Cluster Configuration Spaces of Finite Type
For each Dynkin diagram , we define a ''cluster configuration space'' ℳ and a partial compactification ℳ˜. For = ₙ₋₃, we have ℳₙ₋₃ = ℳ₀,ₙ, the configuration space of points on ℙ¹, and the partial compactification ℳ˜ₙ₋₃ was studied in this case by Brown. The space M˜ is a smooth...
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| description | For each Dynkin diagram , we define a ''cluster configuration space'' ℳ and a partial compactification ℳ˜. For = ₙ₋₃, we have ℳₙ₋₃ = ℳ₀,ₙ, the configuration space of points on ℙ¹, and the partial compactification ℳ˜ₙ₋₃ was studied in this case by Brown. The space M˜ 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 ℳ˜ are generated by coordinates uγ, in bijection with the cluster variables of type , 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.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 092, 41 pages
Cluster Configuration Spaces of Finite Type
Nima ARKANI-HAMED a, Song HE bcde and Thomas LAM f
a) School of Natural Sciences, Institute for Advanced Studies, Princeton, NJ, 08540, USA
E-mail: arkani@ias.edu
b) CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing, 100190, China
E-mail: songhe@itp.ac.cn
c) School of Fundamental Physics and Mathematical Sciences,
Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
d) ICTP-AP International Centre for Theoretical Physics Asia-Pacific,
Beijing/Hangzhou, China
e) School of Physical Sciences, University of Chinese Academy of Sciences,
No.19A Yuquan Road, Beijing 100049, China
f) Department of Mathematics, University of Michigan,
530 Church St, Ann Arbor, MI 48109, USA
E-mail: tfylam@umich.edu
URL: http://math.lsa.umich.edu/~tfylam/
Received January 05, 2021, in final form October 04, 2021; Published online October 16, 2021
https://doi.org/10.3842/SIGMA.2021.092
Abstract. For each Dynkin diagram D, we define a “cluster configuration space” MD and
a partial compactification M̃D. For D = An−3, we have MAn−3 = M0,n, the configuration
space of n points on P1, and the partial compactification M̃An−3
was studied in this case by
Brown. The space M̃D 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 M̃D are generated by coordinates uγ , 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.
Key words: configuration space; cluster algebras; generalized associahedron; string ampli-
tudes
2020 Mathematics Subject Classification: 05E14; 13F60; 14N99; 81T30
1 Introduction
1.1. The configuration space M0,n of n distinct points on P1 is a smooth affine algebraic
variety of dimension n−3, and it has a very well-studied Deligne–Knudsen–Mumford compact-
ification M0,n, which is a smooth projective algebraic variety. The boundary of M0,n consists
of 2n−1−n−1 divisors, satisfying factorization: each divisor is itself a product M0,n1 ×M0,n2 ,
where n1+n2 = n+2.
The real points M0,n(R) have the structure of a smooth real manifold with (n−1)!/2 con-
nected components. Fixing once and for all a (dihedral) ordering on n points, we let (M0,n)>0 ⊂
M0,n(R) denote the connected component where the n points are ordered on P1(R) = S1.
mailto:arkani@ias.edu
mailto:songhe@itp.ac.cn
mailto:tfylam@umich.edu
http://math.lsa.umich.edu/~tfylam/
https://doi.org/10.3842/SIGMA.2021.092
2 N. Arkani-Hamed, S. He and T. Lam
The closure (M0,n)≥0 of (M0,n)>0 in M0,n(R) is a stratified space that is homeomorphic to the
face stratification of the associahedron polytope.
LetW denote the union of those boundary divisors in M0,n whose intersection with (M0,n)≥0
is empty, and let M̃0,n := M0,n \W . The divisors that do intersect (M0,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, M0,5 has ten boundary divisors, and M̃0,5 includes five of them, corresponding
to the five sides of the pentagon (the associahedron of dimension two). Somewhat surprisingly,
the partial compactification M̃0,n is an affine algebraic variety, and its ring of regular functions
has the following description. Let uij be variables labeled by the diagonals (i, j) (not including
sides) of a n-gon Pn. Then C
[
M̃0,n
]
is isomorphic to the polynomial ring C[uij ] modulo the
relations
Rij := uij +
( ∏
(k,ℓ) crossing (i,j)
ukℓ
)
− 1, (i, j) varying over all diagonals, (1.1)
and M0,n ⊂ M̃0,n is the locus where uij ̸= 0. The uij 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 uij are cross-ratios (see (7.1)) onM0,n and appeared
in the study of scattering amplitudes in string theory and for the bi-adjoint ϕ3-theory [1].
1.2. In this paper, we construct in an analogous manner two affine algebraic varietiesMD⊂M̃D
for each Dynkin diagram D of finite type by considering the relations
Rγ := uγ +
∏
ω
u(ω||γ)ω − 1. (1.2)
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 MD the cluster configuration space of type D. In the
case D = An−3, we have MAn−3 = M0,n and M̃An−3 = M̃0,n. Amongst many remarkable
properties of these relations, let us immediately note that uγ = 0 forces uω = 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 M̃D is an example of the notion of “binary
geometry” discussed therein.
Whereas M̃0,n has a stratification indexed by the faces of the associahedron, the space M̃D
has a stratification (Proposition 3.5) indexed by the faces of the Chapoton–Fomin–Zelevinsky
generalized associahedron for D∨ [12, 17]. We show (Theorem 3.3) that MD and M̃D are
smooth affine algebraic varieties and that the boundary stratification of M̃D is simple normal-
crossing. These geometric properties depend on integrality properties of the normal fan N (D∨)
of the generalized associahedron, and an isomorphism (Theorem 5.3) between M̃D and an affine
open subset of the projective toric variety XN (D∨) associated to N (D∨). Like (M0,n)≥0, the
variety M̃D contains a distinguished nonnegative part MD,≥0, which is a stratified space home-
omorphic to the face stratification of the generalized associahedron (Theorem 8.1). The positive
part MD,>0 ⊂ MD(R) is a distinguished connected component in MD(R), and is cut out by the
conditions uγ > 0. Though M̃D is not compact,
(
M̃D, (MD,≥0)≥0
)
satisfies the other properties
of a positive geometry in the sense of [2].
1.3. The configuration space M0,n is isomorphic to the quotient of an open subset G̊r(2, n) ⊂
Gr(2, n) of the Grassmannian of 2-planes by the diagonal torus T ⊂ SLn acting on Gr(2, n).
Let B̃ be a full rank acyclic extended exchange matrix of typeD. Let A
(
B̃
)
be the corresponding
cluster algebra, X
(
B̃
)
= Spec
(
A
(
B̃
))
be the cluster variety, and let X̊
(
B̃
)
⊂ X
(
B̃
)
denote
Cluster Configuration Spaces of Finite Type 3
the locus where all cluster variables are non-vanishing. We show (Theorem 4.2) that MD is
isomorphic to the (free) quotient of X̊
(
B̃
)
by the cluster automorphism group T
(
B̃
)
, generalizing
the construction of M0,n from Gr(2, n). The functions uγ are particular T
(
B̃
)
-invariant rational
functions on X
(
B̃
)
. The uγ 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 uγ 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 betweenMD and cluster X -varieties;
for example, MD does not contain a collection of (cluster) torus charts.
Our approach depends crucially on the flexibility in the choice of B̃. When B̃ = B̃univ 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
(
B̃univ
)
by setting all mutable cluster
variables to 1, and sending the universal frozen variables zγ to uγ . The non-primitive exchange
relations give rise to other relations of the form U + U ′ = 1, where U and U ′ are monomials in
the uγ-s.
When B̃ = B̃prin is the extended exchange matrix for the principal coefficient cluster algebra,
the functions uγ become identified with certain ratios of the F -polynomials Fγ(y). Bazier-Matte,
Douville, Mousavand, Thomas, and Yildrim have shown [8] in the case that D is simply-laced
that the Newton polytope of Fγ(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 M̃D with an open subset of the toric variety XN (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>0MD 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 M0,n, we proceed with studying the topology of MD(C)
and MD(R). We identify MBn with the complement to the Shi-hyperplane arrangement
and thereby compute point counts over finite fields, and the Euler characteristics of MBn(R)
and MBn(C). We give a configuration space style description of MCn (Proposition 7.5) but
were not able to determine whether MCn is a hyperplane arrangement complement. Never-
theless, we were able to compute the point count for MCn(Fq), and the number of connected
components of MCn(R). We found numerically the point counts for types D4, D5 and G2, and
obtained numerically that the point count of MD4(Fq) over a finite field Fq 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,
I =
∫
Rn
>0
∏
i
dxi
xi
xα
′Xi
i
∏
j
pj(x)
−α′cj , (1.3)
where pj(x) is a positive Laurent polynomial. We showed in [3] that the leading order
limα′→0(α
′)nI is a rational function that for fixed cj-s coincides with the canonical rational
function [2] of the Minkowski sum of the Newton polytopes of pj(x). Tree-level n-point open
superstring amplitudes are integrals on M0,n. It turns out that for a suitable parametrization
4 N. Arkani-Hamed, S. He and T. Lam
of M0,n, these amplitudes can be written as an integral IAn−3 in the form (1.3), where the pj(x)
are the F -polynomials for the type An−3 cluster algebra. The importance of the uij-variables
appears in the rewriting (see [3, Section 9] or [10, Section 3])
IAn−3 =
∫
(M0,n)>0
Ω((M0,n)>0)
∏
(i,j)
u
α′Xij
ij
of the open-string amplitude. The poles of IAn−3 are given by Xij = 0, and at this pole, the
factorization of IAn−3 mimics the factorization of the equations (1.1). We define the cluster
string amplitude (where xγ , xi are cluster variables of A
(
B̃
)
of type D)
ID :=
∫
MD,>0
Ω(MD,>0)
∏
γ∈Π
x
α′sγ
γ
n+m∏
i=n+1
xα
′si
i .
The poles of ID are made manifest by rewriting in terms of the uγ-s, and the leading order of ID
is controlled by the combinatorics of the generalized associahedron of D∨.
2 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 = (aij) 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 = (Bij) has type D if
aij =
{
2 if i = j,
−|Bij | if i ̸= j.
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
Bij =
−aij if i→ j,
aij if j → i,
0 otherwise.
For an (n+m)×n extended exchange matrix B̃ extending B, we let A
(
B̃
)
denote the correspon-
ding cluster algebra of geometric type [18]. By convention, A
(
B̃
)
is the C-algebra generated
by all mutable cluster variables, all frozen variables, and the inverses of all frozen variables.
We let X
(
B̃
)
= SpecA
(
B̃
)
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 that B̃ (or A or X) has full rank if B̃ has rank n. We say that B̃ (or A or X)
has really full rank if the rows of B̃ span Zn. If B̃ has full rank, then X
(
B̃
)
is a smooth affine
algebraic variety [24, Theorem 7.7].
Cluster Configuration Spaces of Finite Type 5
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 xγ ∈ A
(
B̃
)
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 {γ1, . . . , γn}.
The set Π can be identified with the following set of pairs of integers:
Π =
⊔
i∈I
{(s, i) | 0 ≤ s ≤ ri}, (2.1)
where ri, 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 {csωi | 0 ≤ s ≤ ri}. 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 An, D2n+1, E6, and in these types ∗ : I → I is the non-trivial automorphism of D (as a
graph). We shall use the notation (−1, i) := (ri∗ , i
∗); see [31, Proposition 1.3].
2.5. Each Dynkin diagram D has a dual denoted D∨ defined by requiring that the Cartan
matrix of D∨ be transpose to that of D. Note that D and D∨ have the same underlying tree.
If B is an exchange matrix of type D, then we let B∨ be the exchange matrix of type D∨
associated to the same orientation of the underlying tree of D and D∨. For dual exchange
matrices B and B∨, the cluster variables Π(B) and Π
(
B∨) are naturally in bijection and under
this bijection the cluster complexes are isomorphic.
2.6. For γ, ω ∈ Π, we let (ω||γ) denote the compatibility degree, defined for example in [31,
Proposition 5.1]. In [31], the dependence of the compatibility degree on the choice of c (equivalent
to our choice of B) is made explicit, but we have suppressed this dependence in our notation.
By [31, Section 5], the compatibility degrees for different choices of B are equivalent under
an appropriate renaming of Π. Examples of the compatibility degree are given in Section 3.2.
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 Aprin = A
(
B̃prin
)
denote the cluster algebra with principal coefficients [19]. Thus
B̃prin 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 x1, x2, . . . , xn and the principal
frozen variables are denoted y1, y2, . . . , yn. We have a Zn-grading on the principal coefficient
cluster algebra Aprin given by
deg(xi) = ei and deg(yi) = −Bei. (2.2)
6 N. Arkani-Hamed, S. He and T. Lam
Each mutable cluster variable is homogeneous with respect to this grading, and we define the
g-vector by gγ := deg(xγ) for γ ∈ Π.
2.8. For γ ∈ Π+, define the F -polynomial Fγ(y) by setting the initial cluster variables to 1 in
the Laurent expansion of the cluster variable xprinγ in the cluster algebra Aprin with principal
coefficients:
Fγ(y) := xprinγ (xi = 1, y1, y2, . . . , yn).
By convention, we have Fγ(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 {gγ1 , . . . ,gγs} as {γ1, . . . , γs}
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 Rn is
called smooth if it is simplicial and for each maximal cone C ∈ N the primitive integer vectors
v1, . . . ,vn spanning C form an integral basis for Zn.
Theorem 2.3 ([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 Pγ denote the Newton polytope of the F -polynomial Fγ(y). By convention,
if γ is initial, we have set Fγ(y) = 1 and Pγ = {0}. Let F (y) =
∏
γ∈Π Fγ(y). Then the
Newton polytope P of F (y) is the Minkowski sum
∑
γ∈Π Pγ . 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 Pγ is equal to N (B∨).
2.11. Let τ : Π → Π be the bijection defined by τ(t, j) = (t − 1, j) for 0 ≤ t ≤ rj , denoted τc
in [31]. Then (τγ||τω) = (γ||ω) and τ induces an automorphism of the cluster complex of D.
An exchange relation for A
(
B̃
)
is called primitive if it is of the form xγxω =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 xτγxγ =M +M ′.
2.12. Let Auniv = A
(
B̃univ
)
denote the cluster algebra with universal coefficients, from [19,
Theorem 12.4], [31, Section 5], and [27, Theorem 10.12 and Remark 10.13]. Thus B̃univ 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 BT , see [27]. The bottom r rows of B̃univ
are again indexed by Π, and we denote the corresponding frozen variables by zγ , for γ ∈ Π.
Proposition 2.5 ([31, Proposition 5.6]). The primitive exchange relations of A
(
B̃univ
)
are
given by
x(t−1,j)x(t,j) = z(t,j)
∏
i→j
x
−aij
(t,i)
∏
j→i
x
−aij
(t−1,i) +
∏
ω
z(ω||(t,j))ω (2.3)
for j ∈ I and 0 ≤ t ≤ rj.
Cluster Configuration Spaces of Finite Type 7
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 xunivγ :
F univ
γ (z) := xunivγ (xi = 1, zω).
By convention, we have F univ
γ (z) = 1 if γ is initial.
2.14. Suppose D is a multiply-laced Dynkin diagram whose underlying tree is oriented. Then
there exists a simply-laced Dynkin diagram D̃ such thatD is obtained from D̃ by folding [13], and
the orientation of D is induced by the orientation of D̃. In this situation, there is a finite group Γ
acting on Ĩ and Π̃ such that I and Π are identified with the Γ-orbits on Ĩ and Π̃. We obtain
surjective quotient maps ν : Ĩ → I and ν : Π̃ → Π. Abusing notation, let ν : R|Π̃| → R|Π| be given
by ν(eγ̃) = eν(γ̃), and ν : R|Ĩ| → R|I| be given by ν(eĩ) = eν (̃i). Similarly, define ν : Z
[
yĩ | ĩ ∈ Ĩ
]
→
Z[yi | i ∈ I] by ν(yĩ) = yν (̃i) and ν : Z
[
zγ̃ | γ̃ ∈ Π̃
]
→ Z[zγ | γ ∈ Π] by ν(zγ̃) = zν(γ̃). The following
results are a consequence of the definitions.
Proposition 2.6.
1. For γ̃, ω̃ ∈ Π̃ and g ∈ Γ, we have (g · γ̃||g · ω̃) = (γ̃||ω̃).
2. For γ, ω ∈ Π, we have (γ||ω)D =
∑
γ̃∈ν−1(γ)(γ̃||ω̃)D̃ for any ω̃ ∈ ω̄.
3. For any γ̃ ∈ Π̃, we have ν(Fγ̃(ỹ)) = Fν(γ̃)(y) and ν
(
F univ
γ̃ (z̃)
)
= F univ
ν(γ̃) (z).
4. For any γ̃ ∈ Π̃, we have ν(gγ̃) = gν(γ̃).
Examples of foldings are given in Section 3.2.
3 The cluster configuration space MD
In this section, we define the cluster configuration space MD and its partial compactifica-
tion M̃D, and we state some geometric properties of these spaces. We also give examples of the
cluster compatibility degree appearing in the defining relations.
3.1. Let C[u] := C[uγ | γ ∈ Π] be the polynomial ring with generators uγ and let C[u±1] denote
the Laurent polynomial ring with the same generators.
Definition 3.1. Let ID denote the ideal
(
in C[u] or C
[
u±1
])
generated by the elements
Rγ := uγ +
∏
ω
u(ω||γ)ω − 1 (3.1)
for γ ∈ Π, and (ω||γ) denotes the compatibility degree.
Definition 3.2. Define the cluster configuration space MD and its partial compactification M̃D
by
MD := Spec
(
C
[
u±1
]
/ID
)
and M̃D := Spec
(
C[u]/ID
)
.
The following result will be proved in Section 5.4.
Theorem 3.3. The two schemes MD and M̃D are smooth, irreducible, affine algebraic varieties
of dimension n. The boundary divisor ∂ := M̃D \ MD is a simple normal-crossing divisor
in M̃D.
8 N. Arkani-Hamed, S. He and T. Lam
If D = ∅, we define MD = M̃D = Spec(C) to be a point. If D = A1, then C[u]/ID =
C[u, u′]/(u+ u′ = 1) so M̃A1 = C and MA1 = C \ {0, 1}.
Remark 3.4. The definition of MD and M̃D depends on the choice of exchange matrix B of
type D only in the indexing of the generators uγ by Π. For two different orientations of D, there
is a natural bijection between the two indexing sets Π(B) that arise, and a natural isomorphism
between the resulting schemes MD(B).
3.2. Let us give the relations Rγ 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.
3.2.1. Type An−3. In this case, the set Π can be identified with the diagonals (not including
sides!) {(i, j)} of an n-gon. The Rγ are the equations (1.1). The compatibility degree is given
by the formula ((i, j)||(k, ℓ)) = 1 if (i, j) and (k, ℓ) cross (in the interior of the polygon) and
((i, j)||(k, ℓ)) = 0 if (i, j) and (k, ℓ) do not cross. The automorphism τ in this case corresponds
to the order n rotation of the polygon. The clusters are exactly the maximal sets of pairwise
compatible diagonals. In other words, clusters are in bijection with triangulations of the n-gon.
3.2.2. Type Cn−1. Let P2n be the 2n-gon with vertices cyclically labeled 1, 2, . . . , n, 1̄, 2̄, . . . , n̄.
The set Π is identified with the union
Π =
{[
i, ī
]
:=
(
i, ī
)
| 1 ≤ i ≤ n
}
∪
{
[i, j] :=
(
(i, j),
(̄
i, j̄
))
| 1 ≤ i < j − 1 < n
}
∪
{[
i, j̄
]
:=
((
i, j̄
)
,
(
j, ī
))
| 1 ≤ i < j ≤ n, (i, j) ̸= (1, n)
}
(3.2)
of the long diagonals, and pairs of centrally symmetric diagonals in P2n. In total we have
|Π| = n2 − 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, 1̄]||[2, 3̄]
)
= 1 but(
[2, 3̄]||[1, 1̄]
)
= 2. For example, for C2, Definition 3.1 gives the two equations
1 = u[11̄] + u[22̄]u[33̄]u
2
[23̄],
1 = u[23̄] + u[11̄]u[13]u[12̄] (3.3)
and the three cyclic rotations of each.
This case is obtained from A2n−3 by folding. There is an action of the two-element group Γ
on P2n mapping i↔ ī. This induces the natural map ν : Π̃ → Π sending diagonals of P2n to Γ-
orbits on the diagonals of P2n. We may verify Proposition 2.6(2): for example
(
[2, 3̄]||[1, 1̄]
)
C2
=(
(2, 3)||(1, 1̄)
)
A3
+
(
(2̄, 3̄)||(1, 1̄)
)
A3
= 1 + 1 = 2. The automorphism τ is inherited from the
rotation of the 2n-gon P2n.
3.2.3. Type Dn. Let PPn denote an n-gon Pn 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 PPn 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 [̃i]
connecting i to 0. We denote the corresponding u-variables by uij , and ui and uĩ. See Figure 1.
(We caution the reader that the notation ĩ here is unrelated to the notation γ̃ used for foldings.)
In this case, the automorphism τ is the composition of the rotation of Pn with “changing the
tagging at 0” (i.e., switching from [i] to [̃i] if the arc is incident to 0).
Cluster Configuration Spaces of Finite Type 9
0
1 2
34
u1̃
u1
u12
u13
Figure 1. The polygon PP4 and some u-variables. Note u21 does not exist.
The compatibility degree (γ||ω) is equal to the (minimal) number of intersection points bet-
ween the arc γ and the arc ω, if at least one of γ and ω do not connect to 0. If both γ and ω
connect to 0, then we have ([i]||[j]) =
(
[̃i]||[j̃]
)
=
(
[i]||[̃i]
)
= 0 but
(
[i]||[j̃]
)
=
(
[j̃]||[i]
)
= 1 if
i ̸= j. For D4, representative equations from Definition 3.1 are
1 = u12 + u3u3̃u4u4̃u
2
34u23u24u41u31,
1 = u13 + u4u4̃u41u42u24u34,
1 = u1 + u2̃u3̃u4̃u23u34u24 (3.4)
and we have 4, 4, 8 equations of these types respectively, for a total of 16 equations.
3.2.4. Type Bn−1. The set Π is the same as for Cn−1 (3.2). However, the compatibility
degree (γ||ω) is equal to the number of crossings of one of the diagonals representing γ with the
diagonals representing ω. Thus,
(
[1, 1̄]||[2, 3̄]
)
= 2 but
(
[2, 3̄]||[1, 1̄]
)
= 1. For B2, Definition 3.1
gives the two equations
1 = u[11̄] + u[22̄]u[33̄]u[23̄],
1 = u[23̄] + u2[11̄]u[13]u[12̄]
and the three cyclic rotations of each. Note that MB2 is isomorphic to MC2 under a non-trivial
re-indexing of the u-variables. However MBn and MCn are not isomorphic for n > 2.
Type Bn−1 can be obtained from type Dn by folding. Let Γ be the two-element group acting
on Π̃ = Π(Dn) by sending [i] ↔ [̃i] and fixing all other (i, j). The map ν : Π̃ → Π sends [i]
and [̃i] to [i, ī], and sends (i, j) to [i, j̄] if i < j and to [j, i] if i > j. For example, for B3, the
images of the equations from (3.4) are
1 = u[12̄] + u2[33̄]u
2
[44̄]u
2
[34̄]u[23̄]u[24̄]u[14]u[13],
1 = u[13̄] + u2[44̄]u[14]u[24]u[24̄]u[34̄],
1 = u[11̄] + u[22̄]u[33̄]u[44̄]u[23̄]u[34̄]u[24̄]. (3.5)
The automorphism τ is inherited from the rotation of the n-gon in type Dn.
3.2.5. Type G2. In type G2, we have |Π| = 8 and we denote u-variables by ai, bi for
i = 1, 2, 3, 4. The u-equations for type G2 are
1 = a1 + a2b2a
2
3b3a4,
1 = b1 + b2a
3
3b
2
3a
3
4b4
10 N. Arkani-Hamed, S. He and T. Lam
and the cyclic rotations under the group Z/4Z. Type G2 can be obtained from D4 by folding.
(Though at present we only consider foldings of simply-laced diagrams, G2 can also be obtained
from B3 by folding.)
3.3. Let F be a face of the generalized associahedron, which we identify with a pairwise
compatible subset {γ1, . . . , γd} of Π. We define M̃D(F ) ⊂ M̃D to be the closed subscheme cut
out by the ideal (uγ1 , . . . , uγd). If F = ∅, then M̃D(∅) := M̃D. We define MD(F ) ⊂ M̃D(F )
to be the open subscheme of M̃D(F ) where all the variables {uγ | γ /∈ F} are non-vanishing.
Proposition 3.5. We have a natural stratification
M̃D =
⊔
F
MD(F ), (3.6)
where F varies over all the faces of the generalized associahedron.
Proof. When ω and γ are incompatible, the coordinate uω is non-vanishing on MD({γ}). This
shows that the subschemes MD(F ) cover M̃D. ■
3.4. Let us analyze M̃D(F ) for F = {γ}. Setting uγ = 0 in the equation
uω +
∏
τ
u(τ ||ω)τ = 1
we find that uω = 1 for all ω incompatible with γ. Let Π(γ) ⊂ Π be the subset of κ ∈ Π that
are compatible with γ. Setting uω = 1 in Rκ for κ ∈ Π(γ), we get
R′
κ := uκ +
∏
τ∈Π(γ)
u(τ ||κ)τ − 1.
It follows that the coordinate ring of M̃D(F ) has the following presentation
C[uκ |κ ∈ Π(γ)]/(R′
κ).
Proposition 3.6. Suppose γ = (t, j) and removing j from D disconnects D into connected
components D1, . . . , Ds. Then we have
M̃D({γ}) = M̃D1 × M̃D2 × · · · × M̃Ds
and
MD({γ}) = MD1 ×MD2 × · · · ×MDs .
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 D1, D2, . . . , Ds. ■
Note that removing a vertex from a finite type Dynkin diagram produces at most three compo-
nents, so in Proposition 3.6 we have s ≤ 3.
By applying Proposition 3.6 repeatedly, we have the following result.
Proposition 3.7. Any M̃D(F ) (resp. MD(F )) is a direct product of M̃D′ (resp. MD′) as D′
varies over a finite set of Dynkin diagrams obtained by removing some vertices from D.
Cluster Configuration Spaces of Finite Type 11
3.5. Recall the bijection τ : Π → Π of Section 2.11. We have automorphisms τ : MD → MD
and τ : M̃D → M̃D induced by uγ 7→ uτγ . The order of the automorphism τ is either h + 2
or (h+2)/2, where h is the Coxeter number of D; see for example [7]. It would be interesting to
compute: (1) the group of automorphisms of the variety M̃D, (2) the group of automorphisms of
the variety MD, and (3) the group of automorphisms of MD that send the positive part MD,>0
(defined in Section 8.1) to itself.
In the case D = An−3, we have MD = M0,n which has a natural action of Sn. The group Sn
acts transitively on the connected components of M0,n(R). The positive part MD,>0 is one of
the connected component of M0,n(R) and it is sent to itself by a dihedral subgroup of Sn of
order 2, and the number of connected components of M0,n(R) is equal to n!/2n, see Section 7.2.
3.6. Let γ ∈ Π. Then as in Proposition 3.6, we can uniquely associate Dynkin diagrams
D1, . . . , Ds(γ) to γ, where s ≤ 3.
Proposition 3.8. Suppose that γ ∈ Π and s(γ) = 1. Then we have a natural morphism
MD −→ MD1 . (3.7)
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 MAn−3 = M0,n. We expect (3.7) to be a fibration,
similar to the M0,n case.
3.7. Let D be a folding of D̃ and let Γ and ν : Π̃ → Π be as in Section 2.14.
Proposition 3.9. The quotient of C[u]/ID̃ by the ideal generated by the equations uγ̃ = ug·γ̃
for γ̃ ∈ Π̃ and g ∈ Γ is canonically isomorphic to C[u]/ID.
Proof. Let ν : C
[
uγ̃ | γ̃ ∈ Π̃
]
→ C[uγ | γ ∈ Π] be the ring homomorphism given by ν(uγ̃) = uν(γ̃).
Then applying Proposition 2.6(1),(2), we have ν(Rγ̃) = Rν(γ̃). The result follows. ■
Thus M̃D can be identified with a closed subscheme of M̃D̃ and it is straightforward to see
that MD is the intersection of M̃D ⊂ M̃D̃ with the open subset MD̃ ⊂ M̃D̃.
3.8. The significance of the following conjecture is unclear to us. We have proved it by a direct,
elementary calculation for D = An, n ≥ 2.
Conjecture 3.10. For D not of type A1, the ring C[u±1]/ID is generated by u−1
γ , for γ ∈ Π.
4 MD as a quotient of a cluster variety
In this section we show thatMD can be obtained as a quotient of an open subspace X̊ of the clus-
ter variety X
(
B̃
)
by the action of the cluster automorphism torus T considered in [23]. An im-
portant role is played by principal and universal coefficients, where B̃ = B̃prin or B̃ = B̃univ.
In particular, the defining relations of MD are obtained from the primitive exchange relations
of X
(
B̃univ
)
.
4.1. Let B̃ be a full rank extended exchange matrix. Let T = T
(
B̃
)
be the cluster automorphism
group [23] of A
(
B̃
)
: this is the group of algebra automorphisms ϕ : A
(
B̃
)
→ A
(
B̃
)
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
(
B̃
)
by scaling the coordinates. By [23,
Proposition 5.1], we have
T = Hom
(
Zn+m/B̃Zn,C∗). (4.1)
12 N. Arkani-Hamed, S. He and T. Lam
Since B̃ has full rank, the group T is a (possibly disconnected) abelian algebraic group of
dimension m. The character group of T is the lattice Zn+m/B̃Zn. By definition, the torus T
acts on each cluster variable by a character, and we denote the weight of the cluster variable
x ∈ A
(
B̃
)
by wt(x) ∈ Zn+m/B̃Zn.
Lemma 4.1. Let B̃ have full rank. Then B̃ has really full rank if and only if Zn+m/B̃Zn has
no torsion, or equivalently, the group T is connected, and thus a torus of dimension m.
Proof. The rows of B̃ span Zn if and only if B̃Qn∩Zn+m = B̃Zn if and only if Zn+m/B̃Zn has
no torsion. ■
Let X = X
(
B̃
)
be the cluster variety, which is a smooth affine algebraic variety. Let X̊ ⊂ X
be the locus where all mutable cluster variables are non-vanishing. In terms of rings, we have
X̊ := Spec
(
A[1/x |x is a mutable cluster variable]
)
.
Thus X̊ 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 preserves X̊, and furthermore the action
of T is free on X̊. The geometric invariant theory quotient
X̊ // T := Spec
(
C
[
X̊
]T )
is again a smooth affine algebraic variety, and furthermore, there is a bijection between closed
points of X̊ // T and T -orbits on X̊. We thus simply denote X̊ // T by X̊/T . Explicitly, the
ring C
[
X̊
]T
consists of all weight zero Laurent polynomials in cluster variables.
4.2. Let τ : Π → Π be the bijection defined by τ(t, j) = (t− 1, j) for 0 ≤ t ≤ rj . The primitive
exchange relations are of the form
xτγxγ =M +M ′,
where M ′ only involves frozen variables. For each primitive exchange relation, we define the
rational function
fγ :=
M
xτγxγ
.
By definition, fγ ∈ C
[
X̊
]
, and it is easy to see that fγ is T -invariant. Thus fγ ∈ C
[
X̊
]T
.
Theorem 4.2. Suppose that B̃ is a full rank extended exchange matrix, acyclic and of finite
type. Let X = X
(
B̃
)
. Then the quotient X̊/T is a smooth affine variety isomorphic to MD,
and the isomorphism C[MD] ∼= C
[
X̊
]T
is given by uγ 7→ fγ.
Theorem 4.3. Suppose that B̃ is a full rank extended exchange matrix, acyclic and of finite
type. Let X = Xprin
(
B̃
)
have principal coefficients. Then MD
∼= X̊prin/T prin is isomorphic
to the locus Xprin(1) ⊂ Xprin, where all initial mutable cluster variables have been set to 1.
The coordinate ring C[MD] is isomorphic to the subring of C(y1, y2, . . . , yn) generated by F±1
γ (y)
and y±1
i .
Theorem 4.4. Suppose that B̃ is a full rank extended exchange matrix, acyclic and of finite type.
Let X = Xuniv
(
B̃
)
have universal coefficients. Then MD
∼= X̊univ/T univ is isomorphic to the
locus Xuniv(1) ⊂ Xuniv, where all mutable cluster variables have been set to 1. The isomorphism
C[MD] ∼= C[Xuniv(1)] is given by uγ 7→ zγ.
Cluster Configuration Spaces of Finite Type 13
4.3. The relations in the following corollary will be discussed in further detail in Section 10.1.
Corollary 4.5. The ideal ID has a natural set of generators of the form U + U ′ − 1, given by
the images of all exchange relations of Xuniv.
The ideal ID also contains the |Π|−n distinguished elements which are images of 1−F univ
γ (z).
4.4. Proof of Theorem 4.4. Recall that the mutable cluster variables of Auniv are denoted xγ
and the frozen variables are denoted zγ , where γ ∈ Π. Let Xuniv(1) ⊂ X̊univ ⊂ Xuniv be the
locus {xγ = 1}, where all mutable cluster variables have been set to 1.
By Proposition 2.5, the primitive exchange relations are of the form
xτγxγ = zγS +
∏
ω∈Π
z(ω||γ)ω ,
where S is a monomial in the mutable cluster variables. So,
fγ =
zγS
xτγxγ
and thus on Xuniv(1) we have (fγ)|Xuniv(1) = zγ and the relation
fγ +
∏
ω∈Π
f (ω||γ)ω = 1. (4.2)
We will now show that the multiplication map gives an isomorphism
T univ ×Xuniv(1) ∼= X̊univ,
or equivalently, every T univ-orbit on X̊univ intersectsXuniv(1) in exactly one point. The character
group of T univ is naturally isomorphic to Zn+r/B̃Zn, which is a free abelian group of rank r = |Π|.
Thus each cluster variable xγ has a weight (or degree) wt(xγ) ∈ Zn+r/B̃Zn (see (6.2) for the
weight of initial and frozen variables). By Proposition 6.5, the set {wt(xγ) | γ ∈ Π} form a basis
of the lattice Zn+r/B̃Zn. Thus we have a projection X̊univ 7→ T univ given by sending x ∈ X̊univ
to the coordinates (xγ)γ∈Π, and the fiber of this projection is Xuniv(1). This is an inverse to the
multiplication map T univ ×Xuniv(1) → X̊univ, and we deduce that T univ ×Xuniv(1) ∼= X̊univ.
We conclude that C
[
X̊univ
]Tuniv∼= C[Xuniv(1)]. Now, any T univ-invariant function in C
[
X̊univ
]
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 zγ-s on C[Xuniv(1)]. It fol-
lows that the functions fγ and their inverses generate C
[
X̊univ
]Tuniv
, and by (4.2) satisfy the
same relations that uγ ∈ C[MD] satisfy. Finally, we check that the generators fγ do not satisfy
any further relations. Suppose we have a polynomial identity p(fγ) = 0 inside C
[
X̊univ
]Tuniv
.
The equality p(fγ) = 0 is equivalent to an equality q(xγ , zγ) = 0 inside C
[
X̊univ
]
, where q(xγ , zγ)
is a Laurent polynomial. We claim that the primitive exchange relations allow us to eliminate
all the non-initial cluster variables, i.e.,
q(xγ , zγ) = r(x1, x2, . . . , xn, zγ) mod ideal generated by primitive exchange relations,
where r(x1, x2, . . . , xn, zγ) is a Laurent polynomial and the ideal is taken inside C
[
X̊univ
]Tuniv
.
To see this, first note that deg(x1), . . . ,deg(xn) and deg(zγ), γ ∈ Π together span Zn+r, and
thus we can always multiply q(xγ , zγ) by a T univ-invariant monomial so that the denominator
involves only initial xi and the zγ . Next, we have
xγ =
M +M ′
xτγ
− xγR,
14 N. Arkani-Hamed, S. He and T. Lam
where R = M+M ′
xτγxγ
−1 is a primitive exchange relation (divided by xτγxγ). This allows us (modulo
the ideal) to replace xγ by an expression involving xτγ 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 xγ will not appear again when this process is
repeated. This proves our claim.
But r(x1, x2, . . . , xn, zγ) = 0 as an element of C
[
X̊univ
]Tuniv
⊂ C
[
X̊univ
]
⊂ C(Xuniv) only if
the polynomial r is 0, since x1, x2, . . . , xn, zγ are algebraically independent. We conclude that
p(fγ) lies in the ideal generated by primitive exchange relations. Thus the ideal of relations
satisfied by the fγ is generated by (4.2). We thus have an isomorphism of rings
C[MD] −→ C
[
X̊univ
]Tuniv
, uγ 7−→ fγ
and an isomorphism of varieties Xuniv(1) ∼= MD.
4.5. Proof of Theorem 4.2. By the defining property of universal coefficients, we have
a homomorphism of rings ϕ : Auniv → A = A
(
B̃
)
such that ϕ(xunivγ ) = xγ and ϕ(zγ) is a Laurent
monomial in the frozen variables xn+1, . . . , xn+m of A. The homomorphism ϕ may not be
surjective, for example this would be the case if B̃ has rows equal to 0, or rows that are repeated.
However, the image A′ := ϕ(Auniv) is itself a cluster algebra: it is generated by ϕ(xunivγ ) and
the monomials ϕ(zγ). The monomials ϕ(zγ) and their inverses generate a Laurent polynomial
subring S ⊂ C
[
x±1
n+1, . . . , x
±1
n+m
]
which is the coefficient ring of A′. For any monomial M
in xn+1, . . . , xn+m, we can find t ∈ T such that t ·M is a Laurent monomial in S. Thus, we have
AT = A′T ′
. 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 : Auniv → A is surjective, and thus we have
a closed immersion ψ : X̊ → X̊univ. The monomials ϕ(zγ), γ ∈ Π together define a surjec-
tive linear map C ′ : Zr → Zm. Extending by the identity in the first n coordinates, we get
a linear map Zn+r → Zn+m, represented by a matrix C satisfying CB̃univ = B̃. Suppose that
t ∈ T = Hom
(
Zn+m/B̃Zn,C∗). Then composing t with C, we get an element t′ ∈ T univ =
Hom
(
Zn+r/B̃univZn,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
[
X̊univ
]Tuniv
→ C
[
X̊
]T
is an isomorphism. For surjectivity,
suppose that f ∈ C
[
X̊
]T
. Then we may assume that f is a Laurent monomial in mutable and
frozen variables. Let g ∈ C
[
X̊univ
]
be such that ϕ(g) = f . It is immediate that g is invariant
under T , i.e., the weight wt(g) ∈ Zn+r/B̃univZn of g satisfies C wt(g) = 0 in Zn+m/B̃Zn. Thus,
there exists u ∈ B̃univZn such that C(wt(g) + u) = 0 ∈ Zn+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 (aγ) are the last m coordinates of wt(g). Then by construction we have wt(gM)+u = 0,
i.e., wt(gM) = 0 ∈ Zn+r/B̃univZn. Furthermore, ϕ(M) = 1 and thus gM ∈ C
[
X̊univ
]Tuniv
satisfies ϕ(gM) = f , proving surjectivity.
For injectivity, suppose that ϕ(g) = 0, where g ∈ C
[
X̊univ
]Tuniv
is nonzero. We have al-
ready shown that Xuniv/T univ is an irreducible affine variety in Section 4.4. The affine variety
Spec
(
C
[
X̊
]T )
is thus identified with a subvariety of Xuniv/T univ of lower dimension. But this
is impossible, since dim(X/T ) = n = dim(Xuniv/T univ).
The isomorphism C[MD] ∼= C
[
X̊
]T
given by uγ 7→ fγ now follows from Section 4.4.
4.6. Proof of Theorem 4.3. The group Z2n/B̃prinZn can be naturally identified with the
subgroup Zn = Z[1,n] ⊂ Z[1,2n] = Z2n consisting of vectors which vanish in the last n-coordinates.
Under this identification, the torus T prin has character lattice Zn, and the grading on Aprin is
Cluster Configuration Spaces of Finite Type 15
given by (2.2). By Theorem 4.2, we have MD
∼= X̊prin/T prin. It follows from wt(xi) = ei that
X̊prin/T prin is identified with the subvariety X̊prin(1) ⊂ X̊prin, where all initial cluster variables
are set to 1.
The function fprinγ on X̊prin(1) restricts to the rational function in y1, . . . , yn given by (see
[31, Theorem 1.5])
f(t,j)(y) =
∏
i→j F
−aij
(t,i)
(y)
∏
j→i F
−aij
(t−1,i)
(y)
F(t−1,j)(y)F(t,j)(y)
, t ̸= 0,
yj
∏
i→j F
−aij
(t,i)
(y)
∏
j→i F
−aij
(t−1,i)
(y)
F(t−1,j)(y)F(t,j)(y)
, t = 0
(4.3)
for γ = (t, j) with 0 ≤ t ≤ rj . By the following result, C
[
X̊prin(1)] ≃ C[MD
]
is isomorphic to
the subring of C(y1, y2, . . . , yn) generated by F±1
γ (y) and y±1
i .
Proposition 4.6. The rational functions {fγ(y) | γ ∈ Π} and {y1, . . . , yn} ∪ {Fγ(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 xγ = xj is an
initial mutable cluster variable. Let B̃ be full rank of type D and let A
(
B̃j
)
denote the cluster
algebra of type D1 that is obtained by freezing the variable xj in A
(
B̃
)
. The extended exchange
matrix B̃j is obtained from B̃ by removing the j-th row and we have A
(
B̃
)[
x−1
j
]
= A
(
B̃j
)
.
Thus Å
(
B̃j
)
⊂ Å
(
B̃
)
. The action of the cluster automorphism group T
(
B̃
)
extends to an action
on A
(
B̃j
)
and we can identify T
(
B̃
)
with a subgroup of T
(
B̃j
)
. The morphism MD → MD1
corresponds to the inclusion of rings Å
(
B̃j
)T (B̃j) ⊂ Å
(
B̃j
)T (B̃) ⊂ Å
(
B̃
)T (B̃)
.
5 M̃D as an affine open in a projective toric variety
In this section, we show that the partial compactification M̃D is an affine open subspace of
the projective toric variety XN (B∨) associated to the cluster fan of B∨. The stratification
(Proposition 3.5) of M̃D is inherited from the natural stratification of XN (B∨) by torus orbits.
Our approach follows that of [3].
5.1. Let C(y) = C(y1, . . . , yn) denote the field of rational functions. Recall that for γ ∈ Π,
we have defined fγ(y) ∈ C(y) in (4.3). By the proof of Theorem 4.3, C[MD] is isomorphic to
the subring of C(y) generated by fγ(y)
±1. Define RB ⊂ C(y) to be the subring generated by
{fγ(y) | γ ∈ Π}. Some examples of fγ(y) are computed in Examples 6.7 and 6.8.
Theorem 5.1. The coordinate ring C
[
M̃D
]
is isomorphic to RB.
Proof. There is a surjective ring homomorphism φ′ : C[u] → RB given by uγ 7→ fγ(y). We al-
ready know that the kernel K of φ contains the ideal ID ⊂ C[u]. We need to show that the
homomorphism φ : C[u]/ID → RB is an isomorphism. From Theorem 4.3, we know this holds
after inverting the {uγ | γ ∈ Π} and {fγ | γ ∈ Π}.
By definition, M̃D({γ}) is cut out of M̃D by the ideal (uγ). Thus the ring C[u]/(ID + (uγ))
is isomorphic to C
[
M̃D({γ})
]
, and by Proposition 3.6, we have M̃D({γ}) = M̃D1 ×M̃D2 × · · ·
× M̃Ds for some Dynkin diagrams Di. By induction on the rank of D, we have that φi : C[u]/
(ID + (uγ)) → RB/(fγ(y)) is an isomorphism. Applying Lemma A.1, we conclude that φ itself
is an isomorphism. ■
16 N. Arkani-Hamed, S. He and T. Lam
5.2. We give another description of RB ⊂ 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 Rn given by the formal substitution
yi 7→ Yi, (+,×,÷) 7→ (min,+,−).
For example, Trop
((
3y21y2 + y22
)
/(y2 + 6y3)
)
= min(2Y1 + Y2, 2Y2)−min(Y2, Y3). Note that the
coefficients are unimportant since, for example, Trop(2y) = Trop(y + y) = min(Y, Y ) = Y .
The domains of linearity of the piecewise-linear function L(Y) = Trop(R(y)) define the struc-
ture of a complete fan on Rn. A piecewise-linear function L(Y) : Rn → R is called nonnegative,
denoted L(Y) ≥ 0, if it takes nonnegative values on Rn.
Proposition 5.2. The ring RB is equal to the subring of C(y) generated by rational func-
tions R(y) satisfying
(1) R(y) =
∏n
i=1 y
ai
i
∏
γ∈Π Fγ(y)
aγ is a Laurent monomial in yi and Fγ(y),
(2) Trop(R(y)) is nonnegative.
Proof. Let R(y) be a Laurent monomial in {yi, Fγ(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 bγ = L(−gγ) as γ varies over Π, and gγ denotes
a g-vector. As in the proof of Proposition 4.6, we have R(y) =
∏
γ fγ(y)
−bτγ . The condition
L(Y) ≥ 0 is equivalent to bγ ≥ 0 for all γ ∈ Π. Thus the subring of rational functions R(y)
satisfying (1) and (2) is exactly the subring RD. ■
5.3. The Laurent polynomial ring C
[
y±1
1 , . . . , y±1
n
]
is the coordinate ring of an n-dimensional
torus Ty. Recall that F (y) =
∏
γ Fγ(y). The following result is an application of [3, Section 10].
Theorem 5.3. The affine scheme M̃D is isomorphic to the affine open {F (y) ̸= 0} in the pro-
jective toric variety XN (B∨) associated to the complete fan N
(
B∨). The subvariety MD ⊂ M̃D
is identified with the intersection of {F (y) ̸= 0} with the open torus orbit Ty in XN (B∨).
Proof. For any g ∈ Rn, the quantity Trop(F (y))(g) is equal to the minimum value that the
linear function Y 7→ 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 {v1, . . . ,vk} = cP ∩ Zn be
the set of all lattice points in cP . For v ∈ Zn, let yv be the monomial with exponent vector v.
Then XN (B∨) can be explicitly realized as the closure of the set of points{[
yv1 : · · · : yvk
]
∈ Pk−1 |y ∈ Ty
}
inside the projective space Pk−1. The polynomial F (y)c can be identified with a hyperplane
section of XN (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
yvi/F (y)c, i = 1, 2, . . . , k. Since Trop
(
yvi/F (y)c
)
is nonnegative, by Proposition 5.2, we have
C[V ] ⊂ RB. It is also not hard to see that fγ(y) ∈ C[V ] (see [3, Section 10]), and we have
C[V ] = RB 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 Fγ(y), very ample? Is P normal?
Question 5.5. Is the polynomial F (y) saturated?
Cluster Configuration Spaces of Finite Type 17
Question 5.6. Is every lattice point in P a sum of lattice points in Pγ?
Fei [14] has shown that Fγ(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 XN (B∨) is a smooth projective toric variety and the torus-orbit
closure stratification of XN (B∨) is simple normal-crossing.
6 Properties of F -polynomials
We establish some technical properties of Fγ(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(fγ(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: Bij > 0 if and only if B∨
ij > 0. Recall that we write i→ j
if Bij > 0.
6.1. Let B be the exchange matrix corresponding to the oriented Dynkin diagram D. Let RΠ
be the vector space with basis indexed by Π, and write (pγ)γ∈Π for a typical vector in RΠ.
Define Π+ := {(s, i) | 1 ≤ s ≤ ri} ⊂ Π and let c = (cγ)γ∈Π+ denote a typical vector in RΠ+
.
Following [8], we consider the c-deformed mesh relations
p(t−1,j) + p(t,j) = c(t,j) +
∑
i→j
|Bij |p(t,i) +
∑
j→i
|Bij |p(t−1,i), (6.1)
where (t, j) ∈ Π+. (Compare with (2.3), and note that if i→ j then Bij > 0, but if j → i then
Bij < 0.) If c = 0, we call (6.1) the 0-mesh relations.
For c = (cγ) ∈ RΠ+
, we let Ec ⊂ RΠ denote the solutions to (6.1), and let Uc := Ec ∩ RΠ
≥0
denote the intersection of Ec with the positive orthant. Let π : RΠ → Rn denote the projection
onto the coordinates pγ , where γ varies over {(ri, 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, eγ denotes the unit basis vector in RΠ+
>0 .
Theorem 6.1.
1. If c = (cγ) ∈ RΠ+
>0 , then the normal fan of π(U(D)c) is equal to N (B). If (cγ) ∈ RΠ+
≥0 ,
then the normal fan of π(U(D)c) is a coarsening of N (B).
2. For γ ∈ Π+, the polytope U
(
D∨)
eγ
is the Newton polytope of F univ
γ (z).
3. For γ ∈ Π+, the polytope π
(
U
(
D∨)
eγ
)
is the Newton polytope of Fγ(y).
Proof of Theorem 2.4. By Theorem 6.1(3) the Newton polytope Pγ of Fγ(y) is π
(
U
(
D∨)
eγ
)
.
The Newton polytope P of
∏
γ Fγ(y) is the Minkowski sum of the Pγ , and by Theorem 6.1(1),
we conclude that P is a generalized associahedron. ■
18 N. Arkani-Hamed, S. He and T. Lam
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.
Let D be a folding of D̃ with folding group Γ, and ν : Π̃ → Π the quotient map on cluster
variables from Section 2.14. Define ν : RΠ̃ → RΠ by ν(eγ̃) = eν(γ̃), and ν∨ : RΠ̃ → RΠ by
ν∨(eγ̃) = 1
|ν−1(ν(γ̃))|eν(γ̃). (The finite set ν−1(γ) has cardinality one, two, or three.) Similarly,
we have ν, ν∨ : RΠ̃+ → RΠ+
.
Lemma 6.2. If (pγ̃)γ̃∈Π̃ ∈ E
(
D̃
)
c̃
then ν∨(pγ̃) ∈ E(D)ν∨(c̃) and ν(pγ̃) ∈ E
(
D∨)
ν(c̃)
.
The g-vectors for D̃ are solutions to the 0-mesh relations in the following sense: for each
i = 1, 2, . . . , n, the i-th coordinates of gγ give a vector g(i) that belongs to E0. This follows
from [19, relation (6.13)], noting that the sign-coherence conjecture [19, Conjecture 6.13] holds
in our case.
The following follows from Lemma 6.2 and Proposition 2.6(3). (The appearance of ν∨ seems
to contradict Lemma 6.2, but it is actually correct: the ν∨ in Lemma 6.2 acts on R|Π̃| while
the ν∨ below acts only on RĨ .)
Proposition 6.3. We have g∨
γ = ν∨
(∑
γ̃∈ν−1(γ) gγ̃
)
.
We say that (pγ̃)γ̃∈Π̃ ∈ RΠ̃ is Γ-invariant and write (pγ̃)γ̃∈Π̃ ∈
(
RΠ̃
)Γ
if for all g ∈ Γ, we
have pγ̃ = pg·γ̃ . Similarly, we define Γ-invariants c̃ ∈
(
RΠ̃+)Γ
. The following result follows from
Lemma 6.2.
Proposition 6.4.
1. Suppose that c̃ ∈
(
RΠ̃+)Γ
. Then the linear map ν∨ (resp. ν) is a bijection between E
(
D̃
)
c̃
∩(
RΠ̃
)Γ
and E(D)ν∨(c̃)
(
resp. E
(
D∨)
ν(c̃)
)
.
2. Suppose that c̃ ∈ (RΠ̃+
≥0 )
Γ. Then the linear map ν∨ (resp. ν) is a bijection between U
(
D̃
)
c̃
∩(
RΠ̃
)Γ
and U(D)ν∨(c̃)
(
resp. U
(
D∨)
ν(c̃)
)
.
Proof of Theorem 6.1. In this proof we write N (D) for N (B) to avoid conflict of notation.
Let D̃ fold onto D. Let c ∈ RΠ+
>0 and pick c̃ ∈
(
RΠ̃+
>0
)Γ
satisfying c = ν∨(c̃). By Proposi-
tion 6.4(2), the map ν∨ is a bijection between U
(
D̃
)
c̃
∩
(
RΠ̃
)Γ
and U(D)c. To prove Theo-
rem 6.1(1) for D, it thus suffices to show that π
(
U
(
D̃
)
c̃
∩
(
RΠ̃
)Γ)
= π
(
U
(
D̃
)
c̃
)
∩
(
RĨ
)Γ ⊂
(
RĨ
)Γ
has normal fan N (D). By Theorem 6.1(1) for D̃, the polytope π
(
U
(
D̃
)
c̃
)
has normal fan N
(
D̃
)
,
and by our choice of c̃, it is Γ-invariant. The faces of π
(
U
(
D̃
)
c̃
)
that intersect
(
RĨ
)Γ
are exactly
those normal to the cones {γ̃1, . . . , γ̃a} ofN
(
D̃
)
consisting of Γ-invariant pairwise compatible col-
lections. Combining with Proposition 2.6(4), we deduce that the normal fan of π
(
U
(
D̃
)
c̃
)
∩
(
RĨ
)Γ
is N (D). This proves the first statement of Theorem 6.1(1) for D, and the second statement is
similar.
Now, let ν(γ̃) = γ. By Proposition 2.6(3), the Newton polytope of F univ
γ (z) is the image of
the Newton polytope of F univ
γ̃ (z) under the map ν. By Proposition 6.4(2) and Theorem 6.1(2)
for D̃, the Newton polytope of
∏
γ̃∈ν−1(γ) F
univ
γ̃ (z) is equal to U
(
D̃
)∑
γ̃∈ν−1(γ) eγ̃
. Thus by Propo-
sition 6.4(2), the Newton polytope of (F univ
γ (z))|ν
−1(γ)| is equal to U
(
D∨)
|ν−1(γ)|eγ , and Theo-
rem 6.1(2) for D follows. Finally, Theorem 6.1(3) follows from Proposition 2.6(3). ■
Cluster Configuration Spaces of Finite Type 19
6.3. The character group of T univ is Zn+r/B̃univZn. Let {e1, . . . , en} ∪ {eγ | γ ∈ Π} be basis
vectors of Zn+r. We have
wt(xi) = ei for i = 1, 2, . . . , n, and wt(zγ) = eγ for γ ∈ Π. (6.2)
For γ ∈ Π+, we have
wt(xγ) = wt
(
F univ
γ
)
mod B̃univZn + span(e1, . . . , en).
Note that all monomials in F univ
γ have the same weight modulo B̃univZn + span(e1, . . . , en).
Proposition 6.5. The sets{
wt
(
F univ
γ
)
| γ ∈ Π+
}
and {wt(xγ) | γ ∈ Π}
are bases of Zn+r/
(
B̃univZn + span(e1, . . . , en)
)
and Zn+r/B̃univZn 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 : RΠ → RΠ+
, sending (pγ)
to (cγ). Let B′ be the last r rows of B̃univ. 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′Zn. We conclude that modulo B′Zn, the last r
entries of wt
(
F univ
γ
)
is equal to the basis vector eγ . Returning to the vector wt
(
F univ
γ
)
∈ Zn+r,
we obtain
wt
(
F univ
γ
)
= eγ mod B̃univZn + span(e1, . . . , en).
Thus
{
wt
(
F univ
γ
)
| γ ∈ Π+
}
form a basis of Zn+r/
(
B̃univZn + span(e1, . . . , en)
)
. ■
6.4. Recall the definition of fγ(y) from (4.3). Let Aγ(g) := Trop(fγ(y)), where we take
(g1, . . . , gn) as the tropicalization of y1, . . . , yn, for example Trop(1 + y1 + y1y2) = min(0, g1,
g1 + g2).
Theorem 6.6. For γ, ω ∈ Π, we have Aγ
(
−g∨
ω
)
= δω,τγ.
Proof. First, assume that D is simply-laced so that gω = g∨
ω . For γ ∈ Π, let Wγ ∈ Db(repQop)
be the object indexed by γ in the bounded derived category of representations of the quiver Qop
corresponding to the reversed orientation of D, see [8, Section 3]. For any g ∈ Rn, the quantity
Trop(Fγ(y))(g) is equal to the minimum value that the linear function Y 7→ Y · g takes on the
Newton polytope Pγ .
Now take γ ∈ Π+. Let G be the n × |Π| matrix whose columns are gω. According to [8,
Proof of Theorem 1], the map Y 7→ Y · (−G) + veγ is a diffeomorphism between Pγ and Ueγ ,
where veγ ∈ Eeγ is the integer vector given by (veγ )ω = dimHom(Wω,Wτγ). Here, Hom is taken
within Db
(
repQop
)
. Furthermore, it follows from [8] that Ueγ has nonempty intersection with
every coordinate hyperplane. Thus,
Trop(Fγ(y))(−gω) = −dimHom(Wω,Wτγ).
Suppose τγ = (t, i) ∈ Π+. Then we have an Auslander–Reiten triangle in Db
(
repQop
)
W(t−1,i) → E →W(t,i) →W(t−1,i)[1], where E =
⊕
i→j
W(t,j) ⊕
⊕
j→i
W(t−1,j).
20 N. Arkani-Hamed, S. He and T. Lam
We have an exact sequence
0 → Hom
(
Wω,W(t−1,i)
)
→ Hom(Wω, E) → Hom
(
Wω,W(t,i)
)
→ Hom
(
Wω,W(t−1,i)[1]
)
→ · · · .
By the definition of Auslander–Reiten triangle, any map from Wω to W(t,i) which is not an
isomorphism factors through E. Thus(
dimHom
(
Wω,W(t−1,i)
)
+ dimHom
(
Wω,W(t,i)
))
− dimHom(Wω, E) = δω,(t,i)
and this is exactly Aγ(−gω). Now if γ ∈ Π+ but τγ is initial, then we have an Auslander–Reiten
triangle of the form
W(ri∗−1,i∗)[−1] → E →W(0,i) →W(ri∗−1,i∗)
and Hom
(
Wω,W(ri∗−1,i∗)[−1]
)
= Ext−1
(
Wω,W(ri∗−1,i∗)
)
= 0 for all ω ∈ Π, agreeing with our
convention that F(0,i)(y) = 1, so again we have Aγ(−gω) = δω,τγ . Finally, suppose that γ = (0, i)
itself is initial. Then we have an Auslander–Reiten triangle of the form
W(ri∗−1,i∗) → E →W(0,i)[1] →W(ri∗−1,i∗)[1].
In this case, our formula for fγ(y) includes a factor of yi, and Trop(yi)(gω) is simply the i-th
coordinate of gω. Our claim then follows from the interpretation [8, Section 6] of g-vectors as
a change of basis between the summands of a tilting object and the indecomposable projectives{
W(0,1), . . . ,W(0,n)
}
(see also [26, Theorem 3.23(ii)]).
Now, suppose that D is multiply-laced and let D̃ be the simply-laced diagram that folds
to D. Note that for u ∈ R|Ĩ| and v ∈
(
R|Ĩ|)Γ, we have u · v = ν(u) · ν∨(v). By Propo-
sition 6.3, Trop(Fγ(y))(−g∨
ω) is equal to the minimum value that the linear function Y 7→
Y ·
(
−
∑
ω̃∈ν−1(ω) gω̃
)
takes on the Newton polytope Pγ̃ (where γ̃ ∈ ν−1(γ)), and we thus
have Trop(Fγ(y))
(
−g∨
ω
)
=
∑
ω̃∈ν−1(ω)Trop(Fγ̃(y))(−gω̃). It follows from the definitions that
fγ(y) = ν(fγ̃(y)) for any γ̃ ∈ ν−1(γ). The equality Aγ
(
−g∨
ω
)
= δω,τγ for D thus follows from
the same equality for D̃. ■
Proof of Proposition 4.6. Let m(y) be a Laurent monomial in {yi, Fγ(y)}, and denote by
G = G(g) := Trop(m(y)) the piecewise-linear function that is the tropicalization of m(y). (Re-
call 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 deter-
mined 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 fγ(y)
G(−g∨
τγ). ■
Example 6.7. We illustrate Theorem 6.6 for the exchange matrix
B =
0 1 0
−1 0 −1
0 1 0
of type A3. In this case, we have Π = {0, 1, 2}× {1, 2, 3} and i∗ = 4− i. We tabulate gγ , Fγ(y),
and fγ(y) below:
Cluster Configuration Spaces of Finite Type 21
γ gγ Fγ(y) fγ(y)
(0, 1) (1, 0, 0) 1
y1(1 + y2)
1 + y1 + y1y2
(0, 2) (0, 1, 0) 1
y2
1 + y2
(0, 3) (0, 0, 1) 1
y3(1 + y2)
1 + y3 + y2y3
(1, 1) (−1, 1, 0) 1 + y1
1
1 + y1
(1, 2) (−1, 1,−1) 1 + y1 + y3 + y1y3 + y1y2y3
(1 + y1)(1 + y3)
1 + y1 + y3 + y1y3 + y1y2y3
(1, 3) (0, 1,−1) 1 + y3
1
1 + y3
(2, 1) (0, 0,−1) 1 + y3 + y2y3
1 + y1 + y3 + y1y3 + y1y2y3
(1 + y1)(1 + y3 + y2y3)
(2, 2) (0,−1, 0) 1 + y2
(1 + y1 + y1y2)(1 + y3 + y2y3)
(1 + y2)(1 + y1 + y3 + y1y3 + y1y2y3)
(2, 3) (−1, 0, 0) 1 + y1 + y1y2
1 + y1 + y3 + y1y3 + y1y2y3
(1 + y3)(1 + y1 + y1y2)
Taking γ = (1, 2) as an example, we have
Aγ(g) = min(0, g1) + min(0, g3)−min(0, g1, g3, g1 + g3, g1 + g2 + g3)
and one can verify that it takes value 0 on all negatives of g-vectors except for −g(0,2), where it
takes value 1.
Example 6.8. Consider the following exchange matrix of type B3:
B =
0 1 0
−1 0 1
0 −2 0
.
We have Π = {(t, j) ∈ [0, 3]× [1, 3]} and τ(t, j) = ((t−1) mod 4, j). We tabulate the g-vectors,
the g∨-vectors, the F -polynomials, and the polynomials fγ(y) in Table 1.
7 Examples of MD as a configuration space
The space MAn−3 can be identified with the configuration space of n distinct points on P1.
In this section, we investigate similar descriptions of MD in the cases D = Bn and D = Cn.
We also consider the question of whether MD is a hyperplane arrangement complement.
So far we have considered MD as a complex algebraic variety. However, the equations (3.1)
make sense over the integers, and we may also consider MD as a scheme over Z. In particular,
in this section we will also consider MD(Fq), the set of Fq-points of MD, where Fq is a finite
field.
Throughout this section, we use the description of Π from Section 3.2.
22 N. Arkani-Hamed, S. He and T. Lam
Table 1. g-vectors, F -polynomials, and fγ(y) in type B3.
γ gγ g∨
γ Fγ(y) fγ(y)
(0, 1)
10
0
10
0
1
y1(1 + y2 + 2y2y3 + y2y
2
3)
1 + y1 + y1y2 + 2y1y2y3 + y1y2y23
(0, 2)
01
0
01
0
1
y2(1 + y3)
2
1 + y2 + 2y2y3 + y2y23
(0, 3)
00
1
00
1
1
y3
1 + y3
(1, 1)
−1
1
0
−1
1
0
1 + y1
1
1 + y1
(1, 2)
−1
0
2
−1
0
1
1 + y1 + y1y2
1 + y1
1 + y1 + y1y2
(1, 3)
−1
0
1
−2
0
1
1 + y1 + y1y2 + y1y2y3
1 + y1 + y1y2
1 + y1 + y1y2 + y1y2y3
(2, 1)
0
−1
2
0
−1
1
1 + y2
1 + y1 + y1y2
(1 + y1)(1 + y2)
(2, 2)
−1
−1
2
−1
−1
1
1 + y1 + y2 + 2y1y2 + y1y
2
2+
2y1y2y3 + 2y1y
2
2y3 + y1y
2
2y
2
3
(1 + y2)(1 + y1 + y1y2 + y1y2y3)
2
(1 + y1 + y1y2)f(2,2)(y)
(2, 3)
0
−1
1
0
−2
1
1 + y2 + y2y3
F(2,2)(y)
(1 + y1 + y1y2 + y1y2y3)(1 + y2 + y2y3)
(3, 1)
−1
0
0
−1
0
0
1 + y1 + y1y2 + 2y1y2y3 + y1y2y
2
3
F(2,2)(y)
(1 + y2)f(3,1)(y)
(3, 2)
0
−1
0
0
−1
0
1 + y2 + 2y2y3 + y2y
2
3
(1 + y2 + y2y3)
2F(3,1)(y)
F(2,2)(y)(1 + y2 + 2y2y3 + y2y23)
(3, 3)
0
0
−1
0
0
−1
1 + y3
1 + y2 + 2y2y3 + y2y
2
3
(1 + y3)(1 + y2 + y2y3)
7.1. Hyperplane arrangements. Let H1, H2, . . . ,Hr be (affine) hyperplanes in Cn, with the
assumption that the hyperplanes are defined over the integers. Let Z(C) := Cn − (H1 ∪H2 ∪
· · · ∪ Hr). By our assumption Z(R) and Z(Fq) are also well-defined. The following result is
well-known [25, 30].
Cluster Configuration Spaces of Finite Type 23
Theorem 7.1.
1. There exists a polynomial χ(t) so that χ(q) = #Z(Fq), where q = pm is a prime power
with sufficiently large p.
2. The number of connected components |π0(Z(R))| of the real hyperplane arrangement com-
plement Z(R) is given by (−1)nχ(−1).
3. The cohomology ring H∗(Z(C),C) is generated by the classes of dlog fi, where Hi =
{fi = 0}, and we have
∑
i dim(H i(Z(C),C)) ti = (−t)nχ(−1/t). Thus the Euler char-
acteristic of Z(C) is equal to χ(1).
7.2. Type An. Let D = An−3 with n ≥ 4. Then Π can be identified with the diagonals of
a n-gon Pn. We write uij for the u-variable indexed by a diagonal (i, j). Then the relations
defining M̃An−3 are given by (1.1). These relations have appeared a number of times in the
literature, for example see [1, 11]. Let M0,n denote the configuration space of n (distinct)
points (z1, z2, . . . , zn) on P1. Then the identification
uij =
(zi − zj+1)(zi+1 − zj)
(zi − zj)(zi+1 − zj+1)
(7.1)
of uij with a cross ratio gives an isomorphism MAn−3
∼= M0,n. There is a well-studied Deligne–
Knudsen–Mumford compactification M0,n, and M̃0,n := M̃An−3 is an affine variety that sits
between M0,n and M0,n, that is, we have open inclusions M0,n ⊂ M̃0,n ⊂ M0,n.
Let Gr(2, n) denote the Grassmannian of 2-planes in Cn. Let Π̊(2, n) ⊂ Gr(2, n) denote the
open subset where the adjacent cyclic minors ∆i,i+1 are non-vanishing. Then Π̊(2, n) is a full
rank cluster variety of type An−3, see [18, Section 12.2]. Let G̊r(2, n) ⊂ Π̊(2, n) be the open
subset where all Plücker coordinates ∆ij are non-vanishing. This is the subset denoted X̊ in
Section 4. Then M0,n can be identified with the quotient of G̊r(2, n) by the diagonal torus T
sitting inside GLn that acts on Gr(2, n). The isomorphism M0,n
∼= G̊r(2, n)/T is an instance
of Theorem 4.2 for D = An−3. In the Gr(2, n) cluster algebra, we have the primitive exchange
relation
∆i,j∆i+1,j+1 = ∆i,j+1∆i+1,j +∆i,i+1∆j,j+1,
where ∆i,i+1 and ∆j,j+1 are frozen variables. Thus (7.1), or equivalently uij =
∆i,j+1∆i+1,j
∆i,j∆i+1,j+1
,
agrees with the formula for uγ in Theorem 4.2.
The geometry and topology of M0,n is very well-studied; see for example [11]. We recall
some basic facts in the context of Theorem 7.1. Gauge-fixing z1, zn−1, zn to 0, 1, ∞, we have
an identification
M0,n(k) =
{
(z2, z3, . . . , zn−2) ∈ kn−3 | zi ̸= zj and zi /∈ {0, 1}
}
for k a field. In particular, M0,n(k) is the complement in kn−3 of the hyperplane arrangement
with hyperplanes
zi − zj = 0, zi = 0, 1− zi = 0.
We may compute that #M0,n(Fq) = (q− 2)(q− 3) · · · (q− n+2). By Theorem 7.1, the number
of connected components of M0,n(R) is given by
|π0(M0,n(R))| = |(−3) · (−4) · · · (−n+ 1)| = (n− 1)!/2.
24 N. Arkani-Hamed, S. He and T. Lam
7.3. Type Bn. By Theorem 4.2, MD can be identified with A
(
B̃
)
/T
(
B̃
)
for any full rank
extended exchange matrix B̃ of type D. One such choice of B̃, and thus of A
(
B̃
)
is given
in [18, Example 12.10]. Let C[Xn+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[Xn+2], labeled by Γ-orbits of sides and
diagonals in the polygon P2n+2 with vertices
{
1, 2, . . . , n+ 1, 1̄, 2̄, . . . , n+ 1
}
:[
a, ā
]
7→ ∆aā = ∆a,n+2 (1 ≤ a ≤ n+ 1),{
[a, b],
[
ā, b̄
]}
7→ ∆ab (1 ≤ a < b ≤ n+ 1),{[
a, b̄
]
,
[
ā, b
]}
7→ ∆ab̄ = ∆a,n+2∆b,n+2 −∆ab (1 ≤ a < b ≤ n+ 1).
Let V̊n be the space of 2× (n+2) matrices such that all the above functions are non-vanishing,
and let Tn+1
∼= (C×)n+1 act on V̊n by scaling the first n + 1 columns. Then the action of
SL2×Tn+1 on V̊n is free.
Proposition 7.2. We have an isomorphism MBn
∼= SL2 \V̊n/Tn+1.
Using the action of SL2 we can gauge-fix the last column of M ∈ V̊n to [0, 1]T , and using
the action of Tn+1, we may gauge-fix the first entry of columns 1, 2, . . . , n+ 1 of M to 1. Thus
modulo the action of SL2×Tn+1, every point in V̊n can be written in the form[
1 1 1 · · · 1 0
z1 z2 z3 · · · zn+1 1
]
,
where zi ∈ C, and two such matrices with parameters z = (z1, . . . , zn+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 MBn with
a subspace of Cn+1/C = (z1, . . . , zn+1)/C · 1. For these matrices, the cluster variables ∆aā are
equal to 1, and we have
∆ab = zb − za (1 ≤ a < b ≤ n+ 1),
∆ab̄ = 1− zb + za (1 ≤ a < b ≤ n+ 1). (7.2)
We recognize the hyperplanes (7.2) as the Shi arrangement [28].
Proposition 7.3. MBn is isomorphic to the complement in Cn+1/C of the Shi arrangement.
Among many well-known properties, we obtain the following as immediate consequences
using Theorem 7.1: (a) #MBn(Fq) = (q − n − 1)n, (b) the number of connected components
|π0(MBn(R))| = (−1)n(#MBn(Fq)|q=−1) is equal to (n+2)n, (c) the cohomology H∗(MBn ,C)
is generated in degree one by dlog∆, as ∆ varies over the hyperplanes (7.2), and (d) |χ(MBn)|
= nn.
7.4. Type Cn. By applying Theorem 4.2 to [18, Example 12.12], we obtain a description
of MCn . Recall that Γ is the two-element group whose non-trivial element maps i↔ ī. Consider
the space Mat2,n+1 of 2× (n+ 1) matrices[
y11 y12 · · · y1,n+1
y21 y22 · · · y2,n+1
]
and the cluster variables{
[a, b], [ā, b̄]
}
7→ ∆ab =
y1ay2b − y1by2a
2i
(1 ≤ a < b ≤ n+ 1),{
[a, b̄], [ā, b]
}
7→ ∆ab̄ =
y1ay2b + y2ay2b
2
(1 ≤ a ≤ b ≤ n+ 1) (7.3)
Cluster Configuration Spaces of Finite Type 25
labeled by Γ-orbits of sides and diagonals in the polygon P2n+2 with vertices {1, 2, . . . , n +
1, 1̄, 2̄, . . . , n+ 1}. These functions generate the ring of invariant functions C[Mat2,n+1]
S , iso-
morphic to a cluster algebra A
(
B̃
)
of type Cn (when frozen variables are inverted). Here,
S = diag(t, 1/t) is the group of 2 × 2 diagonal matrices with determinant 1. The cluster auto-
morphism group is T = (C×)n+1×(Z/2Z), where (C×)n+1 ∼= (C×)n+1/⟨(−1,−1, . . . ,−1)⟩ acts on
Mat2,n+1 by rescaling columns (with the element (−1, . . . ,−1) acting trivially on S\Mat2,n+1),
and the non-trivial element of the group (Z/2Z) acts by swapping the two rows.
Remark 7.4. Note that in contrast to the Bn case ([18, Example 12.10]), the B̃-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 have
B̃ for type B2 =
0 1
−2 0
1 0
1 −1
−1 0
and B̃ for type C2 =
0 2
−1 0
1 0
1 −2
−1 0
respectively, where the rows are labelled by 13, 11̄, 12, 13̄, 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
z1 z2 · · · zn 1
]
(7.4)
and the non-vanishing of the cluster variables is equivalent to the non-vanishing of the linear
forms
zi − zj , zi + zj , 1− zi, 1 + zi, zi. (7.5)
Let Z̊n denote the space of matrices of the form (7.4), where the linear forms (7.5) are non-
vanishing. There is still a free action of Z/2Z on Z̊n, acting by swapping the two rows, which
induces (z1, . . . , zn) 7→ (1/z1, . . . , 1/zn). By Theorem 4.2, we obtain
Proposition 7.5. We have an isomorphism MCn
∼= Z̊n/(Z/2Z).
This isomorphism is valid over the integers even though (7.3) involves the scalars 1/2 and 1/2i:
this is because the scalars cancel in any T -invariant ratio of cluster variables.
Proposition 7.6. We have
#MCn(Fq) = (q − n− 1)(q − 3)(q − 5) · · · (q − 2n+ 1)
for char(q) > 2 and
|π0(MCn(R))| =
1
2
(
2n(n+ 1)! + 2nn!
)
= 2n−1(n+ 2)n!.
Note that |π0(MCn(R))| = |(#MCn(Fq))|q=−1|, agreeing with Theorem 7.1, even though we
do not know whether MCn is isomorphic to a hyperplane arrangement complement.
26 N. Arkani-Hamed, S. He and T. Lam
Proof. For a field k, the points of MCn(k) in general come from Z̊n(k̄), where k̄ denotes the
algebraic closure of k.
First, suppose that k = Fq with char(q) > 2. Let F̄q denote the algebraic closure of Fq.
Then the Galois group is topologically generated by one generator σ, called the Frobenius auto-
morphism. It acts as the field automorphism σ(x) = xq, and furthermore, we have σ(x) = x
if and only if x ∈ Fq. The map π : Z̊n(F̄q) → MCn(F̄q) commutes with the action of σ. Thus
if π(z) = u ∈ MCn(Fq), we have
u ∈ MCn(Fq) ⇔ 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
#Z̊n(Fq). Imposing the conditions (7.5), we get
#Z̊n(Fq) = (q − 3)(q − 5) · · · (q − 2n− 1).
For case (2), the equation σ(z) = 1/z is equivalent to zq+1
i = 1 for i = 1, 2, . . . , n. There are
q + 1 solutions to the polynomial equation xq+1 − 1 in F̄q. Two of the solutions are x = ±1.
The other q − 1 solutions lie in Fq2 , since x
q+1 = 1 =⇒ xq
2−1 = 1 =⇒ xq
2
= x. Thus in
case (2), we are counting n-tuples (z1, z2, . . . , zn) ∈ Fq2 satisfying the condition xq+1 = 1 and
the conditions imposed by (7.5). The conditions zi ̸= ±1 are automatically satisfied, so we get
(q − 1)(q − 3) · · · (q − 2n+ 1).
In sum, taking into account the two-to-one covering Z̊n → MCn , we have
#MCn(Fq) =
1
2
((2q − 2n− 2)(q − 3)(q − 5) · · · (q − 2n+ 1))
= (q − n− 1)(q − 3)(q − 5) · · · (q − 2n+ 1).
(Note that this point count is also the same as the hyperplane arrangement complement with
hyperplanes zi + zj , zi − zj , 1 + zi, 1− zi, though we do not know an explanation for this.)
Next let us consider k = R, so k̄ = C. In this case, σ is replaced by complex conjugation.
So the same argument says that we should consider the two cases (1) z̄ = z, or (2) z̄ = 1/z. For
case (1), we are looking at Z̊n(R) and by Theorem 7.1 we get
|π0(Z̊n(R))| = 2n(n+ 1)!
since #Z̊n(Fq) = (q − 3)(q − 5) · · · (q − 2n− 1).
For case (2), we must have zi = exp(iθi) on the unit circle, where θi ∈ (π, π]. The condi-
tions (7.5) give
θi ̸= θj , θi ̸= 0, θi ̸= π, θi + θj ̸= 0.
Thus the quantities |θ1|, . . . , |θn| lie in (0, π) and satisfy |θi| ̸= |θj |. There are n! regions in |θ|
space, and 2n sign choices going from |θi| to θi, giving∣∣π0(Z̊n(S
1)
)∣∣ = 2nn!.
In sum, taking into account the two-to-one covering Z̊n → MCn , we have
|π0(MCn(R))| =
1
2
(2n(n+ 1)! + 2nn!) = 2n−1(n+ 2)n!,
as claimed. ■
Cluster Configuration Spaces of Finite Type 27
The variety MCn can be identified with a subvariety of MA2n−1 . Sending (z1, . . . , zn) to[
1 1 · · · 1 1 1 · · · 1 1
z1 z2 · · · zn 1 −z1 · · · −zn 1
]
(7.6)
maps MCn into MA2n−1 . Note that the non-vanishing of (7.5) is equivalent to the non-vanishing
of all minors in (7.6), and that (z1, . . . , zn) and (1/z1, . . . , 1/zn) represent the same point
in MA2n−1 .
7.5. Type Dn. We do not know a simple description of MD in this case. Indeed, the point
counts we have obtained show that MD cannot be a hyperplane arrangement complement.
In type D4, numerical computations indicate that for p ̸= 2, we have
|MD4(Fp)| =
{
206− 231p+ 93p2 − 16p3 + p4 if p = 2 mod 3,
208− 231p+ 93p2 − 16p3 + p4 if p = 1 mod 3.
(7.7)
Substituting p = −1 in (7.7), we get 547. We do not know for sure that MD4(R) has 547
connected components, but see Section 11.4.
For type D5, numerical computations indicate that for p ̸= 2, 3, we have
|MD5(Fp)| = −2318 + 2644p− 1156p2 + 244p3 − 25p4 + p5 + (−36 + 5p)δ3(p)− δ4(p),
where we define δ3(p) = 0 for p = 2 mod 3 or 2 for p = 1 mod 3 and similarly δ4(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 MD5(R) has 6388 connected
components, but see Section 11.4.
7.6. Type G2. Numerical computations give
|MG2(Fp)| =
{
(p− 4)2 if p = 2 mod 3,
(p− 4)2 + 4 if p = 1 mod 3.
Substituting p = −1, we get 25, which we expect to be the number of connected components
of MG2(R).
8 Positive part
In this section, we define the nonnegative subspace MD,≥0 of MD(R) and show that it is
diffeomorphic to the generalized associahedron of D∨.
8.1. We define the positive part MD,>0 ⊂ MD(R) as the subspace
MD,>0 :=
{
x ∈ MD(R) |uγ(x) > 0 for all γ ∈ Π
}
and the nonnegative part MD,≥0 ⊂ M̃D(R) by
MD,≥0 := MD,>0 ⊂ M̃D(R).
Intersecting with the stratification (3.6), we obtain
MD,≥0 =
⊔
F
MF,>0, where MF,>0 = MD,>0 ∩MF (R).
Let P be an integer polytope andN (P ) denote its normal fan. It is well-known [20, Chapter 4]
that the nonnegative part XN (P ),≥0 of the projective toric variety XN (P ) is diffeomorphic to the
polytope P . The following result thus follows from Theorem 5.3.
28 N. Arkani-Hamed, S. He and T. Lam
Theorem 8.1. There is a face-preserving diffeomorphism between MD,≥0 and the generalized
associahedron of D∨.
The positive part MD,>0 is identical to the positive part XN (B∨),>0 of the ambient pro-
jective toric variety, and MD,>0 is equal to one of the connected components of the smooth
manifold MD(R).
8.2. The space MD has a distinguished rational top-form Ω(MD), called the canonical form,
that can be described in a number of ways. Suppose B̃ is a full rank extended exchange
matrix of type D. The cluster algebra A
(
B̃
)
has a natural top-form Ω which in any cluster
(x1, x2, . . . , xn+m) can be written (up to sign):
Ω =
dx1
x1
∧ · · · ∧ dxn+m
xn+m
,
which is the natural top-form on the corresponding cluster torus (C×)n+m. The cluster auto-
morphism group T
(
B̃
)
can be identified with a subgroup of (C×)n+m, and the quotient group
(C×)n+m/T
(
B̃
)
is again an algebraic torus S = (C×)n. The inclusion (C×)n+m ⊂ X
(
B̃
)
iden-
tifies S birationally with MD (note that neither S nor MD contains the other, but the two
share a common dense open subset). The torus S has a natural top-form, and the canonical
form Ω(MD) is the image of this form under the birational isomorphism between S and MD.
Another way to obtain the canonical form of MD is via Theorem 5.3: any toric variety XP
has a canonical rational top-form Ω(XP ), which is simply the (extension of the) natural top-form
of the dense algebraic torus in XP . Restricting Ω
(
XN (B∨)
)
to MD gives a top-form on MD
which equals Ω(MD).
The pair
(
M̃D,MD,≥0
)
is nearly a positive geometry in the sense of [2], with Ω
(
M̃D
)
=
Ω(MD) as canonical form: the residue of Ω(MD) along a divisor M̃D(F ) is equal to the
canonical form Ω
(
M̃D(F )
)
, which is simply the product of canonical forms corresponding to
the factorization of Proposition 3.6. This follows from the similar statement concerning XN (B∨),
proven in [2, Appendix G]. However, M̃D is an affine variety rather than a projective variety,
so it is not a positive geometry in the strict sense.
In the case D = An−3, we have MD = M0,n, and the canonical form can be written as
Ω
(
(M0,n)>0
)
=
dz2 · · · dzn−2
(z2 − z1)(z3 − z2) · · · (z1 − zn)
,
where (z1, zn−1, zn) = (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((M0,n)>0) only has poles along the boundary divisors of M̃0,n
(
and not elsewhere in M0,n
)
is [9, Proposition 2.7]. Combining with the above discussion, we have
Proposition 8.2. The pair
(
M0,n, (M0,n)≥0
)
is a positive geometry with canonical form
Ω
(
(M0,n)>0
)
.
9 Positive tropicalization
In this section, we consider the positive tropicalization of MD. 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=1R
((
t1/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 αtr, where
Cluster Configuration Spaces of Finite Type 29
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) ∈ MD(R>0) is a collection u(t) = {uγ(t) ∈ R>0, γ ∈ Π} of (positive) Puiseux se-
ries satisfying the relations from Definition 3.1. We define the positive tropicalization Trop>0MD
as the closure of valuations
Trop>0MD :=
{
val(uγ(t))γ∈Π ∈ RΠ |u(t) ∈ MD(R>0)
}
⊂ RΠ.
Lemma 9.1. The subspace Trop>0MD is a (not complete) polyhedral fan inside the linear
space RΠ.
Proof. We use the identification MD
∼= X̊prin/T prin from Theorem 4.3. According to Proposi-
tion 4.6, there is an invertible monomial transformation between the functions uγ = fγ and the
set of functions {y1, . . . , yn} ∪ {Fγ(y) | γ ∈ Π+}. Since each Fγ(y) is a positive Laurent poly-
nomial in the yi-s, it follows that each uγ(y) is a subtraction-free rational function in the yi-s.
Thus the map (uγ)γ∈Π 7→ (y1, y2, . . . , yn) induces an isomorphism MD(R>0) ∼= Rn
>0. It follows
that we have a homeomorphism Trop>0MD
∼= Rn. The embedding of Trop>0MD in the (larger
dimensional) linear space RΠ endows it with the structure of a polyhedral fan. ■
Theorem 9.2. The fan Trop>0MD is isomorphic to the cluster fan N (B∨).
Proof. Under the isomorphism Trop>0MD
∼= Rn, the fan structure of Trop>0MD gives a com-
plete fan in Rn whose maximal cones are the common domains of linearity of the piecewise linear
functions Trop(uγ(y)). By Proposition 4.6, we can equivalently take the common domains of lin-
earity of the functions Trop(Fγ(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+ Fγ(y). By Theorem 2.4, we deduce the isomorphism of fans Trop>0MD
∼= N (B∨). ■
Now let A
(
B̃
)
denote a cluster algebra of finite type, and X
(
B̃
)
the corresponding cluster
variety. Using the set of cluster variables xγ , γ ∈ Π and coefficient variables xn+1, . . . , xn+m,
we have an embedding X
(
B̃
)
↪→ C|Π|+m. We define Trop>0X
(
B̃
)
⊂ R|Π|+m as the closure of the
image of X
(
B̃
)
(R>0) under the map val : R>0 → R. Note that the tropicalization Trop>0X
(
B̃
)
depends only on the cluster algebra A = A
(
B̃
)
and not on the choice of initial cluster.
The projection map (xγ , xi) 7→ (x1, x2, . . . , xn+m) from R|Π|+m to Rn+m (onto the initial
cluster variables) identifies Trop>0X
(
B̃
)
with a complete polyhedral fan in Rn+m.
Proposition 9.3. Suppose that B̃ is of full rank and has finite type D. Then Trop>0X
(
B̃
)
modulo its lineality space L is isomorphic to Trop>0MD.
Proof. The translation action of the lineality space L on Rn+m is simply the tropicalization
of the action of the automorphism torus T
(
B̃
)
(4.1). Thus B̃ is of full rank if and only if L
has dimension m if and only if T
(
B̃
)
is of dimension m. In particular, Trop>0X
(
B̃
)
/L is
a polyhedral fan of dimension n.
Via the isomorphism of Theorem 4.2, each uγ = fγ ∈ C[MD] can be identified with a mono-
mial in the cluster and coefficient variables xγ , γ ∈ Π and xn+1, . . . , xn+m. We thus have
a linear projection map p : R|Π|+m → R|Π| (the tropicalization of the rational map sending x-s
to u-s) mapping Trop>0X
(
B̃
)
surjectively to Trop>0MD. The fibers of p are exactly the orbits
of the lineality space L acting on Trop>0X
(
B̃
)
. It follows that the fans Trop>0X
(
B̃
)
/L and
Trop>0MD are isomorphic. ■
Noting that the fans N
(
B∨) and N (B) are combinatorially isomorphic, we deduce from
Theorem 9.2 and Proposition 9.3 the following conjecture of Speyer and Williams [29, Conjec-
ture 8.1]. The principal coefficient case was established in [22] and we thank Christian Stump
for drawing our attention to this work.
30 N. Arkani-Hamed, S. He and T. Lam
Corollary 9.4. Suppose that B̃ is of full rank and has finite type. Then Trop>0X
(
B̃
)
/L is
combinatorially isomorphic to the complete fan N (B).
10 Extended and local u-equations
In this section, we first study two additional sets of equations satisfied by u-variables, the exten-
ded u-equations and the local u-equations. In other words, we give some further distinguished
elements in the ideal ID.
In type A, the extended u-equations were used by Brown [11] to define what we call M̃An−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.
10.1. Extended u-equations. An extended u-equation is an equation which holds in C[MD]
of the form∏
γ
u
αγ
γ +
∏
γ
u
βγ
γ = 1, (10.1)
where αγ , βγ are nonnegative integer parameters. All the primitive u-equations Rγ in Defini-
tion 3.1 are examples. Corollary 4.5 gives a class of extended u-equations for each D. It would
be desirable to have a uniform (instead of case-by-case) description of the extended u-equations
coming from Corollary 4.5 similar to the description of the relations Rγ . This would follow
from a solution to the following problem, which we believe is open. (Recall that the primitive
exchange relations are described in Proposition 2.3.)
Problem 10.1. Give a uniform, root-system theoretic, description of all the exchange relations
in a cluster algebra of finite type with universal coefficients.
We now present explicitly extended u-equations for all the classical types A, B, C, D, and also
for type G2. 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 A2n−3 (resp. Dn) gives one for type Cn−1 (resp. Bn−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
UI,J :=
∏
i∈I,j∈J
ui,j . (10.2)
Note that UI,J and UJ,I are not necessarily equal.
10.1.1. Type A. Let {1, 2, . . . , n} = A ⊔ B ⊔ C ⊔ D be a decomposition of [n] into cyclic
intervals. Then we have the extended u-equation
UA,C + UB,D = 1. (10.3)
Each equation depends on the choice of four cyclically ordered points a, b, c, d: A = {a+1, . . . , b},
B = {b+1, . . . , c}, C = {c+1, . . . , d} and D = {d+1, . . . , a}, and thus there are
(
n
4
)
equations
in total, in bijection with the exchange relations of the type A cluster algebra. These equations
Cluster Configuration Spaces of Finite Type 31
arise from Corollary 4.5. See [31, Proposition 7.2] for the exchange relations of the universal
coefficient cluster algebra of type A.
For A = {i}, C = {j}, the equation (10.3) becomes the primitive u-equations, Rij = 0.
As discussed in [4, 11], it is natural to interpret these U ’s as cross-ratios of n points on P1:
denote UA,C = [a, b|c, d], and (10.3) becomes [a, b|c, d]+[b, c|d, a] = 1. Together with the identity
by definition [a, b|c, e][a, b|e, d] = [a, b|c, d], the equalities [a, b|c, d] + [b, c|d, a] = 1 invariantly
characterize cross-ratios of n points, namely [a, b|c, d] = ∆ad∆bc
∆ac∆bd
.
10.1.2. Type C. Extended u-equations for type Cn−1 arise via folding A2n−3. For a decom-
position
{
1, 2, . . . , n, 1̄, . . . , n̄
}
≃ [2n] = A ⊔B ⊔ C ⊔D into cyclic intervals, the image of (10.3)
gives the extended u-equation for Cn−1 (which become the primitive ones for A = {i}, C = {j}).
For example, if we choose A = {1} and C = {n+1}, we have
1− u[11̄] = U{2,...,n},{2̄,...,n̄} =
n∏
i=2
u[īi]
∏
2≤i<j≤n
u2[ij̄],
which is the primitive u-equation R[11̄] = 0. For C2, in addition to the 6 primitive u-equations
given in (3.3), we have 3 more equations:
u[11̄]u[12̄] + u[23̄]u[33̄] = 1, (10.4)
and its cyclic rotations.
Let us count the number of extended u-equations for Cn−1 we have obtained. There are
(
2n
4
)
equations (10.3) in type A2n−3. Of those,
(
n
2
)
are equal to its mirror image. For the remainder,
both the equation and its mirror image map to the same equation in type Cn−1. Thus we have
obtained n(n−1)(2n2−4n+3)
6 extended u-equations for Cn−1. Note that this number is greater than
the number of exchange relations of type Cn−1. For example, for C2, there are 6 exchange
relations, but we have found 9 extended u-equations.
10.1.3. Type D. We now consider type Dn. We use the notation (10.2) and also define (here
a ≺ b means a precedes b in I)
UI :=
∏
a≺b∈I
ua,b
∏
i∈I
ui, ŨI :=
∏
a≺b∈I
ua,b
∏
i∈I
uĩ.
We now describe two types of extended u-equations. (These equalities were discovered empri-
cally, but it should be relatively straightforward to prove them by induction.) First, similar
to (10.3), for a cyclically ordered partition A ⊔B ⊔ C ⊔D = {1, 2, . . . , n}, we have
UC,A + UD,BUA,BUB,CUB,DUBŨB = 1. (10.5)
We allow D to be empty here, in which case (10.5) becomes UC,A+UA,BUB,CUBŨB = 1. Second,
for a cyclically ordered partition A ⊔B ⊔ C = {1, 2, . . . , n}, we have
UAUA,B + ŨCUB,C = 1. (10.6)
Note that B can be empty here, in which case we have UA+ŨC = 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)(n2+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 n2 cluster variables, 1
4(n
4 + n2 − 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.
32 N. Arkani-Hamed, S. He and T. Lam
It is straightforward to obtain the primitive u-equations for type Dn. In (10.5), choosing
A = {i}, C = {j} (including the degenerate case with D = ∅, thus j = i−1), we have
1− uj,i = UD∪{i},BUB,j∪{D}UBŨB,
where B = {i+1, . . . , j−1} and D = {j+1, . . . , i−1}. In (10.6), take B = ∅, and choosing A = i
or C = i we have
1− ui = Ũ{i+1,...,i−1}, 1− uĩ = U{i+1,...,i−1}.
For example, for n = 4, we have 52 extended u-equations of the form (10.5) and (10.6), including
the 16 primitive u-equations in (3.4).
10.1.4. Type B. Finally, we consider type Bn−1 by folding Dn. We identify ui = uĩ, and the
two types of equations become
UC,A + UD,BUA,BUB,CUB,DU
2
B = 1, UAUA,B + UCUB,C = 1.
We have n(n−1)(n2+n−3)
6 such extended u-equations in total: note that each equation (10.5) has
a distinct image in Bn−1 but two of the equations (10.6) map to a single one in Bn−1. Again,
we note that this number is greater than the number of exchange relations in type Bn−1.
The primitive u-equations can be recovered by setting A = {i}, C = {j} (also B = ∅ in
the second one). For example, for n = 4, we obtain 34 extended u-equations including the 12
primitive ones of (3.5). Let’s write the additional 22 equations as dihedral orbits of sizes 8,
8, 4, 2:
8 eqs : u3,1u4,1 + u1,2u2,3u
2
2 = 1,
8 eqs : u1,2u1u2u1,3u2,3 + u4u3,4 = 1,
4 eqs : u1u1,2u1,3 + u4u2,4u3,4 = 1,
2 eqs : u1,2u1u2 + u3,4u3u4 = 1.
There are 22 exchange relations in type B3.
10.1.5. Type G. As in Section 3.2.5, let us call the 8 u-variables ai, bi for i = 1, . . . , 4 that
can be thought of as labelling the edges of an octagon (a1, b1, . . . , a4, b4). Folding D4, we obtain
18 extended u-equations for G2, the primitive u-equations
a1 + a2a
2
3a4b2b3 = 1, b1 + a33a
3
4b2b
2
3b4 = 1, +cyclic,
together with
a1b1 + a23a4b2b3 = 1, a2b1 + a3a
2
4b3b4 = 1, a1a2b1 + a3a4b3 = 1, +cyclic.
10.2. Local u-equations. The relations Rγ , and the extended u-equations are global in
nature: they involve many uγ 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 theX-coordinates for Fock and Goncharov’s clusterX-variety. For an exchange
relation xx′ =M+M ′, we have a clusterX-variableX =M/M ′. Now, let us consider a primitive
exchange relation in X
(
B̃
)
:
xτγxγ =M +M ′,
Cluster Configuration Spaces of Finite Type 33
where M ′ only involves frozen variables. We recall that the isomorphism of Theorem 4.2 iden-
tifies the rational function M
xτγxγ
with uγ . Thus the cluster X-variable Xγ := M/M ′ can be
identified with uγ/(1 − uγ) ∈ C[MD] which is equal to a Laurent monomial in the uω-s using
the relations Rγ . By [17] or [19, Proposition 3.9 or equation (8.11)] the variables Xγ satisfy the
relation
X(t−1,j)X(t,j) =
∏
i→j
(
1 +X(t,i)
)−aij
∏
j→i
(
1 +X(t−1,i)
)−aij (10.7)
or, equivalently, the variables uγ satisfy the relation
u(t−1,j)
1− u(t−1,j)
u(t,j)
1− u(t,j)
=
∏
i→j
(
1− u(t,i)
)aij ∏
j→i
(
1− u(t−1,i)
)aij .
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.
10.2.1. Type A. For type An−3 we have u-variables ui,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:
jj + 1
i i+ 1
Xi,jXi+1,j+1 = (1 +Xi,j+1)(1 +Xi+1,j) (10.8)
or
ui,j
1− ui,j
ui+1,j+1
1− ui+1,j+1
=
1
1− ui,j+1
1
1− ui+1,j
.
10.2.2. Type C. By folding A2n−3, we obtain local u-equations for type Cn−1. The local
u-equations take the form of (10.8); for the special case of j = ī, it reads
X[i,̄i]X[i+1,i+1] =
(
1 +X[i,i+1]
)2
.
10.2.3. Type D. We use the notation for u-variables from Section 3.2.3. The n2 local u-
equations for type Dn read
Xi,jXi+1,j+1 = (1 +Xi,j+1)(1 +Xi+1,j),
Xi−1,iXi,i+1 = (1 +Xi−1,i+1)(1 +Xi)
(
1 +Xĩ
)
,
XiXĩ+1
= 1 +Xi,i+1,
XĩXi+1 = 1 +Xi,i+1.
10.2.4. Type B. By folding type Dn, we obtain local u-equations for type Bn−1 (see Sec-
tion 3.2.4). The local u-equations are
X[i,j]X[i+1,j+1] =
(
1 +X[i,j+1]
)(
1 +X[i+1,j]
)
,
34 N. Arkani-Hamed, S. He and T. Lam
X[i−1,̄i]X[i,i+1] =
(
1 +X[i−1,i+1]
)(
1 +X[i,̄i]
)2
,
X[i,̄i]X[i+1,i+1] = 1 +X[i,i+1]
together with cyclic rotations.
10.2.5. Type E. Finally, for En with n = 6, 7, 8, we use the identification (2.1) to index u
and X variables. When B̃ is bipartite, i.e., every vertex is either a source or a sink in the induced
orientation of D, we have ri = h/2 + 1 does not depend on i, where h is the Coxeter number.
The Coxeter number is even in types E6, E7, E8, and Π is identified with m = 7, 10, 16 copies
of I respectively. We index the nodes of En as shown below:
1 2 3 4 n−2 n−1
n
The local u-equations take the following form:
X(t,i)X(t+1,i) = (1 +X(t+1,i+1))(1 +X(t+1,i−1))(1 +X(t+1,n))
δi,3 , for odd i < n,
X(t,i)X(t+1,i) = (1 +X(t,i))(1 +X(t,i+1)), for even i < n,
X(t,n)X(t+1,n) = 1 +X(t,3),
for t = 1, 2, . . . ,m in all these cases. We have n×m equations in total.
10.2.6. Types F4 and G2. By folding E6 we obtain local u-equations for F4, and by foldingD4
we obtain those for G2.
11 Connected components and sign patterns
The permutation group Sn acts on the moduli space M0,n(R), permuting the n points and per-
muting the connected components. The presentation ofM0,n(R) using uij and the relations (1.1)
depends on the choice of a dihedral ordering and the action of the symmetry group Sn is ob-
scured. In this section, we use the extended u-equations to investigate the connected components
of MD(R), with the hope of uncovering an appropriate symmetry group for MD 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 MD(R) is the application to
string amplitudes where it is important to consider canonical forms of different connected com-
ponents of M0,n(R); see Section 12.2 for a brief discussion.
We also define an analogue of an oriented matroid for MD, 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 (sγ) ∈
{+,−}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, (sγ) are possible signs
for some solution (uγ) 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.
Conjecture 11.1. The number of connected components of MD(R) is given by the number of
sign patterns of u-variables consistent with the extended u-equations for D.
Cluster Configuration Spaces of Finite Type 35
11.2. Type A. We consider D = An−3. The space M0,n(R) has (n−1)!/2 connected compo-
nents, corresponding to the dihedral orderings of n points. In the positive connected component
(M0,n)>0, all the cross ratios uij are positive, indeed, we have 0 < uij < 1. In other connected
components of M0,n, some of the uij-s are negative. We find that the extended u-equations ex-
clude 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 M0,n(R).
Let us now consider the problem of finding new u-variables for other components of M0,n(R),
that is, we seek cross-ratios that are positive on that component. It suffices to consider the orde-
ring that is obtained from the standard one by an adjacent transposition, e.g., (1′, 2′, 3′, . . . , n′) =
(2, 1, 3, . . . , n). Using the following identities for the cross ratio [a, b|c, d] (see Section 10.1.1):
[a, b|d, c] = 1
[a, b|c, d]
, [a, c|b, d] = − [a, b|c, d]
[b, c|d, a]
we find that the u-variables in this new ordering include
u′1,3 = [n′, 1′|2′, 3′] = [n, 2|1, 3] = −[n, 1|2, 3]
1, 2|3, n]
=
−[n, 1|2, 3]∏n−1
i=3 [1, 2|i, i+1]
=
−u1,3
u2,4 · · ·u2,n
,
u′1,i = [n′, 1′|i′−1, i′] = [n, 2|i− 1, i] = [n, 1|i−1, i][1, 2|i−1, i] = u1,iu2,i,
u′2,i = [1′, 2′|i′−1, i′] = [2, 1|i−1, i] =
1
[1, 2; i−1, i]
=
1
u2,i
,
u′3,i = [2′, 3′|i′−1, i′] = [1, 3|i−1, i] = [1, 2|i−1, i][2, 3|i−1, i] = u2,iu3,i, (11.1)
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 uij to a permutation of the extended u-equations for the u′i,j , and thus
exposing a hidden Sn-symmetry of these equations.
11.3. Type C. From the analysis of Section 7.4, we know that MCn(R) has 2n−1n!(n + 1)
connected components. Computationally, we find that this agrees with the number of consistent
sign patterns.
There are two types of components inMCn(R) corresponding to two types of configurations of
the (2n+2)-gon with labels i, ī for i = 1, 2, . . . , n+1: (A) 2n−1n! components for polygons with
central symmetry, e.g., 1, 2, . . . , n+1, 1̄, 2̄, . . . , n+1; (B) 2n−1(n+1)! components for polygons
with reflection symmetry avoiding vertices, e.g., the ordering 1, 2, . . . , n+1, n+1, . . . , 2̄, 1̄. 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,
C2 = B2.
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, 1̄, 2̄, 3̄, 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, 2̄, 1̄, 3̄, we find that the 6 new variables can be
obtained by a monomial change of variables:
u′[12̄] =
1
u[21̄]u[22̄]
, u′[23̄] = −
u[13]
u[1̄2]u[22̄]u[23̄]
, u[31̄] =
1
u[22̄]u[23̄]
,
u′[11̄] = u[22̄], u′[22̄] = u[11̄]u
2
[21̄]u[22̄], u[33̄] = u[22̄]u
2
[23̄]u[33̄].
36 N. Arkani-Hamed, S. He and T. Lam
It is straightforward to check that these 6 new variables satisfy identical extended u-equations
for the ordering 2, 1, 3, 2̄, 1̄, 3̄. More generally, under this kind of transformation, similar to the
type An−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.
Let us now consider orderings in (B). For example, for the ordering 1, 2, 3, 3̄, 2̄, 1̄, we find
that the 6 new variables are given in terms of the old ones by
u′[13̄] = −
u[11̄]
u[22̄]u
2
[23̄]
u[33̄]
, u′[1̄2] =
1
u[22̄]
, u′[2̄3] =
1
u[33̄]
,
u′[12̄] = u[13]u[23̄]u[33̄], u′[23̄] = u[21̄]u[22̄]u[23̄], u′[22̄] =
1
u[23̄]
.
The 9 extended u-equations become the following ones for the 6 new variables:
u′[12̄]u
′
[13̄] + u′[22̄]u
′
[1̄2] = 1, u′[13̄]u
′
[23̄] + u′[22̄]u
′
[2̄3] = 1, u′[22̄](u
′
[12̄] + u′[23̄]) = 1,
u′[23̄] + u′[12̄]u
′
[22̄]u
′
[2̄3] = 1, u′[13̄] + u′2[22̄]u
′
[1̄2]u
′
[2̄3] = 1, u′[12̄] + u′[22̄]u
′
[23̄]u
′
[1̄2] = 1,
u′[1̄2] + u′2[12̄]u
′
[13̄] = 1, u′[23̄] + u′[13̄]u
′2
[23̄] = 1, u′[22̄](1 + u′[12̄]u
′
[13̄]u
′
[23̄]) = 1.
Note that u′
[22̄]
is special: from the third and the last equations, it is easy to see that u′
[22̄]
cannot
take the value 0, and thus u′
[22̄]
= 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 D4 and D5. We were unable to determine the number of connected components
of MDn(R). However, we can obtain a consistency check by comparing the number of consistent
sign patterns with the point count over Fq.
Recall from the point count (7.7) in type D4, we predicted that MD(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 D5 case computationally agrees with the number of consistent sign
patterns of u-variables.
11.5. Type G2. As we have discussed in Section 7.6, we expect that there are 25 different
connected components for G2. Furthermore, the point count MG2(Fq) is not polynomial, and
thus MG2 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 MG2(R), and note some new features.
The positive component has 0 < ai, bi < 1 as usual. But suppose b1 is made negative; to wit
we put b1 = −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 (monomial1) + (monomial2) = 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
x1 = b−1
4 , x2 = a1a3a
2
4b3b4, x3 = b′1a
−3
3 a−3
4 b−1
2 b−2
3 b−1
4 ,
x4 = a2a
2
3a4b2b3, x5 = b−1
2 ,
y1 = b3, y2 = a−1
4 b−1
3 , y3 = a−1
3 b−1
3
Cluster Configuration Spaces of Finite Type 37
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
x1 + x3x
2
4 = 1, x2 + x4x5y
2
1y2y
2
3 = 1, x3 + x1x5y
4
1y
3
2y
2
3 = 1,
x4 + x1x2y
2
1y
2
2y3 = 1, x5 + x32x3 = 1,
y1 + y41x
3
2x
2
3x
3
4y
3
2y
2
3 = 1, y2 + y22x2x3x
2
4y
2
1y
2
3 = 1, y3 + y23x
2
2x3x4y
2
1y
2
2 = 1.
As usual these equations tell us that if the x, y ≥ 0, then we also have x, y ≤ 1. But note an
interesting feature of the equations for the yi that we also saw for Cn: both monomials contain
a factor of yi, and therefore we cannot set any of the y’s to zero. Thus the only boundaries of
this connected component are associated with the xi → 0 for i = 1, . . . , 5; and we have found
an unusual binary realization (in the sense of [4]) of a pentagon.
Suppose instead we now set a1 to be negative, that is, a1 = −a′1 with a′1 > 0. Repeating the
same analysis something more interesting happens. The set of 36 exponent vectors associated
with the monomials of the extended u-equations span a cone with 12 generators. The monomials
associated with these 12 generators are
x1 = a−1
4 , x2 = a32a
3
3b1b
2
2b3, x3 = b−1
3 , x4 = b−1
2 , x5 = a′1a
−1
2 a−2
3 a−1
4 b−1
2 b−1
3 ,
x6 = a33a
3
4b2b
2
3b4, x7 = a′1a
2
4b3b4a
−1
2 b−1
2 , x8 = a−2
3 a−1
4 b−1
2 b−1
3 ,
x9 = a′1a
2
2b1b2a
−1
4 b−1
3 , x10 = a3b2, x11 = a3b3, x12 = a−1
2 a−2
3 b−1
2 b−1
3
and all the extended u-equations can be written (albeit not uniquely) as a sum of two monomials
in the x’s with positive coefficients. Of course, twelve of the equations are of the form xi +
(monomial) = 1, so all the x’s are restricted to lie between 0 and 1. But the twelve exponent
vectors in eight dimensions satisfy four relations which can be expressed in many equivalent
ways, for instance
x22x3x
2
5x8 = x1x4x
2
9, x26x4x
2
5x8 = x1x3x
2
7, x210x4x8 = x1x3, x211x3x8 = x1x4.
So in the language of these x-variables, we have monomial equations with positive exponents,
but also satisfying non-trivial constraint relations. Further study of this particular region reveals
it to be a binary realization of a hexagon. It is natural to conjecture that the phenomenon we
have seen in this G2 example is generic–when studying different connected components using u-
equations, the set of exponent vectors will be a pointed cone, and that the equations will always
force all variables to lie between (0, 1). But the variables will satisfy additional monomial
constraints.
12 Outlook
We close with a few comments on open directions for future exploration.
12.1. Understanding real components of cluster configuration space. An open ques-
tion immediately suggested by our investigations (see Section 11) is understanding all the con-
nected components of the real points of cluster configuration spaces. For An−3, there is a beau-
tiful picture, where the complete space is tiled by “binary associahedra” corresponding to all
(n−1)!/2 orderings of n points on the projective line, and there is a similar complete picture of
all the orderings for Cn by folding. These examples are especially easy to understand since we
have a “linear model” for M0,n in terms of a hyperplane arrangement. The connected compo-
nents can also be easily understood in these examples, directly studying the space of solutions of
the u-equations with different sign patterns. In general, MD is not a hyperplane arrangement
38 N. Arkani-Hamed, S. He and T. Lam
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 configu-
rations 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 uγ
are in bijection with all the cluster variables. Consider any acyclic cluster (xγ1 , . . . , xγn). Then,
the canonical form is simply given by taking the wedge product
Ω =
n∧
i=1
duγi
uγi(1− uγi)
. (12.1)
As we noted in Section 10.2, acyclic seeds have the following feature: all the uγ 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 G2
case, where b1 = −b′1 < 0. We can readily check that all of the (x, y) variables can be rationally
solved for in terms of either (x2, x3) or in terms of (x3, x4). The canonical form is then given as
Ω = dlog
x2
1− x2
dlog
x3
1− x3
= dlog
x3
1− x3
dlog
x4
1− x4
.
12.2. Open and closed cluster string amplitudes. The stringy canonical forms of [3] can
be applied to the cluster configuration space MD, and we obtain the cluster string amplitude.
For a cluster algebra A
(
B̃
)
of full rank and of type D, the cluster string integral is defined to be
ID(s) :=
∫
MD,>0
Ω(MD,>0)
∏
γ∈Π
x
α′sγ
γ
n+m∏
i=n+1
xα
′si
i , (12.2)
where {sγ | γ ∈ Π}∪{si | 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 MD,>0. Choosing B̃ = B̃prin, we may use
Theorem 4.3 to rewrite (12.2) as
ID(X, {c}) =
∫
Rn
>0
∏
i
dyi
yi
yα
′Xi
i
∏
γ∈Π+
Fγ(y)
−α′cγ , (12.3)
where (X, {c}) are related linearly to (si, sγ). By [3, Claim 2], (12.3) converges when the point X
belongs to the generalized associahedron P (c) =
∑
γ∈Π+ cγPγ , where Pγ is the Newton polytope
of Fγ(y). By [3, Claim 2] the leading order of ID(X, {c}) is the canonical function Ω(P (c))
of P (c) evaluated at X:
lim
α′→0
(α′)nID(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 ID(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
ID(U) =
∫
MD,>0
Ω(MD,>0)
∏
γ∈Π
u
α′Uγ
γ (12.4)
Cluster Configuration Spaces of Finite Type 39
and the convergence condition is the very simple condition Uγ > 0. By Proposition 4.6, the uγ
and {yi, Fγ} are related by an invertible monomial transformation, and thus {Uγ | γ ∈ Π} and
{X1, . . . , Xn} ∪ {cγ | γ ∈ Π+} are related by an invertible linear transformation. (The matrix
of this linear transformation has entries given by the integers Trop(Fγ(y))
(
−g∨
ω
)
that appeared
in Section 6.) We see from (12.4) that the u-variables uγ 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 MD,>0) will span a space of integral functions
of dimension equal to the absolute value of the Euler characteristic |χ(MD(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
Iclosed
D
({
U, Ū
})
=
∫
MD(C)
Ω(MD,>0)
∏
γ∈Π
u
α′Uγ
γ ∧ Ω̄(MD,>0)
∏
γ∈Π
ū
α′Ūγ
γ ,
where in order for the integrand to be single-valued, we must have that the exponents Ūγ differ
from Uγ at most by integers.
As we have remarked, it is plausible that real components of cluster configuration space other
than the region associated with uγ ≥ 0 provide us with many different positive geometries M(i)
D ,
with associated canonical forms Ω
(
M(i)
D
)
. In this case we can extend the closed string integrals
to be more generally labelled by pairs of these positive geometries,
Iclosed
M(i)
D ,M(j)
D
({
U, Ū
})
=
∫
MD(C)
Ω
(
M(i)
D
) ∏
γ∈Π
u
α′Uγ
γ ∧ Ω̄
(
M(j)
D
) ∏
γ∈Π
ū
α′Ūγ
γ .
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 reverse-
engineering 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 x1, x2, . . . , xp such that
f(x1), f(x2), . . . , f(xp) are not zero-divisors in B. (A.1)
40 N. Arkani-Hamed, S. He and T. Lam
Suppose that
(1) the localized homomorphism S−1f : S−1A→ S−1B is an isomorphism, and
(2) for each i = 1, 2, . . . , p the induced homomorphism fi : A/(xi) → B/(f(xi)) is an isomor-
phism.
Then f is an isomorphism.
Proof. Let K denote the kernel of f . Let a ∈ K be a nonzero element. Suppose that xia ̸= 0
in A for all i. Then the image of a in S−1A is nonzero and it is in the kernel of S−1f : S−1A
→ S−1B. This contradicts (1). Thus Ma = 0 for some monomial M in the xi-s. Replacing a
by M ′a for some other monomial M ′, and using (A.1), we may assume that a ∈ K and xia = 0
for some i = 1, 2, . . . , p.
If a ∈ (xi), then by (A.1), we have a = xia1 for a nonzero element a1 ∈ K. Repeating,
we either find a nonzero element a′ ∈ K such that a′ /∈ (xi), or we have an ascending chain of
ideals (a) ⊂ (a1) ⊂ (a2) ⊂ · · · . In the former case, the image of a′ in A/(xi) is nonzero and
in the kernel of fi, contradicting (2). Thus we are in the latter case. Since xi 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′′ = xia
′ and xna′ = 0 for some n > 0. This is impossible:
letting m be minimal such that xmi a
′ = 0 we find that xm−1
i a′′ = 0 which implies xm−1
i a′ = 0,
a contradiction. ■
Acknowledgements
We thank Mark Spradlin and Hugh Thomas for many discussions related to this work and for
closely related collaborations. We thank the anonymous referees for a number of corrections and
helpful suggestions to the exposition. T.L. was supported by NSF DMS-1464693, NSF DMS-
1953852, and by a von Neumann Fellowship from the Institute for Advanced Study. N.A-H. was
supported by DOE grant DE-SC0009988. S.H. was supported in part by the National Natural
Science Foundation of China under Grant No. 11935013, 11947301, 12047502, 12047503.
References
[1] Arkani-Hamed N., Bai Y., He S., Yan G., Scattering forms and the positive geometry of kinematics, color
and the worldsheet, J. High Energy Phys. 2018 (2018), no. 5, 096, 76 pages, arXiv:1711.09102.
[2] Arkani-Hamed N., Bai Y., Lam T., Positive geometries and canonical forms, J. High Energy Phys. 2017
(2017), no. 11, 039, 122 pages, arXiv:1703.04541.
[3] Arkani-Hamed N., He S., Lam T., Stringy canonical forms, J. High Energy Phys. 2021 (2021), no. 2, 069,
59 pages, arXiv:1912.08707.
[4] Arkani-Hamed N., He S., Lam T., Thomas H., Binary geometries, generalized particles and strings, and
cluster algebras, arXiv:1912.11764.
[5] Arkani-Hamed N., Lam T., Spradlin M., Non-perturbative geometries for planar N = 4 SYM amplitudes,
J. High Energy Phys. 2021 (2021), no. 3, 065, 14 pages, arXiv:1912.08222.
[6] Arkani-Hamed N., Lam T., Spradlin M., Positive configuration space, Comm. Math. Phys. 384 (2021),
909–954, arXiv:2003.03904.
[7] Assem I., Schiffler R., Shramchenko V., Cluster automorphisms, Proc. Lond. Math. Soc. 104 (2012), 1271–
1302, arXiv:1009.0742.
[8] Bazier-Matte V., Douville G., Mousavand J., Thomas H., Yildirim E., ABHY associahedra and Newton
polytopes of F -polynomials for finite type cluster algebras, arXiv:1808.09986.
[9] Brown F., Carr S., Schneps L., The algebra of cell-zeta values, Compos. Math. 146 (2010), 731–771,
arXiv:0910.0122.
[10] Brown F., Dupont C., Single-valued integration and superstring amplitudes in genus zero, arXiv:1910.01107.
https://doi.org/10.1007/jhep05(2018)096
https://arxiv.org/abs/1711.09102
https://doi.org/10.1007/jhep11(2017)039
https://arxiv.org/abs/1703.04541
https://doi.org/10.1007/jhep02(2021)069
https://arxiv.org/abs/1912.08707
https://arxiv.org/abs/1912.11764
https://doi.org/10.1007/jhep03(2021)065
https://arxiv.org/abs/1912.08222
https://doi.org/10.1007/s00220-021-04041-x
https://arxiv.org/abs/2003.03904
https://doi.org/10.1112/plms/pdr049
https://arxiv.org/abs/1009.0742
https://arxiv.org/abs/1808.09986
https://doi.org/10.1112/S0010437X09004540
https://arxiv.org/abs/0910.0122
https://arxiv.org/abs/1910.01107
Cluster Configuration Spaces of Finite Type 41
[11] Brown F.C.S., Multiple zeta values and periods of moduli spaces M0,n, Ann. Sci. Éc. Norm. Supér. (4) 42
(2009), 371–489, arXiv:math.AG/0606419.
[12] Chapoton F., Fomin S., Zelevinsky A., Polytopal realizations of generalized associahedra, Canad. Math.
Bull. 45 (2002), 537–566, arXiv:math.CO/0202004.
[13] Dupont G., An approach to non-simply laced cluster algebras, J. Algebra 320 (2008), 1626–1661,
arXiv:math.RT/0512043.
[14] Fei J., Combinatorics of F -polynomials, arXiv:1909.10151.
[15] Fei J., Tropical F -polynomials and general presentation, arXiv:1911.10513.
[16] Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm.
Supér. (4) 42 (2009), 865–930, arXiv:math.AG/0311245.
[17] Fomin S., Zelevinsky A., Y -systems and generalized associahedra, Ann. of Math. 158 (2003), 977–1018,
arXiv:hep-th/0111053.
[18] Fomin S., Zelevinsky A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63–121,
arXiv:math.RA/0208229.
[19] Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112–164,
arXiv:math.RA/0602259.
[20] Fulton W., Introduction to toric varieties, Annals of Mathematics Studies, Vol. 131, Princeton University
Press, Princeton, NJ, 1993.
[21] Hohlweg C., Lange C.E.M.C., Thomas H., Permutahedra and generalized associahedra, Adv. Math. 226
(2011), 608–640, arXiv:0709.4241.
[22] Jahn D., Löwe R., Stump C., Minkowski decompositions for generalized associahedra of acyclic type,
arXiv:2005.14065.
[23] Lam T., Speyer D.E., Cohomology of cluster varieties. I. Locally acylic case, Algebra Number Theory, to
appear, arXiv:1604.06843.
[24] Muller G., Locally acyclic cluster algebras, Adv. Math. 233 (2013), 207–247, arXiv:1111.4468.
[25] Orlik P., Terao H., Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften,
Vol. 300, Springer-Verlag, Berlin, 1992.
[26] Padrol A., Palu Y., Pilaud V., Plamondon P.-G., Associahedra for finite type cluster algebras and minimal
relations between g-vectors, arXiv:1906.06861.
[27] Reading N., Universal geometric cluster algebras, Math. Z. 277 (2014), 499–547, arXiv:1209.3987.
[28] Shi J.Y., The Kazhdan–Lusztig cells in certain affine Weyl groups, Lecture Notes in Math., Vol. 1179,
Springer-Verlag, Berlin, 1986.
[29] Speyer D., Williams L., The tropical totally positive Grassmannian, J. Algebraic Combin. 22 (2005), 189–
210, arXiv:math.CO/0312297.
[30] Stanley R.P., An introduction to hyperplane arrangements, in Geometric Combinatorics, IAS/Park City
Math. Ser., Vol. 13, Amer. Math. Soc., Providence, RI, 2007, 389–496.
[31] Yang S.-W., Zelevinsky A., Cluster algebras of finite type via Coxeter elements and principal minors, Trans-
form. Groups 13 (2008), 855–895, arXiv:0804.3303.
https://doi.org/10.24033/asens.2099
https://arxiv.org/abs/math.AG/0606419
https://doi.org/10.4153/CMB-2002-054-1
https://doi.org/10.4153/CMB-2002-054-1
https://arxiv.org/abs/math.CO/0202004
https://doi.org/10.1016/j.jalgebra.2008.03.018
https://arxiv.org/abs/math.RT/0512043
https://arxiv.org/abs/1909.10151
https://arxiv.org/abs/1911.10513
https://doi.org/10.24033/asens.2112
https://doi.org/10.24033/asens.2112
https://arxiv.org/abs/math.AG/0311245
https://doi.org/10.4007/annals.2003.158.977
https://arxiv.org/abs/hep-th/0111053
https://doi.org/10.1007/s00222-003-0302-y
https://arxiv.org/abs/math.RA/0208229
https://doi.org/10.1112/S0010437X06002521
https://arxiv.org/abs/math.RA/0602259
https://doi.org/10.1515/9781400882526
https://doi.org/10.1515/9781400882526
https://doi.org/10.1016/j.aim.2010.07.005
https://arxiv.org/abs/0709.4241
https://arxiv.org/abs/2005.14065
https://arxiv.org/abs/1604.06843
https://doi.org/10.1016/j.aim.2012.10.002
https://arxiv.org/abs/1111.4468
https://doi.org/10.1007/978-3-662-02772-1
https://arxiv.org/abs/1906.06861
https://doi.org/10.1007/s00209-013-1264-4
https://arxiv.org/abs/1209.3987
https://doi.org/10.1007/BFb0074968
https://doi.org/10.1007/s10801-005-2513-3
https://arxiv.org/abs/math.CO/0312297
https://doi.org/10.1090/pcms/013/08
https://doi.org/10.1007/s00031-008-9025-x
https://doi.org/10.1007/s00031-008-9025-x
https://arxiv.org/abs/0804.3303
1 Introduction
2 Background on cluster algebras and generalized associahedra
3 The cluster configuration space MD
4 MD as a quotient of a cluster variety
5 MD as an affine open in a projective toric variety
6 Properties of F-polynomials
7 Examples of MD as a configuration space
8 Positive part
9 Positive tropicalization
10 Extended and local u-equations
11 Connected components and sign patterns
12 Outlook
A A lemma in commutative algebra
References
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| id | nasplib_isofts_kiev_ua-123456789-211435 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T18:31:28Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Arkani-Hamed, Nima He, Song Lam, Thomas 2026-01-02T08:32:52Z 2021 Cluster Configuration Spaces of Finite Type. Nima Arkani-Hamed, Song He and Thomas Lam. SIGMA 17 (2021), 092, 41 pages 1815-0659 2020 Mathematics Subject Classification: 05E14; 13F60; 14N99; 81T30 arXiv:2005.11419 https://nasplib.isofts.kiev.ua/handle/123456789/211435 https://doi.org/10.3842/SIGMA.2021.092 For each Dynkin diagram , we define a ''cluster configuration space'' ℳ and a partial compactification ℳ˜. For = ₙ₋₃, we have ℳₙ₋₃ = ℳ₀,ₙ, the configuration space of points on ℙ¹, and the partial compactification ℳ˜ₙ₋₃ was studied in this case by Brown. The space M˜ 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 ℳ˜ are generated by coordinates uγ, in bijection with the cluster variables of type , 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. We thank Mark Spradlin and Hugh Thomas for many discussions related to this work and for closely related collaborations. We thank the anonymous referees for a number of corrections and helpful suggestions to the exposition. T.L. was supported by NSF DMS-1464693, NSF DMS1953852, and by a von Neumann Fellowship from the Institute for Advanced Study. N.A-H. was supported by DOE grant DE-SC0009988. S.H. was supported in part by the National Natural Science Foundation of China under Grants No. 11935013, 11947301, 12047502, 12047503. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Cluster Configuration Spaces of Finite Type Article published earlier |
| spellingShingle | Cluster Configuration Spaces of Finite Type Arkani-Hamed, Nima He, Song Lam, Thomas |
| title | Cluster Configuration Spaces of Finite Type |
| title_full | Cluster Configuration Spaces of Finite Type |
| title_fullStr | Cluster Configuration Spaces of Finite Type |
| title_full_unstemmed | Cluster Configuration Spaces of Finite Type |
| title_short | Cluster Configuration Spaces of Finite Type |
| title_sort | cluster configuration spaces of finite type |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211435 |
| work_keys_str_mv | AT arkanihamednima clusterconfigurationspacesoffinitetype AT hesong clusterconfigurationspacesoffinitetype AT lamthomas clusterconfigurationspacesoffinitetype |