These observations disfavor scenarios where energy loss is a significant
contributor to the nuclear modification of high $p_{T}$ particle production in
small systems. The counter-intuitive centrality dependence is likely linked to
a mismatch of the centrality selection of events using charged particle
multiplicity and mapping them to a number of binary collisions using the
standard Glauber model. In this picture, events with a high $p_{T}$ $\pi^{0}$
are biased towards smaller underlying event multiplicity. This might be due to
physical fluctuations of the proton size or simply due to energy conservation
if high $p_{T}$ jets are present.

For lower $p_{T}$, $R_{xA}$ for all systems initially increases with $p_{T}$
and reaches a peak near 4–6 GeV/$c$ for central and semi-central collisions.
For peripheral collisions, $R_{xA}$ levels off to a constant at approximately
the same high $p_{T}$ value for all systems. For inelastic collisions and more
central collisions, $R_{xA}$ resembles what has been referred to as the Cronin
effect in fixed target experiments - a rise, followed by a peak, followed by a
plateau. However, unlike at lower energies, $p$$+$$p$ and $x$$+$A $\pi^{0}$
cross sections are not related by a power $(xA)^{n(\mbox{$p_{T}$})}$ with a
common $n(\mbox{$p_{T}$})$. Furthermore, the peak height value around 4–6
GeV/$c$ shows a clear system size dependence
$\mbox{$p$$+$Au}>\mbox{$d$$+$Au}>\mbox{${}^{3}$He$+$Au}>\mbox{$p$$+$Al}$,
where the $R_{xA}$ peak height value is well above unity for $p$$+$Au and
drops to close to unity for $p$$+$Al collisions.

While the shape of $R_{xA}$ roughly resembles what is expected from the
nuclear modification of PDFs, the observed system size dependence of the peak
height of $R_{xA}$ shows exactly the reverse ordering of what was predicted by
nPDF calculations. Therefore it is likely that nPDFs alone are insufficient to
explain the nuclear modifications in small systems.

In the same $p_{T}$ region, $\langle R_{xA}\rangle$ was used to study the
dependence on centrality. For all systems, $\langle R_{xA}\rangle$ in the
range 4–6 GeV/$c$ follows a common trend with $N_{\rm coll}$. At high $p_{T}$,
$\langle R_{xA}\rangle$ scales with $N_{\rm coll}$/$N_{\rm proj}$ for Au
target nuclei. While at lower $p_{T}$, $d$$+$Au and 3He$+$Au are not a
superposition of $p$$+$Au-like collisions. Consequently, different mechanisms
must contribute to the nuclear modification at high and low $p_{T}$. For high
$p_{T}$, the apparent centrality dependence is likely due to a bias in the
event selection. At lower $p_{T}$, final state effects related to the presence
of interacting hadrons may be at play. If a QGP droplet is indeed produced
during the collision, radial flow may be one possible mechanism to explain
this trend, although further investigation is needed that is outside the scope
of this paper.

###### Acknowledgements.

We thank the staff of the Collider-Accelerator and Physics Departments at
Brookhaven National Laboratory and the staff of the other PHENIX participating
institutions for their vital contributions. We also thank I. Helenius and J.
Rojo for the nPDF calculations plus M. van Leeuwen for the NLO calculations.
We acknowledge support from the Office of Nuclear Physics in the Office of
Science of the Department of Energy, the National Science Foundation, Abilene
Christian University Research Council, Research Foundation of SUNY, and Dean
of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry
of Education, Culture, Sports, Science, and Technology and the Japan Society
for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento
Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São
Paulo (Brazil), Natural Science Foundation of China (People’s Republic of
China), Croatian Science Foundation and Ministry of Science and Education
(Croatia), Ministry of Education, Youth and Sports (Czech Republic), Centre
National de la Recherche Scientifique, Commissariat à l’Énergie Atomique, and
Institut National de Physique Nucléaire et de Physique des Particules
(France), Bundesministerium für Bildung und Forschung, Deutscher Akademischer
Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), J. Bolyai
Research Scholarship, EFOP, the New National Excellence Program (ÚNKP), NKFIH,
and OTKA (Hungary), Department of Atomic Energy and Department of Science and
Technology (India), Israel Science Foundation (Israel), Basic Science Research
and SRC(CENuM) Programs through NRF funded by the Ministry of Education and
the Ministry of Science and ICT (Korea). Physics Department, Lahore University
of Management Sciences (Pakistan), Ministry of Education and Science, Russian
Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and
Wallenberg Foundation (Sweden), the U.S. Civilian Research and Development
Foundation for the Independent States of the Former Soviet Union, the
Hungarian American Enterprise Scholarship Fund, the US-Hungarian Fulbright
Foundation, and the US-Israel Binational Science Foundation.

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$\operatorname{\mathbf{E}}[\mathrm{e}^{\eta
X}]=1+\eta\operatorname{\mathbf{E}}[X]+\frac{1}{2}\eta^{2}\operatorname{\mathbf{E}}[X^{2}\mathrm{e}^{\eta^{\prime}X}]$
(58)

for some $\eta^{\prime}$ with $\eta\leq\eta^{\prime}<\eta^{*}$.

This is just the standard Taylor approximation of the moment generating
function for random variable $X$ at $\eta=0$. However, in this article we need
this approximation also for the case that
$\operatorname{\mathbf{E}}[\mathrm{e}^{\eta X}]=\infty$ for all $\eta<0$ (e.g.
if $X$ has polynomial left tail), in which the standard Taylor’s theorem does
not apply any more, since the standard (two-sided) derivative at $\eta=0$ is
undefined. The lemma shows that nevertheless, everything still works as one
would expect.

######  Proof C.6.

Fix $\eta_{0}\in(0,\eta^{*})$. Then all derivatives of
$\operatorname{\mathbf{E}}[\exp(\eta X)]$ exist at $\eta$ with
$\eta_{0}\leq\eta\leq\eta^{*}$, so that:

$\displaystyle\operatorname{\mathbf{E}}[\mathrm{e}^{\eta
X}]=\operatorname{\mathbf{E}}[\mathrm{e}^{\eta_{0}X}]+(\eta-\eta_{0})\operatorname{\mathbf{E}}[X\mathrm{e}^{\eta_{0}X}]+\frac{1}{2}(\eta-\eta_{0})^{2}\operatorname{\mathbf{E}}[X^{2}\mathrm{e}^{s(\eta_{0},\eta)X}]$
(59)

with $s$ some function $s:[0,\eta^{*}]^{2}\rightarrow[\eta_{0},\eta]$. Since

$\displaystyle|\mathrm{e}^{\eta X}X|\leq|\mathrm{e}^{\eta
X_{+}}X|\leq|\mathrm{e}^{\eta^{*}X_{+}}X|\leq|X_{-}|+|\mathop{{\bf
1}_{\\{X\geq
0\\}}}\mathrm{e}^{\eta^{*}X_{+}}X|\leq|X_{-}|+|\mathrm{e}^{\eta^{*}X}X|$

and we know $\operatorname{\mathbf{E}}[|\mathrm{e}^{\eta^{*}X}X|]<\infty$, we
can use the dominated convergence theorem to conclude that
$\lim_{\eta_{0}\downarrow
0}\operatorname{\mathbf{E}}[X\mathrm{e}^{\eta_{0}X}]=\operatorname{\mathbf{E}}[X]$.
Analogously one shows that $\lim_{\eta_{0}\downarrow
0}\operatorname{\mathbf{E}}[X^{2}\mathrm{e}^{s(\eta_{0},\eta)X}]=\operatorname{\mathbf{E}}[X^{2}\mathrm{e}^{\eta^{\prime}X}]$
for an $\eta^{\prime}\in[\eta_{0},\eta]$. The result now follows by using
these two limiting results in taking the limit for $\eta_{0}\downarrow 0$ in
(59).

##  Appendix D Proofs for Section 5

######  Proof D.1.

(of Theorem 5.2) Inequality (38) follows from Proposition 2.3, applied with
$X=\hat{\eta}X_{\hat{\eta}},Y=\hat{\eta}Y_{\hat{\eta}},\eta=1$. (40) follows
from Proposition 2.6, with $X,Y$ and $\eta$ set in the same way (see the
remark at the end of the proposition statement).

Now we prove (39). Define the random variable $W_{\hat{\eta}}\coloneqq
X_{\hat{\eta}}-Z_{\hat{\eta}}$, and for $(a,k)\in(0,\infty)\times\mathbb{N}$
define the event
$\mathcal{E}_{a,k}\coloneqq\left\\{a\cdot(k-1)\leq\hat{\eta}W_{\hat{\eta}}\leq
ak\right\\}$. With $Z=\hat{\eta}W_{\hat{\eta}}$, we will first show that

$\displaystyle\limsup_{a\downarrow 0}\sum_{k=1}^{\infty}ak\cdot
P(\mathcal{E}_{a,k})\leq\int_{0}^{\infty}P(Z\geq z)\mathrm{d}z.$ (60)

For $a,b>0$ and $k_{a,b}\coloneqq\lfloor b/a\rfloor$, we have

$\displaystyle\sum_{k=1}^{k_{a,b}}ak\cdot P(\mathcal{E}_{a,k})$
$\displaystyle=\sum_{k=1}^{k_{a,b}}ak\cdot\left(P(Z>a\cdot(k-1))-P(Z\geq
ak)\right),$ $\displaystyle\leq\sum_{k=1}^{k_{a,b}}ak\cdot\left(P(Z\geq
a\cdot(k-1))-P(Z\geq ak)\right),$
$\displaystyle\leq\sum_{k=0}^{k_{a,b}-1}a\cdot P(Z\geq ak),$
$\displaystyle\leq aP(Z\geq 0)+\int_{0}^{k_{a,b}}aP(Z\geq
at)\mathrm{d}t,\quad(\text{$t\mapsto a\cdot P(Z\geq at)$ is nonincreasing})$
$\displaystyle=aP(Z\geq 0)+\int_{0}^{ak_{a,b}}P(Z\geq
z)\mathrm{d}z,\quad(\text{change of variable $y=at$})$ $\displaystyle\leq
a+\int_{0}^{b}P(Z\geq z)\mathrm{d}z.$ (61)

Since (61) holds for all $a,b>0$, we have

$\displaystyle\limsup_{a\downarrow 0}\sum_{k=1}^{\infty}ak\cdot
P(\mathcal{E}_{a,k})=\limsup_{a\downarrow
0}\left\\{\sup_{b>0}\sum_{k=1}^{k_{a,b}}ak\cdot
P(\mathcal{E}_{a,k})\right\\}\stackrel{{\scriptstyle\eqref{eq:lowertail}}}{{\leq}}\int_{0}^{\infty}P(Z\geq
z)\mathrm{d}z,$ (62)

and thus, the desired inequality (60) follows. We now have, for all $a>0$,

$\displaystyle\operatorname{\mathbf{E}}\left[W_{\hat{\eta}}\right]$
$\displaystyle\leq\sum_{k=1}^{\infty}P\left(\mathcal{E}_{a,k}\right)\cdot\operatorname{\mathbf{E}}\left[\left.W_{\hat{\eta}}\right\lvert\mathcal{E}_{a,k}\right],$
$\displaystyle\leq\sum_{k=1}^{\infty}P\left(\mathcal{E}_{a,k}\right)\cdot\operatorname{\mathbf{E}}\left[\frac{ak}{\hat{\eta}}\right]\quad(\text{by
the definition of $\mathcal{E}_{a,k}$})$
$\displaystyle=\left(\sum_{k=1}^{\infty}P\left(\mathcal{E}_{a,k}\right)\cdot
ak\right)\cdot\operatorname{\mathbf{E}}\left[\frac{1}{\hat{\eta}}\right].$
Since this inequality holds for all $a>0$, using (60) implies that
$\displaystyle\operatorname{\mathbf{E}}\left[W_{\hat{\eta}}\right]$
$\displaystyle\leq\operatorname{\mathbf{E}}\left[\frac{1}{\hat{\eta}}\right]\cdot\int_{0}^{\infty}P(\hat{\eta}W_{\hat{\eta}}\geq
t)\mathrm{d}t$
$\displaystyle\leq\operatorname{\mathbf{E}}\left[\frac{1}{\hat{\eta}}\right]\cdot\operatorname{\mathbf{E}}\left[\mathrm{e}^{\hat{\eta}W_{\hat{\eta}}}\right]\int_{0}^{\infty}\mathrm{e}^{-t}\mathrm{d}t,(\text{Markov's
Inequality})$
$\displaystyle\leq\operatorname{\mathbf{E}}\left[\frac{1}{\hat{\eta}}\right],$
(63)

where the last step follows from the fact that
$W_{\hat{\eta}}\leqclosed_{\hat{\eta}}0$.

######  Proof D.2.

(of Proposition 5.4) Let’s denote $X_{i,\eta}(Z;z^{n\setminus i})\coloneqq
X_{i,\eta}(z_{1},\dots,z_{i-1},Z,z_{i+1},\dots,z_{n})$, for all
$\eta\in\mathcal{G}$, $i\in[n]$, $z^{n\setminus i}\in\mathcal{Z}^{n-1}$, and
$Z\in\mathcal{Z}$. In this way, (41) can be written as

$\displaystyle X_{i,\eta}(Z_{i};z^{n\setminus
i})\leqclosed_{\eta}0,\quad\text{for all $i\in[n]$ and $z^{n\setminus
i}\in\mathcal{Z}^{n-1}$.}$ (64)

In particular, since this holds for all $z^{n\setminus
i}\in\mathcal{Z}^{n-1}$, we can also write

$\displaystyle X_{i,\eta}(Z_{i};Z^{n\setminus i})\leqclosed_{\eta}0.$ (65)

Now let $Z_{1}^{n},\dots,Z_{n}^{n}\in\mathcal{Z}^{n}$ be $n$ i.i.d copies of
$Z^{n}$. From (65), we have, for each $i\in[n]$,

$\displaystyle Y_{i,\eta}\coloneqq X_{i,\eta}(Z_{i,i};Z_{i}^{n\setminus
i})\leqclosed_{\eta}0.$ (66)

Since $(Y_{i,\eta})_{i\in[n]}$ are i.i.d, we can chain (66), for
$i=1,\dots,n$, using Proposition 2.7 to get

$\displaystyle\sum_{i=1}^{n}Y_{i,\eta}\leqclosed_{\eta}0.$ (67)

By applying Proposition 5.5, we have, for any random $\hat{\eta}$ in
$\mathcal{G}$,

$\displaystyle\operatorname{\mathbf{E}}\left[\frac{\log|\mathcal{G}|+1}{\hat{\eta}}\right]$
$\displaystyle\geq\operatorname{\mathbf{E}}\left[\sum_{i=1}^{n}Y_{i,\hat{\eta}}\right],$
$\displaystyle=\operatorname{\mathbf{E}}\left[\sum_{i=1}^{n}X_{i,\hat{\eta}}(Z_{i,i},Z^{n\setminus
i}_{i})\right],$
$\displaystyle=\operatorname{\mathbf{E}}\left[\sum_{i=1}^{n}X_{i,\hat{\eta}}(Z^{n})\right],$
(68)

where the last equality follows from the fact that $(Z_{i}^{n})_{i\in[n]}$ are
i.i.d copies of $Z^{n}$.


$\operatorname{Trop}_{>0}{\mathcal{M}}_{D}$. The fibers of $p$ are exactly the
orbits of the lineality space $L$ acting on
$\operatorname{Trop}_{>0}X\big{(}{\tilde{B}}\big{)}$. It follows that the fans
$\operatorname{Trop}_{>0}X\big{(}{\tilde{B}}\big{)}/L$ and
$\operatorname{Trop}_{>0}{\mathcal{M}}_{D}$ are isomorphic. ∎

Noting that the fans ${\mathcal{N}}\big{(}B^{\vee}\big{)}$ and
${\mathcal{N}}(B)$ are combinatorially isomorphic, we deduce from Theorem 9.2
and Proposition 9.3 the following conjecture of Speyer and Williams [29,
Conjecture 8.1]. The principal coefficient case was established in [22] and we
thank Christian Stump for drawing our attention to this work.

######  Corollary 9.4.

Suppose that ${\tilde{B}}$ is of full rank and has finite type. Then
$\operatorname{Trop}_{>0}X\big{(}{\tilde{B}}\big{)}/L$ is combinatorially
isomorphic to the complete fan ${\mathcal{N}}(B)$.

##  10 Extended and local $\boldsymbol{u}$-equations

In this section, we first study two additional sets of equations satisfied by
$u$-variables, the extended $u$-equations and the local $u$-equations. In
other words, we give some further distinguished elements in the ideal $I_{D}$.

In type $A$, the extended $u$-equations were used by Brown [11] to define what
we call ${\widetilde{\mathcal{M}}}_{A_{n-3}}$; we see here that they can be
interpreted as arising from all the exchange relations of the cluster algebra,
rather than just the primitive exchange relations.

10.1. Extended $\boldsymbol{u}$-equations. An _extended $u$-equation_ is an
equation which holds in ${\mathbb{C}}[{\mathcal{M}}_{D}]$ of the form

$\displaystyle\prod_{\gamma}u_{\gamma}^{\alpha_{\gamma}}+\prod_{\gamma}u_{\gamma}^{\beta_{\gamma}}=1,$
(10.1)

where $\alpha_{\gamma}$, $\beta_{\gamma}$ are nonnegative integer parameters.
All the primitive $u$-equations $R_{\gamma}$ in Definition 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 $R_{\gamma}$. 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 $G_{2}$. In types $A$ and $D$, the only
extended $u$-equations we know come from Corollary 4.5, and we conjecture that
in simply-laced types these are the only ones. In types $B$ and $C$, we find
more extended $u$-equations than those from Corollary 4.5. Indeed, any
extended $u$-equation for type $A_{2n-3}$ (resp. $D_{n}$) gives one for type
$C_{n-1}$ (resp. $B_{n-1}$), but not all of these come from Corollary 4.5. We
conjecture that all extended $u$-equations in multiply-laced type come from
folding.

A similar analysis of extended $u$-equations for the types $E$ and $F$ can be
found by a lengthy, but finite computation which we do not present here.

In the following, we will use the indexing of $\Pi$ from Section 3.2. For two
disjoint subsets $I$ and $J$, define

$\displaystyle U_{I,J}:=\prod_{i\in I,j\in J}u_{i,j}.$ (10.2)

Note that $U_{I,J}$ and $U_{J,I}$ are not necessarily equal.

10.1.1. Type $\boldsymbol{A}$. Let $\\{1,2,\dots,n\\}=A\sqcup B\sqcup C\sqcup
D$ be a decomposition of $[n]$ into cyclic intervals. Then we have the
extended $u$-equation

$\displaystyle U_{A,C}+U_{B,D}=1.$ (10.3)

Each equation depends on the choice of four cyclically ordered points $a$,
$b$, $c$, $d$: $A=\\{a{+}1,\dots,b\\}$, $B=\\{b{+}1,\dots,c\\}$,
$C=\\{c{+}1,\dots,d\\}$ and $D=\\{d{+}1,\dots,a\\}$, and thus there are
$\binom{n}{4}$ equations in total, in bijection with the exchange relations of
the type $A$ cluster algebra. These equations 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, $R_{ij}=0$. As discussed in [4, 11], it is natural to interpret
these $U$’s as cross-ratios of $n$ points on ${\mathbb{P}}^{1}$: denote
$U_{A,C}=[a,b|c,d]$, 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 $[a,b|c,d]=\frac{\Delta_{ad}\Delta_{bc}}{\Delta_{ac}\Delta_{bd}}$.

10.1.2. Type $\boldsymbol{C}$. Extended $u$-equations for type $C_{n-1}$ arise
via folding $A_{2n-3}$. For a decomposition
$\big{\\{}1,2,\dots,n,\bar{1},\dots,\bar{n}\big{\\}}\simeq[2n]=A\sqcup B\sqcup
C\sqcup D$ into cyclic intervals, the image of (10.3) gives the extended
$u$-equation for $C_{n{-}1}$ (which become the primitive ones for $A=\\{i\\}$,
$C=\\{j\\}$). For example, if we choose $A=\\{1\\}$ and $C=\\{n{+}1\\}$, we
have

$\displaystyle
1-u_{[1\bar{1}]}=U_{\\{2,\dots,n\\},\\{\bar{2},\dots,\bar{n}\\}}=\prod_{i=2}^{n}u_{[i\bar{i}]}\prod_{2\leq
i<j\leq n}u_{[i\bar{j}]}^{2},$

which is the primitive $u$-equation $R_{[1\bar{1}]}=0$. For $C_{2}$, in
addition to the $6$ primitive $u$-equations given in (3.3), we have $3$ more
equations:

$\displaystyle u_{[1\bar{1}]}u_{[1\bar{2}]}+u_{[2\bar{3}]}u_{[3\bar{3}]}=1,$
(10.4)

and its cyclic rotations.

Let us count the number of extended $u$-equations for $C_{n-1}$ we have
obtained. There are $\binom{2n}{4}$ equations (10.3) in type $A_{2n-3}$. Of
those, $\binom{n}{2}$ are equal to its mirror image. For the remainder, both
the equation and its mirror image map to the same equation in type $C_{n-1}$.
Thus we have obtained $\frac{n(n{-}1)(2n^{2}-4n+3)}{6}$ extended $u$-equations
for $C_{n-1}$. Note that this number is greater than the number of exchange
relations of type $C_{n-1}$. For example, for $C_{2}$, there are $6$ exchange
relations, but we have found $9$ extended $u$-equations.

10.1.3. Type $\boldsymbol{D}$. We now consider type $D_{n}$. We use the
notation (10.2) and also define (here $a\prec b$ means $a$ precedes $b$ in
$I$)

$\displaystyle U_{I}:=\prod_{a\prec b\in I}u_{a,b}\prod_{i\in
I}u_{i},\qquad\tilde{U}_{I}:=\prod_{a\prec b\in I}u_{a,b}\prod_{i\in
I}u_{\tilde{i}}.$

We now describe two types of extended $u$-equations. (These equalities were
discovered emprically, but it should be relatively straightforward to prove
them by induction.) First, similar to (10.3), for a cyclically ordered
partition $A\sqcup B\sqcup C\sqcup D=\\{1,2,\dots,n\\}$, we have

$\displaystyle U_{C,A}+U_{D,B}U_{A,B}U_{B,C}U_{B,D}U_{B}\tilde{U}_{B}=1.$
(10.5)

We allow $D$ to be empty here, in which case (10.5) becomes
$U_{C,A}+U_{A,B}U_{B,C}U_{B}\tilde{U}_{B}=1$. Second, for a cyclically ordered
partition $A\sqcup B\sqcup C=\\{1,2,\dots,n\\}$, we have

$\displaystyle U_{A}U_{A,B}+\tilde{U}_{C}U_{B,C}=1.$ (10.6)

Note that $B$ can be empty here, in which case we have
$U_{A}+\tilde{U}_{C}=1$. In the first and second type we have $4{n\choose
4}+3{n\choose 3}$ and $6{n\choose 3}+2{n\choose 2}$ equations, respectively,
thus in total there are $\frac{n(n{-}1)(n^{2}+4n-6)}{6}$ extended
$u$-equations. It is not difficult to see that this is equal to the number of
(unordered) pairs of exchangeable cluster variables. We have $n^{2}$ cluster
variables, $\frac{1}{4}(n^{4}+n^{2}-2n)$ pairs of compatible cluster
variables, and $2\binom{n}{4}$ pairs of cluster variables where the
compatibility degree is greater than one. The remaining pairs of cluster
variables are exchangeable.

It is straightforward to obtain the primitive $u$-equations for type $D_{n}$.
In (10.5), choosing $A=\\{i\\}$, $C=\\{j\\}$ (including the degenerate case
with $D=\varnothing$, thus $j=i{-}1$), we have

$\displaystyle
1-u_{j,i}=U_{D\cup\\{i\\},B}U_{B,j\cup\\{D\\}}U_{B}\tilde{U}_{B},$

where $B=\\{i{+}1,\dots,j{-}1\\}$ and $D=\\{j{+}1,\dots,i{-}1\\}$. In (10.6),
take $B=\varnothing$, and choosing $A=i$ or $C=i$ we have

$\displaystyle 1-u_{i}=\tilde{U}_{\\{i{+}1,\dots,i{-}1\\}},\qquad
1-{u}_{\tilde{i}}=U_{\\{i{+}1,\dots,i{-}1\\}}.$

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 $\boldsymbol{B}$. Finally, we consider type $B_{n{-}1}$ by
folding $D_{n}$. We identify $u_{i}=u_{\tilde{i}}$, and the two types of
equations become

$\displaystyle U_{C,A}+U_{D,B}U_{A,B}U_{B,C}U_{B,D}U_{B}^{2}=1,\qquad
U_{A}U_{A,B}+U_{C}U_{B,C}=1.$

We have $\frac{n(n{-}1)(n^{2}+n-3)}{6}$ such extended $u$-equations in total:
note that each equation (10.5) has a distinct image in $B_{n{-}1}$ but two of
the equations (10.6) map to a single one in $B_{n{-}1}$. Again, we note that
this number is greater than the number of exchange relations in type
$B_{n{-}1}$.

The primitive $u$-equations can be recovered by setting $A=\\{i\\}$,
$C=\\{j\\}$ (also $B=\varnothing$ 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:

$\displaystyle 8~{}{\rm eqs}\colon\quad
u_{3,1}u_{4,1}+u_{1,2}u_{2,3}u_{2}^{2}=1,$ $\displaystyle 8~{}{\rm
eqs}\colon\quad u_{1,2}u_{1}u_{2}u_{1,3}u_{2,3}+u_{4}u_{3,4}=1,$
$\displaystyle 4~{}{\rm eqs}\colon\quad
u_{1}u_{1,2}u_{1,3}+u_{4}u_{2,4}u_{3,4}=1,$ $\displaystyle 2~{}{\rm
eqs}\colon\quad u_{1,2}u_{1}u_{2}+u_{3,4}u_{3}u_{4}=1.$

There are $22$ exchange relations in type $B_{3}$.

10.1.5. Type $\boldsymbol{G}$. As in Section 3.2.5, let us call the 8
$u$-variables $a_{i}$, $b_{i}$ for $i=1,\dots,4$ that can be thought of as
labelling the edges of an octagon ($a_{1},b_{1},\dots,a_{4},b_{4})$. Folding
$D_{4}$, we obtain 18 extended $u$-equations for $G_{2}$, the primitive
$u$-equations

$\displaystyle a_{1}+a_{2}a_{3}^{2}a_{4}b_{2}b_{3}=1,\qquad
b_{1}+a_{3}^{3}a_{4}^{3}b_{2}b_{3}^{2}b_{4}=1,\qquad{\rm+\,cyclic},$

together with

$\displaystyle a_{1}b_{1}+a_{3}^{2}a_{4}b_{2}b_{3}=1,\qquad
a_{2}b_{1}+a_{3}a_{4}^{2}b_{3}b_{4}=1,\qquad
a_{1}a_{2}b_{1}+a_{3}a_{4}b_{3}=1,\qquad{\rm+\,cyclic}.$

10.2. Local $\boldsymbol{u}$-equations. The relations $R_{\gamma}$, and the
extended $u$-equations are global in nature: they involve many $u_{\gamma}$
variables which are “far away from each other”. We now describe a class of
local $u$-equations. Using them one can show that all $u$-variables can be
solved rationally in terms of $u$-variables of any acyclic seed; see also [4].
This has implications for canonical forms; see (12.1).

We first recall the $X$-coordinates for Fock and Goncharov’s cluster
$X$-variety. For an exchange relation $xx^{\prime}=M+M^{\prime}$, we have a
cluster $X$-variable $X=M/M^{\prime}$. Now, let us consider a primitive
exchange relation in $X\big{(}{\tilde{B}}\big{)}$:

$\displaystyle x_{\tau\gamma}x_{\gamma}=M+M^{\prime},$

where $M^{\prime}$ only involves frozen variables. We recall that the
isomorphism of Theorem 4.2 identifies the rational function
$\frac{M}{x_{\tau\gamma}x_{\gamma}}$ with $u_{\gamma}$. Thus the cluster
$X$-variable $X_{\gamma}:=M/M^{\prime}$ can be identified with
$u_{\gamma}/(1-u_{\gamma})\in{\mathbb{C}}[{\mathcal{M}}_{D}]$ which is equal
to a Laurent monomial in the $u_{\omega}$-s using the relations $R_{\gamma}$.
By [17] or [19, Proposition 3.9 or equation (8.11)] the variables $X_{\gamma}$
satisfy the relation

$\displaystyle X_{(t-1,j)}X_{(t,j)}=\prod_{i\to
j}\big{(}1+X_{(t,i)}\big{)}^{-a_{ij}}\prod_{j\to
i}\big{(}1+X_{(t-1,i)}\big{)}^{-a_{ij}}$ (10.7)

or, equivalently, the variables $u_{\gamma}$ satisfy the relation

$\displaystyle\frac{u_{(t-1,j)}}{1-u_{(t-1,j)}}\frac{u_{(t,j)}}{1-u_{(t,j)}}=\prod_{i\to
j}\big{(}1-u_{(t,i)}\big{)}^{a_{ij}}\prod_{j\to
i}\big{(}1-u_{(t-1,i)}\big{)}^{a_{ij}}.$

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 $\boldsymbol{A}$. For type $A_{n-3}$ we have $u$-variables
$u_{i,j}$ for $1\leq i<j{-}1<n$, 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:

$j$$j+1$$i$$i+1$ $\displaystyle
X_{i,j}X_{i{+}1,j{+}1}=(1+X_{i,j{+}1})(1+X_{i{+}1,j})$ (10.8)

or

$\displaystyle\frac{u_{i,j}}{1-u_{i,j}}\frac{u_{i{+}1,j{+}1}}{1-u_{i{+}1,j{+}1}}=\frac{1}{1-u_{i,j{+}1}}\frac{1}{1-u_{i{+}1,j}}.$

10.2.2. Type $\boldsymbol{C}$. By folding $A_{2n-3}$, we obtain local
$u$-equations for type $C_{n-1}$. The local $u$-equations take the form of
(10.8); for the special case of $j=\bar{i}$, it reads

$\displaystyle
X_{[i,\bar{i}]}X_{[i{+}1,\overline{i+1}]}=\big{(}1+X_{[i,i{+}1]}\big{)}^{2}.$

10.2.3. Type $\boldsymbol{D}$. We use the notation for $u$-variables from
Section 3.2.3. The $n^{2}$ local $u$-equations for type $D_{n}$ read

$\displaystyle X_{i,j}X_{i{+}1,j{+}1}=(1+X_{i,j{+}1})(1+X_{i{+}1,j}),$
$\displaystyle
X_{i{-}1,i}X_{i,i{+}1}=(1+X_{i{-}1,i{+}1})(1+X_{i})\big{(}1+X_{\tilde{i}}\big{)},$
$\displaystyle X_{i}X_{\widetilde{i{+}1}}=1+X_{i,i{+}1},$ $\displaystyle
X_{\tilde{i}}X_{i{+}1}=1+X_{i,i{+}1}.$

10.2.4. Type $\boldsymbol{B}$. By folding type $D_{n}$, we obtain local
$u$-equations for type $B_{n-1}$ (see Section 3.2.4). The local $u$-equations
are

$\displaystyle
X_{[i,j]}X_{[i{+}1,j{+}1]}=\big{(}1+X_{[i,j{+}1]}\big{)}\big{(}1+X_{[i{+}1,j]}\big{)},$
$\displaystyle
X_{[i{-}1,\bar{i}]}X_{[i,\overline{i{+}1}]}=\big{(}1+X_{[i{-}1,\overline{i{+}1}]}\big{)}\big{(}1+X_{[i,\bar{i}]}\big{)}^{2},$
$\displaystyle X_{[i,\bar{i}]}X_{[i+1,\overline{i+1}]}=1+X_{[i,i{+}1]}$

together with cyclic rotations.

10.2.5. Type $\boldsymbol{E}$. Finally, for $E_{n}$ with $n=6,7,8$, we use the
identification (2.1) to index $u$ and $X$ variables. When ${\tilde{B}}$ is
bipartite, i.e., every vertex is either a source or a sink in the induced
orientation of $D$, we have $r_{i}=h/2+1$ does not depend on $i$, where $h$ is
the Coxeter number. The Coxeter number is even in types $E_{6}$, $E_{7}$,
$E_{8}$, and $\Pi$ is identified with $m=7$, $10$, $16$ copies of $I$
respectively. We index the nodes of $E_{n}$ as shown below:

$1$$2$$3$$4$${n{-}2}$$n{-}1$$n$

The local $u$-equations take the following form:

$\displaystyle
X_{(t,i)}X_{(t+1,i)}=(1+X_{(t+1,i{+}1)})(1+X_{(t+1,i{-}1)})(1+X_{(t+1,n)})^{\delta_{i,3}},\qquad\text{for~{}odd}\quad
i<n,$ $\displaystyle
X_{(t,i)}X_{(t+1,i)}=(1+X_{(t,i)})(1+X_{(t,i{+}1)}),\qquad\text{for~{}even}\quad
i<n,$ $\displaystyle X_{(t,n)}X_{(t+1,n)}=1+X_{(t,3)},$

for $t=1,2,\dots,m$ in all these cases. We have $n\times m$ equations in
total.

10.2.6. Types $\boldsymbol{F_{4}}$ and $\boldsymbol{G_{2}}$. By folding
$E_{6}$ we obtain local $u$-equations for $F_{4}$, and by folding $D_{4}$ we
obtain those for $G_{2}$.

##  11 Connected components and sign patterns

The permutation group $S_{n}$ acts on the moduli space
${\mathcal{M}}_{0,n}({\mathbb{R}})$, permuting the $n$ points and permuting
the connected components. The presentation of
${\mathcal{M}}_{0,n}({\mathbb{R}})$ using $u_{ij}$ and the relations (1.1)
depends on the choice of a dihedral ordering and the action of the symmetry
group $S_{n}$ is obscured. In this section, we use the extended $u$-equations
to investigate the connected components of ${\mathcal{M}}_{D}({\mathbb{R}})$,
with the hope of uncovering an appropriate symmetry group for
${\mathcal{M}}_{D}$ in other types. While our results here are more
speculative, we are able to construct new classes of $u$-equations.

A further motivation for studying connected components of
${\mathcal{M}}_{D}({\mathbb{R}})$ is the application to string amplitudes
where it is important to consider canonical forms of different connected
components of ${\mathcal{M}}_{0,n}({\mathbb{R}})$; see Section 12.2 for a
brief discussion.

We also define an analogue of an oriented matroid for ${\mathcal{M}}_{D}$,
called a “consistent sign pattern”, and it is conjectured that the number of
consistent sign patterns is equal to the number of connected components.

11.1. Consistent sign patterns. A _consistent sign pattern_ for type $D$ is an
element $(s_{\gamma})\in\\{+,-\\}^{\Pi}$ such that for each extended
$u$-equation (10.1)
$\prod_{\gamma}u_{\gamma}^{\alpha_{\gamma}}+\prod_{\gamma}u_{\gamma}^{\beta_{\gamma}}=1$,
we have that at least one of the signs
$\prod_{\gamma}s_{\gamma}^{\alpha_{\gamma}}$ and
$\prod_{\gamma}s_{\gamma}^{\beta_{\gamma}}$ is positive. In other words,
$(s_{\gamma})$ are possible signs for some solution $(u_{\gamma})$ 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 ${\mathcal{M}}_{D}({\mathbb{R}})$ is
given by the number of sign patterns of $u$-variables consistent with the
extended $u$-equations for $D$.

11.2. Type $\boldsymbol{A}$. We consider $D=A_{n-3}$. The space
${\mathcal{M}}_{0,n}({\mathbb{R}})$ has $(n{-}1)!/2$ connected components,
corresponding to the dihedral orderings of $n$ points. In the positive
connected component $({\mathcal{M}}_{0,n})_{>0}$, all the cross ratios
$u_{ij}$ are positive, indeed, we have $0<u_{ij}<1$. In other connected
components of ${\mathcal{M}}_{0,n}$, some of the $u_{ij}$-s are negative. We
find that the extended $u$-equations exclude those sign patterns for which
both $[a,b|c,d]$ and $[b,c|d,a]$ are negative. Empirically, we find that
precisely $(n{-}1)!/2$ consistent sign patterns are allowed by the extended
$u$-equations (10.3), and this count agrees with the number of connected
components of ${\mathcal{M}}_{0,n}({\mathbb{R}})$.

Let us now consider the problem of finding new $u$-variables for other
components of ${\mathcal{M}}_{0,n}({\mathbb{R}})$, that is, we seek cross-
ratios that are positive on that component. It suffices to consider the
ordering that is obtained from the standard one by an adjacent transposition,
e.g., $(1^{\prime},2^{\prime},3^{\prime},\dots,n^{\prime})=(2,1,3,\dots,n)$.
Using the following identities for the cross ratio $[a,b|c,d]$ (see Section
10.1.1):

$\displaystyle[a,b|d,c]=\frac{1}{[a,b|c,d]},\qquad[a,c|b,d]=-\frac{[a,b|c,d]}{[b,c|d,a]}\,$

we find that the $u$-variables in this new ordering include

$\displaystyle
u^{\prime}_{1,3}=[n^{\prime},1^{\prime}|2^{\prime},3^{\prime}]=[n,2|1,3]=\frac{-[n,1|2,3]}{1,2|3,n]}=\frac{-[n,1|2,3]}{\prod_{i=3}^{n{-}1}[1,2|i,i{+}1]}=\frac{-u_{1,3}}{u_{2,4}\cdots
u_{2,n}},$ $\displaystyle
u^{\prime}_{1,i}=[n^{\prime},1^{\prime}|i^{\prime}{-}1,i^{\prime}]=[n,2|i-1,i]=[n,1|i{-}1,i][1,2|i{-}1,i]=u_{1,i}u_{2,i},$
$\displaystyle
u^{\prime}_{2,i}=[1^{\prime},2^{\prime}|i^{\prime}{-}1,i^{\prime}]=[2,1|i{-}1,i]=\frac{1}{[1,2;i{-}1,i]}=\frac{1}{u_{2,i}},$
$\displaystyle
u^{\prime}_{3,i}=[2^{\prime},3^{\prime}|i^{\prime}{-}1,i^{\prime}]=[1,3|i{-}1,i]=[1,2|i{-}1,i][2,3|i{-}1,i]=u_{2,i}u_{3,i},$
(11.1)

and all other $u$’s are unchanged. These new $u^{\prime}_{ij}$-s are positive
in the connected component given by the ordering $(2,1,3,\dots,n)$, and
furthermore they satisfy the extended $u$-equations (for this ordering). In
other words, the (invertible) signed monomial transformation (11.1) sends the
extended $u$-equations for $u_{ij}$ to a permutation of the extended
$u$-equations for the $u^{\prime}_{i,j}$, and thus exposing a hidden
$S_{n}$-symmetry of these equations.

11.3. Type $\boldsymbol{C}$. From the analysis of Section 7.4, we know that
${\mathcal{M}}_{C_{n}}({\mathbb{R}})$ has $2^{n{-}1}n!(n+1)$ connected
components. Computationally, we find that this agrees with the number of
consistent sign patterns.

There are two types of components in ${\mathcal{M}}_{C_{n}}({\mathbb{R}})$
corresponding to two types of configurations of the $(2n{+}2)$-gon with labels
$i$, $\bar{i}$ for $i=1,2,\dots,n{+}1$: ($A$) $2^{n{-}1}n!$ components for
polygons with central symmetry, e.g.,
$1,2,\dots,n{+}1,\bar{1},\bar{2},\dots,\overline{n{+}1}$; ($B$)
$2^{n{-}1}(n{+}1)!$ components for polygons with reflection symmetry avoiding
vertices, e.g., the ordering
$1,2,\dots,n{+}1,\overline{n{+}1},\dots,\bar{2},\bar{1}$. Unlike type $A$, our
investigations indicate that the compactification (arising from $u$-equations)
of these two types of components have differing boundary combinatorics: for
any component in ($A$), combinatorially it is a cyclohedron (the generalized
associahedron of type $C$), while for any component in ($B$), combinatorially
it is an associahedra. We expect that this can be proven via a careful
analysis of the extended $u$-equations, and here we illustrate it for the
simplest example, $C_{2}=B_{2}$.

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 $1,2,3,\bar{1},\bar{2},\bar{3}$, which cuts out a hexagon. We can see
the other $3$ orderings in $(A)$ by making a signed monomial transformation of
the $u$-variables. For example, for the ordering
$2,1,3,\bar{2},\bar{1},\bar{3}$, we find that the $6$ new variables can be
obtained by a monomial change of variables:

$\displaystyle\begin{split}&u^{\prime}_{[1\bar{2}]}=\frac{1}{u_{[2\bar{1}]}u_{[2\bar{2}]}},\qquad
u^{\prime}_{[2\bar{3}]}=-\frac{u_{[13]}}{u_{[\bar{1}2]}u_{[2\bar{2}]}u_{[2\bar{3}]}},\qquad
u_{[3\bar{1}]}=\frac{1}{u_{[2\bar{2}]}u_{[2\bar{3}]}},\\\
&u^{\prime}_{[1\bar{1}]}=u_{[2\bar{2}]},\qquad
u^{\prime}_{[2\bar{2}]}=u_{[1\bar{1}]}u^{2}_{[2\bar{1}]}u_{[2\bar{2}]},\qquad
u_{[3\bar{3}]}=u_{[2\bar{2}]}u^{2}_{[2\bar{3}]}u_{[3\bar{3}]}.\end{split}$

It is straightforward to check that these $6$ new variables satisfy identical
extended $u$-equations for the ordering $2,1,3,\bar{2},\bar{1},\bar{3}$. More
generally, under this kind of transformation, similar to the type $A_{n-3}$
case, we find that for any ordering in $(A)$ the new variables satisfy
identical extended $u$-equations, and the corresponding component is
combinatorially a cyclohedron.

Let us now consider orderings in $(B)$. For example, for the ordering
$1,2,3,\bar{3},\bar{2},\bar{1}$, we find that the $6$ new variables are given
in terms of the old ones by

$\displaystyle
u^{\prime}_{[1\bar{3}]}=-\frac{u_{[1\bar{1}]}}{u_{[2\bar{2}]}u^{2}_{[2\bar{3}]}u_{[3\bar{3}]}},\qquad
u^{\prime}_{[\bar{1}2]}=\frac{1}{u_{[2\bar{2}]}},\qquad
u^{\prime}_{[\bar{2}3]}=\frac{1}{u_{[3\bar{3}]}},$ $\displaystyle
u^{\prime}_{[1\bar{2}]}=u_{[13]}u_{[2\bar{3}]}u_{[3\bar{3}]},\qquad
u^{\prime}_{[2\bar{3}]}=u_{[2\bar{1}]}u_{[2\bar{2}]}u_{[2\bar{3}]},\qquad
u^{\prime}_{[2\bar{2}]}=\frac{1}{u_{[2\bar{3}]}}.$

The $9$ extended $u$-equations become the following ones for the $6$ new
variables:

$\displaystyle
u^{\prime}_{[1\bar{2}]}u^{\prime}_{[1\bar{3}]}+u^{\prime}_{[2\bar{2}]}u^{\prime}_{[\bar{1}2]}=1,\qquad
u^{\prime}_{[1\bar{3}]}u^{\prime}_{[2\bar{3}]}+u^{\prime}_{[2\bar{2}]}u^{\prime}_{[\bar{2}3]}=1,\qquad
u^{\prime}_{[2\bar{2}]}(u^{\prime}_{[1\bar{2}]}+u^{\prime}_{[2\bar{3}]})=1,$
$\displaystyle
u^{\prime}_{[2\bar{3}]}+u^{\prime}_{[1\bar{2}]}u^{\prime}_{[2\bar{2}]}u^{\prime}_{[\bar{2}3]}=1,\qquad
u^{\prime}_{[1\bar{3}]}+u^{\prime
2}_{[2\bar{2}]}u^{\prime}_{[\bar{1}2]}u^{\prime}_{[\bar{2}3]}=1,\qquad
u^{\prime}_{[1\bar{2}]}+u^{\prime}_{[2\bar{2}]}u^{\prime}_{[2\bar{3}]}u^{\prime}_{[\bar{1}2]}=1,$
$\displaystyle u^{\prime}_{[\bar{1}2]}+u^{\prime
2}_{[1\bar{2}]}u^{\prime}_{[1\bar{3}]}=1,\qquad
u^{\prime}_{[2\bar{3}]}+u^{\prime}_{[1\bar{3}]}u^{\prime
2}_{[2\bar{3}]}=1,\qquad
u^{\prime}_{[2\bar{2}]}(1+u^{\prime}_{[1\bar{2}]}u^{\prime}_{[1\bar{3}]}u^{\prime}_{[2\bar{3}]})=1.$

Note that $u^{\prime}_{[2\bar{2}]}$ is special: from the third and the last
equations, it is easy to see that $u^{\prime}_{[2\bar{2}]}$ cannot take the
value 0, and thus $u^{\prime}_{[2\bar{2}]}=0$ does not correspond to a facet.
The other $5$ variables do correspond to facets, and from the equations we see
that requiring all $u^{\prime}\geq 0$ 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 $u^{\prime}$ cannot reach zero, and (an
appropriate closure) of the component has the combinatorics of a (type $A$)
associahedron.

11.4. Types $\boldsymbol{D_{4}}$ and $\boldsymbol{D_{5}}$. We were unable to
determine the number of connected components of
${\mathcal{M}}_{D_{n}}({\mathbb{R}})$. However, we can obtain a consistency
check by comparing the number of consistent sign patterns with the point count
over ${\mathbb{F}}_{q}$.

Recall from the point count (7.7) in type $D_{4}$, we predicted that
${\mathcal{M}}_{D}({\mathbb{R}})$ has $547$ connected components. By a direct
computation, we checked that this is equal to the number of consistent sign
patterns of $u$-variables with respect to the extended $u$-equations in
Section 10.1.3. Similarly, the prediction of $6388$ in the $D_{5}$ case
computationally agrees with the number of consistent sign patterns of
$u$-variables.

11.5. Type $\boldsymbol{G_{2}}$. As we have discussed in Section 7.6, we
expect that there are 25 different connected components for $G_{2}$.
Furthermore, the point count ${\mathcal{M}}_{G_{2}}({\mathbb{F}}_{q})$ is not
polynomial, and thus ${\mathcal{M}}_{G_{2}}$ is not a hyperplane arrangement
complement. Computationally, we find that there are 25 consistent sign
patterns for the 18 extended $u$-equations from Section 10.1.5.

We now investigate the connected components of
${\mathcal{M}}_{G_{2}}({\mathbb{R}})$, and note some new features. The
positive component has $0<a_{i},b_{i}<1$ as usual. But suppose $b_{1}$ is made
negative; to wit we put $b_{1}=-b^{\prime}_{1}$ with $b^{\prime}_{1}>0$. 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

$\displaystyle x_{1}=b_{4}^{-1},\qquad
x_{2}=a_{1}a_{3}a_{4}^{2}b_{3}b_{4},\qquad
x_{3}=b^{\prime}_{1}a_{3}^{-3}a_{4}^{-3}b_{2}^{-1}b_{3}^{-2}b_{4}^{-1},$
$\displaystyle x_{4}=a_{2}a_{3}^{2}a_{4}b_{2}b_{3},\qquad x_{5}=b_{2}^{-1},$
$\displaystyle y_{1}=b_{3},\quad y_{2}=a_{4}^{-1}b_{3}^{-1},\quad
y_{3}=a_{3}^{-1}b_{3}^{-1}$

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

$\displaystyle x_{1}+x_{3}x_{4}^{2}=1,\qquad
x_{2}+x_{4}x_{5}y_{1}^{2}y_{2}y_{3}^{2}=1,\qquad
x_{3}+x_{1}x_{5}y_{1}^{4}y_{2}^{3}y_{3}^{2}=1,$ $\displaystyle
x_{4}+x_{1}x_{2}y_{1}^{2}y_{2}^{2}y_{3}=1,\qquad x_{5}+x_{2}^{3}x_{3}=1,$
$\displaystyle
y_{1}+y_{1}^{4}x_{2}^{3}x_{3}^{2}x_{4}^{3}y_{2}^{3}y_{3}^{2}=1,\qquad
y_{2}+y_{2}^{2}x_{2}x_{3}x_{4}^{2}y_{1}^{2}y_{3}^{2}=1,\qquad
y_{3}+y_{3}^{2}x_{2}^{2}x_{3}x_{4}y_{1}^{2}y_{2}^{2}=1.$

As usual these equations tell us that if the $x,y\geq 0$, then we also have
$x,y\leq 1$. But note an interesting feature of the equations for the $y_{i}$
that we also saw for $C_{n}$: both monomials contain a factor of $y_{i}$, and
therefore we cannot set any of the $y$’s to zero. Thus the only boundaries of
this connected component are associated with the $x_{i}\to 0$ for
$i=1,\dots,5$; and we have found an unusual binary realization (in the sense
of [4]) of a pentagon.

Suppose instead we now set $a_{1}$ to be negative, that is,
$a_{1}=-a^{\prime}_{1}$ with $a^{\prime}_{1}>0$. 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

$\displaystyle
x_{1}=a_{4}^{-1},\qquad\\!x_{2}=a_{2}^{3}a_{3}^{3}b_{1}b_{2}^{2}b_{3},\qquad
x_{3}=b_{3}^{-1},\qquad\\!x_{4}=b_{2}^{-1},\qquad\\!x_{5}=a^{\prime}_{1}a_{2}^{-1}a_{3}^{-2}a_{4}^{-1}b_{2}^{-1}b_{3}^{-1},$
$\displaystyle x_{6}=a_{3}^{3}a_{4}^{3}b_{2}b_{3}^{2}b_{4},\qquad
x_{7}=a^{\prime}_{1}a_{4}^{2}b_{3}b_{4}a_{2}^{-1}b_{2}^{-1},\qquad
x_{8}=a_{3}^{-2}a_{4}^{-1}b_{2}^{-1}b_{3}^{-1},$ $\displaystyle
x_{9}=a^{\prime}_{1}a_{2}^{2}b_{1}b_{2}a_{4}^{-1}b_{3}^{-1},\qquad
x_{10}=a_{3}b_{2},\qquad x_{11}=a_{3}b_{3},\qquad
x_{12}=a_{2}^{-1}a_{3}^{-2}b_{2}^{-1}b_{3}^{-1}$

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 $x_{i}+\text{(monomial)}=1$, 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

$\displaystyle x_{2}^{2}x_{3}x_{5}^{2}x_{8}=x_{1}x_{4}x_{9}^{2},\qquad
x_{6}^{2}x_{4}x_{5}^{2}x_{8}=x_{1}x_{3}x_{7}^{2},\qquad
x_{10}^{2}x_{4}x_{8}=x_{1}x_{3},\qquad x_{11}^{2}x_{3}x_{8}=x_{1}x_{4}.$

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 $G_{2}$ 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
question immediately suggested by our investigations (see Section 11) is
understanding all the connected components of the real points of cluster
configuration spaces. For $A_{n-3}$, there is a beautiful 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 $C_{n}$ by folding. These
examples are especially easy to understand since we have a “linear model” for
${\mathcal{M}}_{0,n}$ in terms of a hyperplane arrangement. The connected
components can also be easily understood in these examples, directly studying
the space of solutions of the $u$-equations with different sign patterns. In
general, ${\mathcal{M}}_{D}$ is not a hyperplane arrangement complement and it
would thus be interesting to systematically study the question of connected
components directly from the $u$-equations defining the space, as we have done
in some examples in Section 11.

It is natural to conjecture that some or all of the other real components of
cluster configurations spaces (suitably compactified using the $u$-equations)
are also positive geometries, and it would be interesting to determine their
canonical forms.

In connection with determining canonical forms for general components, we
state here without proof, a simple expression for the canonical forms of the
positive component we have studied above, not in terms of cluster variables,
but directly in terms of $u$-variables. Recall that the $u_{\gamma}$ are in
bijection with all the cluster variables. Consider any acyclic cluster
$(x_{\gamma_{1}},\dots,x_{\gamma_{n}})$. Then, the canonical form is simply
given by taking the wedge product

$\displaystyle\Omega=\bigwedge_{i=1}^{n}\frac{{\rm
d}u_{\gamma_{i}}}{u_{\gamma_{i}}(1-u_{\gamma_{i}})}.$ (12.1)

As we noted in Section 10.2, acyclic seeds have the following feature: all the
$u_{\gamma}$ variables can be expressed rationally in terms of those in the
initial seed. This idea can be extended to give canonical forms for other
connected components of cluster configuration spaces. As a simple example, let
us consider the description of the component discussed in Section 11 for the
$G_{2}$ case, where $b_{1}=-b^{\prime}_{1}<0$. We can readily check that all
of the $(x,y)$ variables can be rationally solved for in terms of either
$(x_{2},x_{3})$ or in terms of $(x_{3},x_{4})$. The canonical form is then
given as

$\displaystyle\Omega={\rm dlog}\frac{x_{2}}{1-x_{2}}{\rm
dlog}\frac{x_{3}}{1-x_{3}}={\rm dlog}\frac{x_{3}}{1-x_{3}}{\rm
dlog}\frac{x_{4}}{1-x_{4}}.$

12.2. Open and closed cluster string amplitudes. The _stringy canonical forms_
of [3] can be applied to the cluster configuration space ${\mathcal{M}}_{D}$,
and we obtain the _cluster string amplitude_. For a cluster algebra
${\mathcal{A}}\big{(}{\tilde{B}}\big{)}$ of full rank and of type $D$, the
cluster string integral is defined to be

$\displaystyle{\mathcal{I}}_{D}(s):=\int_{{\mathcal{M}}_{D,>0}}\Omega({\mathcal{M}}_{D,>0})\prod_{\gamma\in\Pi}x_{\gamma}^{\alpha^{\prime}s_{\gamma}}\prod_{i=n+1}^{n+m}x_{i}^{\alpha^{\prime}s_{i}},$
(12.2)

where $\\{s_{\gamma}\,|\,\gamma\in\Pi\\}\cup\\{s_{i}\,|\,i\in[n+1,n+m]\\}$ are
parameters chosen so that the product
$\prod_{\gamma\in\Pi}x_{\gamma}^{\alpha^{\prime}s_{\gamma}}$ is $T$-invariant,
and thus descends to a function on ${\mathcal{M}}_{D,>0}$. Choosing
${\tilde{B}}={\tilde{B}}^{\operatorname{prin}}$, we may use Theorem 4.3 to
rewrite (12.2) as

$\displaystyle{\mathcal{I}}_{D}({\mathbf{X}},\\{c\\})=\int_{{\mathbb{R}}_{>0}^{n}}\prod_{i}\frac{{\rm
d}y_{i}}{y_{i}}y_{i}^{\alpha^{\prime}X_{i}}\prod_{\gamma\in\Pi^{+}}F_{\gamma}({\mathbf{y}})^{-\alpha^{\prime}c_{\gamma}},$
(12.3)

where $({\mathbf{X}},\\{c\\})$ are related linearly to $(s_{i},s_{\gamma})$.
By [3, Claim 2], (12.3) converges when the point ${\mathbf{X}}$ belongs to the
generalized associahedron $P(c)=\sum_{\gamma\in\Pi^{+}}c_{\gamma}P_{\gamma}$,
where $P_{\gamma}$ is the Newton polytope of $F_{\gamma}({\mathbf{y}})$. By
[3, Claim 2] the leading order of ${\mathcal{I}}_{D}({\mathbf{X}},\\{c\\})$ is
the canonical function $\Omega(P(c))$ of $P(c)$ evaluated at $X$:

$\displaystyle\lim_{\alpha^{\prime}\to
0}(\alpha^{\prime})^{n}{\mathcal{I}}_{D}({\mathbf{X}},\\{c\\})=\underline{\Omega}(P(c))({\mathbf{X}}).$

(We refer the reader to [2, 3] for background on canonical functions and
canonical forms.) In particular, the poles of
${\mathcal{I}}_{D}({\mathbf{X}},\\{c\\})$ as $\alpha^{\prime}\to 0$, all of
which are simple, correspond bijectively to the facets of the generalized
associahedron of $D^{\vee}$. By [3, Section 9] and Theorem 6.6, we may also
rewrite

$\displaystyle{\mathcal{I}}_{D}({\bf
U})=\int_{{\mathcal{M}}_{D,>0}}\Omega({\mathcal{M}}_{D,>0})\prod_{\gamma\in\Pi}u_{\gamma}^{\alpha^{\prime}U_{\gamma}}$
(12.4)

and the convergence condition is the very simple condition $U_{\gamma}>0$. By
Proposition 4.6, the $u_{\gamma}$ and $\\{y_{i},F_{\gamma}\\}$ are related by
an invertible monomial transformation, and thus
$\\{U_{\gamma}\,|\,\gamma\in\Pi\\}$ and
$\\{X_{1},\dots,X_{n}\\}\cup\\{c_{\gamma}\,|\,\gamma\in\Pi^{+}\\}$ are related
by an invertible linear transformation. (The matrix of this linear
transformation has entries given by the integers
$\operatorname{Trop}(F_{\gamma}({\mathbf{y}}))\big{(}{-}{\mathbf{g}}^{\vee}_{\omega}\big{)}$
that appeared in Section 6.) We see from (12.4) that the $u$-variables
$u_{\gamma}$ 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
${\mathcal{M}}_{D,>0}$) will span a space of integral functions of dimension
equal to the absolute value of the Euler characteristic
$|\chi({\mathcal{M}}_{D}({\mathbb{C}}))|$. 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

$\displaystyle{\mathcal{I}}_{D}^{\rm{closed}}\big{(}\big{\\{}U,\bar{U}\big{\\}}\big{)}=\int_{{\mathcal{M}}_{D}({\mathbb{C}})}\Omega({\mathcal{M}}_{D,>0})\prod_{\gamma\in\Pi}u_{\gamma}^{\alpha^{\prime}U_{\gamma}}\wedge\bar{\Omega}({\mathcal{M}}_{D,>0})\prod_{\gamma\in\Pi}\bar{u}_{\gamma}^{\alpha^{\prime}\bar{U}_{\gamma}},$

where in order for the integrand to be single-valued, we must have that the
exponents $\bar{U}_{\gamma}$ differ from $U_{\gamma}$ at most by integers.

As we have remarked, it is plausible that real components of cluster
configuration space other than the region associated with $u_{\gamma}\geq 0$
provide us with many different positive geometries ${\mathcal{M}}^{(i)}_{D}$,
with associated canonical forms $\Omega\big{(}{\mathcal{M}}^{(i)}_{D}\big{)}$.
In this case we can extend the closed string integrals to be more generally
labelled by pairs of these positive geometries,

$\displaystyle{\mathcal{I}}_{{\mathcal{M}}^{(i)}_{D},{\mathcal{M}}^{(j)}_{D}}^{\rm{closed}}\big{(}\big{\\{}U,\bar{U}\big{\\}}\big{)}=\int_{{\mathcal{M}}_{D}({\mathbb{C}})}\Omega\big{(}{\mathcal{M}}^{(i)}_{D}\big{)}\prod_{\gamma\in\Pi}u_{\gamma}^{\alpha^{\prime}U_{\gamma}}\wedge\bar{\Omega}\big{(}{\mathcal{M}}^{(j)}_{D}\big{)}\prod_{\gamma\in\Pi}\bar{u}_{\gamma}^{\alpha^{\prime}\bar{U}_{\gamma}}.$

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.

##  Appendix A A lemma in commutative algebra

######  Lemma A.1.

Let $f\colon A\to B$ be a surjective homomorphism of Noetherian commutative
rings with identity. Let $S\subset A$ be the multiplicative set generated by
elements $x_{1},x_{2},\dots,x_{p}$ such that

$\displaystyle f(x_{1}),f(x_{2}),\dots,f(x_{p})\quad\text{are not zero-
divisors in $B$}.$ (A.1)

Suppose that

  1. $(1)$

the localized homomorphism $S^{-1}f\colon S^{-1}A\to S^{-1}B$ is an
isomorphism, and

  2. $(2)$

for each $i=1,2,\dots,p$ the induced homomorphism $f_{i}\colon A/(x_{i})\to
B/(f(x_{i}))$ is an isomorphism.

Then $f$ is an isomorphism.

###### Proof.

Let $K$ denote the kernel of $f$. Let $a\in K$ be a nonzero element. Suppose
that $x_{i}a\neq 0$ in $A$ for all $i$. Then the image of $a$ in $S^{-1}A$ is
nonzero and it is in the kernel of $S^{-1}f\colon S^{-1}A\allowbreak\to
S^{-1}B$. This contradicts (1). Thus $Ma=0$ for some monomial $M$ in the
$x_{i}$-s. Replacing $a$ by $M^{\prime}a$ for some other monomial
$M^{\prime}$, and using (A.1), we may assume that $a\in K$ and $x_{i}a=0$ for
some $i=1,2,\dots,p$.

If $a\in(x_{i})$, then by (A.1), we have $a=x_{i}a_{1}$ for a nonzero element
$a_{1}\in K$. Repeating, we either find a nonzero element $a^{\prime}\in K$
such that $a^{\prime}\notin(x_{i})$, or we have an ascending chain of ideals
$(a)\subset(a_{1})\subset(a_{2})\subset\cdots$. In the former case, the image
of $a^{\prime}$ in $A/(x_{i})$ is nonzero and in the kernel of $f_{i}$,
contradicting (2). Thus we are in the latter case. Since $x_{i}$ is not a unit
and $A$ is Noetherian, the chain of ideals stabilizes to a proper ideal
$(a^{\prime})=I\subsetneq A$, and we thus have
$(a^{\prime})=(a^{\prime\prime})$, where $a^{\prime\prime}=x_{i}a^{\prime}$
and $x^{n}a^{\prime}=0$ for some $n>0$. This is impossible: letting $m$ be
minimal such that $x_{i}^{m}a^{\prime}=0$ we find that
$x_{i}^{m-1}a^{\prime\prime}=0$ which implies $x_{i}^{m-1}a^{\prime}=0$, 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.

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