Quantum deformation of planar amplitudes

In maximally supersymmetric four-dimensional gauge theories planar on-shell diagrams are closely related to the positive Grassmannian and the cell decomposition of it into the union of so called positroid cells \cite{A}. We establish that volume forms on positroids used to express scattering amplitudes can be $q$-deformed to Hochschild homology classes of corresponding quantum algebras. The planar amplitudes are represented in \cite {A} as sums of contributions of some set of positroid cells; we quantize these contributions. In classical limit our considerations allow us to obtain explicit formulas for contributions of positroid cells to scattering amplitudes.

The scattering amplitude can be described as a sum of contributions of some positroid cells. The nicest description of the amplitude can be given in terms of amplituhedron [2].
We hope that one can "quantize" results of [2] , [3] applying our results to partial flag varieties. (It was shown in [10] that partial flag varieties can be considered as cluster varieties, see also [13].) The partial flag varieties arise naturally in the framework of [1] either as flags of subspaces of dimensions 2, m, n − 2 or , in twistor formalism, as flags of subspaces of dimension 4, m.
The physical interpretation of our results is not clear. It is possible that they are related to non-commutative gauge theories in the spirit of [8], [18], [26]. Another conjecture: quantizing planar on-shell diagrams with q = e i equal to the N -th root of unity (in other words = 2π N ) we can obtain physical quantities associated with gauge theories with the gauge group U (N ). Here q is the parameter entering the definition of the quantum Grassmannian. (Recall that the planar diagrams correspond to the limit N → ∞.) Due to existence of Seiberg-Witten map [26] these two conjectures can be true simultaneously; if this is the case the U (N ) gauge theory should interact with constant B-field. ( Interaction with B-field can explain the violation of Lorentz invariance that should be present in U (N ) gauge theory.) The second conjecture is supported by the remark that a similar statement is true for Chern-Simons theory. It is well known that Chern-Simons theory with gauge group U (N ) is equivalent to topological string with string coupling constant g s = 2π k+N [31], [22]. (Here k stands for the coefficient in front of Chern-Simons action functional-the level.) From the other side the topological string for any genus can be obtained from genus zero topological string by means of quantization and the string coupling constant plays the role of Planck constant [32]. This agrees with our conjecture. ( Genus zero string theory corresponds to N = ∞ and planar diagrams.) One more reason to expect that after the quantization of Grassmannian we should get U (N ) gauge theory is based on the following remark. The scattering amplitudes of s particles in N = ∞ case are functions on some manifold. (For example, in twistor formalism one can consider them as functions on the configuration space of s points on the manifold CP 3|4 .) After quantization of this manifold with q obeying q N = 1 we should get functions depending of the points of the same manifold and additionally of some discrete variables. (This follows from the fact that for q N = 1 the quantized algebra has a large center isomorphic to the original commutative algebra.) This agrees with the picture of U (N ) gauge theory where these discrete variables can be identified with N 2 "colors" for every particle. It is a non-trivial check of our conjecture that the number of discrete variables is precisely N 2s where s is the number of particles; under some assumptions we were able to confirm this statement.
Our considerations are based on consideration of quantum cluster algebras in the sense of [6] and of quantum spaces in the sense of [9], [12]. However, we do not need the full strength of the theory of cluster algebras and our exposition does not depend on this theory. We introduce much simpler notions of poor man cluster algebra and poor man quantum space that are sufficient for our goals.
We will work with unital associative algebras over C. Prime ideal of an algebra (or of a ring) A is a two-sided ideal I = A such that for two ideals X, Y obeying XY ⊂ I at least one of these ideals is contained in I. Completely prime ideal is a two-sided ideal I = A such that for two elements x, y of the algebra xy ∈ I implies that at least one of these elements belongs to I. For commutative algebras these two notions coincide, prime ideals of the algebras of polynomials correspond to subvarieties. In general every completely prime ideal is prime, but prime ideal is not necessarily completely prime. However, in algebras we are interested in these two notions coincide.
Every commutative ring R without zero divisors (entire ring) can be embedded in the so called field of fractions F (formal expressions of the form ab −1 with some identifications and natural operations). It is easy to check that any homomorphism of R into a field has a unique extension to F . For non-commutative rings without zero divisors we can take this property as a definition. We say that a skew field F is a skew field of fractions of the ring A if A is a subring of F and every homomorphism of A into a skew field can be extended in unique way to a homomorphism of F. (Skew field is a non-commutative ring, where every non-zero element is invertible.) It is not always possible to construct the skew field of fractions, however, in many interesting situations it does exist. For example, it is sufficient to assume that A has no zero divisors and has at most polynomial growth, i.e. its Gelfand-Kirillov dimension (see [4] for definition ) is finite ( [5] Lemma A1). In particular, one can consider the fraction skew field of non-commutative torus T defined as the algebra with generators

Cluster algebras
Let us consider an associative unital algebra A with generators {A i }. We denote the set of generators {A i } by S; we use the same notation for the corresponding set of indices. Let us fix a subset F ⊂ S . We assume that the generators in F quasi-commute, i.e.
The set F is called the set of frozen variables. We say that a finite set K ⊂ S is a cluster if the generators A i ∈ K quasi-commute, K ⊃ F and all other quasi-commuting systems of generators containing F have cardinality ≤ |K|. We denote |K| (the number of elements in a cluster) by k. Algebra generated by A i ∈ K will be denoted by K.
We can consider an extensionÃ of the algebra A where all generators A i are invertible elements. Then for every cluster K = {A i 1 , ..., A i k } we can consider a subalgebra ofÃ generated by A i 1 , ..., A i k , A −1 i 1 , ..., A −1 i k (a non-commutative torus denoted by T ). Let us us suppose that the subset L of cluster K 1 contains k − 1 elements, hence one can obtain K 1 from L adding a generator A n . We assume that L ⊃ F. We will describe how one can obtain a new cluster K 2 from L adding a generator A m (in other words We will consider monomials with respect to L, i.e. expressions of the form r A α i i where r ∈ C. (We assume that the product as well as the products below run over L. We should fix the order of factors in some way, the change of the order can be absorbed into the change of the constant factor r.) We say that the transition from K 1 to K 2 is a mutation (more precisely an A-mutation) in the direction A n if A m A n is a sum of two monomials with respect to elements of L and the following conditions are satisfied a) α i , β i are non-negative integers obeying α i β i = 0 (i.e. each factor A i enters only in one of the monomials), b) For j ∈ K 1 , j = n (i.e. j ∈ L) It is easy to check that these conditions guarantee that the element A m defined by the formula (1) in terms of the elements of the cluster K 1 quasicommutes with A i , i ∈ L. If A m is a generator of A we get a new cluster K 2 . We will be writing B = V U −1 for an element A α i i ( A β i i ) −1 ∈ T 1 in the noncommutative torus . We can express conditions (2) saying that B commutes with all A i ∈ L, but does not commute with A n (i.e. B does not belong to the center of the non-commutative torus T 1 ).
If the clusters K 1 and K 2 are related by mutation then they are embedded in the same non-commutative torus: T 1 = T 2 . We will embed this torus into its skew field of fractions denoted by F. There exists a unique homomorphism φ B of the torus into The homomorphism is unique, because it is defined on generators of T 1 , it exists because the above formulas preserve the commutation relations between generators.) It can be extended in unique way to an automorphism of F denoted by the same symbol.
It is easy to check that We say that A is a poor man cluster algebra if a) Every quasi-commuting family of generators containing F belongs to a cluster; b) One can get any cluster from any other cluster by means of a sequence of mutations. Let us assume that the the defining relations of the algebra A depend on parameter q (e.g. q ij are powers of q) in such a way that for q = 1 we obtain a commutative algebra.
It is convenient to assume that as we vary q linear spaces A a := A| q=a form a continuous family. Technically it means that A is flat over C[q, q −1 ]. Under certain conditions A 1 can be identified with the algebra of polynomial homogeneous functions on a projective variety P . Then we write the quasi-commutation relations in the form A j A l = q λ jl A l A j where λ jl = −λ lj ∈ Z and the mutation should have the form The conditions imposed on the monomials in this formula will be specified later ( formula (4)). It is convenient to work in non-commutative torus generated by A i 1 , ..., A i k , A −1 i 1 , ..., A −1 i k . We will be writing M (ρ), ρ = (ρ 1 , . . . , ρ k ) ∈ Z k for the elements These elements coincide with A i j when ρ = e j is an element of the standard basis for Z k and satisfy the following relations β i e i , α n = β n = 0, α i ≥ 0, β i ≥ 0 and for every i either α i or β i vanishes.
We introduce the notation b = α − β and impose the conditions Here d(b) denotes the minimal positive value for Λ(b, ρ) where ρ runs over Z k .
In the construction of the automorphism φ B : F → F we use the same formulas with B = M (b).
Notice that in the limit q → 1 we obtain Poisson structure on the variety P and on classical limits of clusters, these Poisson structures agree.( The embedding of the algebra generated by cluster into the algebra A induces a Poisson map.) A quantum cluster algebra in the sense of [6] can be described as poor man cluster algebra where a mutation exists in the direction of any non-frozen element of the cluster. In the construction of such an algebra Berenstein and Zelevinsky start with integer-valued skew-symmetric form Λ on Z k . Then they assume that there exists a solution b j ∈ Z k of the equation for every non-frozen variable. (As earlier d(b) denotes the minimal positive value for Λ(b, ρ) where ρ runs over Z k .) This allows them to define a mutation in every non-frozen direction e j using the formula where α j stands for the positive part of b j and β j stands for its negative part. Let us consider prime ideals in A that are generated by subsets of the set S (set of generators of A). For every ideal we consider the subset of S belonging to this ideal; such subsets will be called ideal subsets (of course, an ideal subset generates the corresponding ideal, but the ideal can be generated also by smaller subset). One can hope that taking a quotient of a cluster algebra with respect to the ideal of this kind we obtain a cluster algebra.

Quantum spaces
We have introduced the notion of non-commutative torus T as an algebra generated by We say that an algebra T is a based non-commutative torus if the system of generators is fixed. An isomorphism of two non-commutative tori is called base change if it transforms generators of one torus into monomials of generators of another torus: where a ij is an invertible matrix over Z, i.e. both the matrix and its inverse have integer entries.(One can prove that all isomorphisms of non-commutative tori have this form.) Every isomorphism of non-commutative tori induces an isomorphism of their skew-fields of fractions, hence we can talk about base change in these skew fields. (Keep in mind that skew fields have also other isomorphisms. ) Let us consider a family of non-commutative based tori T i and their skew fields of fractions F i , labeled by index i. We define an A-automorphism of a skew field corresponding to a torus with generators We say that an isomorphism F i → F i ′ is an A-mutation if it can be represented as a composition of an A-automorphism and base change. The family is called (a poor man) quantum A-space if every two elements of it can be connected by a sequence of A-mutations.
Every poor man cluster algebra ( hence every quantum cluster algebra in the sense of [6]) generates a quantum A-space: tori T i correspond to clusters. The representation of mutation of clusters as a composition of an A-automorphism and base change follows immediately from (3).
We say that automorphism ρ of F i is an X-automorphism if it leaves one of the generators (say x n ) intact and multiplies other generators by a polynomial of x n or by an inverse of such a polynomial: composition of an X-automorphism with base change is called X-mutation (in the direction n). By definition, a family of based tori T i and their skew fields of fractions F i is a poor man quantum X-space if every two fields of fractions in this family are connected by a sequence of mutations.
The definitions above are downgraded versions of the definitions in [9], [12] : we singled out only the properties of quantum spaces that are necessary for our proofs. Fock and Goncharov gave a construction of quantum spaces starting with some algebraic data. They [12] where N is the least common multiple of the {d i }. We introduce a notation q k := q 1/d k ∈ Z[q 1/N , q −1/N ] An X-automorphism ν k of a fraction field F of a torus with generators x 1 , ..., x n can be defined by the formula : ( formula (60) from [12] ). The first line of this formula corresponds to ǫ ik > 0, the second line corresponds to ǫ ik < 0. The X-mutation is a composition of X-automorphism with base change.
The corresponding quantum X-space can be obtained from the initial feed by means of consecutive application of X-mutations. The same initial data can be used also to construct a quantum A-space.

Quantum Grassmannian
The quantum Grassmannian C q [G m,n ] can be defined as an algebra with generators ∆ I where I is an ordered subset of the set [n] = {1, 2, ..., n} consisting of m elements. ( Hence the set of generators S can be identified with {I ⊂ [n]||I| = m}).These generators (quantum minors) satisfy quantum Plücker relations In the limit q → 1 quantum Plücker relations give conventional Plücker relations. Hence the underlying projective variety P coincides with the classical Grassmannian in this case.
Two minors ∆ I 1 , ∆ I 2 are quasi-commuting iff I 1 = R⊔A 1 , I 2 = R⊔A 2 where A 1 , A 2 are cyclically separated (i.e. after some cyclic shift they belong to non-overlapping intervals).
This condition (called weak separation) can be represented as a requirement that either In these notations ∆ I 1 ∆ I 2 = q ǫ 12 ∆ I 2 ∆ I 1 where ǫ 12 = |C| − |B| in the first case and ǫ 12 = |D| − |E| in the second case [25].
A maximal collection of quasi-commuting minors ( a cluster) consists of m(n − k) + 1 elements. (We assume that the set F of frozen variables is empty.) Every collection of quasi-commuting minors can be extended to a maximal collection [23].
In the case when m = 2, n = 4 we have two clusters (1, 2), (2, 3), (3,4), (4, 1), (1, 3) and (1, 2), (2, 3), (3,4), (4, 1), (2,4) The mutation can be written in the form Let us consider the general case. We will use the notation Rab for a union of some set R ⊂ [n] and two-element set {a, b}. Let us assume now that a cluster contains minors with I = Rab, Rbc, Rcd, Rda, Rac where a, b, c, d are cyclically ordered elements of [n] \ R. Then we can get a new cluster replacing the minor ∆ Rac with the minor ∆ Rbd . The new cluster is related with the old one by mutation It is easy to check that the conditions (4) are satisfied. Every two clusters are connected by a sequence of mutations of this kind [23], hence C q [G m,n ] can be considered as poor man cluster algebra.
For generic q the prime ideals generated by a subset of the set of generators of quantum Grassmannian were classified in [19]. (This classification was stated there as a conjecture, the proof was given in [20].) Recall that for every ideal of this kind we define the ideal set as the subset of S consisting of generators belonging to this ideal. It was shown in [19] that for quantum Grassmannian the ideal sets are complements to positroid subsets of S = [n] m defined by Postnikov [24]. (Positroid subsets correspond to the cells of positive Grassmannians -positroid cells.)This means that the algebra obtained from C q [G m,n ] by means of factorization with respect to prime ideal generated by a subset of family of minors quantizes the algebra of functions on a positroid cell. (All these ideals are completely prime, hence the quotient algebra is an entire ring.) The positroid subsets (as well as prime ideals ) can be labeled by decorated permutations (permutations where every fixed point is decorated by an element of a two-point set). For a decorated permutation σ we denote the corresponding prime ideal by I(σ), the corresponding ideal set by {I(σ)} and the positroid S \ {I(σ)} by P os(σ) The dimension of corresponding positroid cell is denoted by l(σ). It is natural to assume that both the the quotient algebra C q [G m,n ]/I(σ) and the subalgebra of quantum Grassmannian generated by P os(σ) are cluster algebras with the size of cluster equal to l(σ) + 1. The first of these statements was not proved yet. The second one is proved under assumption that the set of frozen variables is defined as Grassmann necklace corresponding to the permutation σ [23].(One can prove also that there exists a family of minors Inn(σ) ⊂ P os(σ) generating a cluster algebra with the same clusters and empty F [15].)

Hochschild homology and cohomology
In the section we give some basic definitions and formulate some results about Hochschild homology and cohomology of non-commutative tori. The proofs are relegated to Appendix.
The standard complex used for definition of Hochschild homology HH * (A) (see e.g. [21]) is a direct sum of C n = A ⊗ A ⊗n , n ≥ 0. It is common practice to denote an element a 0 ⊗ a 1 ⊗ · · · ⊗ a n ∈ C n by a 0 |a 1 | · · · |a n . The differential is defined by the formula ∂a 0 |a 1 | · · · |a n = a 0 a 1 | · · · |a n + n−1 i=1 (−1) i a 0 |a 1 | · · · |a i a i+1 | · · · |a n + (−1) n a n a 0 |a 1 | · · · |a n−1 Similarly one can define Hochschild homology with coefficients in bimodule M (in this case the differential acts on the direct sum of M ⊗ A ⊗n , n ≥ 0).
The Hochschild cohomology of the algebra A with coefficients in bimodule M can be defined by means of a complex represented as a direct sum of C n (A, M ) where elements of C n (A, M ) (cochains) are n-linear functions on A with values in M. The differential d on the chain D is defined by the formula (dD)(a 0 , . . . , a n ) = a 0 D(a 1 , . . . , a n )+ n−1 i=0 (−1) i+1 D(a 0 , . . . , a i a i+1 . . . , a n )+(−1) n+1 D(a 0 , . . . , a n )a n Notice that in the case when M = A the multiplication of cochains induces graded commutative multiplication of Hochschild cohomology classes [14]. Proposition 1. We assume that associative algebra A admits a skew field of fractions F There is an action of cohomology on homology See [28] for the analysis of this action that generalizes the pairing of polyvector fields and differential forms with values in differential forms. It can be described in the following way. Fix D ∈ Hom(A ⊗i , A) a Hochschild cochain. Then on the level of chains the pairing (15) up to a sign is given by the formula ( [27], formula 2.32) For smooth commutative algebras Hochschild homology classes can be identified with differential forms and Hochschild cohomology classes with polyvector fields.
Hochschild homology and cohomology of an algebra with quasi-commuting generators were calculated in [16]. One can use the considerations of [16] to construct some special elements of the Hochschild homology of non-commutative torus, i.e. of the algebra A generated by (Later in this section A stands for this algebra.) We will give a direct construction of these elements. Proposition 2. We will be writing c σ(1),...,σ(k) , σ ∈ S k for x σ(1) · · · x σ(n) −1 |x σ(1) | · · · |x σ(k) .
Here S k stands for symmetric group on k letters.The element is a Hochschild cycle. (One can say that this cycle is obtained from x σ(1) · · · x σ(n) −1 |x σ(1) | · · · |x σ(k) by means on antisymmetrization.) Its homology class is denoted by [c].
This statement can be derived from [16] (see Appendix for the derivation) or proved by direct calculation.
Notice that in the case of commutative torus the class [c] can be identified with the standard volume form ω. In other words [c] should be considered as q-deformed volume form.

Remark 3.
There is a simple way to construct more cycles. Fix a subalgebra B ⊂ A isomorphic to a quantum torus with generators x ′ 1 , . . . , x ′ n ′ . Then the image of the antisym- in HH * (A) will be a nontrivial homological class. Applying this construction to subalgebras generated by a subset of generators of A we get a family of cycles c R labelled by subsets R ⊂ [k] = {1, . . . , k}. It follows from the results of [16] that every class of the Hochschild homology HH * (A) contains precisely one cycle represented as a linear combination of cycles c R with coefficient from the center of the algebra A.
Hochschild cohomology and homology of A are related: Proof. See Corollary 14.

Hochschild homology of quantum spaces
We have defined quantum A-spaces and X-spaces as families of based tori T i and their skew fields of fractions F i connected by mutations.
Proposition 2 gives a special element in Hochschild homology of every torus T i and therefore an element [c i ] in the Hochschild homology of F i . Proposition 7 gives a special element [µ i ] in Hochschild cohomology. (This theorem is a "quantization" of the statements about behavior of volume elements with respect to A-and X-mutation in [1], formula (15.18)) We start with a proof of this statement for µ i .
First of all we calculate the effect of base change on the derivations D i . From the relation D i (x a 1 1 ...x an n ) = a i x a 1 1 ...x an n we obtain that the base change transforms these derivations into linear combinations of them with coefficients given by the matrix specifying the base change. To calculate the transformation of [µ] we use the fact that cup-product is graded symmetric. We see that this transformation is given by the multiplication by the determinant of the matrix of base change. This matrix has integer entries and is invertible over integers, hence the determinant of it is equal to ±1.
The next step is the consideration of the behavior of µ with respect to A-automorphism φ = φ B . Again we start with action of this automorphism on the derivations D i . To The most important consequence of these relations is the remark thatD j (x j ) = D j (x j ) for all j, henceμ(x 1 , ..., x n ) = µ(x 1 , ..., x n ). (Hereμ stands forD 1 ∪ · · · ∪D n .) Notice that all terms that we add by anti-symmetrization vanish becauseD j (x i ) = D j (x i ) = 0 for j = i. Therefore the relation [μ] = [µ] follows from Proposition 9.
The same proof works for X-automorphism ρ. We defineD j = ρD j ρ −1 and check that Cup-product with µ defines a nonzero Z(F)-linear map from one-dimensional linear space H k (F, F) over center Z(F) (Corollary 14) to H 0 (F, F). One-dimensionality and (8) imply Notice that working in terms of power series with respect to 1 − q one can represent the X-automorphism as a conjugation with quantum dilogarithm [12]. This allows us to simplify the proof in this setting (Hochschild (co)homology is invariant with respect to inner automorphisms).

Poisson and Hochschild homology
If a non-commutative algebra A q depends on parameter q and becomes commutative for q = 1 then the algebra A 1 has a structure of Poisson algebra. We are interested in the case when A 1 is an algebra of functions on a Poisson manifold (or an algebra of homogeneous functions on a projective Poisson variety).
One can define Poisson homology HP i and cohomology HP i of Poisson algebra basically "taking the limit q → 1" in the definition of Hochschild homology and cohomology ( a limit of bimodule over A q as q → 1 is a Poisson module over A 1 hence one can talk about Poisson homology and Poisson cohomology with coefficients in Poisson module).
The limit of Hochschild homology class c constructed in Sec 3 as q → 1 is closely related to the differential form constructed in [9] and used in [1]. It follows that this form is a Poisson cycle. Of course, this can be checked directly.
It follows from the results of Section 3 that in the case when A q is a non-commutative torus and q is generic the Hochschild homology HH i (A q ) is canonically isomorphic to Poisson homology of commutative torus A 1 . Similar results are valid for Hochschild and Poisson cohomology. ( Non-canonical isomorphism between Hochschild and Poisson is known in more general situations.) The results of Section 4 imply in the limit q → 1 that for two commutative tori related by mutation there exists a q-dependent family of isomorphisms of their Poisson homologies.
It was shown in [9] there there exists a canonical quantization of Poisson cluster algebras, it seems that using this result one can give an independent construction of the q-dependent family of isomorphisms we have mentioned.

Contribution of positroid cell to scattering amplitudes
It was proved in [1] that the contribution of positroid cell σ ⊂ G(k, n) to the amplitude can be described in momentum -twistor formalism as an integral (8.27) containing the delta-function ofĈZ whereĈ ∈σ,σ stands for the positroid cell in G(k − 2, n) and Z denotes a a column of n supertwistors. (A supertwistor is considered as a pair (z, η) where z is momentum-twistor variable and η is an odd variable; it was shown in [3] that η should be identified with dz.) We will give reformulation of this construction that allows us to quantize it. Moreover, this reformulation opens new ways for calculation of amplitudes.
To calculate the integral (8.27) one should solve the equationĈZ = 0 . It is sufficient to solve this equation for purely bosonic Z (the equation for the odd part has precisely the same form). If there exists a uniqueĈ obeying the equation for generic Z the solution specifies a map α of the top cell G(4, n + 4) into the orthogonal positroid cellσ ⊥ ( this top cell can be identified with the top cell of the space of 4 × n matrices). Then the amplitude is equal to α * ω ( the pullback of the standard volume form ω on the positroid ) for appropriate choice of α. If the equation has finite number of solutions (the map α is multivalued), then we can lift α to a single-valued map of some variety M intoσ ⊥ , the natural projection of M onto G(4, n + 4) is a branched covering. ( The variety M can be defined as the set of pairsĈ, Z where Z is purely bosonic obeyingĈZ = 0.) Then we can construct the scattering amplitude taking first the pullback of the volume form ω to M and then the push-forward to G(4, n + 4). (Denoting by α the projection of M ontoσ ⊥ and by π the projection onto G(4, n + 4) we represent the differential form corresponding to the scattering amplitude as π ! α * ω.) The construction of scattering amplitude presented in this form can be easily quantized by means of the theorems we have proven. Instead of the positroid cellσ ⊥ we consider the corresponding noncommutative torus T , instead of top positroid cell in the Grassmannian G(4, n + 4) we consider the noncommutative torus U corresponding to this cell. If there exists a homomorphism T → U the q-deformed scattering amplitude can be defined as an image of q-deformed volume form (of Hochschild homology class [c] constructed in Theorem 2). Moreover, it is easy to describe explicitly all homomorphisms of noncommutative tori; if the homomorphism we need does exist we get an explicit expression for a singlevalued map α in the classical limit. If such a homomorphism does not exist we should consider intermediate noncommutative variety V that models the variety M and construct homomorphisms T → V , U → V . All homomorphisms can be easily calculated. Taking the limit q → 1 we obtain an explicit expression for the contribution of the positroid cell into scattering amplitude in the situation of [1].

Appendix
There is a remarkable complex associated with the algebra whose generators satisfy y i y j = q ij y j y i [16], see also [29]. In fact the authors construct the complex in greater generality.
Introduce noncommutative algebras X * and A over C[q] := C[q ij ]/(q ij q ji − 1), 1 ≤ i, j ≤ n. X * is a quotient of a free associative algebra on generators y i , z j , e l , y −1 i , z −1 j 1 ≤ i, j, l ≤ k. The ideal is generated by the relations e j e i = −q ij e i e j , e 2 i = 0, e j y i = q ij y i e j , e j z i = q ij z i e j In positively graded algebra X * y i , z j have degree zero, e k has degree one. The differential ∂ satisfies graded Leibniz rule and ∂y i = ∂z j = 0, ∂e l = y l − z l .
Let A be a noncommutative torus . X * is A-bimodule: x i × a × x j := y i az j for any a ∈ X * . Algebra A op shares linear space with A but has the product defined by the formula a × b := ba. Ant A-bimodule M is the same thing as a left module over the algebra A e := A ⊗ A op . (a ⊗ b)m = amb. It is also a right module over the same algebra m(a ⊗ b) := bma. A is naturally an A-bimodule Proposition 11. X * is a resolution of A by free A-bimodules. The augmentation map ǫ : X * → A sends y i , z i to x i and e i to zero.
We can use this complex for computation of Hochschild homology. To this end we take the tensor product of A-bimodules X * (A) := X * ⊗ A e A and compute the cohomology of the resulting complex. Denote by Λ q the subalgebra in X * generated by e i . The space X * (A), which not an algebra any more, is spanned by a ⊗ b, a ∈ Λ q , b ∈ A. X * (A) in addition to relations that come from (9,10) has relation a ⊗ b = b ⊗ a. It comes from the tensor product of bimodules.
Note that elements e i y −1 i e i z −1 i belong to the (graded) center of X * Proposition 12. Let P ∈ X * be a nonzero polynomial in e i y −1 i The image of P ⊗ 1 defines a nonzero ∂-cycle in X * (A). It also defines a nontrivial homology class in HH * (A).
The cycles corresponding to products of e i y −1 i are described in the Proposition 2. Let us prove this Proposition.
The complex X * (A) := Hom A e (X * , A) computes Hochschild cohomology HH i (A, A).
The following two-term complex P will be useful in our computations Its only nontrivial nonzero cohomology is in degree zero and is to C. We interpret P ⊗n as a projective resolution of C over The complexes are isomorphic under the map ψ : B is the group algebra of the group Z k . We can use K * and K * for computation of homology and cohomology of Z n : It is convenient to choose η i = e i y −1 i , y i , z i as a generating set for X * . The algebra A is a projective representation of Z n : x j x i a = q ij x i x j a. Because of that it wouldn't be a surprise to learn that in adjoint action x i × a = x i ax −1 i cocycle factors q ij drop out and we have an ordinary (as opposed to projective) representation of Z n . We denote it by A ad Proposition 13. A map aη i 1 · · · η is → aξ i 1 · · · ξ is defines an isomorphism of complexes X * (A) and K * (A ad ). Fix g ∈ Hom A e (X * , A). We will be writing g i 1 ,...,is for the value of g on η i 1 · · · η is . The map g → g i 1 ,...,is ξ i 1 · · · ξ is defines an isomorphism of complexes X * (A) and K * (A ad ). The last two statements are still valid if we replace A ad by F ad .
Denote by F(x 1 , . . . , x k ) the noncommutative field of fractions of A(x 1 , . . . , x k ). Fix a ring B and an injective automorphism φ of a B. Skew Laurent series ring B((x, φ)) (see [7] for details) consists of formal series b = ∞ i=k b i x i of an indeterminate x with an integer k (which could be negative) and coefficients b i ∈ B. Addition in B((x, φ)) is defined naturally and multiplication is defined by the rule (a ∈ B). The ring B((x, id)) = B((x)) is the ordinary Laurent series ring of B.
According to [7] Theorem 2.3.1 if B is a skew field then B((x, φ)) is also a skew field. We can construct this way examples of skew fields inductively. Start with C((x 1 )) and define an automorphism φ that acts by φ(x 1 ) = q −1 12 x 1 . The skew field C((x 1 ))((x 2 , φ)) will contain subalgebra A(x 1 , x 2 ) generated by x ± 1 , x ± 2 , which satisfy x 2 x 1 = q 12 x 1 x 2 . C((x 1 ))((x 2 , φ)) will also contain the field of fractions F (x 1 , x 2 ). Suppose C((x 1 ))((x 2 , φ)) · · · ((x n−1 , φ)) is given. Extend φ by the rule:φ(x i ) = q −1 1n x i . The resulting skew field SF (x 1 , . . . , x n ) := C((x 1 ))((x 2 , φ)) · · · ((x n , φ)) contains A(x 1 , . . . , x n ) and its field of fractions F(x 1 , . . . , x n ). Proof. Let us carry the proof for the left structures. SF is a linear space over F. It has a Hamel basis indexed by some set Y . Let Z be a set of finite subsets of Y ordered by inclusion. Then SF = lim → Z F z , z ∈ Z. We already know that F is flat over A and by Lazard, Govorov theorem F = lim → Z A z , z ∈ U . From this we conclude that SF is a direct limit of free A-modules indexed by the set Z × U and SF is flat A-module. Proof. We do the proof only for F. The proof for SF is similar. Flatness in this case follows from Proposition 15. F e is A e -flat. This is true because by Lazard, Govorov theorem F = lim → F α where F α free finitely generated A-modules indexed by directed system. F op β is a similar system of modules for F op over A op . F αβ = F α ⊗ C F op β is a bi-system over A e for F e . F αβ are obviously A e -free. Thus lim → F αβ = F e is flat.
We use this to identify F e ⊗ A e A with F.

Proof of Proposition 5
Proof. By Proposition 16 F ⊗ In the following we will use freely these isomorphisms. ? in our case)(see [17] for details).
More explicitly the pairings have been described by [28] in the context of noncommutative calculus.  (17) has no solutions. Solution of this equation in SF is the same as a sequence a i (r) ∈ A, r = 1, . . . , ∞, which converge in SF to a i and such that n i=1 [x i , a i (k)] − 1 converge to zero,