By Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich
Abstract
In an earlier work (Publ. Inst. Hautes Études Sci., 122 (2015), 65–168) the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi–Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalizations of holomorphic discs).
Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock–Goncharov dual basis conjecture (Publ. Inst. Hautes Études Sci., 103 (2006), 1–211). In particular, under suitable hypotheses, for each $Y$ the partial compactification of an affine cluster variety $U$ given by allowing some frozen variables to vanish, we obtain canonical bases for $H^0(Y,\mathcal{O}_Y)$ extending to a basis of $H^0(U,\mathcal{O}_U)$. Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell $U$ in the basic affine space $Y,$ we obtain a canonical basis of each irreducible representation of $\operatorname {SL}_r$, parameterized by a set which each choice of seed identifies with the integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation-theoretic considerations.
Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.
Introduction
0.1. Statement of the main results
Fock and Goncharov conjectured that the algebra of functions on a cluster variety has a canonical vector space basis parameterized by the tropical points of the mirror cluster variety. Unfortunately, as shown in Reference GHK13 by the first three authors of this paper, this conjecture is usually false: in general the cluster variety may have far too few global functions. One can only expect a power series version of the conjecture, holding in the “large complex structure limit”, and honest global functions parameterized by a subset of the mirror tropical points. For the conjecture to hold as stated, one needs further affineness assumptions. Here we apply methods developed in the study of mirror symmetry, in particular scattering diagrams, introduced by Kontsevich and Soibelman in Reference KS06 for two dimensions and by Gross and Siebert in Reference GS11 for all dimensions, broken lines, introduced by Gross in Reference G09 and developed further by Carl, Pumperla, and Siebert in Reference CPS, and theta functions, introduced by Gross, Hacking, Keel, and Siebert, see Reference GHK11, Reference CPS, Reference GS12, and Reference GHKS, to prove the conjecture in this corrected form. We give in addition a formula for the structure constants in this basis, nonnegative integers given by counts of broken lines. Definitions of all these objects, essentially combinatorial in nature, in the context of cluster algebras will be given in later sections. Here are more precise statements of our results.
For basic cluster variety notions we follow the notation of Reference GHK13, §2, for convenience, as we have collected there a number of definitions across the literature; nothing there is original. We recall some of this notation in Appendices A and B. The various flavors of cluster varieties are all varieties of the form $V=\bigcup _{\mathbf{s}} T_{L,\mathbf{s}}$, where $T_{L,\mathbf{s}}$ is a copy of the algebraic torus
over a field $\mathbb{k}$ of characteristic $0$, and $L = \mathbb{Z}^n$ is a lattice, indexed by $\mathbf{s}$ running over a set of seeds (a seed being roughly an ordered basis for $L$). The birational transformations induced by the inclusions of two different copies of the torus are compositions of mutations. Fock and Goncharov introduced a simple way to dualize the mutations, and using this define the Fock–Goncharov dual,Footnote1$V^{\vee }=\bigcup _{\mathbf{s}} T_{L^*,\mathbf{s}}$. We write $\mathbb{Z}^T$ for the tropical semifield of integers under $\max ,+$. There is a notion of the set of $\mathbb{Z}^T$-valued points of $V$, written as $V(\mathbb{Z}^T)$. This can also be viewed as being canonically in bijection with $V^{\operatorname {trop}}(\mathbb{Z})$, the set of divisorial discrete valuations on the field of rational functions of $V$ where the canonical volume form has a pole; see §2. Each choice of seed $\mathbf{s}$ determines an identification $V(\mathbb{Z}^T) = L$.
1
Roughly one can view the Fock–Goncharov dual as the mirror variety, but this is not always precisely the case. With some additional effort, one can make this precise “at the boundary”, but we shall not do so here.
Our main object of study is the $\mathcal{A}$ cluster variety with principal coefficients, $\mathcal{A}_{\operatorname {prin}}=\bigcup _{\mathbf{s}} T_{\widetilde{N}^{\circ },\mathbf{s}}$; see Appendices A and B for notation. This comes with a canonical fibration over a torus $\pi : \mathcal{A}_{\operatorname {prin}} \to T_M$ and a canonical free action by a torus $T_{N^\circ }$. We let $\mathcal{A}_t := \pi ^{-1}(t)$. The fiber $\mathcal{A}_e \subset \mathcal{A}_{\operatorname {prin}}$($e \in T_M$ the identity) is the Fock–Goncharov $\mathcal{A}$ variety (whose algebra of regular functions is the Fomin–Zelevinsky upper cluster algebra). The quotient $\mathcal{A}_{\operatorname {prin}}/T_{N^\circ }$ is the Fock–Goncharov $\mathcal{X}$ variety.
For example, $\operatorname {ord}(\mathcal{A})$ is the original cluster algebra defined by Fomin and Zelevinsky in Reference FZ02a, and $\operatorname {up}(\mathcal{A})$ is the corresponding upper cluster algebra as defined in Reference BFZ05.
Given a global monomial $f$ on $V$, there is a seed $\mathbf{s}$ such that $f|_{T_{L,\mathbf{s}}}$ is a character $z^m$,$m\in L^*$. Because the seed $\mathbf{s}$ gives an identification of $V^{\vee }(\mathbb{Z}^T)$ with $L^*$, we obtain an element $\mathbf{g}(m)\in V^{\vee }(\mathbb{Z}^T)$, which we show is well-defined (independent of the open set $T_{L,\mathbf{s}}$); see Lemma 7.10. This is the $g$-vector of the global monomial $f$. We show this notion of $g$-vector coincides with the notion of $g$-vector from Reference FZ07 in the $\mathcal{A}$ case; see Corollary 5.9. Let $\Delta ^+(\mathbb{Z}) \subset V^{\vee }(\mathbb{Z}^T)$ be the set of $g$-vectors of all global monomials on $V$. Finally, we write $\mathrm{can}(V)$ for the $\mathbb{k}$-vector space with basis $V^{\vee }(\mathbb{Z}^T)$, i.e.,
(where $\vartheta _q$ for the moment indicates the abstract basis element corresponding to $q \in V^{\vee }(\mathbb{Z}^T)$).
Fock and Goncharov’s dual basis conjecture says that $\mathrm{can}(V)$ is canonically identified with the vector space $\operatorname {up}(V)$, and so in particular $\mathrm{can}(V)$ should have a canonical $\mathbb{k}$-algebra structure. Note that such an algebra structure is determined by its structure constants, a function
There is an analogue to Theorem 0.3 for $\mathcal{A}_t$ (the main difference is that the theta functions, i.e., the canonical basis for $\operatorname {mid}(\mathcal{A}_t)$, are only defined up to scaling each individual element, and the structure constants will not in general be integers). Injectivity in (7) holds for very general $\mathcal{A}_t$; see Theorem 7.16.
Note that (5)–(6) immediately imply:
This was conjectured by Fomin and Zelevinsky in their original paper Reference FZ02a. Positivity was obtained independently in the skew-symmetric case by [LS13] by an entirely different argument. In our proof the positivity in (1) and (6) both come from positivity in the scattering diagram, a powerful tool fundamental to the entire paper; see Theorem 1.13.
We conjecture that injectivity in (7) holds for all $\mathcal{A}_t$ (without the convexity assumption). Note (7) includes the linear independence of cluster monomials, which has already been established (without convexity assumptions) for skew-symmetric cluster algebras in Reference CKLP by a very different argument. The linear independence of cluster monomials in the principal case also follows easily from our scattering diagram technology, as pointed out to us by Greg Muller; see Theorem 7.20.
When there are frozen variables, one obtains a partial compactification $V \subset \overline{V}$ (where the frozen variables are allowed to take the value $0$) for $V = \mathcal{A}$,$\mathcal{A}_{\operatorname {prin}},$ or $\mathcal{A}_t$. The notions of $\operatorname {ord}$,$\operatorname {up}$,$\mathrm{can}$, and $\operatorname {mid}$ extend naturally to $\overline{V}$; see Construction B.9.
Of course if $\operatorname {ord}(V) = \operatorname {up}(V)$, and we have injectivity in (7), $\operatorname {ord}(V) = \operatorname {mid}(V) = \operatorname {up}(V)$ has a canonical basis $\Theta$ with the given properties. Also, $\operatorname {ord}(V) = \operatorname {up}(V)$ implies, under certain hypotheses, $\operatorname {ord}(\overline{V}) = \operatorname {up}(\overline{V})$; see Lemma 9.10. Such partial compactifications are essential for representation-theoretic applications:
The tools necessary for the proof of Theorem 0.3 are developed in the first six sections of the paper, with the proof given in §7. This material is summarized in more detail in §0.2.
The second part of the paper turns to criteria for the full Fock–Goncharov conjecture to hold. Precisely:
We prove a number of criteria which guarantee the full Fock–Goncharov conjecture holds. One such condition, which seems to be very natural in our setup and is implied, say, by the existence of a maximal green sequence, is:
Many of the results in the second part of the paper are proved using a generalized notion of convex function or convex polytope; see §§0.3 and 0.4 for more details.
In §8.5, we turn to results on partial compactifications. We first explain how convex polytopes in our sense give rise, under suitable hypotheses, to compactifications of $\mathcal{A}$-type cluster varieties and toric degenerations of such. This connects our constructions to the mirror symmetry picture described in Reference GHK11, and in particular describes a partial compactification of $\mathcal{A}_{\operatorname {prin}}$ as giving a degeneration of a family of log Calabi–Yau varieties to a toric variety. Partial compactifications via frozen variables are also important in representation theoretic applications, as already indicated in Example 0.5. We prove results for such partial compactifications which, combined with recent results of T. Magee Reference Ma15, Reference Ma17, yield strong representation-theoretic results; see §0.4 for more details.
We now turn to a more detailed summary of the contents of the paper.
0.2. Toward the main theorem
Section 1 is devoted to the construction of the fundamental tool of the paper, scattering diagrams. While Reference GS11 defined these in much greater generality, here they are collections of walls living in a vector space with attached functions constructed canonically from a choice of seed data. A precise definition can be found in §1.1. Here we simply highlight the main new result, Theorem 1.13, whose proof, being fairly technical, is deferred to Appendix C. This says that the functions attached to walls of a scattering diagram associated to seed data have positive coefficients. All positivity results in this paper flow from this fundamental observation, and indeed many of our arguments use this in an essential way. For the reader’s convenience, we give in §1.2 an elementary construction of the relevant scattering diagrams, drawing on the method given in Reference KS13. Since a scattering diagram depends on a choice of seed, §1.3 shows how scattering diagrams associated to mutation equivalent seeds are related. This shows that a scattering diagram has a chamber structure indexed by seeds mutation equivalent to the initial choice of seed.
In §2 we review some notions of tropicalizations of cluster varieties, showing that scattering diagrams naturally live in such tropicalizations. Indeed, the scattering diagram which is associated to a cluster variety $V$ lives naturally in the tropical space of the Fock–Goncharov dual $V^{\vee }(\mathbb{R}^T)$. These tropicalizations, crucially, can only be viewed as piecewise linear, rather than linear, spaces, with a choice of seed giving an identification of the tropicalization with a linear space. Already the mutation combinatorics becomes apparent:
The collection of cones $\Delta ^+$ was introduced by Fock and Goncharov, who conjectured they formed a fan. It is not at all obvious from the definition that the interiors of the cones cannot overlap. Our description of the chamber structure induced by a scattering diagram in fact shows that part of the chamber structure coincides with the collection of cones $\Delta ^+$. This shows the fact that they form a fan directly. In addition, the set $\Delta ^+(\mathbb{Z})$ of Theorem 0.3 consists of the integral points of the union of cones in $\Delta ^+$.
Section 3 gives the definition of broken line, the second principal combinatorial tool of the paper. These were originally introduced in Reference G09 and developed further in Reference CPS as tropical replacements for Maslov index two disks. In Reference GHK11, they were used to define theta functions, which are, in principle, formal sums over all broken lines with fixed boundary conditions. The relevance of theta functions for us comes in §4. Here we show the direct relationship between scattering diagrams and the $\mathcal{A}$ cluster algebra. We show that if we associate a suitable torus $T_L$ to each chamber of the scattering diagram associated to a mutation of the initial seed, then the walls separating the chambers can be interpreted as giving birational maps between these tori. Gluing together these copies of $T_L$ gives the $\mathcal{A}$ cluster variety; see Theorem 4.4. Further, a theta function $\vartheta _p$ depends on a point $p\in \mathcal{A}^{\vee }(\mathbb{Z}^T)$. If for a given choice of $p$,$\vartheta _p$ is in fact a finite sum, then $\vartheta _p$ is a global function on $\mathcal{A}$. We show that this holds in particular when $p$ lies in the cluster complex $\Delta ^+$, and in this case $\vartheta _p$ agrees with the cluster monomial with $g$-vector given by $p$. Because of the positivity result Theorem 1.13, $\vartheta _p$ is in any event always a power series with positive coefficients. Thus we get positivity of the Laurent phenomenon, Theorem 4.10, as an easy consequence of our formalism.
In §5 we begin with what is another essential observation for our approach. A choice of initial seed $\mathbf{s}$ provides a partial compactification $\overline{\mathcal{A}}_{\operatorname {prin}}^{\mathbf{s}}$ of $\mathcal{A}_{\operatorname {prin}}$ by allowing the variables $X_1,\ldots , X_n$ (the principal coefficients) to be zero. These variables induce a flat map $\pi : \overline{\mathcal{A}}_{\operatorname {prin}}^{\mathbf{s}}\rightarrow \mathbb{A}^n_{X_1,\ldots ,X_n}$, with $\mathcal{A}$ being the fiber over $(1,\ldots ,1)$. Our methods easily show:
Though immediate from our scattering diagram methods, the result is not obvious from the original definitions; indeed, it is equivalent to the sign-coherence of $c$-vectors (see Corollary 5.5).
The last major ingredient in the proof of Theorem 0.3 is a formal version of the Fock–Goncharov conjecture. As mentioned above, this conjecture does not hold in general, but in §6, we show that the Fock–Goncharov conjecture holds in a formal neighborhood of the torus fiber of $\overline{\mathcal{A}}^{\mathbf{s}}_{\operatorname {prin}}\rightarrow \mathbb{A}^n$. We show the structure constants given in Theorem 0.3(1) have a tropical interpretation and determine an associative product on $\mathrm{can}(\mathcal{A}_{\operatorname {prin}})$, except that $\vartheta _p \cdot \vartheta _q$ will in general be an infinite sum of theta functions. Further, canonically associated to each universal Laurent polynomial $g \in \operatorname {up}(\mathcal{A}_{\operatorname {prin}})$ is a formal power series $\sum _{q \in \mathcal{A}_{\operatorname {prin}}^{\vee }(\mathbb{Z}^T)} \alpha _q \vartheta _q$ which converges to $g$ in a formal neighborhood of the central fiber. For the precise statement see Theorem 6.8, which we interpret as saying that the Fock–Goncharov dual basis conjecture always holds in the large complex structure limit. This is all one should expect from log Calabi–Yau mirror symmetry in the absence of further affineness assumptions. A crucial point, shown in the proof of Theorem 6.8, is that the expansion of $g\in \operatorname {up}(\mathcal{A}_{\operatorname {prin}})$ is independent of the choice of seed $\mathbf{s}$ determining the compactification $\overline{\mathcal{A}}_{\operatorname {prin}}^{\mathbf{s}}$; i.e., it is independent of which degeneration is used to perform the expansion.
In §7 we introduce the middle cluster algebra $\operatorname {mid}(\mathcal{A}_{\operatorname {prin}})$. The idea is that while we do not know that every regular function on $\mathcal{A}_{\operatorname {prin}}$ can be written as a linear combination of theta functions, there is a set $\Theta \subset \mathcal{A}_{\operatorname {prin}}^{\vee }(\mathbb{Z}^T)$ indexing those $p$ for which $\vartheta _p$ is a regular function on $\mathcal{A}_{\operatorname {prin}}$. These in fact yield a vector space basis for a subalgebra of $\operatorname {up}(\mathcal{A}_{\operatorname {prin}})$ which necessarily includes all cluster monomials, hence includes the ordinary cluster algebra. With this in hand, Theorem 0.3 becomes a summary of the results proved up to this point. We then deduce the result for $\mathcal{X}$ and $\mathcal{A}$-type cluster varieties from the $\mathcal{A}_{\operatorname {prin}}$ case.
0.3. Convexity conditions
We now turn to the use of convexity conditions to prove the Fock–Goncharov conjecture in a number of different situations, as covered in §8. To motivate the concepts, let us define a partial minimal model of a log Calabi–Yau variety $V$. This is an inclusion $V\subset Y$ as an open subset such that the canonical volume form on $V$ has a simple pole along each irreducible divisor of the boundary $Y \setminus V$. For example, a partial minimal model for an algebraic torus is the same as a toric compactification. We wish to extend elementary constructions of toric geometry to the cluster case. For example, the partial compactification $\mathcal{A}\subset \overline{\mathcal{A}}$ determined by frozen variables is a partial minimal model.
The generalization of the cocharacter lattice $N \subset N_{\mathbb{R}}$ of the algebraic torus $T_N := N \otimes \mathbb{G}_m$ is the tropical set $V(\mathbb{Z}^T) \subset V(\mathbb{R}^T)$ of $V$. The main difference between the torus and the general case is that $V(\mathbb{R}^T)$ is not in general a vector space. Indeed, the identification of $V(\mathbb{Z}^T)$ with the cocharacter lattices of various charts of $V$ induce piecewise linear (but not linear) identifications between the cocharacter lattices. As a result, a piecewise straight path in $V(\mathbb{R}^T)$ which is straight under one identification $V(\mathbb{R}^T) = N_{\mathbb{R}}$ will be bent under another. Thus the usual notions of straight lines, convex functions, or convex sets do not make sense on $V(\mathbb{R}^T)$.
The idea for generalizing the notion of convexity is to instead make use of broken lines, which are piecewise linear paths in $V(\mathbb{R}^T)$. Using broken lines in place of straight lines, we can say which piecewise linear functions, and thus which polytopes, are convex; see Definition 8.2. Each regular function $W: V \to \mathbb{A}^1$ has a canonical piecewise linear tropicalization $w:=W^T: V(\mathbb{R}^T) \to \mathbb{R}$, which we conjecture is convex in the sense of Definition 8.2; see Conjecture 8.11. The conjecture is easy for $W \in \operatorname {ord}(V) \subset \operatorname {up}(V)$; see Proposition 8.13. Each convex piecewise linear $w$ gives a convex polytope $\Xi _w = \{x\,|\, w(x) \geq -1 \}$ and a convex cone $\{x \in V^{\vee }(\mathbb{R}^T)\,|\, w(x) \geq 0 \}$, where italics indicates convexity in our broken line sense. We believe the existence of a bounded polytope is equivalent to the full Fock–Goncharov conjecture:
The examples of Reference GHK13, §7, show that for the full Fock–Goncharov conjecture to hold, we need to assume $V$ has enough global functions. In that case tropicalizing a general function gives (conjecturally) a bounded convex polytope. As we are unable to prove Conjecture 8.11 except in the monomial case, we use a restricted version (which happily still has wide application):
The condition that $V$ has EGM is equivalent to the existence of $W \in \operatorname {ord}(V)$ whose associated convex polytope $\Xi _{W^T}$ is bounded; see Lemma 8.15.
The following theorem demonstrates the value of the EGM condition:
We note that $\mathcal{A}_{\operatorname {prin}}$ has EGM in many cases:
Returning to the role of convexity notions, we note that our formula for the structure constants $\alpha$ of Theorem 0.3(1) is given by counting broken lines. As a result, our notion of convexity interacts nicely with the multiplication rule. This allows us to generalize basic polyhedral constructions from toric geometry in a straightforward way.
A polytope $\Xi \subset V^{\vee }(\mathbb{R}^T)$ convex in our sense determines (by familiar Rees-type constructions for graded rings) a compactification of $V$. Furthermore, for any choice of seed, $V^{\vee }(\mathbb{R}^T)$ is identified with a linear space $\mathbb{R}^n$ and $\Xi$ with an ordinary convex polytope. Our construction also gives a flat degeneration of this compactification of $V$ to the ordinary polarized toric variety for $\Xi \subset \mathbb{R}^n$; see §8.5. We expect this specializes to a uniform construction of many degenerations of representation theoretic objects to toric varieties; see, e.g., Reference C02, Reference AB, and Reference KM05. Applied to the Fock–Goncharov moduli spaces of $G$-local systems, this will give for the first time compactifications of character varieties with nice (e.g., toroidal anticanonical) boundary; see Remark 8.34. The polytope can be chosen so that the boundary of the compactification is very simple, a union of toric varieties. For example, let $\operatorname {Gr}^o(k,n) \subset \operatorname {Gr}(k,n)$ be the open subset where the frozen variables for the standard cluster structure are nonvanishing. Then the boundary $\operatorname {Gr}(k,n)\setminus \operatorname {Gr}^o(k,n)$ consists of a union of certain Schubert cells. Using a polytope, we obtain an alternative compactification where the Schubert cells (which are highly nontoric) are replaced by toric varieties; see Theorem 8.35.
The Fock–Goncharov conjecture is the cluster special case of Reference GHK11, Conjecture 0.6, which says (roughly) that affine log Calabi–Yau varieties with maximal boundary come in canonical dual pairs with the tropical set of one parameterizing a canonical basis of functions on the other. We can view the conjecture as having two parts: First, the vector space, $\mathrm{can}$, with this basis $V^{\operatorname {trop}}(\mathbb{Z})$ is naturally an algebra in a such a way that $V^{\vee } := \operatorname {Spec}(\mathrm{can})$ is an affine log CY. And then furthermore, this log CY is the mirror—in the cluster case the Fock–Goncharov dual (it is natural to further ask if this is the mirror in the sense of homological mirror symmetry but we do not consider this question here). Our deepest mirror theoretic result is the following weakening of the first part:
We turn to §9. Here we study features of partial compactifications coming from frozen variables. As explained in Example 0.5, these partial compactifications are often the relevant ones in representation-theoretic examples. In particular, for a partial minimal model $\mathcal{A}\subset \overline{\mathcal{A}}$, often the vector subspace $\operatorname {up}(\overline{\mathcal{A}}) \subset \operatorname {up}(\mathcal{A})$ is more important than $\operatorname {up}(\mathcal{A})$ itself. For example there is a cluster structure with frozen variables for the open double Bruhat cell $U$ in a semisimple group $G$. Then $\operatorname {up}(\mathcal{A})$ is the ring of functions on the open double Bruhat cell and $\operatorname {up}(\overline{\mathcal{A}})=H^0(G,\mathcal{O}_G)$. Of course $H^0(G,\mathcal{O}_G)$ is the most important representation of $G$. However, one cannot expect a canonical basis of $\operatorname {up}(\overline{\mathcal{A}})$, i.e., one determined by the intrinsic geometry of $\overline{\mathcal{A}}$. For example, $G$ has no nonconstant global functions which are eigenfunctions for the action of $G$ on itself. But we expect, and in the myriad cases above can prove, that the affine log Calabi–Yau open subset $\mathcal{A}\subset \overline{\mathcal{A}}$ has a canonical basis $\Theta$, and we believe that $\Theta \cap \operatorname {up}(\overline{\mathcal{A}})$, the set of theta functions on $\mathcal{A}$ that extends regularly to all of $\overline{\mathcal{A}}$, is a basis for $\operatorname {up}(\overline{\mathcal{A}})$, canonically associated to the choice of log Calabi–Yau open subset $\mathcal{A}\subset \overline{\mathcal{A}}$; see Reference GHK13, Remark 1.10. This is not a basis of $G$-eigenfunctions, but they are eigenfunctions for the associated maximal torus, which is the subgroup of $G$ that preserves $U$. This is exactly what one should expect: the basis is not intrinsic to $G$, instead it is (we conjecture) intrinsic to the pair $U \subset G$; see Remark 0.16.
We shall now describe in more detail what can be proved for partial compactifications of cluster varieties coming from frozen variables. A key point is a technical but combinatorial hypothesis that each variable has an optimized seed; see Definition 9.1 and Lemmas 9.2 and 9.3. The main need for this hypothesis is Proposition 9.7, which states that if a linear combination of theta functions extends across a boundary divisor, then each theta function in the sum extends across the divisor. Thus the middle cluster algebra, in this case, behaves well with respect to boundary divisors. Happily, this condition holds for the cluster structures on the Grassmannian, and, for $G=\operatorname {SL}_r$, for the cluster structure on a maximal unipotent subgroup $N \subset G$, the basic affine space $\mathcal{A}= G/N$, and the Fock–Goncharov cluster structure on $(\mathcal{A}\times \mathcal{A}\times \mathcal{A})/G$; see Remark 9.5.
Let us now work with the principal cluster variety $\mathcal{A}_{\operatorname {prin}}$. Consider the partial compactification