eγ . The equations (6.1) define a linear map L : R Π → R Π + , sending (p γ ) to (c γ ). Let B be the last r rows ofB univ . By [27], see also [8,Section 8], the matrix B has rows given by −g ∨ γ . By [19, relation (6.13)], the g-vectors are solutions to the 0-mesh relations, and thus the kernel of L is exactly B Z n . We conclude that modulo B Z n , the last r entries of wt F univ γ is equal to the basis vector e γ . Returning to the vector wt F univ γ ∈ Z n+r , we obtain wt F univ γ = e γ modB univ Z n + span(e 1 , . . . , e n ).
Proof . First, assume that D is simply-laced so that g ω = g ∨ ω . For γ ∈ Π, let W γ ∈ D b (rep Q op ) be the object indexed by γ in the bounded derived category of representations of the quiver Q op corresponding to the reversed orientation of D, see [8,Section 3]. For any g ∈ R n , the quantity Trop(F γ (y))(g) is equal to the minimum value that the linear function Y → Y · g takes on the Newton polytope P γ . Now take γ ∈ Π + . Let G be the n × |Π| matrix whose columns are g ω . According to [8, Proof of Theorem 1], the map Y → Y · (−G) + v eγ is a diffeomorphism between P γ and U eγ , where v eγ ∈ E eγ is the integer vector given by (v eγ ) ω = dim Hom(W ω , W τ γ ). Here, Hom is taken within D b rep Q op . Furthermore, it follows from [8] that U eγ has nonempty intersection with every coordinate hyperplane. Thus, Trop(F γ (y))(−g ω ) = − dim Hom(W ω , W τ γ ).
Proof of Proposition 4.6. Let m(y) be a Laurent monomial in {y i , F γ (y)}, and denote by G = G(g) := Trop(m(y)) the piecewise-linear function that is the tropicalization of m(y). (Recall that by convention the variables g are the tropicalizations of the variables y.) The domains of linearity of the function G is a coarsening of −N B ∨ , so the function G is uniquely determined by the integer vector G −g ∨ γ | γ ∈ Π ∈ Z |Π| . By Theorem 6.6, any vector in Z |Π| can arise in this way. It follows that m(y) is uniquely determined by its tropicalization G by the formula m(y) = γ f γ (y) G(−g ∨ τ γ ) .

Examples of M D as a configuration space
The space M A n−3 can be identified with the configuration space of n distinct points on P 1 . In this section, we investigate similar descriptions of M D in the cases D = B n and D = C n . We also consider the question of whether M D is a hyperplane arrangement complement.
So far we have considered M D as a complex algebraic variety. However, the equations (3.1) make sense over the integers, and we may also consider M D as a scheme over Z. In particular, in this section we will also consider M D (F q ), the set of F q -points of M D , where F q is a finite field.
Throughout this section, we use the description of Π from Section 3.2. Table 1. g-vectors, F -polynomials, and f γ (y) in type B 3 .

1.
There exists a polynomial χ(t) so that χ(q) = #Z(F q ), where q = p m is a prime power with sufficiently large p.

3.
The cohomology ring H * (Z(C), C) is generated by the classes of dlog f i , where H i = {f i = 0}, and we have i dim(H i (Z(C), C)) t i = (−t) n χ(−1/t). Thus the Euler characteristic of Z(C) is equal to χ(1).

7.2.
Type A n . Let D = A n−3 with n ≥ 4. Then Π can be identified with the diagonals of a n-gon P n . We write u ij for the u-variable indexed by a diagonal (i, j). Then the relations defining M A n−3 are given by (1.1). These relations have appeared a number of times in the literature, for example see [1,11]. Let M 0,n denote the configuration space of n (distinct) points (z 1 , z 2 , . . . , z n ) on P 1 . Then the identification . This is the subset denotedX in Section 4. Then M 0,n can be identified with the quotient ofGr(2, n) by the diagonal torus T sitting inside GL n that acts on Gr(2, n). The isomorphism M 0,n ∼ =Gr(2, n)/T is an instance of Theorem 4.2 for D = A n−3 . In the Gr(2, n) cluster algebra, we have the primitive exchange relation where ∆ i,i+1 and ∆ j,j+1 are frozen variables. Thus (7.1), or equivalently u ij = ∆ i,j+1 ∆ i+1,j ∆ i,j ∆ i+1,j+1 , agrees with the formula for u γ in Theorem 4.2.
The geometry and topology of M 0,n is very well-studied; see for example [11]. We recall some basic facts in the context of Theorem 7.1. Gauge-fixing z 1 , z n−1 , z n to 0, 1, ∞, we have an identification M 0,n (k) = (z 2 , z 3 , . . . , z n−2 ) ∈ k n−3 | z i = z j and z i / ∈ {0, 1} for k a field. In particular, M 0,n (k) is the complement in k n−3 of the hyperplane arrangement with hyperplanes We may compute that #M 0,n (F q ) = (q − 2)(q − 3) · · · (q − n + 2). By Theorem 7.1, the number of connected components of M 0,n (R) is given by 7.3. Type B n . By Theorem 4.2, M D can be identified with A B /T B for any full rank extended exchange matrixB of type D. One such choice ofB, and thus of A B is given in [18,Example 12.10]. Let C[X n+2 ] be the ring generated by the Plücker coordinates of the Grassmannian Gr(2, n + 2). Recall that Γ is the two-element group whose non-trivial element maps i ↔ī. Consider the following functions in C[X n+2 ], labeled by Γ-orbits of sides and diagonals in the polygon P 2n+2 with vertices 1, 2, . . . , n + 1,1,2, . . . , n + 1 : LetV n be the space of 2 × (n + 2) matrices such that all the above functions are non-vanishing, and let T n+1 ∼ = (C × ) n+1 act onV n by scaling the first n + 1 columns. Then the action of SL 2 ×T n+1 onV n is free.
Using the action of SL 2 we can gauge-fix the last column of M ∈V n to [0, 1] T , and using the action of T n+1 , we may gauge-fix the first entry of columns 1, 2, . . . , n + 1 of M to 1. Thus modulo the action of SL 2 ×T n+1 , every point inV n can be written in the form where z i ∈ C, and two such matrices with parameters z = (z 1 , . . . , z n+1 ) and z = (z 1 , . . . , z n+1 ) are equivalent if z − z = c1, where 1 is the all 1-s vector. We may thus identify M Bn with a subspace of C n+1 /C = (z 1 , . . . , z n+1 )/C · 1. For these matrices, the cluster variables ∆ aā are equal to 1, and we have We recognize the hyperplanes (7.2) as the Shi arrangement [28].
Proposition 7.3. M Bn is isomorphic to the complement in C n+1 /C of the Shi arrangement.
Remark 7.4. Note that in contrast to the B n case ([18, Example 12.10]), theB-matrix of [18,Example 12.12] is full rank but not really full rank. For example, for n = 2 we may choose an initial cluster so that we havẽ respectively, where the rows are labelled by 13,11,12,13,23. This explains why the cluster automorphism group T in our discussion is disconnected.
On the locus where all cluster variables are non-vanishing, such 2 × (n + 1) matrices can be gauge-fixed, using S and (C × ) n+1 ⊂ T to the form: 1 1 · · · 1 1 z 1 z 2 · · · z n 1 (7.4) and the non-vanishing of the cluster variables is equivalent to the non-vanishing of the linear forms LetZ n denote the space of matrices of the form (7.4), where the linear forms (7.5) are nonvanishing. There is still a free action of Z/2Z onZ n , acting by swapping the two rows, which induces (z 1 , . . . , z n ) → (1/z 1 , . . . , 1/z n ). By Theorem 4.2, we obtain Proposition 7.5. We have an isomorphism M Cn ∼ =Z n /(Z/2Z).
Proof . For a field k, the points of M Cn (k) in general come fromZ n (k), wherek denotes the algebraic closure of k.
First, suppose that k = F q with char(q) > 2. LetF q denote the algebraic closure of F q . Then the Galois group is topologically generated by one generator σ, called the Frobenius automorphism. It acts as the field automorphism σ(x) = x q , and furthermore, we have σ(x) = x if and only if x ∈ F q . The map π :Z n (F q ) → M Cn (F q ) commutes with the action of σ. Thus if π(z) = u ∈ M Cn (F q ), we have u ∈ M Cn (F q ) ⇔ σ(u) = u ⇔ σ(π(z)) = π(z) ⇔ π(σ(z)) = π(z), and we have two possibilities: (1) σ(z) = z, or (2) σ(z) = 1/z. For case (1), we are just counting #Z n (F q ). Imposing the conditions (7.5), we get For case (2), the equation σ(z) = 1/z is equivalent to z q+1 i = 1 for i = 1, 2, . . . , n. There are q + 1 solutions to the polynomial equation x q+1 − 1 inF q . Two of the solutions are x = ±1. The other q − 1 solutions lie in F q 2 , since x q+1 = 1 =⇒ x q 2 −1 = 1 =⇒ x q 2 = x. Thus in case (2), we are counting n-tuples (z 1 , z 2 , . . . , z n ) ∈ F q 2 satisfying the condition x q+1 = 1 and the conditions imposed by (7.5). The conditions z i = ±1 are automatically satisfied, so we get In sum, taking into account the two-to-one coveringZ n → M Cn , we have (Note that this point count is also the same as the hyperplane arrangement complement with hyperplanes z i + z j , z i − z j , 1 + z i , 1 − z i , though we do not know an explanation for this.) Next let us consider k = R, sok = C. In this case, σ is replaced by complex conjugation. So the same argument says that we should consider the two cases (1)z = z, or (2)z = 1/z. For case (1), we are looking atZ n (R) and by Theorem 7.1 we get |π 0 (Z n (R))| = 2 n (n + 1)!
7.5. Type D n . We do not know a simple description of M D in this case. Indeed, the point counts we have obtained show that M D cannot be a hyperplane arrangement complement. In type D 4 , numerical computations indicate that for p = 2, we have Substituting p = −1 in (7.7), we get 547. We do not know for sure that M D 4 (R) has 547 connected components, but see Section 11.4. For type D 5 , numerical computations indicate that for p = 2, 3, we have where we define δ 3 (p) = 0 for p = 2 mod 3 or 2 for p = 1 mod 3 and similarly δ 4 (p) = 0 for p = 3 mod 4 or 2 for p = 1 mod 4. Substituting p = −1, we get 6388. We do not know whether M D 5 (R) has 6388 connected components, but see Section 11.4. 7.6. Type G 2 . Numerical computations give Substituting p = −1, we get 25, which we expect to be the number of connected components of M G 2 (R).

Positive part
In this section, we define the nonnegative subspace M D,≥0 of M D (R) and show that it is diffeomorphic to the generalized associahedron of D ∨ . Let P be an integer polytope and N (P ) denote its normal fan. It is well-known [20, Chapter 4] that the nonnegative part X N (P ),≥0 of the projective toric variety X N (P ) is diffeomorphic to the polytope P . The following result thus follows from Theorem 5.3. 8.2. The space M D has a distinguished rational top-form Ω(M D ), called the canonical form, that can be described in a number of ways. SupposeB is a full rank extended exchange matrix of type D. The cluster algebra A B has a natural top-form Ω which in any cluster (x 1 , x 2 , . . . , x n+m ) can be written (up to sign): which is the natural top-form on the corresponding cluster torus (C × ) n+m . The cluster automorphism group T B can be identified with a subgroup of (C × ) n+m , and the quotient group In the case D = A n−3 , we have M D = M 0,n , and the canonical form can be written as where (z 1 , z n−1 , z n ) = (0, 1, ∞) as in Section 7.2, and denominator factors equal to ∞ are understood to be omitted. This form is also called a cell-form in [9] and the condition that Ω((M 0,n ) >0 ) only has poles along the boundary divisors of M 0,n and not elsewhere in M 0,n is [9, Proposition 2.7]. Combining with the above discussion, we have

Positive tropicalization
In this section, we consider the positive tropicalization of M D . We use our results to resolve a conjecture of Speyer and Williams [29] on positive tropicalizations of cluster algebras of finite type. We refer the reader to [29] for background on positive tropicalizations. Let R = ∞ n=1 R t 1/n denote the field of Puiseux series over R. We define val : R → R ∪ {∞} by val(0) = ∞ and val(x(t)) = r if the lowest term of x(t) is equal to αt r , where α ∈ R × . We define R >0 ⊂ R to be the semifield consisting of Puiseux series x(t) that are non-zero and such that coefficient of the lowest term is a positive real number.
A point u(t) ∈ M D (R >0 ) is a collection u(t) = {u γ (t) ∈ R >0 , γ ∈ Π} of (positive) Puiseux series satisfying the relations from Definition 3.1. We define the positive tropicalization Trop >0 M D as the closure of valuations Lemma 9.1. The subspace Trop >0 M D is a (not complete) polyhedral fan inside the linear space R Π .
Proof . We use the identification M D ∼ =X prin /T prin from Theorem 4.3. According to Proposition 4.6, there is an invertible monomial transformation between the functions u γ = f γ and the set of functions {y 1 , . . . , y n } ∪ {F γ (y) | γ ∈ Π + }. Since each F γ (y) is a positive Laurent polynomial in the y i -s, it follows that each u γ (y) is a subtraction-free rational function in the y i -s. Thus the map (u γ ) γ∈Π → (y 1 , y 2 , . . . , y n ) induces an isomorphism M D (R >0 ) ∼ = R n >0 . It follows that we have a homeomorphism Trop >0 M D ∼ = R n . The embedding of Trop >0 M D in the (larger dimensional) linear space R Π endows it with the structure of a polyhedral fan. Proof . Under the isomorphism Trop >0 M D ∼ = R n , the fan structure of Trop >0 M D gives a complete fan in R n whose maximal cones are the common domains of linearity of the piecewise linear functions Trop(u γ (y)). By Proposition 4.6, we can equivalently take the common domains of linearity of the functions Trop(F γ (y)), γ ∈ Π + . It is well-known (see for example [6, Section 11.1]) that the resulting fan is the normal fan of the Newton polytope of the Laurent polynomial γ∈Π + F γ (y). By Theorem 2.4, we deduce the isomorphism of fans Trop >0 M D ∼ = N (B ∨ ). Now let A B denote a cluster algebra of finite type, and X B the corresponding cluster variety. Using the set of cluster variables x γ , γ ∈ Π and coefficient variables x n+1 , . . . , x n+m , we have an embedding X B → C |Π|+m . We define Trop >0 X B ⊂ R |Π|+m as the closure of the image of X B (R >0 ) under the map val : R >0 → R. Note that the tropicalization Trop >0 X B depends only on the cluster algebra A = A B and not on the choice of initial cluster.
The projection map (x γ , x i ) → (x 1 , x 2 , . . . , x n+m ) from R |Π|+m to R n+m (onto the initial cluster variables) identifies Trop >0 X B with a complete polyhedral fan in R n+m .

Extended and local u-equations
In this section, we first study two additional sets of equations satisfied by u-variables, the extended u-equations and the local u-equations. In other words, we give some further distinguished elements in the ideal I D .
In type A, the extended u-equations were used by Brown [11] to define what we call M A n−3 ; we see here that they can be interpreted as arising from all the exchange relations of the cluster algebra, rather than just the primitive exchange relations. We now present explicitly extended u-equations for all the classical types A, B, C, D, and also for type G 2 . In types A and D, the only extended u-equations we know come from Corollary 4.5, and we conjecture that in simply-laced types these are the only ones. In types B and C, we find more extended u-equations than those from Corollary 4.5. Indeed, any extended u-equation for type A 2n−3 (resp. D n ) gives one for type C n−1 (resp. B n−1 ), but not all of these come from Corollary 4.5. We conjecture that all extended u-equations in multiply-laced type come from folding.
A similar analysis of extended u-equations for the types E and F can be found by a lengthy, but finite computation which we do not present here.
In the following, we will use the indexing of Π from Section 3.2. For two disjoint subsets I and J, define Note that U I,J and U J,I are not necessarily equal. 3) becomes the primitive u-equations, R ij = 0. As discussed in [4,11], it is natural to interpret these U 's as cross-ratios of n points on P 1 : which is the primitive u-equation R [11] = 0. For C 2 , in addition to the 6 primitive u-equations given in (3.3), we have 3 more equations: u [11] u [12] + u [23] u Note that B can be empty here, in which case we have U A +Ũ C = 1. In the first and second type we have 4 n 4 +3 n 3 and 6 n 3 +2 n 2 equations, respectively, thus in total there are n(n−1)(n 2 +4n−6) 6 extended u-equations. It is not difficult to see that this is equal to the number of (unordered) pairs of exchangeable cluster variables. We have n 2 cluster variables, 1 4 (n 4 + n 2 − 2n) pairs of compatible cluster variables, and 2 n 4 pairs of cluster variables where the compatibility degree is greater than one. The remaining pairs of cluster variables are exchangeable.
It is straightforward to obtain the primitive u-equations for type D n . In 10.1.4. Type B. Finally, we consider type B n−1 by folding D n . We identify u i = uĩ, and the two types of equations become We have n(n−1)(n 2 +n−3) 10.1.5. Type G. As in Section 3.2.5, let us call the 8 u-variables a i , b i for i = 1, . . . , 4 that can be thought of as labelling the edges of an octagon (a 1 , b 1 , . . . , a 4 , b 4 ). Folding D 4 , we obtain 18 extended u-equations for G 2 , the primitive u-equations together with

10.2.
Local u-equations. The relations R γ , and the extended u-equations are global in nature: they involve many u γ variables which are "far away from each other". We now describe a class of local u-equations. Using them one can show that all u-variables can be solved rationally in terms of u-variables of any acyclic seed; see also [4]. This has implications for canonical forms; see (12.1). We first recall the X-coordinates for Fock and Goncharov's cluster X-variety. For an exchange relation xx = M +M , we have a cluster X-variable X = M/M . Now, let us consider a primitive exchange relation in X B : where M only involves frozen variables. We recall that the isomorphism of Theorem 4.2 identifies the rational function M xτγ xγ with u γ . Thus the cluster X-variable X γ := M/M can be identified with u γ /(1 − u γ ) ∈ C[M D ] which is equal to a Laurent monomial in the u ω -s using the relations R γ . By [17] or [19,Proposition 3.9 or equation (8.11)] the variables X γ satisfy the relation (10.7) or, equivalently, the variables u γ satisfy the relation For the convenience of the reader, we give some examples of (10.7), noting that the passage between X-variables and u-variables is completely compatible with folding.

Type A.
For type A n−3 we have u-variables u i,j for 1 ≤ i < j−1 < n, corresponding to the n(n−3)/2 diagonals of n-gon. We have the same number of local u-equations, one for each "skinny" quadrilateral: 10.2.2. Type C. By folding A 2n−3 , we obtain local u-equations for type C n−1 . The local u-equations take the form of (10.8); for the special case of j =ī, it reads 10.2.3. Type D. We use the notation for u-variables from Section 3.2.3. The n 2 local uequations for type D n read 10.2.4. Type B. By folding type D n , we obtain local u-equations for type B n−1 (see Section 3.2.4). The local u-equations are together with cyclic rotations.
10.2.5. Type E. Finally, for E n with n = 6, 7, 8, we use the identification (2.1) to index u and X variables. WhenB is bipartite, i.e., every vertex is either a source or a sink in the induced orientation of D, we have r i = h/2 + 1 does not depend on i, where h is the Coxeter number. The Coxeter number is even in types E 6 , E 7 , E 8 , and Π is identified with m = 7, 10, 16 copies of I respectively. We index the nodes of E n as shown below: The local u-equations take the following form: for odd i < n, i+1) ), for even i < n, for t = 1, 2, . . . , m in all these cases. We have n × m equations in total.
10.2.6. Types F 4 and G 2 . By folding E 6 we obtain local u-equations for F 4 , and by folding D 4 we obtain those for G 2 .

Connected components and sign patterns
The permutation group S n acts on the moduli space M 0,n (R), permuting the n points and permuting the connected components. The presentation of M 0,n (R) using u ij and the relations (1.1) depends on the choice of a dihedral ordering and the action of the symmetry group S n is obscured. In this section, we use the extended u-equations to investigate the connected components of M D (R), with the hope of uncovering an appropriate symmetry group for M D in other types. While our results here are more speculative, we are able to construct new classes of u-equations. A further motivation for studying connected components of M D (R) is the application to string amplitudes where it is important to consider canonical forms of different connected components of M 0,n (R); see Section 12.2 for a brief discussion.
We also define an analogue of an oriented matroid for M D , called a "consistent sign pattern", and it is conjectured that the number of consistent sign patterns is equal to the number of connected components. 11.1. Consistent sign patterns. A consistent sign pattern for type D is an element (s γ ) ∈ {+, −} Π such that for each extended u-equation (10.1) γ u αγ γ + γ u βγ γ = 1, we have that at least one of the signs γ s αγ γ and γ s βγ γ is positive. In other words, (s γ ) are possible signs for some solution (u γ ) of extended u-equations. We could also call a consistent sign pattern a "uniform oriented matroid" for the u-variables.
In [4], we made the following conjecture. 11.2. Type A. We consider D = A n−3 . The space M 0,n (R) has (n−1)!/2 connected components, corresponding to the dihedral orderings of n points. In the positive connected component (M 0,n ) >0 , all the cross ratios u ij are positive, indeed, we have 0 < u ij < 1. In other connected components of M 0,n , some of the u ij -s are negative. We find that the extended u-equations exclude those sign patterns for which both [a, b|c, d] and [b, c|d, a] are negative. Empirically, we find that precisely (n−1)!/2 consistent sign patterns are allowed by the extended u-equations (10.3), and this count agrees with the number of connected components of M 0,n (R).
Let us now consider the problem of finding new u-variables for other components of M 0,n (R), that is, we seek cross-ratios that are positive on that component. It suffices to consider the ordering that is obtained from the standard one by an adjacent transposition, e.g., (1 , 2 , 3 , . . . , n ) = (2, 1, 3, . . . , n) = −u 1,3 u 2,4 · · · u 2,n , and all other u's are unchanged. These new u ij -s are positive in the connected component given by the ordering (2, 1, 3, . . . , n), and furthermore they satisfy the extended u-equations (for this ordering). In other words, the (invertible) signed monomial transformation (11.1) sends the extended u-equations for u ij to a permutation of the extended u-equations for the u i,j , and thus exposing a hidden S n -symmetry of these equations.

11.
3. Type C. From the analysis of Section 7.4, we know that M Cn (R) has 2 n−1 n!(n + 1) connected components. Computationally, we find that this agrees with the number of consistent sign patterns. There are two types of components in M Cn (R) corresponding to two types of configurations of the (2n+2)-gon with labels i,ī for i = 1, 2, . . . , n+1: (A) 2 n−1 n! components for polygons with central symmetry, e.g., 1, 2, . . . , n+1,1,2, . . . , n+1; (B) 2 n−1 (n+1)! components for polygons with reflection symmetry avoiding vertices, e.g., the ordering 1, 2, . . . , n+1, n+1, . . . ,2,1. Unlike type A, our investigations indicate that the compactification (arising from u-equations) of these two types of components have differing boundary combinatorics: for any component in (A), combinatorially it is a cyclohedron (the generalized associahedron of type C), while for any component in (B), combinatorially it is an associahedra. We expect that this can be proven via a careful analysis of the extended u-equations, and here we illustrate it for the simplest example, The extended u-equations are given by (3.3) and (10.4), plus cyclic rotations. The positive part with all u-s positive corresponds to the ordering 1, 2, 3,1,2,3, which cuts out a hexagon. We can see the other 3 orderings in (A) by making a signed monomial transformation of the u-variables. For example, for the ordering 2, 1, 3,2,1,3, we find that the 6 new variables can be obtained by a monomial change of variables: u [12] = 1 u [21] u [22] , u [23] = − u [13] u [12] u [22] u [23] , u [31] = 1 u [22] u [23] , u [11] = u [22] , u [22] = u [11] u 2 [21] u [22] , It is straightforward to check that these 6 new variables satisfy identical extended u-equations for the ordering 2, 1, 3,2,1,3. More generally, under this kind of transformation, similar to the type A n−3 case, we find that for any ordering in (A) the new variables satisfy identical extended u-equations, and the corresponding component is combinatorially a cyclohedron.
Note that u [22] is special: from the third and the last equations, it is easy to see that u [22] cannot take the value 0, and thus u [22] = 0 does not correspond to a facet. The other 5 variables do correspond to facets, and from the equations we see that requiring all u ≥ 0 cuts out a pentagon instead. In general, we expect that for any ordering in (B), such a transformation give equations of this type, where certain u cannot reach zero, and (an appropriate closure) of the component has the combinatorics of a (type A) associahedron.

11.4.
Types D 4 and D 5 . We were unable to determine the number of connected components of M Dn (R). However, we can obtain a consistency check by comparing the number of consistent sign patterns with the point count over F q . Recall from the point count (7.7) in type D 4 , we predicted that M D (R) has 547 connected components. By a direct computation, we checked that this is equal to the number of consistent sign patterns of u-variables with respect to the extended u-equations in Section 10.1.3. Similarly, the prediction of 6388 in the D 5 case computationally agrees with the number of consistent sign patterns of u-variables. 11.5. Type G 2 . As we have discussed in Section 7.6, we expect that there are 25 different connected components for G 2 . Furthermore, the point count M G 2 (F q ) is not polynomial, and thus M G 2 is not a hyperplane arrangement complement. Computationally, we find that there are 25 consistent sign patterns for the 18 extended u-equations from Section 10.1.5.
We now investigate the connected components of M G 2 (R), and note some new features. The positive component has 0 < a i , b i < 1 as usual. But suppose b 1 is made negative; to wit we put b 1 = −b 1 with b 1 > 0. As in our discussion for types A and C, we rearrange all the extended u-equations to put them again in the form of (monomial 1 ) + (monomial 2 ) = 1. Quite nicely, the 36 exponent vectors of these monomials lie in an 8-dimensional cone, that is, all 36 vectors can be expressed as a positive linear combination of eight of them. The 8 generators can be associated with the new variables and using these variables, we get 18 equations (monomial) 1 + (monomial) 2 = 1, written in terms of the x's and y's, all with positive exponents. Eight of these are "primitive" u-equations x 1 + x 3 x 2 4 = 1, x 2 + x 4 x 5 y 2 1 y 2 y 2 3 = 1, x 3 + x 1 x 5 y 4 1 y 3 2 y 2 3 = 1, x 4 + x 1 x 2 y 2 1 y 2 2 y 3 = 1, x 5 + x 3 2 x 3 = 1, y 1 + y 4 1 x 3 2 x 2 3 x 3 4 y 3 2 y 2 3 = 1, y 2 + y 2 2 x 2 x 3 x 2 4 y 2 1 y 2 3 = 1, y 3 + y 2 3 x 2 2 x 3 x 4 y 2 1 y 2 2 = 1.
complement and it would thus be interesting to systematically study the question of connected components directly from the u-equations defining the space, as we have done in some examples in Section 11. It is natural to conjecture that some or all of the other real components of cluster configurations spaces (suitably compactified using the u-equations) are also positive geometries, and it would be interesting to determine their canonical forms.
In connection with determining canonical forms for general components, we state here without proof, a simple expression for the canonical forms of the positive component we have studied above, not in terms of cluster variables, but directly in terms of u-variables. Recall that the u γ are in bijection with all the cluster variables. Consider any acyclic cluster (x γ 1 , . . . , x γn ). Then, the canonical form is simply given by taking the wedge product . (12.1) As we noted in Section 10.2, acyclic seeds have the following feature: all the u γ variables can be expressed rationally in terms of those in the initial seed. This idea can be extended to give canonical forms for other connected components of cluster configuration spaces. As a simple example, let us consider the description of the component discussed in Section 11 for the G 2 case, where b 1 = −b 1 < 0. We can readily check that all of the (x, y) variables can be rationally solved for in terms of either (x 2 , x 3 ) or in terms of (x 3 , x 4 ). The canonical form is then given as

12.2.
Open and closed cluster string amplitudes. The stringy canonical forms of [3] can be applied to the cluster configuration space M D , and we obtain the cluster string amplitude. where {s γ | γ ∈ Π}∪{s i | i ∈ [n+1, n+m]} are parameters chosen so that the product γ∈Π x α sγ γ is T -invariant, and thus descends to a function on M D,>0 . ChoosingB =B prin , we may use Theorem 4.3 to rewrite (12.2) as where (X, {c}) are related linearly to (s i , s γ ). By [3, Claim 2], (12.3) converges when the point X belongs to the generalized associahedron P (c) = γ∈Π + c γ P γ , where P γ is the Newton polytope of F γ (y). By [3, Claim 2] the leading order of I D (X, {c}) is the canonical function Ω(P (c)) of P (c) evaluated at X: lim α →0 (α ) n I D (X, {c}) = Ω(P (c))(X).
(We refer the reader to [2,3] for background on canonical functions and canonical forms.) In particular, the poles of I D (X, {c}) as α → 0, all of which are simple, correspond bijectively to the facets of the generalized associahedron of D ∨ . By [3, Section 9] and Theorem 6.6, we may also rewrite Ω(M D,>0 ) γ∈Π u α Uγ γ (12.4) and the convergence condition is the very simple condition U γ > 0. By Proposition 4.6, the u γ and {y i , F γ } are related by an invertible monomial transformation, and thus {U γ | γ ∈ Π} and {X 1 , . . . , X n } ∪ {c γ | γ ∈ Π + } are related by an invertible linear transformation. (The matrix of this linear transformation has entries given by the integers Trop(F γ (y)) −g ∨ ω that appeared in Section 6.) We see from (12.4) that the u-variables u γ are reverse-engineered from the cluster string integral: they are those monomials in cluster variables making the domain of convergence explicit.
As explained in [3,Section 7], for generic exponents X, we expect that varying the cycle of integration (to something other than the cycle M D,>0 ) will span a space of integral functions of dimension equal to the absolute value of the Euler characteristic |χ(M D (C))|. Indeed, it is especially natural to integrate over any of the other real connected components of the cluster configuration space, directly generalizing the basis of all (tree-level) open string amplitudes associated with type A.
We can also define the analog of "closed string" cluster amplitudes. The simplest object we can define (as in [3]) is the "mod square" of the open string integral where in order for the integrand to be single-valued, we must have that the exponentsŪ γ differ from U γ at most by integers.
As we have remarked, it is plausible that real components of cluster configuration space other than the region associated with u γ ≥ 0 provide us with many different positive geometries M It is clear that a complete understanding of the space of open and closed string integrals will go hand-in-hand with a similarly complete understanding of the space of all connected real components of the cluster configuration space.
12.3. Beyond finite type. Finally, the most obvious open question is whether the notions of cluster configuration space presented in this paper can naturally be extended beyond finite-type cluster algebras. It is interesting to note that, as we have seen in (12.4) above, in the finite type case, the introduction of the u-variables is naturally reverse engineered, starting from the definition of the cluster string amplitude, see also [3]. This definition can be extended in various ways to define natural "compactifications" of the infinite-type configuration spaces, as recently been explored for the case of Grassmannian cluster algebras [5]. In these examples, the reverseengineering of u-variables does not work as it does in finite type: amongst other things the polytope capturing the combinatorics of the boundary structure in these cases is typically not simple. But there may be other choices of stringy integral that are more natural from the perspective of finding good u-variables and "binary" realizations of general cluster configuration spaces.

A A lemma in commutative algebra
Lemma A.1. Let f : A → B be a surjective homomorphism of Noetherian commutative rings with identity. Let S ⊂ A be the multiplicative set generated by elements x 1 , x 2 , . . . , x p such that f (x 1 ), f (x 2 ), . . . , f (x p ) are not zero-divisors in B. (A.1)
Then f is an isomorphism.
Proof . Let K denote the kernel of f . Let a ∈ K be a nonzero element. Suppose that x i a = 0 in A for all i. Then the image of a in S −1 A is nonzero and it is in the kernel of S −1 f : S −1 A → S −1 B. This contradicts (1). Thus M a = 0 for some monomial M in the x i -s. Replacing a by M a for some other monomial M , and using (A.1), we may assume that a ∈ K and x i a = 0 for some i = 1, 2, . . . , p.
If a ∈ (x i ), then by (A.1), we have a = x i a 1 for a nonzero element a 1 ∈ K. Repeating, we either find a nonzero element a ∈ K such that a / ∈ (x i ), or we have an ascending chain of ideals (a) ⊂ (a 1 ) ⊂ (a 2 ) ⊂ · · · . In the former case, the image of a in A/(x i ) is nonzero and in the kernel of f i , contradicting (2). Thus we are in the latter case. Since x i is not a unit and A is Noetherian, the chain of ideals stabilizes to a proper ideal (a ) = I A, and we thus have (a ) = (a ), where a = x i a and x n a = 0 for some n > 0. This is impossible: letting m be minimal such that x m i a = 0 we find that x m−1 i a = 0 which implies x m−1 i a = 0, a contradiction.