Symmetry, Integrability and Geometry: Methods and Applications 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. J. 126 (2005), 1-52].


Introduction
The idea of mutations of potentials was introduced in [9] and the Laurent phenomenon was established 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 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 functions 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].
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].

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 code 1 .

Combinatorial data
Let (·, ·) : L * × L → Z be the canonical pairing between a pair of dual lattices L Z r and L * = Hom(L, Z) Z r .
The bilinear form ω gives rise to a map i = i ω : L → L * that sends an element v ∈ L into a linear form i ω (v) ∈ L * such that (i ω (v), v ) = ω(v, v ) for any v ∈ L. The map i ω 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 det ω, finally if ω is degenerate then both the kernel and the cokernel of the map i ω 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 morphisms Hom((L , ω ), (L, ω)) are linear maps f : Any linear map f : L → L defines an adjoint f * : L * → L * and if it respects the bilinear forms, then For a vector u ∈ L we define a symplectic reflection R u and a piecewise linear mutation µ u to be the (piecewise)linear automorphisms of the set L given by the formulae For any morphism f ∈ Hom((L , ω ), (L, ω)) and any vector u ∈ L we have R ω,f u f = f R ω ,u and µ ω, Note that R aω,bu = R ab 2 ω,u for all a, b ∈ Z and µ aω,bu = µ ab 2 ω,u for all a, b ∈ Z + . However Therefore, changing max by min and + by −, simultaneously, corresponds to changing the form ω to the opposite −ω. The exchange collections could be pushed forward by morphisms f ∈ Hom((L , ω ), (L, ω)): v 1 , . . . , v n ∈ L n will go to f v 1 , . . . , f v n ∈ L n . This gives rise to a natural diagonal action of Aut(L, ω) = Sp(L, ω) on L n . This action commutes with the permuting action of S n .
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 L 1 . Similarly one can define primitive vectors in the dual lattice L * . Note that if det ω = ±1 then i ω (n) may be a non-primitive element of L * even for primitive elements n ∈ L 1 . A vector u ∈ L defines a birational transformation of K L (and its various subfields and subrings) as follows µ u,ω : X m → X m 1 + X iω(u) (u,m) .

Birational transformations
If f : T 1 → T 2 is a rational map between two tori, and u : G m → T 1 is a one-parameter subgroup of T then its image f u : G m → T 2 is not necessarily a one-parameter subgroup, but asymptotically behaves like one, this defines a tropicalization map T (F ) : Hom(G m , T 1 ) → Hom(G m , T 2 ). The tropicalization of the birational map µ u,ω : T 1 → T 2 is the piecewise-linear map µ u,ω : L 1 → L 2 defined in the previous subsection.
One can easily see most of the relations of the previous subsection on the birational level.
u,ω for any a, b ∈ Z, and µ au,ω = (µ u,aω ) a for any a ∈ Z, however neither of them is a power of µ u,ω . 2 In particular, For any morphism f ∈ Hom((L , ω ), (L, ω)) and a vector u ∈ L we have a homomorphism f * : Then the mutation µ u commutes with the action of the group G and with the projections: πµ u,T = µ u,T π and gµ u,T = µ u,T g for any g ∈ G.

Rank two case
Let us see the mutations explicitly in case rank L = 2. Let e 1 , e 2 be a base of L and f 1 , f 2 be the dual base of L * , so (e i , f j ) = δ i,j . Also let x i = X f i be the respective monomials in Z[L * ]. For the skew-symmetric bilinear form ω k defined by ω k (e 1 , e 2 ) = k and a vector u in particular the inverse map to µ u,ω 1 is given by . Conjugation by this automorphism acts on the set of mutations: 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 (x 1 , x 2 ) and (x 1 , x 2 = x 1 x 2 ) the mutation µ (0,1) is given by the same formula. Also every mutation commutes with 1dimensional subtorus of T , in case of u = (0, 1) the action of the subtorus is given by (x 1 , x 2 ) → (x 1 , αx 2 ).

Mutations of exchange collections and seeds
Let L be a lattice equipped with a bilinear skew-symmetric form ω. A cluster y ∈ K m L is a collection y = (y 1 , . . . , y m ) of m rational functions y i ∈ K L . We call y a base cluster if y = (y 1 , , . . . , y r ) is a base of the ambient field K L . A C-seed (supported on (L, ω)) is a pair (y, V ) of a cluster y ∈ K m L and an exchange collection V = (v 1 , . . . , v n ) ∈ L n . A V -seed (supported on (L, ω)) is a pair (W, V ) of a rational function W ∈ K L and an exchange collection The mutation of a C-seed (y, V ) in the direction 1 j n is a new C-seed (y j , V j ) where V j = µ j V is a mutation of the exchange collection, and y j = µ v j ,ω y where each variable is transformed by the birational transformation µ v j ,ω .
The identity µ −u µ u = R u implies that µ j (µ j (V )) and V are related by the Sp(L, ω)-transformation R u .

Upper bounds and property (V )
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 K L given by 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 ).

Laurent phenomenon
In what follows we restrict ourselves to the case rank L = 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. 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 e 1 and e 2 , then the dual lattice L * = Hom(L, Z) has the dual base f 1 , f 2 determined by (f i , e j ) = δ i,j . The form ω is uniquely determined by its value k = ω(e 1 , e 2 ), and further we denote this isomorphism class of forms by ω k . We assume that k = 0, i.e. the form ω is non-degenerate 6 , by swapping e 1 and e 2 one can exchange k to −k. A base e i of L corresponds to a base 3. In particular, if for a vector v ∈ L we define V v = m V (v) × v to be an exchange collection that consists of a single vector v with multiplicity m V (v) and Σ v = (L, ω; V v ) be the respective seed, then 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.

4.
Let V consist of a vector v 1 with multiplicity m + 1, a vector v 2 = −v 1 with multiplicity m − 0, and vectors v k (k 3) that are non-collinear to v 1 with some multiplicities 5. Let V = µ 1 V be an exchange collection obtained by mutation of V in v 1 ; it consists of vector −v 1 with multiplicity m − + 1 1, vector v 1 with multiplicity m + − 1 0 and vectors v k = µ v 1 v k (k 3) with multiplicities m k . Similarly to the previous step define 6. Hence, to proof Theorem 3.1 it is necessary and sufficient to show that We will prove these equalities in Proposition 3.3 and Lemma 3.3.
Since map m * is invertible by Proposition 2.2 it gives the equality Lemma 3.2. Assume a seed Σ = (L, ω; m 1 × v 1 ) consists of a unique vector v 1 with multiplicity m 1 1.

1.
If v 1 = e 2 = (0, 1) then the upper bound U(Σ) consists of all Laurent polynomials W of the form W = l c l (x 1 )x l 2 where c l ∈ Q[x ± 1 ] and for l 0 we have that c l is divisible by Proof . Recall that mutation in the direction e 2 is given by we have a Laurent polynomial W = l∈Z c l (x 1 )x l 2 . Then W can be expressed in terms of x 1 and x 2 as W = l c l (x 1 )(1 + x k 1 ) l (x 2 ) l . This function is a Laurent polynomial in terms of (x 1 , x 2 ) ⇐⇒ c l (x 1 )(1 + x k 1 ) l is a Laurent polynomial of x 1 for all l. This is equivalent to c l being divisible by (1 + 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 Proof . By Proposition 3.2 the upper bounds are expressed as: we have the desired equality of the upper bounds.

2.
If v 1 = ae 1 + be 2 and v 2 = ce 1 + de 2 with ad − bc = 1 then the upper bound U(Σ) equals Proof . For the first case, by Lemmas 3.1 and 3.2 we have U(Σ) = Q x ± 1 , x 2 , can be easily modified to include the case we need since . Part (2) follows from Proposition 3.1.
Proof . First of all note that, Spec Z[L * ], and two regular systems of coordinates , since x 2 = x 2 and x 1 = (they are the mutations of x 1 and x 2 with respect to v 1 ), thus what we need to show is that the rings Q x 1 , x 2 , are equal. We will first show that We have that . Clearly the expression in the right side belongs to Q x 1 , x 2 , We have that . Again, clearly the expression in the right side belongs to Q x 1 , x 2 , . Thus, we have the equality between the rings. Similarly, if the multiplicity of v 2 is m 2 > 1, we have that Q x 1 , x 2 , If v 1 and v 2 are another basis of Z 2 the result follows from Proposition 3.1. In case v 1 and v 2 is a pair of non-collinear vectors which are not a basis for Z 2 , consider the sublattice L ⊂ L generated by e 1 = v 1 and e 2 = v 2 with the form ω = ω| L . As we just saw upper bounds with respect to the sublattice coincide: . Now the statement follows from the Proposition 2.3.
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 = a i x i then C = gcd(a i ). Clearly C(P ) ∈ Z ⇐⇒ P ∈ Z[x ± ]. Gauss's lemma says that C(P · P ) = C(P ) · C(P ). Since C(1 + x k 1 ) = gcd(1, 1) = 1 we see that

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: Conjecture 4.1 (which will be proved in [5]) gives a partial answer for the above question.
Conjecture 4.1 is useful for symplectic geometry as long as one knows two (non-trivial) properties of the FOOO's potentials m 0 [8] (here W = m 0 ): 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 phenomenon 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 Def initions of exchange matrix, coef f icients, cluster and seed
Fix n-dimensional lattice L Z n . 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 = (b ij ): for any i and j, either b ij = b ji = 0 or b ij b ji < 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 ) ∈ P 2 . Finally the non-combinatorial object of the theory is a cluster. 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 S n ). As noticed in [3] the symmetric group S n naturally acts on exchange matrices, coefficients, clusters, and hence seeds by permutating indices i.

A.2 Mutations
For each 1 k 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 µ i of an exchange matrix B). Given an exchange matrix B = (b ij ) and an index 1 k n define µ k B = B = (b ij ) as follows: It is easy to check that µ k (µ k (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 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 k n is a new seed Σ = (x k , p , B ) where B = µ k B 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: The next definition is a technicality required by [3] for the proof.

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 The next theorem is a manifestation of the Laurent phenomenon in terms of upper bounds.

A.4 Relations between BFZ with [9] and this paper
Given an exchange collection V = (v 1 , . . . , v n ) one can associate a skew-symmetric n × n matrix Lemma A.1. For any V and 1 k n we have B(µ k V ) = µ k B(V ).
Remark A.2. We note that in case rank L = 2 the matrix B(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 ∈ L n the sublattice L V in L denotes the sublattice generated by v i . It can be seen that L V is preserved under mutations of V , and actually can be reconstructed from B(V ) if ω is non-degenerate on L V .
Remark A.4. Roughly the setup of BFZ corresponds to a special class of C-seeds with v 1 , . . . , v n 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 multiplicities and exploiting nice functorial properties with respect to the maps of the lattices and subcollections.

B Def initions from [9]
Definition B.1 (U -seed). A U -seed is a quadruple (W, V, F, X) where V ∈ L n 1 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 where Σ t n t D t is the part corresponding to toric divisors.
Definition B.2 (property (U )). We say a U -seed satisfies property (U ) if the following conditions 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 · D t ) n t ; 5) for a toric divisor D t the intersection index (A·D t ) equals the number of i such that v i = 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 ).