Upper Bounds for Mutations of Potentials
In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math....
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irk-123456789-1492102019-02-20T01:27:34Z Upper Bounds for Mutations of Potentials Cruz Morales, J.A. Galkin, S. In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. 2013 Article Upper Bounds for Mutations of Potentials / J.A. Cruz Morales, S. Galkin // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 11 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13F60; 14J33; 53D37 DOI: http://dx.doi.org/10.3842/SIGMA.2013.005 http://dspace.nbuv.gov.ua/handle/123456789/149210 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. |
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Cruz Morales, J.A. Galkin, S. |
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Cruz Morales, J.A. Galkin, S. Upper Bounds for Mutations of Potentials Symmetry, Integrability and Geometry: Methods and Applications |
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Cruz Morales, J.A. Galkin, S. |
| author_sort |
Cruz Morales, J.A. |
| title |
Upper Bounds for Mutations of Potentials |
| title_short |
Upper Bounds for Mutations of Potentials |
| title_full |
Upper Bounds for Mutations of Potentials |
| title_fullStr |
Upper Bounds for Mutations of Potentials |
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Upper Bounds for Mutations of Potentials |
| title_sort |
upper bounds for mutations of potentials |
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Інститут математики НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/149210 |
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Upper Bounds for Mutations of Potentials / J.A. Cruz Morales, S. Galkin // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 11 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT cruzmoralesja upperboundsformutationsofpotentials AT galkins upperboundsformutationsofpotentials |
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2025-07-12T21:05:09Z |
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2025-07-12T21:05:09Z |
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1837476680061419520 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 005, 13 pages
Upper Bounds for Mutations of Potentials?
John Alexander CRUZ MORALES †
1
and Sergey GALKIN †2†3†4†5
†1 Department of Mathematics and Information Sciences, Tokyo Metropolitan University,
Minami-Ohsawa 1-1, Hachioji, Tokyo 192-037, Japan
E-mail: cruzmorales-johnalexander@ed.tmu.ac.jp, alekosandro@gmail.com
†2 Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
†3 Independent University of Moscow, 11 Bolshoy Vlasyevskiy per., 119002, Moscow, Russia
†4 Moscow Institute of Physics and Technology, 9 Institutskii per.,
Dolgoprudny, 141700, Moscow Region, Russia
E-mail: Sergey.Galkin@phystech.edu
†5 Universität Wien, Fakultät für Mathematik, Garnisongasse 3/14, A-1090 Wien, Austria
Received May 31, 2012, in final form January 16, 2013; Published online January 19, 2013
http://dx.doi.org/10.3842/SIGMA.2013.005
Abstract. In this note we provide a new, algebraic proof of the excessive Laurent phe-
nomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU
10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Beren-
stein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1–52].
Key words: cluster algebras; Laurent phenomenon; mutation of potentials; mirror symmetry
2010 Mathematics Subject Classification: 13F60; 14J33; 53D37
1 Introduction
The idea of mutations of potentials was introduced in [9] and the Laurent phenomenon was estab-
lished in the two dimensional case by means of birational geometry of surfaces. More precisely,
in op. cit. the authors considered a toric surface X with a rational function W (a potential), and
using certain special birational transformations (mutations), they established the (excessive)
Laurent phenomenon which roughly says that if W is a Laurent polynomial whose mutations
are Laurent polynomials, then all subsequent mutations of these polynomials are also Laurent
polynomials (see Theorem B.1 in Appendix B for a precise statement of the excessive Laurent
phenomenon as established in [9]). The motivating examples of such potentials come from the
mirror images of special Lagrangian tori on del Pezzo surfaces [8] and Auroux’s wall-crossing
formula relating invariants of different tori [2].
The cluster algebras theory of Fomin and Zelevinsky [7] provides an inductive way to construct
some birational transformations of n variables as a consecutive composition of elementary ones
(called elementary mutations) with a choice of N = n directions at each step.
The theory developed in [9] can be seen as an extension of the theory of cluster algebras [7]
when the number of directions of mutations N is allowed to be (much) bigger than the number
of variables n, but at least one function remains to be a Laurent polynomial after all mutations.
So, it is natural to try to extend the machinery of the theory of cluster algebras for this new
setup. The main goal of this paper is to give the first step in such an extension by means of
?This paper is a contribution to the Special Issue “Mirror Symmetry and Related Topics”. The full collection
is available at http://www.emis.de/journals/SIGMA/mirror symmetry.html
mailto:cruzmorales-johnalexander@ed.tmu.ac.jp
mailto:alekosandro@gmail.com
mailto:Sergey.Galkin@phystech.edu
http://dx.doi.org/10.3842/SIGMA.2013.005
http://www.emis.de/journals/SIGMA/mirror_symmetry.html
2 J.A. Cruz Morales and S. Galkin
the introduction of the upper bounds (in the sense of [3]) and establishing the excessive Laurent
phenomenon [9] in terms of them. It is worth noticing that a further generalization can be done
and in a forthcoming work [5] we plan to study the quantization of the mutations of potentials
and their upper bounds. Naturally, this quantization can be seen as an extension of the theory
of quantum cluster algebras developed in [4, 11] and the theory of cluster ensembles in [6].
The upper bounds introduced in this paper can be described as a collection of regular func-
tions that remain regular after one elementary mutation in any direction. Thus, we can establish
the main result of this paper in the following terms (see Theorem 3.1 for the exact formulation).
Theorem (Laurent phenomenon in terms of the upper bounds). The upper bounds are preserved
by mutations.
Aside from providing a new proof for the excessive Laurent phenomenon and the already
mentioned generalization in the quantized setup, the algebraic approach that we are introducing
here is helpful for tackling the following two problems:
1. Develop a higher dimensional theory (i.e. dimension higher than 2) for the mutations of
potentials. Some work in that direction is carried out in [1].
2. Present an explicit construction to compactify Landau–Ginzburg models (Problem 44
of [9]).
In the present paper we do not deal with the above two problems (only a small comment
on 2 will be made at the end of the paper). We plan to give a detailed discussion of them
in [5] too. We just want to mention that the new algebraic approach has interesting geometrical
applications.
Some words about the organization of the text are in order. In Section 2 we extend the
theory developed in [9] to lattices of arbitrary rank and general bilinear forms (i.e., we can
consider even degenerate and not unimodular forms) and introduce the notion of upper bounds
in order to establish our main theorem. In Section 3 we actually establish the main theorem and
present its proof when the rank of the lattice is two and the form is non-degenerate which is the
case of interest for the geometrical setup of [9]. In the last section some questions and future
developments are proposed. For the sake of completeness of the presentation we include two
appendices. In Appendix A we review some definitions of [3] and briefly compare their theory
with ours. Appendix B is dedicated to presenting the Laurent phenomenon in terms of [9].
2 Mutations of potentials and upper bounds
Now we present an extension of the theory of mutations of potentials [9] (as formulated by
the second author and Alexandr Usnich) and introduce our modified definitions with the new
definition of upper bound. Notice that a slightly different theory (which fits into the framework
of this paper, but not [9]) is used in our software code1.
2.1 Combinatorial data
Let (·, ·) : L∗ × L → Z be the canonical pairing between a pair of dual lattices L ' Zr and
L∗ = Hom(L,Z) ' Zr.
In what follows the lattice L is endowed with a skew-symmetric bilinear integral form ω :
L × L → Z (we use the notation 〈v, v′〉 = ω(v, v′)). In the most important (both technically,
and from the point of view of applications) case r = rankL = 2, we have Λ2L ' Z, so all integer
skew-symmetric bilinear forms are integer multiples ωk = kω1 (k ∈ Z) where a generator ω1 is
1http://member.ipmu.jp/sergey.galkin/degmir.gp.
http://member.ipmu.jp/sergey.galkin/degmir.gp
Upper Bounds for Mutations of Potentials 3
fixed by the choice of orientation on L⊗R so that ω1((1, 0), (0, 1)) = 1. We would occasionally
use notations 〈·, ·〉1 = ω1(·, ·) and 〈·, ·〉k = ωk(·, ·).
The bilinear form ω gives rise to a map i = iω : L → L∗ that sends an element v ∈ L into
a linear form iω(v) ∈ L∗ such that (iω(v), v′) = ω(v, v′) for any v′ ∈ L. The map iω is an
isomorphism ⇐⇒ the form ω is non-degenerate and unimodular, when ω is non-degenerate
but not unimodular the map i identifies the lattice L with a full sublattice in L∗ of index detω,
finally if ω is degenerate then both the kernel and the cokernel of the map iω has positive rank.
We would like to have some functoriality, so we consider a category whose objects are
given by pairs (L, ω) of the lattice L and a skew-symmetric bilinear form ω, and the mor-
phisms Hom((L′, ω′), (L, ω)) are linear maps f : L′ → L such that ω′ = f∗ω, i.e. ω(v1, v2) =
ω′(f(v1), f(v2)) for all v1, v2 ∈ L′. Any linear map f : L′ → L defines an adjoint f∗ : L∗ → L′∗
and if it respects the bilinear forms, then iω′ = f∗iωf .
For a vector u ∈ L we define a symplectic reflection Ru and a piecewise linear mutation µu
to be the (piecewise)linear automorphisms of the set L given by the formulae
Rω,u(v) = v + ω(u, v)u,
µω,uv = v + max(0, ω(u, v))u.
For any morphism f ∈ Hom((L′, ω′), (L, ω)) and any vector u ∈ L′ we have Rω,fuf = fRω′,u
and µω,fuf = fµω′,u. Indeed, fµuv = f(v + max(0, ω′(u, v))u) = fv + max(0, ω′(u, v))(fu) =
fv + max(0, ω(fu, fv))(fu) = µfu(fv).
Note that Raω,bu = Rab
2
ω,u for all a, b ∈ Z and µaω,bu = µab
2
ω,u for all a, b ∈ Z+. However µω,−uv =
−µu(−v) = v + min(0, ω(u, v))u, hence µω,−uµω,u = Rω,u. Both Rω,u and µω,u are invertible:
R−1ω,uv = R−ω,uv = v − ω(u, v)u, µ−1ω,uv = µ−ω,−uv = R−1ω,uµω,−uv = v −max(0, ω(u, v))u. Note
that µ−1ω,−uv = R−1ω,−uµω,uv = R−1ω,uµω,u(v) = v − min(0, ω(u, v))u. Therefore, changing max
by min and + by −, simultaneously, corresponds to changing the form ω to the opposite −ω.
Further we omit ω from the notations of Ru and µu where the choice of the form is clear.
The underlying combinatorial gadget of our story is a collection of n vectors in L:
Definition 2.1. An exchange collection V is an element of Ln, i.e. an n-tuple (v1, . . . , vn) of
vectors vi ∈ L. Some vi may coincide. For a vector v its multiplicity mV (v) in the exchange
collection V equals the number of vectors in V that coincide with v: mV (v) = #{1 6 i 6 n :
vi = v}. We say that an exchange collection V ′ is a subcollection of exchange collection V if
mV ′ 6 mV . Equivalently, one may define an exchange collection V by its (non-negative integer)
multiplicity function mV : L→ Z>0. In this case n =
∑
v∈L
mV (v).
The exchange collections could be pushed forward by morphisms f ∈ Hom((L′, ω′), (L, ω)):
v′1, . . . , v
′
n ∈ L′n will go to fv1, . . . , fvn ∈ Ln. This gives rise to a natural diagonal action of
Aut(L, ω) = Sp(L, ω) on Ln. This action commutes with the permuting action of Sn.
A vector n ∈ L is called primitive if it is nonzero and its coordinates are coprime, i.e. n does
not belong to the sublattice kL for any k > 1, in other words n is not a multiple of other vector
in L. We denote the set of all primitive vectors in L as L1. Similarly one can define primitive
vectors in the dual lattice L∗. Note that if detω 6= ±1 then iω(n) may be a non-primitive
element of L∗ even for primitive elements n ∈ L1.
2.2 Birational transformations
Consider the group ring Z[L∗] – ring of Laurent polynomials of r variables. Its spectrum
T = Spec Z[L∗] ' Gr
m(Z) is the r-dimensional torus over the integers, in particular T (C) =
Hom(L∗,C∗), L∗ = Hom(T,Gm) is the lattice of characters of T and L = Hom(Gm, T ) is the
4 J.A. Cruz Morales and S. Galkin
lattice of 1-parameter subgroups in T . Define the ambient field K = KL = Q(L∗) as the fraction
field of Z[L∗] extended by all roots of unity (Q = Q(exp(2πiQ)).
A vector u ∈ L defines a birational transformation of KL (and its various subfields and
subrings) as follows
µu,ω : Xm → Xm
(
1 +Xiω(u)
)(u,m)
.
If f : T1 → T2 is a rational map between two tori, and u : Gm → T1 is a one-parameter
subgroup of T then its image fu : Gm → T2 is not necessarily a one-parameter subgroup,
but asymptotically behaves like one, this defines a tropicalization map T (F ) : Hom(Gm, T1)→
Hom(Gm, T2). The tropicalization of the birational map µu,ω : T1 → T2 is the piecewise-linear
map µu,ω : L1 → L2 defined in the previous subsection.
One can easily see most of the relations of the previous subsection on the birational level.
For example, µ−uµu = µuµ−u = Ru and RvµuR
−1
v = µRvu, where Ru is the homomorphism
of the torus T given by Ru,ω : Xm → Xm+(u,m)iωu. Also Rau,bω = Ra
2b
u,ω for any a, b ∈ Z, and
µau,ω = (µu,aω)a for any a ∈ Z, however neither of them is a power of µu,ω.2 In particular,
(µu,ω)−1 = µ−u,−ω.
Note that if M ⊂ L∗ is some sublattice of L∗ that contains iω(u) then µu preserves the
fraction field of Z[M ] ⊂ Z[L∗]. For any morphism f ∈ Hom((L′, ω′), (L, ω)) and a vector
u ∈ L′ we have a homomorphism f∗ : Z[L∗] → Z[L′∗] and two birational transformations
µu ∈ AutKL′ , µfu ∈ AutKL that commute: µuf
∗ = f∗µfu.
Remark 2.1. We have the following functoriality of the mutations with respect to the lattice L:
let L′ ⊂ L be a sublattice of index k in the lattice L, so L∗ = Hom(L,Z) is a sublattice of
index k in L′∗ = Hom(L′,Z), and assume that the vector u lies in the sublattice L′. Then the
Abelian group G = (L/L′) of order k acts on Q[L′∗],3 and its invariants is the subring Q[L∗],
so G acts on the torus T ′ = Spec Q[L′∗] and the torus T = Spec Q[L∗] is the quotient-torus
T = T ′/G, let π : T ′ → T be the projection to the quotient. The vector u defines the birational
transformation µu,T of the torus T and the birational transformation µu,T ′ of the torus T ′.
Then the mutation µu commutes with the action of the group G and with the projections:
πµu,T ′ = µu,Tπ and gµu,T ′ = µu,T ′g for any g ∈ G.
2.2.1 Rank two case
Let us see the mutations explicitly in case rankL = 2. Let e1, e2 be a base of L and f1, f2 be the
dual base of L∗, so (ei, fj) = δi,j . Also let xi = Xfi be the respective monomials in Z[L∗]. For
the skew-symmetric bilinear form ωk defined by ωk(e1, e2) = k and a vector u = u1e1 +u2e2 ∈ L
we have iωk(u1e1 + u2e2) = (−ku2)f1 + (ku1)f2 and so
µu,ωk : (x1, x2)→
(
x1 ·
(
1 + x−ku21 xku12
)u1 , x2 · (1 + x−ku21 xku12
)u2),
in particular the inverse map to µu,ω1 is given by µ−u,−ω1 : (x1, x2)→ (x1 · (1 +xu21 x
−u1
2 )−u1 , x2 ·
(1 + xu21 x
−u1
2 )−u2). In particular, µ∗(0,1)f = f
(
x1,
x2
1+x1
)
.
For any matrix A =
(
a b
c d
)
∈ SL(2,Z) = Sp(L, ω) there is a regular automorphism of the
torus t∗A(x1, x2) = (xa1x
b
2, x
c
1x
d
2). Conjugation by this automorphism acts on the set of mutations:
µ∗Au = (tAµut
−1
A )∗. So any mutation commutes with an infinite cyclic group given by the
stabilizer of u in Sp(L, ω), explicitly if u = (0, 1) then in coordinates (x1, x2) and (x1, x
′
2 = x1x2)
2Since (1 + xa) is not a power of (1 + x).
3An element n in L multiplies monomial Xm′
by the root of unity exp((2πi)(n,m′)), here (n,m′) is bilinear
pairing between L and L′∗ with values in Q extended by linearity from the pairing L′ ⊗ L′∗ → Z.
Upper Bounds for Mutations of Potentials 5
the mutation µ(0,1) is given by the same formula. Also every mutation commutes with 1-
dimensional subtorus of T , in case of u = (0, 1) the action of the subtorus is given by (x1, x2)→
(x1, αx2).
2.3 Mutations of exchange collections and seeds
Let L be a lattice equipped with a bilinear skew-symmetric form ω. A cluster y ∈ Km
L is
a collection y = (y1, . . . , ym) of m rational functions yi ∈ KL. We call y a base cluster if
y = (y1, , . . . , yr) is a base of the ambient field KL. A C-seed (supported on (L, ω)) is a pair
(y, V ) of a cluster y ∈ Km
L and an exchange collection V = (v1, . . . , vn) ∈ Ln. A V -seed
(supported on (L, ω)) is a pair (W,V ) of a rational function W ∈ KL and an exchange collection
V = (v1, . . . , vn) ∈ Ln.
Given two exchange collections V ′ = (v′1, . . . , v
′
n) ∈ L′n and V = (v1, . . . , vn) ∈ Ln we say
that V ′ is a mutation of V in the direction 1 6 j 6 n and denote it by V ′ = µjV if under the
given identification sj : L ' L′ we have v′j = sj(−vj) and v′i = sj(µvjvi) for i 6= k.
The mutation of a C-seed (y, V ) in the direction 1 6 j 6 n is a new C-seed (yj , Vj) where
Vj = µjV is a mutation of the exchange collection, and yj = µvj ,ωy where each variable is
transformed by the birational transformation µvj ,ω.
The identity µ−uµu = Ru implies that µj(µj(V )) and V are related by the Sp(L, ω)-transfor-
mation Ru.
2.4 Upper bounds and property (V )
Definition 2.2 (property (V )). We say a V -seed (W,V ) satisfies property (V ) if W is a Laurent
polynomial and for all v ∈ L the functions (µ∗v)
mV (v)W are also Laurent polynomials.
In this paper we introduce the upper bound of an exchange collection.
Definition 2.3 (upper bounds). For a C-seed Σ = (y, V ) define its upper bound U(Σ) to be
the Q-subalgebra of KL given by
U(Σ) = Q
[
y±1
]
∩
(
∩v∈L Q
[
(µ∗v)
mV (v)y±1
])
.
In case y is a base cluster (by abuse of notation) we denote U(Σ) just by U(V ).
The upper bounds defined here are a straightforward generalization of the upper bounds
in [3], but also they can be thought of as the gatherings of all potentials satisfying property (V ).
Proposition 2.1 (relation between property (V ) and upper bounds). The upper bound U(V )
of an exchange collection V consists of all functions W ∈ KL such that the V -cluster (W,V )
satisfies property (V ).
Proposition 2.2. Any morphism f : (L, ω)→ (L′, ω′) induces a dual morphism f∗ : L′∗ → L∗,
a homomorphism of algebras f∗ : Z[L′∗] → Z[L∗]. Assume that this homomorphism has no
kernel4. Then it induces a homomorphism of upper bounds f∗ : U(L′, ω′; fV )→ U(L, ω;V ). In
particular, if f is an isomorphism, then maps f∗ and (f−1)∗ establish the isomorphisms between
the upper bounds f∗ : U(L′, ω′; fV ) ' U(L, ω;V ).
Proposition 2.3. Consider a seed Σ = (L, ω; v1, . . . , vn). For a sublattice L′ ⊂ L that contains
all vectors vi ∈ L′ ⊂ L consider the seed Σ′ = (L′, ω|L′ ; v1, . . . , vn). By Remark 2.1 there is
a natural action of G = L/L′ on KL′ with KL = KG
L′. Moreover, the action of G obviously
4One can bypass this assumption by defining the upper bound U(L, ω;V ) as a subalgebra in some localization
of Z[L∗] determined by the exchange collection V .
6 J.A. Cruz Morales and S. Galkin
preserves the property of being a Laurent polynomial (i.e. it preserves the subalgebras Q[L′∗]),
and the mutations µvi commute with the G-action. Thus the upper bound with respect to the
overlattice L is the subring of G-invariants of the upper bound with respect to the sublattice L′:
U(Σ) = U(Σ′)G = U(Σ′) ∩Q[L∗].
3 Laurent phenomenon
In what follows we restrict ourselves to the case rankL = 2, ω is a non-degenerate form and the
vectors of exchange collection are primitive, however none of these conditions is essential.
Next theorem is the analogue of Theorem 1.5 in [3], presented here as Theorem A.1.
Theorem 3.1 (Laurent phenomenon in terms of upper bounds). Consider two C-seeds: Σ =
(L, ω; v1, . . . , vn) and Σ′ = (L′, ω′; v′1, . . . , v
′
n). If Σ′ = µiΣ is a mutation of Σ in direction
1 6 i 6 n then the upper bounds for Σ and Σ′ coincide: U(Σ) = µ∗viU(Σ′). As a corollary, if
a seed Σ′ is obtained from a seed Σ by a sequence of mutations, then the upper bound U(Σ′)
equals to the upper bound U(Σ) under identification of the ambient field by composition of the
birational mutations.
By Proposition 2.1 Theorem 3.1 is equivalent to the next corollary, which is easier to check
in practice and has almost the same consequences as the main theorem of [9], presented here as
Theorem B.1.
Corollary 3.1 (V -lemma). If V -seeds Σ and Σ′ are related by a mutation then the seed Σ
satisfies property (V ) ⇐⇒ the seed Σ′ satisfies property (V ).
In the rest of this section we prove Theorem 3.1. Our proof is quite similar to that of [3]5:
The set-theoretic argument reduces the problem to exchange collection V with small number of
vectors (1 or 2) without counting of multiplicities. Actually, when the collection V has only one
vector the equality of the upper bounds is obvious from the definitions. When the exchange
collection consists of two base vectors one can explicitly compute the upper bounds and compare
them. Finally, the case of two non-base non-collinear vectors is thanks to functoriality.
First of all, let us fix the notations. If the rank two lattice L is generated by a pair of
vectors e1 and e2, then the dual lattice L∗ = Hom(L,Z) has the dual base f1, f2 determined
by (fi, ej) = δi,j . The form ω is uniquely determined by its value k = ω(e1, e2), and further
we denote this isomorphism class of forms by ωk. We assume that k 6= 0, i.e. the form ω is
non-degenerate6, by swapping e1 and e2 one can exchange k to −k. A base ei of L corresponds
to a base xi = Xfi of Z[L∗].
Lemma 3.1. Let V be an exchange collection in (L, ω) and Σ = (L, ω;V ) be the respective seed.
1. If V is empty, then obviously U(L, ω;V ) = Q[L∗].
2. Otherwise, let Vα be a set of exchange collections such that for any v ∈ L we have mV (v) =
maxαmVα(v). Then
U(V ) = ∩αU(Vα).
3. In particular, if for a vector v ∈ L we define Vv = mV (v)× v to be an exchange collection
that consists of a single vector v with multiplicity mV (v) and Σv = (L, ω;Vv) be the re-
spective seed, then U(Σ) = ∩v∈L(Q[y±] ∩ Q[(µ∗v)
mV (v)y±]) = ∩v∈LU(Σv). In other words,
the upper bound of a C-seed Σ = (L, ω; y, V ) can be expressed as the intersection of the
upper bounds for its 1-vector subseeds.
5See Appendix A and Remark A.4 for the detailed comparison.
6If k = 0 then ω = 0 and all mutations are trivial.
Upper Bounds for Mutations of Potentials 7
4. Let V consist of a vector v1 with multiplicity m+ > 1, a vector v2 = −v1 with multiplicity
m− > 0, and vectors vk (k > 3) that are non-collinear to v1 with some multiplicities
mk > 0. Consider exchange subcollections V0 = {m+× v1,m−× (−v1)} and Vk = {1× v1,
mk × vk} (k > 3). Then
U(V ) = U(V0) ∩ U(V3) ∩ U(V4) ∩ · · · .
5. Let V ′ = µ1V be an exchange collection obtained by mutation of V in v1; it consists of
vector −v1 with multiplicity m− + 1 > 1, vector v1 with multiplicity m+ − 1 > 0 and
vectors v′k = µv1vk (k > 3) with multiplicities mk. Similarly to the previous step define
V ′0 = {(m− + 1)× (−v1), (m+ − 1)× v1} and V ′k = {1× (−v1),mk × v′k} (k > 3). Then
U(V ′) = U(V ′0) ∩ U(V ′3) ∩ U(V ′4) ∩ · · · .
6. Hence, to proof Theorem 3.1 it is necessary and sufficient to show that
U(V0) = µ∗v1U(V ′0) and U(Vk) = µ∗v1U(V ′k) (for all k > 3).
We will prove these equalities in Proposition 3.3 and Lemma 3.3.
Proposition 3.1. Let v1, v2 ∈ L be a pair of vectors v1 = ae1 + be2, v2 = ce1 + de2 such that
ad − bc = 1. Consider the lattice L′ with the base e′1, e′2 and the form ω′(e′1, e
′
2) = ω(v1, v2);
let f ′1, f ′2 be the dual base of L′∗. Consider a map m : L→ L′ given by m(e1) = de′1−be′2,m(e2) =
−ce′1 + ae′2; note that m(v1) = m(ae1 + be2) = e′1 and m(v2) = m(ce1 + de2) = e′2. The
dual isomorphism m∗ : L′∗ → L∗ is given by the transposed map m∗(f ′1) = df1 − cf2 and
m∗(f ′2) = −bf1 +af2. Let z1 = Xf ′1 = xd1x
−c
2 and z2 = Xf ′2 = x−b1 xa2. Since map m∗ is invertible
by Proposition 2.2 it gives the equality
U(L, ω;m1 × v1,m2 × v2) = U
(
L′, ω′;m1 × e′1,m2 × e′2
)∣∣
z1=xd1x
−c
2 , z2=x
−b
1 xa2
.
Lemma 3.2. Assume a seed Σ = (L, ω;m1× v1) consists of a unique vector v1 with multiplicity
m1 > 1.
1. If v1 = e2 = (0, 1) then the upper bound U(Σ) consists of all Laurent polynomials W of
the form W =
∑
l cl(x1)x
l
2 where cl ∈ Q[x±1 ] and for l 6 0 we have that cl is divisible by
(1 + xk1)−m1l. Moreover, U(Σ) = Q
[
x±1 , x2,
1
x′′2
=
(1+xk1)
m1
x2
]
.
2. If v1 = ae1 + be2 = (a, b) is an arbitrary primitive vector then U(Σ) = Q
[
z±, z1,
(1+zk)m1
z1
]
where z =
xa1
xb2
, z1 = xr1x
s
2 and (r, s) ∈ Z2 satisfies rb+ sa = 1.
Proof. Recall that mutation in the direction e2 is given by x′1 = x and x′2 = x2
1+xk1
. Assume
we have a Laurent polynomial W =
∑
l∈Z cl(x1)x
l
2. Then W can be expressed in terms of x1
and x′2 as W =
∑
l cl(x1)(1 + xk1)l(x′2)
l. This function is a Laurent polynomial in terms of
(x1, x
′
2) ⇐⇒ cl(x1)(1 + xk1)l is a Laurent polynomial of x1 for all l. This is equivalent to cl
being divisible by (1 + xk1)−l for l 6 0. Similarly if we do m1 mutations then x′′2 = x2
(1+xk1)
a and
W =
∑
cl(1 + xk1)m1l(x′′2)l so for l 6 0 we have that cl is divisible by (1 + xk1)−m1l. Let cl(x1) =
(1 + xk1)−m1lc′−l(x1) for l < 0, c′l are also Laurent polynomials. Denote W+ =
∑
l>0 cl(x1)x
l
2
and W− =
∑
l<0 cl(x1)x
l
2 =
∑
l>0 c
′
l(x1)(x
′′
2)−l. Then obviously both W+ and W− belong to
Q
[
x±1 , x2,
1
x′′2
]
. The reverse inclusion is straightforward.
Part (2) follows from Proposition 3.1. �
8 J.A. Cruz Morales and S. Galkin
Proposition 3.2. Let exchange collection V consists of a vector v1 = e2 = (0, 1) with multi-
plicity m1 > 0 and its inverse v2 = −v1 = −e2 = (0,−1) with multiplicity m2 > 0,
1. The upper bound U(L, ω;m1 × e2,m2 × (−e2)) consists of all Laurent polynomials W of
the form W =
∑
l cl(x1)x
l
2 where cl ∈ Q[x±1 ] and for l 6 0 we have that cl is divisible by
(1 + xk1)−m1l and for l > 0 we have that cl is divisible by (1 + xk1)m2l.
2. U(L, ω;m1 × e2,m2 × (−e2)) = Q
[
x±1 , x2(1 + xk1)m2 ,
(1+xk1)
m1
x2
]
.
Proof. The first statement is a straightforward corollary of Lemmas 3.1 and 3.2(1). The proof
of the second statement is similar to the end of the proof of Lemma 3.2(2): separate the Laurent
polynomial W into positive and negative parts W+ and W−; then both parts lie in the ring
Q
[
x±1 , x2(1 + xk1)m2 ,
(1+xk1)
m1
x2
]
. �
Proposition 3.3. Assume a seed Σ consists of a vector v1 = (0, 1) with multiplicity m1 and
its inverse −v1 = (0,−1) with multiplicity m2. Then its mutation Σ′ = µ1(Σ) consists of v1
and −v1 with respective multiplicities m1 − 1 and m2 + 1. Then U(Σ) = U(Σ′).
Proof. By Proposition 3.2 the upper bounds are expressed as: U(Σ) = Q
[
x±1 , x2(1 + xk1)m2 ,
(1+xk1)
m1
x2
]
, U(Σ′) = Q
[
x′±1 ,
(1+x′k1 )m1−1
x′2
, x′2(1 + x′k1 )m2+1
]
. Since x′1 = x1 and x′2 = x2
1+xk1
we have
the desired equality of the upper bounds. �
Proposition 3.4. Assume that the seed Σ = (L, ω;m1× v1,m2× v2) consists of vectors v1 with
multiplicity m1 > 0 and v2 with multiplicity m2 > 0.
1. If v1 = e1 and v2 = e2, then the upper bound U(Σ) equals Q
[
x1, x2,
(1+xk2)
m1
x1
,
(1+xk1)
m2
x2
]
.
2. If v1 = ae1 + be2 and v2 = ce1 + de2 with ad − bc = 1 then the upper bound U(Σ) equals
Q
[
z1, z2,
(1+zk2 )
m1
z1
,
(1+zk1 )
m2
z2
]
with z1 = xd1x
−c
2 and z2 = x−b1 xa2.
Proof. For the first case, by Lemmas 3.1 and 3.2 we have U(Σ) = Q
[
x±1 , x2,
(1+xk1)
m2
x2
]
∩
Q
[
x±2 , x1,
(1+xk2)
m1
x1
]
. If m1 = m2 = 1, by Proposition 4.3 of [3] (with |b12| = |b21| = b = c = k
and q1 = q2 = r1 = r2 = 1) this intersection equals Q
[
x1, x2,
1+xk2
x1
,
1+xk1
x2
]
. Lemma 3.2 covers
cases with m1 = 0 or m2 = 0. If m1 and m2 are greater than 1, the proof of Proposition 4.3 in [3]
can be easily modified to include the case we need since x2
1+xk1
x2
(1 + xk1)m2−1 = (1 + xk1)m2 and
x1
1+xk2
x1
(1 + xk2)m1−1 = (1 + xk2)m1 , then the intersection equals Q
[
x1, x2,
(1+xk2)
m1
x1
,
(1+xk1)
m2
x2
]
.
Part (2) follows from Proposition 3.1. �
Lemma 3.3. Let Σ = (L, ω; 1×v1,m2×v2) be a seed of two non-collinear vectors v1 and v2 with
m(v1) = 1 and m(v2) = m2 > 0 and Σ′ = Σ1 = (L′ = L, ω′ = ω; v′1 = −v1,m2 × (v′2 = µv1v2))
be the mutation of the seed Σ in v1. Then U(Σ) = µ∗v1U(Σ′).7
Proof. First of all note that, ω′(v′1, v
′
2) = −ω(v1, v2) and since µ−v1µv1 = Rv1 it is sufficient
to consider only the case ω(v1, v2) > 0. We first consider the case when v1 = e1 = (1, 0) and
v2 = e2 = (0, 1); denote k = ω(e1, e2) > 0. Let e′1, e
′
2 be the base of L′ that corresponds
to e1, e2 under the natural identification of L′ ' L; finally consider a base e′′1, e′′2 of L′ given by
e′′1 = v′2 = ke′1 + e′2, e
′′
2 = v′1 = −e′1. Let f ′1, f
′
2 and f ′′1 , f ′′2 be the respective dual bases of L′∗.
Thus we have one natural regular system of coordinates x1 = Xf1 , x2 = Xf2 on the torus T =
7Denote Σ2 = {(µv2)m2v1,−v2} – m2-multiple mutation of Σ in v2, and Σ′2 = {(µv′2)m2v′1,−v′2} – m2-multiple
mutation of Σ′ in v′2. We are going to prove that U(Σ) = Q[y]∩Q[y′ = y1]∩Q[y2] equals to U(Σ′) = Q[y]∩Q[y′ =
y1] ∩Q[y′2].
Upper Bounds for Mutations of Potentials 9
Spec Z[L∗], and two regular systems of coordinates x′1 = Xf ′1 , x′2 = Xf ′2 ; x′′1 = Xf ′′1 , x′′2 = Xf ′′2
on the torus T ′ = Spec Z[L′∗]. Taking a = k, b = 1, c = −1 and d = 0 in Proposition 3.1 we
have that x′′1 = x′2 and x′′2 =
x′k2
x′1
, since x′2 = x2 and x′1 =
x1xk2
1+xk2
(they are the mutations of x1
and x2 with respect to v1), thus what we need to show is that the rings Q
[
x1, x2,
1+xk2
x1
,
1+xk1
x2
]
and
Q
[
x1, x2,
1+xk2
x1
,
xk1+(1+xk2)
k
xk1x2
]
are equal. We will first show that
xk1+(1+xk2)
k
xk1x2
∈ Q
[
x1, x2,
1+xk2
x1
,
1+xk1
x2
]
.
We have that
xk1+(1+xk2)
k
xk1x2
=
(1+xk1
x2
)( (1+xk2)k
xk1
)
−
k∑
j=1
k!
j!(k−j)!x
kj−1
2 . Clearly the expression in the right
side belongs toQ
[
x1, x2,
1+xk2
x1
,
1+xk1
x2
]
. Now, we will show that
1+xk1
x2
∈ Q
[
x1, x2,
1+xk2
x1
,
xk1+(1+xk2)
k
xk1x2
]
.
We have that
1+xk1
x2
= xk1
xk1+(1+xk2)
k
xk1x2
−
k∑
j=1
k!
j(k−j)!x
kj−1
2 . Again, clearly the expression in the
right side belongs to Q
[
x1, x2,
1+xk2
x1
,
xk1+(1+xk2)
k
xk1x2
]
. Thus, we have the equality between the
rings. Similarly, if the multiplicity of v2 is m2 > 1, we have that Q
[
x1, x2,
(1+xk2)
x1
,
(1+xk1)
m2
x2
]
=
Q
[
x1, x2,
1+xk2
x1
,
(xk1+(1+xk2)
k)m2
x
m2k
1 x2
]
.8 If v1 and v2 are another basis of Z2 the result follows from
Proposition 3.1. In case v1 and v2 is a pair of non-collinear vectors which are not a basis for Z2,
consider the sublattice L′ ⊂ L generated by e′1 = v1 and e′2 = v2 with the form ω′ = ω|L′ .
As we just saw upper bounds with respect to the sublattice coincide: U(L′, ω′; v1,m2 × v2) =
µ∗v1U(L′, ω′;−v1,m2 × µv1v2). Now the statement follows from the Proposition 2.3. �
Remark 3.1. If a mutation of a Laurent polynomial with integer coefficients happened to be
a Laurent polynomial, then its coefficients are also integer. Let u ∈ L be a primitive vector
and W,W ′ ∈ Q[L∗] be a pair of Laurent polynomials with arbitrary coefficients such that
W = µ∗uW
′. Then W ∈ Z[L∗] ⇐⇒ W ′ ∈ Z[L∗].
Proof. Choose coordinates on L so that u = e2. Assume W has integer coefficients. By
Lemma 3.2(1) W =
∑
cl(x1)x
l
2 and W ′ =
∑
l∈Z c
′
l(x1)x
′l
2 with c′l = cl(1 + xk1)−l for all l ∈ Z.
Clearly W ∈ Z[L∗] ⇐⇒ all coefficients of W are integer ⇐⇒ for all l ∈ Z all coefficients
of cl are integer. Since (1 + xk1) ∈ Q[x±1 ] it is clear that c′l ∈ Q[x±1 ]. Recall that for a Laurent
polynomial P ∈ Q[x] its Gauss’s content C(P ) ∈ Q is defined as the greatest common divisor
of all its coefficients: if P =
∑
aix
i then C = gcd(ai). Clearly C(P ) ∈ Z ⇐⇒ P ∈ Z[x±].
Gauss’s lemma says that C(P ·P ′) = C(P ) ·C(P ′). Since C(1 + xk1) = gcd(1, 1) = 1 we see that
C(c′l) = C(cl) · 1−l = C(cl), hence c′l ∈ Z[x±1 ] ⇐⇒ cl ∈ Z[x±1 ]. �
4 Questions and future developments
In the introduction was pointed out that our definition of upper bounds makes plausible to
consider a quantum version of mutations of potentials and the corresponding quantum Laurent
phenomenon. On the other hand, in [10] a non-commutative version of the Laurent phenomenon
is discussed. Thus, we would like to ask:
Question 4.1. Is it possible to consider a non-commutative version of the Laurent phenomenon
for mutation of potentials and develop a theory of upper bounds in this context?
In [9] the following problem (Problem 44) was proposed
Question 4.2. Construct a fiberwise-compact canonical mirror of a Fano variety as a gluing of
open charts given by (all) different toric degenerations.
8The argument for showing the equality of these two rings is the same of that when m2 = 1, but the compu-
tations are slightly longer, so we omit them.
10 J.A. Cruz Morales and S. Galkin
Conjecture 4.1 (which will be proved in [5]) gives a partial answer for the above question.
Conjecture 4.1. For the 10 potentials W (i.e., (W1,W2, . . . ,W9,WQ)) listed in [9] (or rather the
exchange collections V (resp. V1, . . . , V9, VQ)) the upper bound U(V ) is the algebra of polynomials
in one variable. Moreover, this variable is W .
Conjecture 4.1 is useful for symplectic geometry as long as one knows two (non-trivial) prop-
erties of the FOOO’s potentials m0 [8] (here W = m0):
1. W is a Laurent polynomial (this is some kind of convergence/finiteness property).
2. W is transformed according to Auroux’s wall-crossing formula [2], and more specifically by
the mutations described in Section 2. The directions of the mutations/walls are encoded
by an exchange collection V .
What we believe is that once one knows these assumptions, one should be able to prove that
some disc-counting potential equals some particularly written W (formally) without any actual
disc counting. Needless to say this is a speculative idea.
A Review of the classical cluster algebras, upper bounds
and Laurent phenomenon
In this appendix we review some results of the first section of [3]: approach to Laurent phe-
nomenon via upper bounds by Berenstein, Fomin and Zelevinsky, and make a brief comparison
between their theory and the one presented here. We will denote the framework of cluster
algebras developed by Berenstein, Fomin and Zelevinsky in [3] by BFZ.
A.1 Definitions of exchange matrix, coefficients, cluster and seed
Fix n-dimensional lattice L ' Zn. The underlying combinatorial gadget in the theory of cluster
algebras is a n× n matrix.
Definition A.1 (exchange matrix B). An exchange matrix is a sign-skew-symmetric n × n
integer matrix B = (bij): for any i and j, either bij = bji = 0 or bijbji < 0.
Obviously a skew-symmetric matrix is sign-skew-symmetric, and for simplicity we assume
further that B is skew-symmetric.
Any matrix B can be considered as an element of L∗ ⊗ L∗. Skew-symmetric matrices are
then identified with ∧2(L).
Let P be the coefficient group – an Abelian group without torsion written multiplicatively.
Fix an ambient field F of rational functions on n independent variables with coefficients in (the
field of fractions of) the integer group ring ZP.
Definition A.2 (coefficients). A coefficient tuple p is an n-tuple of pairs (p+i , p
−
i ) ∈ P2.
Finally the non-combinatorial object of the theory is a cluster.
Definition A.3 (BFZ-cluster). A cluster x = (x1, . . . , xn) is a transcendence basis of F over
the field of fractions of ZP. Let ZP[x±1] denote the ring of Laurent polynomials of x1, . . . , xn
with coefficients in ZP.
Definition A.4 (BFZ-seed). A seed (or BFS-seed) is a triple (x,p, B) of a cluster, coefficients
tuple and exchange matrix.
Remark A.1 (action of the symmetric group Sn). As noticed in [3] the symmetric group Sn
naturally acts on exchange matrices, coefficients, clusters, and hence seeds by permutating
indices i.
Upper Bounds for Mutations of Potentials 11
A.2 Mutations
For each 1 6 k 6 n we can define the mutation of exchange matrix B, of a pair (B,p) and of
a seed (B,p,x).
Definition A.5 (mutation µi of an exchange matrix B). Given an exchange matrix B = (bij)
and an index 1 6 k 6 n define µkB = B′ = (b′ij) as follows: b′ik = −bik, b′kj = −bkj , and
otherwise b′ij = bij +
|bik|bkj+bik|bkj |
2 .
It is easy to check that µk(µk(B)) = B.
Definition A.6 (mutations of coefficients). Given an exchange matrix B and a coefficients
tuple p define a mutation of the coefficients in direction k as any new n-tuple (p′+i , p
′−
i ) that
satisfies
p′+i
p′−i
= (p+k )bki
p+i
p−i
if bki > 0 and
p′+i
p′−i
= (p−k )bki
p+i
p−i
if bki 6 0.
In this definition the choice of a new n-tuple has (n− 1) degrees of freedom. This ambiguity
is not important, however one of the ways of curing this ambiguity is by considering tuples with
p− = 1. Also one can get rid of coefficients by considering the trivial tuples p+ = p− = 1.
Definition A.7 (mutations of seeds). The mutation of a seed Σ = (x,p, B) in the direction
1 6 k 6 n is a new seed Σ′ = (xk,p
′, B′) where B′ = µkB is a mutation of the exchange
matrix B in the direction k, p′ is a mutation of p using B in the direction k (Definition A.6),
and x′ is defined as follows: x′kxk = Pk(x) = p+j
∏
bik>0
xbiki + p−k
∏
bik<0
x−biki and x′i = xi for i 6= k.
The next definition is a technicality required by [3] for the proof.
Definition A.8. A seed Σ is called coprime if polynomials P1, . . . , Pn are pairwise coprime
in ZP[x].
A.3 Upper bounds and Laurent phenomenon
Definition A.9 (upper bound U(Σ)). For a BFZ-seed Σ its upper bound is the ZP-subalgebra
of F given by
U(Σ) = ZP
[
x±1
]
∩ ZP
[
x1
±1] ∩ · · · ∩ ZP
[
xn
±1].
The next theorem is a manifestation of the Laurent phenomenon in terms of upper bounds.
Theorem A.1 ([3, Theorem 1.5]). If two seeds Σ and Σ′ are related by a seed mutation and
both are coprime, then the corresponding upper bounds coincide: U(Σ) = U(Σ′).
A.4 Relations between BFZ with [9] and this paper
Given an exchange collection V = (v1, . . . , vn) one can associate a skew-symmetric n×n matrix
B(V ) = (bij):
bi,j = ω(vi, vj).
Lemma A.1. For any V and 1 6 k 6 n we have B(µkV ) = µkB(V ).
Proof. Indeed, let B(V ) = (bij) and B(µkV ) = (b′ij). Then b′ij = ω(vi+max(0, ω(vk, vi))vk, vj+
max(0, ω(vk, vj))vk) = ω(vi, vj) + max(0, ω(vk, vi))ω(vk, vj) + max(0, ω(vk, vj))ω(vi, vk) = bij +
max(0,−bik)bkj + max(0, bkj)bik = bij + a · bikbkj , where a =
sgn(bik)+sgn(bkj)
2 , i.e. 1 if both bik
and bkj are positive, −1 if they are both negative, and 0 otherwise. It is easy to check that this
coincides with Definition A.5. �
12 J.A. Cruz Morales and S. Galkin
Remark A.2. We note that in case rankL = 2 the matrixB(V ) is a very special skew-symmetric
matrix: it is non-zero only if the collection V has at least two non-collinear vectors, and in this
case its rank equals two.
Remark A.3. For an exchange collection V ∈ Ln the sublattice LV in L denotes the sublattice
generated by vi. It can be seen that LV is preserved under mutations of V , and actually can be
reconstructed from B(V ) if ω is non-degenerate on LV .
Remark A.4. Roughly the setup of BFZ corresponds to a special class of C-seeds with v1, . . . , vn
being a base of the lattice L with all multiplicities equal to 1. Thus the proof of Theorem 3.1
mostly reduces to Theorem A.1 and its proof, with extra care of keeping track of all the mul-
tiplicities and exploiting nice functorial properties with respect to the maps of the lattices and
subcollections.
Lemmas 3.1 and 3.2 are analogues and almost immediate consequences of Lemmas 4.1 and 4.2
in [3]. Propositions 3.2 and 3.3 are analogues of Case 1 in the proof of Proposition 4.3 in [3].
Proposition 3.4 is analogue of the Case 2 in the proof of Proposition 4.3 of [3]. Lemma 3.3 is
similar to Lemma 4.6 in [3].
B Definitions from [9]
Definition B.1 (U -seed). A U -seed is a quadruple (W,V, F,X) where V ∈ Ln1 is an exchange
collection, F is a fan in M , X is a toric surface associated with the fan F and W is a rational
function on X. In addition, given a U -seed we can define a curve C by the equation
C − ΣtntDt = (W ),
where ΣtntDt is the part corresponding to toric divisors.
Definition B.2 (property (U)). We say a U -seed satisfies property (U) if the following condi-
tions hold:
1) C is an effective divisor, i.e. W is a Laurent polynomial;
2) C = A + B, where A is an irreducible non-rational curve and B is supported on rational
curves;
3) the intersection of C with toric divisors has canonical coordinates −1;
4) if t ∈ V , then the intersection index (C ·Dt) > nt;
5) for a toric divisor Dt the intersection index (A ·Dt) equals the number of i such that vi = t.
In [9] the Laurent phenomenon is established in the following terms
Theorem B.1 (U -lemma). If two U -seeds Σ and Σ′ are related by a mutation then Σ satisfies
property (U) ⇐⇒ Σ′ satisfies property (U).
Acknowledgements
We want to thank Bernhard Keller for interesting discussions and in particular for suggesting
the idea of considering upper bounds in the context of [9]. We thank Denis Auroux, Arkady
Berenstein, Alexander Efimov, Alexander Goncharov and especially Alexandr Usnich for gene-
rously sharing their insights on mutations or mirror symmetry. We would like to thank Martin
Guest for reading the draft of this paper and his valuable comments. We also want to thank the
anonymous referees for their helpful comments and suggestions. First author was supported by
a Japanese Government (Monbukagakusho:MEXT) Scholarship. Also this work was supported
Upper Bounds for Mutations of Potentials 13
by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan,
Grant-in-Aid for Scientific Research (10554503) from Japan Society for Promotion of Science,
Grant of Leading Scientific Schools (N.Sh. 4713.2010.1), grants ERC GEMIS and FWF P 24572-
N25. The authors thank the IPMU for its hospitality and support during their visits in May
and November 2012 which helped to finish this work.
References
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http://dx.doi.org/10.3842/SIGMA.2012.094
http://arxiv.org/abs/1212.1785
http://arxiv.org/abs/0706.3207
http://dx.doi.org/10.1215/S0012-7094-04-12611-9
http://dx.doi.org/10.1215/S0012-7094-04-12611-9
http://arxiv.org/abs/math.RT/0305434
http://dx.doi.org/10.1016/j.aim.2004.08.003
http://arxiv.org/abs/math.QA/0404446
http://dx.doi.org/10.1007/978-0-8176-4745-2_15
http://dx.doi.org/10.1007/978-0-8176-4745-2_15
http://arxiv.org/abs/math.AG/0311245
http://dx.doi.org/10.1090/S0894-0347-01-00385-X
http://arxiv.org/abs/math.RT/0104151
http://arxiv.org/abs/1009.1648
http://arxiv.org/abs/1109.2469
http://arxiv.org/abs/math.QA/0502260
1 Introduction
2 Mutations of potentials and upper bounds
2.1 Combinatorial data
2.2 Birational transformations
2.2.1 Rank two case
2.3 Mutations of exchange collections and seeds
2.4 Upper bounds and property (V)
3 Laurent phenomenon
4 Questions and future developments
A Review of the classical cluster algebras, upper bounds and Laurent phenomenon
A.1 Definitions of exchange matrix, coefficients, cluster and seed
A.2 Mutations
A.3 Upper bounds and Laurent phenomenon
A.4 Relations between BFZ with GU and this paper
B Definitions from GU
References
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