Algebraic branch points at all loop orders from positive kinematics and wall crossing
PPrepared for submission to JHEP
LCTP-21-04
Algebraic branch points at all loop orders frompositive kinematics and wall crossing
Aidan Herderschee
Leinweber Center for Theoretical Physics,Randall Laboratory of Physics, Department of Physics,University of Michigan, Ann Arbor, MI 48109, USA
E-mail: [email protected]
Abstract:
There is a remarkable connection between the boundary structure of the pos-itive kinematic region and branch points of integrated amplitudes in planar N = 4 SYM.A long standing question has been precisely how algebraic branch points emerge from thispicture. We use wall crossing and scattering diagrams to systematically study the boundarystructure of the positive kinematic regions associated with MHV amplitudes. The notionof asymptotic chambers in the scattering diagram naturally explains the appearance of al-gebraic branch points. Furthermore, the scattering diagram construction also motivates anew coordinate system for kinematic space that rationalizes the relations between algebraicletters in the symbol alphabet. As a direct application, we conjecture a complete list of allalgebraic letters that could appear in the symbol alphabet of the 8-point MHV amplitude. a r X i v : . [ h e p - t h ] F e b ontents g -vector fan 102.2 Scattering diagrams and wall crossing 112.3 Asymptotic chambers and limiting walls 16 A , A , A , , Gr (4 , /T and algebraic letters 273.3 Beyond A , sub-algebras 31 A Introduction to cluster algebras 42B Differentiating x -variables with frozen nodes 45C Review: A , cluster algebra 47D Algorithm for finding asymptotic chambers from A , sub-algebra 48E Review: cluster polytopes 50 Scattering amplitudes are one of the most fundamental observables in modern high energyphysics. Significant strides have been made in both understanding the underlying structureof amplitudes and developing new computation methods. Many techniques for computing– 1 –mplitudes are under active research, such as the double copy [1–3], positive geometry [4–6], intersection theory [7–9] and much more [10–17]. In particular, a number of techniqueshave emerged that leverage non-perturbative properties of the S-matrix, such as unitarityand cluster decomposition. For example, cutting rules for integrands that hold at all orderscan be derived from unitarity [18, 19], and bounds on couplings can be derived by testingwhether the theory allows for macroscopic superluminal signal transmission [20]. Recently,significant focus in this program has been placed on studying the analytic structure ofamplitudes, which encode the causal dynamics of the underlying theory. Imposing that theamplitude has the correct analytic structure imposes many non-trivial properties, such asbounds on higher dimension operators [21–24] and Steinmann relations [25–27].In this paper, we study what singularities and branch cuts can appear in integratedMaximal Helicity Violating (MHV) amplitudes at all loop orders in N = 4 planar SYM(pSYM). Remarkably, all evidence to date suggests that the integrated MHV amplitudesonly have logarithmic branch points on the boundary of the positive kinematic region. Werestrict ourselves to MHV amplitudes due to their simplicity and plethora of computationalevidence, but similar results seem to hold for general N k MHV amplitudes [27, 30]. There-fore, by studying the boundary structure of the positive kinematic region, one can makepredictions for branch points of the amplitude at any loop-order. However, the boundarystructure of the positive kinematic region is very difficult to study due to subtleties involvedin choosing particular compactifications of the positive kinematic region. Cluster algebrasprovide a precise understanding of the boundary structure of the positive kinematic regionat 6-point and 7-point [31–33]. At 8-point and beyond though, many new features appearthat are still under active investigation and not well understood from the cluster algebraperspective. Foremost among these features is the appearance of algebraic letters in thesymbol alphabet [30, 34, 35].To approach these questions, we use scattering diagrams [36–41], a natural general-ization of the cluster algebra framework, to study different compactifications of the positivekinematic region of the 8-point MHV amplitude. We show how the boundary structure ofthe positive kinematic region can be systematically studied using scattering diagrams andfind that algebraic letters naturally emerge from the notion of asymptotic chambers in thescattering diagram. We ultimately find a list of 72 multiplicatively independent letters,of which at most 50 are algebraic, associated with the asymptotic chambers relevant forthe 8-point MHV amplitude. We argue this alphabet includes all algebraic letters thatcould possibly appear in the 8-point MHV symbol alphabet. Furthermore, we also discusshow scattering diagrams provide a new approach to studying rational letters in the symbolalphabet. N = 4 Super-Yang Mills
Amplitudes in N = 4 pSYM are an ideal testing ground for exploring the analytic struc-ture of planar scattering amplitudes. For instance, amplitudes in N = 4 pSYM have “Integrated amplitude” refers to the BDS-like normalized amplitude [28, 29]. The term “scattering diagrams” in this context has nothing to do with Feynman diagrams. – 2 – finite number of branch points associated with solutions to the Landau equations andare expected to have a finite radius of convergence in perturbation theory [42]. Significantprogress has been made in understanding the structure of N = 4 pSYM amplitudes beyondFeynman diagrams. At weak coupling, deep geometric structures, such as the amplituhe-dron, have emerged that provide both powerful computational techniques for computingintegrands at any loop order and a radically different perspective on the nature of local-ity and unitarity [43–47]. At strong coupling, holographic calculations provide non-trivialpredictions for the behavior of N = 4 pSYM amplitudes in the form of the BDS-ansatz[48–50] and its generalizations [28, 51]. Other formalisms motivated by the duality betweenWilson loops and scattering amplitudes have also emerged [52–56].MHV amplitudes in N = 4 pSYM are particularly simple and have been useful litmustests for conjectures. The MHV n -point amplitudes are transcendental functions of fixedweight at each loop order that can be expressed in terms of multi-polylogarithms (MPL) atall orders calculated to date [31, 46, 57–61]. These transcendental functions of weight W , F W , are the generalizations of logarithms that obey extremely nice properties. Primarily,the symbol provides a map from the amplitude to a sum of W -fold tensor products: F → (cid:88) F φ α ,φ α ,...,φ αW [log( φ α ) ⊗ log( φ α ) ⊗ . . . ⊗ log( φ α W )] , (1.1)where F are rational numbers. Each factor in the tensor product behaves similarly to alogarithm, leading to properties like:[ . . . ⊗ log( φ φ ) ⊗ . . . ] = [ . . . ⊗ log( φ ) ⊗ . . . ] + [ . . . ⊗ log( φ ) ⊗ . . . ] . (1.2)The φ i in eq. (1.1) are functions of external kinematic data and correspond to branchpoints of F W . The space of all φ that can appear in eq. (1.1) is called the symbol alphabetof F W .The symbol provides a very transparent understanding of the analytic structure of F W .We will focus on finding a minimal symbol alphabet, a set of multiplicatively independentletters that all letters in the original symbol alphabet factor into. For example, considerthe initial symbol alphabet { φ , φ , φ φ } . One minimal symbol alphabet is { φ , φ } as φ φ factors in φ and φ . An alternative minimal symbol alphabet is { φ , φ φ } , as φ = ( φ φ ) /φ . Given a minimal symbol alphabet, one can use eq. (1.2) to construct acomplete basis of possible tensors. Finding a minimal symbol alphabet of the 8-point MHVamplitude would be a major achievement and open up the possibility of bootstrapping the8-point MHV higher loop amplitudes. We take an important step towards this goal byproposing a minimal symbol alphabet for the algebraic letters. The Positive Region of Kinematic Space
Scattering amplitudes are functions on kinematic space. The positive kinematic regionis a region of kinematic space where planar gauge theory amplitudes are conjectured to Using the Grassmannian form of N = 4 pSYM loop integrands, one can directly show all integrals inthe MHV (and NMHV) sector can be written as iterated integrals of d log-forms [46]. Unfortunately, thisdoes not necessarily mean they integrate to a function that can be written in terms of MPL [62]. – 3 –ave no poles or branch-cuts. More precisely, in all examples studied to date, the Landauequations admit no solutions when the external data is taken to be in the positive kinematicregion [63]. The positive region for a given ordering of externals, α ∈ Perm[1 , , . . . , n ], isassociated with the region where all planar variables are positive definite: X i,j = (cid:32) j − (cid:88) a = i p α ( a ) (cid:33) > , (1.3)along with additional constraints [31, 44, 63]. Although not well understood, the positivekinematic region implicitly appears in many computations. For example, the integrandsthat appear in open superstring scattering amplitudes are generically divergent unless eval-uated in the positive kinematic region [6, 64]. Therefore, when evaluating 4-D superstringintegrands without using string field theory techniques or taking sophisticated Pochham-mer contours in the string moduli space [65], one must implicitly work in the positiveregion, only taking an analytic continuation to generic momentum configurations at theend of the calculation. Again, we emphasize that the physical significance of the positiveregion remains mysterious and its importance has been found through direct computation.Since we are studying the positive kinematic region of a massless planar gauge theoryamplitudes in 4-D, we parameterize our external kinematic data using momentum twistors[31]; Z Ai is the momentum twistor of state i and the A index transforms in the fundamentalrepresentation of the dual conformal algebra, SU (2 , Z Ai ∼ t i Z Ai . (1.4)Therefore, the kinematic space of the n -point amplitude can be interpreted as a quotientof the Grassmannian, Gr (4 , n ) /T , where T acts on columns by a re-scaling. The positivekinematic region is then a quotient of the positive Grassmannian, Gr + (4 , n ) /T , cut out bythe inequalities: 0 < (cid:104) i, j, k, l (cid:105) when i < j < k < l . (1.5)where (cid:104) . . . (cid:105) corresponds to a minor of columns “ . . . ”. We are particularly interested in the boundary structure of the positive region since theboundary is where planar gauge theory amplitudes can have singularities and branch-cuts[42, 63]. This is most easily seen at tree-level, where poles of the form X i,j → Gr + (4 , n ) /T , lets consider a simpler version of the problem by studying boundaries of Gr + (2 , /T . A naive parameterization of this space is C αi ∼ (cid:32) z z (cid:33) , (1.6) This is only true in the MHV sector. Beyond MHV, the kinematic region is most naturally interpretedas bundles over Gr (4 , n ) /T [44, 63]. – 4 – igure 1 . Different parameterizations of Gr + (2 , /T . In the z -variable parameterization, onlythree boundaries are manifest. However, the u -variable parameterization makes all five boundariesmanifest. where 0 < z < z < u -variables [4, 66, 67] u i,j = (cid:104) i, j − (cid:105)(cid:104) i − , j (cid:105)(cid:104) i, j (cid:105)(cid:104) i − , j − (cid:105) , (1.7)where (cid:104) a, b (cid:105) denotes a minor of C αi . These u i,j variables obey the non-linear relations: u , = 1 − u , u , , (Cyclic permutations) , (1.8)and the bounds, 0 < u i,j <
1, in the positive region. The second plot of the positiveregion in fig. 1 using u -variables shows that there are 5 boundaries, not 3. The underlyingproblem with eq. (1.6) is that a single set of coordinates, unless chosen very carefully, willnot manifest all possible boundaries of the positive region. In other words, to study theboundary structure of the positive region, we need to be careful in how we compactify ourkinematic space, Gr (4 , n ) /T → Gr (4 , n ) /T . Critically Positive Coordinates and Cluster Algebras
Previous research into the connection between the positive region and N = 4 pSYM am-plitudes has generally focused on the cluster algebra structure of the positive kinematicregion for N = 4 pSYM amplitudes [31–33, 68–78]. More concretely, the kinematic regionof N = 4 pSYM amplitudes corresponds to a X -type cluster algebra which associates to Note that alternate approaches have also been very successful without directly referencing the clusteralgebra structure of the positive kinematic region. The ¯ Q approach in particular has been extremely usefulin probing n ≥ A quick introduction to cluster algebras is provided in appendix A. – 5 –he positive region a set of critically positive coordinates called ˆ y -variables because theyvanish on at least one boundary of the positive region. Although each individual clus-ter parameterization makes only a sub-set of boundaries manifest, considering all clusterparameterizations together allows one to study all the possible boundaries. At 6-pointand 7-point, the symbol alphabet, the φ i in eq. (1.1), consists solely of the ˆ y -variables,implying that ˆ y -variables correspond to logarithmic branch cuts! Calculations are furthersimplified by considering a minimal multiplicative basis of ˆ y -variables instead of the spaceof ˆ y -variables themselves. The minimal multiplicative basis of ˆ y -variables can be writtenas Laurent polynomials of ˆ y -variables of some initial cluster. We denote the set of suchfunctions as O ( X ).Starting at 8-point, two problematic features appear in the cluster algebra approach: • There are an infinite number of ˆ y -variables in the cluster algebra. • Algebraic letters start to appear in the symbol alphabet.A number of approaches have appeared in the literature to tackling these problems andsignificant progress has been made.The first problem is troublesome because a key restriction for calculations at 6-pointand 7-point is that the symbol alphabet is finite. Upon finding that the cluster algebra isinfinite at 8-point, one might be tempted to assume that the symbol alphabet at 8-point isalso infinite. However, it has been proven that the n -point amplitude in N = 4 pSYM hasa finite number of branch points associated with solutions to the Landau equations [42],implying that the symbol alphabet could also be finite. Following this train of thought, anumber of truncation procedures have been proposed, motivated by connections betweenstringy canonical forms and compactifications of configuration spaces [63, 67, 85, 86].The second problem has proven a major obstacle for interpreting letters as cluster vari-ables because cluster variables are rational by construction. Multiple methods have beendeveloped to extract algebraic functions from the cluster algebra and then match thesefunctions with algebraic letters that appear in direct calculations [63, 81, 82, 87, 88]. Weuse the term cluster algebraic letters as an umbrella term for all such cluster-like variablesthat are algebraic. However, no unified picture has emerged that provides a systematicunderstanding of these cluster algebraic functions.
Scattering Diagrams and Asymptotic Chambers
In this paper, we propose wall crossing, and scattering diagrams more specifically, as auseful framework to address these issues [36–41]. Wall crossing has found applications in anumber research areas, such as the moduli spaces of N = 2 gauge theories and black holeentropy formulas [89–94]. However, we are not studying any kind of entropy formula ormoduli space, but instead compactifications of the positive kinematic region, Gr (4 , n ) /T . Notably, the initial definition of cluster algebraic functions in ref. [63] only included 2 algebraic lettersfor each limiting ray in the Gr (4 , /T g -vector fan. However, at least 18 algebraic letters seem to appearin the MHV amplitude at 8-point [34]. – 6 – igure 2 . Pictures of the scattering diagrams corresponding to Gr (2 , /T (left) and Gr (2 , /T (right). The application of wall crossing and scattering diagrams to partial compactifications of isbest understood in the context of mirror symmetry [36, 41, 95], but such a discussion isunfortunately beyond the scope of this paper. Instead, we take a more practical approach,giving the computational definition of a scattering diagram with examples and then makingthe connection to cluster algebras. We argue that scattering diagrams, which represent amore general mathematical framework than cluster algebras, are useful for studying clusteralgebraic functions that appear in the symbol alphabet of N = 4 pSYM.The scattering diagram of a rank N cluster algebra corresponds to a fan in R N , whereeach cone in the fan corresponds to a different coordinate system for X . Cones of thescattering diagram correspond to quivers of the cluster algebra. In the case of finite clusteralgebras, crossing between adjacent cones in the scattering diagram always corresponds to acluster mutation. For example, the scattering diagrams dual to Gr (2 , /T and Gr (2 , /T are provided in fig. 2. Crucially, the scattering diagram perspective motivates an alternateset of coordinates for X , denoted as ˆ y γ -variables. For a given cone/cluster, the ˆ y γ -variablescan be written as a monomial of the ˆ y -variables and vice-versa. Therefore, the ˆ y γ -variablesand ˆ y -variables have the same multiplicative basis.In the finite case, the walls corresponding to cluster mutations define a complete scat-tering diagram. In some sense, the finite scattering diagram is simply a re-writing of thecluster algebra and contains no new information. In the infinite case, where there are aninfinite number of cones, scattering diagrams are a genuine generalization of the clusteralgebra framework. In particular, infinite sequences of cones appear in the scattering di- The schematic connection between mirror symmetry and cluster algebras is as follows. We can interpret X as the blow-up of an associated toric geometry. Cluster transformations correspond to changing the blowup description by an elementary transformation. Scattering diagrams provide a framework to systematically“sew” these different parameterizations together using a fan defined by tropical points of the dual mirrormanifold, A ∨ . This framework is famous for giving a geometric interpretation of the connection betweentropical points of A ∨ and regular functions on X using mirror symmetry. – 7 – igure 3 . A schematic representation of the cone structure near the limiting ray in some 3-dimensional scattering diagram. We are looking down on the limiting ray, which corresponds to thegreen dot. agrams that asymptotically approach limiting rays, as schematically drawn in fig. 3. Weuse these infinite sequences of cones to define the notion of asymptotic chambers: conesthat are asymptotically close to the limiting ray. Although there are always an infinite number of walls as you approach the limiting ray, we argue that walls not intersecting thelimiting ray can be ignored for the purposes of calculating relations between ˆ y γ -variablesin this asymptotic limit. For example, there are 6 asymptotic chambers in fig. 3 as only3 walls intersect the limiting ray. We can calculate relations between the ˆ y γ -variables ofdistinct asymptotic chambers using the wall crossing framework.The initial motivation for asymptotic chambers actually came from N = 2 super-symmetric gauge theories. For specific N = 2 gauge theories on R × S , the moduli spacecorresponds to a X -type cluster algebra [89]. This connection between cluster algebrasand N = 2 gauge theories leads to a number of interesting results, such as a connectionbetween canonical bases of the cluster algebra and the set of simple line defects in thetheory [92]. The concept of an asymptotic chamber was first proposed in ref. [89] althoughinitial calculations were performed in Section 5.9 of ref. [90] using different terminology.Later generalizations made connections between asymptotic chambers and Fenchel-Nielsencoordinates of (higher) Teichmuller spaces [96–98]. However, to our knowledge, the notionof asymptotic chambers in the context of higher dimension scattering diagrams for general X spaces has been largely unstudied.Crucially, although the ˆ y -variables often diverge in the asymptotic limit, the ˆ y γ -variables themselves remain finite. These “asymptotic” ˆ y γ -variables correspond to thealgebraic letters that appear in the 8-point symbol alphabet! Using scattering diagramsand the notion of asymptotic chambers, we conjecture a complete multiplicative basis forall algebraic letters that could appear in the N = 4 pSYM symbol alphabet at 8-point.Remarkably, we find at most 50 multiplicatively independent algebraic letters associatedwith the asymptotic chambers. This result systematizes the techniques in ref. [87, 88],which effectively analyzed a particular sub-set of asymptotic chambers and did not studyrelations between the algebriac letters of different chambers.The scattering diagram approach also offers a new perspective on proposed truncationprocedures for ˆ y -variables. We take a similar philosophy to ref. [63, 87, 88], arguing that– 8 –he positive kinematic region is not maximally compactified so not all boundaries appear.However, in contrast to ref. [63, 87, 88], which argue for a truncation of x -variables, weinstead argue for a truncation of clusters in the cluster algebra, or equivalently cones in thescattering diagram. We further argue that such a truncation naturally leads to the notionof asymptotic chambers and algebraic critical coordinates. The paper is structured as follows: • Section 2: We first introduce the notion of scattering diagrams and wall crossing forfinite cluster algebras before defining the notion of an asymptotic chamber for infinitecluster algebras. We show that the cluster algebraic functions follow naturally fromthe notion of asymptotic chambers. The core result of this section is the conjecturedbound in eq. (2.32) that is a necessary condition for asymptotic chambers to be welldefined. • Section 3: We study asymptotic chambers in a number of examples, eventually study-ing the asymptotic chambers associated with Gr (4 , /T . The core results of thissection are eqs. (3.17), (3.18), (3.19) and (3.24), which together give an explicitalphabet for the algebraic letters at 8-point in terms of momentum twistors. We con-clude with some comments about possible obstructions to applying these techniquesto more general cluster algebras. • Section 4: We introduce the notion of degenerate scattering diagrams, motivatingtheir construction using tropicalization of ˆ y -variables in the dual cluster algebra. Wethen motivate the importance of asymptotic chambers using the degenerate scatteringdiagrams. • Section 5: We conclude the paper with a short summary and a list of future directions.A short introduction to cluster algebras is provided in appendix A.
Notation : We denote the cluster variables associated with A and X as x and ˆ y respec-tively. This notation differs from ref. [63, 87], which denote cluster variables associatedwith A and X as a and x respectively. Furthermore, we denote mutations of the k th nodeas µ k . For example, x i = µ k x i if i (cid:54) = k . Finally, we often denote cluster algebras usingthe notation A p ,p ,...,p n . This notation is based on how these cluster algebras correspondto the Teichmuller space of bordered Riemann surfaces. The cluster algebra A p ,p ,...,p n corresponds to the Teichmuller space of a Riemann surface with n borders and p i punc-tures on border i . For those unfamiliar with the connection between cluster algebra andsurfaces, this notation is unimportant for our applications to N = 4 pSYM but is nice fororganizational purposes. – 9 – Wall crossing, cluster algebras and asymptotic chambers
In this section, we develop the notion of scattering diagrams and asymptotic chambers.We begin with a short introduction to g -vectors before giving a relation between the scat-tering diagram and the g -vector fan of the cluster algebra. We then develop the notion ofasymptotic chambers, using the A , cluster algebra as our guide. g -vector fan Our goal is to study the multiplicative basis of ˆ y -variables that parameterize the positivekinematic region. Unfortunately, for generic cluster algebras, the multiplicative basis ofˆ y -variables is very difficult to study in complete generality. For example, ˆ y -variables willnot always be independent. To see the problem, consider the initial quiver, x x x so ˆ y = 1 x , ˆ y = x x , ˆ y = 1 x . (2.1)Without any frozen nodes, we trivially see that ˆ y = ˆ y . However, suppose we include thefrozen node y x x x , so ˆ y = y /x and ˆ y (cid:54) = ˆ y . From this example, it is clear that the frozen nodes playa crucial role in distinguishing ˆ y -variables. One approach to this problem is to simplyadd frozen nodes until the ˆ y -variables are maximally disambiguated [99, 100]. Only afinite, albeit large, number of frozen nodes are necessary to maximally disambiguate theˆ y -variables.However, we are not interested in the space of all ˆ y -variables, but instead finding amultiplicatively independent basis of the ˆ y -variables. Given that any ˆ y -variable can bewritten as a monomial of x -variables, we only need to maximally disambiguate x -variablesof the cluster algebra, not the ˆ y -variables. We are therefore motivated to consider a clusteralgebra with a principal quiver [99]. To construct a principal quiver, first consider an initialquiver without any frozen nodes. Then add a frozen node, y i , to each non-frozen node, x i ,with an edge pointing from the frozen node to the mutable node. For example, a principlequiver of the A cluster algebra is y y x x . – 10 –emarkably, the frozen nodes of a principal quiver are enough to maximally disambiguateall x -variables! Details of this statement are provided in appendix B. We therefore studycluster algebras with a principle quiver to study the multiplicative basis of ˆ y -variables ofcluster algebras with arbitrary frozen nodes. Furthermore, we can choose any quiver of ourcluster algebra to be the principal quiver.We now turn to the problem of understanding the relation between ˆ y -variables and x -variables for a cluster algebra with a principal quiver. Although we cannot write a directmap from ˆ y to x , attempting to do so allows us to associate a canonical vector to each x -variable. Suppose we start with the principal quiver. Any cluster variable x in the clusteralgebra can be written as a Laurent polynomial of x i and y i variables of the principlequiver. It is not generically possible to re-write this Laurent polynomial entirely in termsof ˆ y -variables. However, it can be written as a polynomial of ˆ y -variables of the principlequiver up to a monomial of x i : x = x (cid:126)g F (ˆ y i ) , x (cid:126)g = (cid:89) i x g i i , (2.2)where F (ˆ y i ) is a Laurent polynomial in ˆ y -variables of the principal quiver, which we denoteas ˆ y i . No two x -cluster variables share the same g -vector, allowing us to associate acanonical g -vector to each element of the cluster algebra. As an illustrative example, againconsider the A cluster algebra. The ˆ y i variables of the above A principal quiver are:ˆ y = y x − , ˆ y = y x . (2.3)Upon mutating x , we find x = y + x x = x x (1 + ˆ y ) → (cid:126)g = ( − , , F (ˆ y i ) = 1 + ˆ y . (2.4)Mutating through all quivers yields all F (ˆ y γ i ) polynomials and g -vectors of the A clusteralgebra, which are provided in Table 1. Each cluster defines a cone bounded by the g -vectors of the x -variables in the cluster. Remarkably, the cones associated with distinctcluster are non-overlapping, which isn’t at all obvious from the above definition. Thecollection of these cones defines a (sometimes incomplete) fan.In summary, we reduced the problem of finding a multiplicative basis of ˆ y -variablesfor a cluster algebra with generic frozen variables to finding the multiplicative basis forˆ y -variables of a cluster algebra with a principal quiver. We then used the ˆ y -variables ofthe principle quiver to find a map from x -variables to g -vectors. In this section, we introduce the notion of scattering diagrams and wall crossing, followingthe review in ref. [101]. We then show how cluster algebras fit into the wall crossingframework, using the A cluster algebra as our primary example.An arbitrary scattering diagram is defined on a lattice, R N , where vectors living in R N – 11 – i F i g i x , x , x y ( − , x y + ˆ y ˆ y ( − , x y (0 , − Table 1 . The F (ˆ y i ) polynomials and g -vectors of the A cluster algebra. are denoted as γ and basis vectors specifically as γ i . A scattering diagram requires threepieces of input data: • A collection of cones bounded by co-dimension-1 walls. Each wall in the scatteringdiagram is associated with a function, f ( y ). • N coordinates ˆ y γ i on X associated with vectors in the dual of vector space of R N . • Some skew-symmetric matrix, B i,j , that defines an skew-symmetric product: (cid:104) γ i , γ j (cid:105) = γ i · B · γ j (2.5)Each cone in the fan is associated with a particular parameterization of X , denoted ˆ y γ i ,similar to how the ˆ y -variables of a cluster correspond to a particular parameterization of X .We can interpret crossing a wall between two cones as performing a coordinate transforma-tion that takes a particuraly simple form. Each co-dimension one wall is associated with avector, γ ⊥ , perpendicular to the wall. Naively, the wall only defines γ ⊥ up to a constant.The sign of γ ⊥ is chosen so γ ⊥ points opposite the direction one is mutating across thewall. Furthermore, the magnitude of γ ⊥ is chosen so that all of its components are integerswhose least common denominator is 1. Finally, we associate a unique monomial, ˆ y γ ⊥ , toeach γ ⊥ : γ ⊥ = a i γ i , ˆ y γ ⊥ = ( (cid:89) ˆ y a i γ i ) Sign( γ ⊥ · (cid:126)N ) ,(cid:126)N = (1 , , . . . , . (2.6)For example, for a wall with the perpendicular vector γ ⊥ = (0 , , y (0 , , = ˆ y γ ˆ y γ . (2.7) For example, if N = 3, then γ = (1 , , γ = (0 , ,
0) and γ = (0 , , We use the notation ˆ y γ i , instead of ˆ y i , to distinguish them from ˆ y -variables. – 12 – + ˆ y γ y γ y γ y γ C C C C (a) Inconsistent scattering diagram. y γ ˆ y γ y γ y γ y γ y γ C C C C C (b) Self-consistent scattering diagram. Figure 4 . Two examples of scattering diagrams. The scattering diagram on the left is inconsistentas it is not path independent, as shown in eq. (2.12). The scattering diagram on the right is pathindependent and associated with the A cluster algebra. Due to the Sign( γ ⊥ · (cid:126)N ) exponent, the perpendicular vector γ ⊥ = (0 , − , −
1) is associatedwith the same monomial: ˆ y (0 , − , − = ˆ y γ ˆ y γ . (2.8)This makes sense as (0 , − , −
1) and (0 , ,
1) correspond to the same wall and shouldtherefore be associated with the same monomial. Although γ ⊥ flips sign depending onthe direction you are mutating across the wall, ˆ y γ ⊥ is the same due to the Sign( γ ⊥ · (cid:126)N )exponent. The mutation relation for ˆ y γ i across a wall is then defined asˆ y γ i → ˆ y γ i f (ˆ y γ ⊥ ) (cid:104) γ i ,γ ⊥ (cid:105) , (2.9)which gives ˆ y γ i of the new cone in terms of ˆ y γ i of the initial cone. To see eq. (2.9) in anexplicit example, suppose we are crossing from cone C to cone C in fig. 4a, where we fix: B i,j = (cid:34) − (cid:35) , (2.10)and f (ˆ y γ ⊥ ) = 1 + ˆ y γ ⊥ for all walls. The perpendicular vector for the relevant wall is γ ⊥ = (0 , y γ ⊥ = ˆ y γ . Applying eq. (2.9), the ˆ y γ i of chamber C are then:ˆ y γ = ˆ y I γ (1 + ˆ y I γ ) (1 , · B · (0 , = ˆ y I γ (1 + ˆ y I γ ) , ˆ y γ = ˆ y I γ (1 + ˆ y I γ ) (0 , · B · (0 , = ˆ y I γ , (2.11)where ˆ y I γ i corresponds to ˆ y γ i of chamber C .For a scattering diagram to be self-consistent, the relations between ˆ y γ i of two distinctcones should be path independent. To see why self-consistency is non-trivial, again considerthe scattering diagram in fig. 4a with the same f (ˆ y γ ⊥ ) and B i,j . Applying eq. (2.9) to– 13 –ach path in fig. 4a, we find (cid:32) ˆ y I γ ˆ y I γ (cid:33) → ˆ y I γ ˆ y I γ ˆ y I γ +1 → ˆ y I γ (cid:18) ˆ y I γ ˆ y I γ +1 + 1 (cid:19) ˆ y I γ ˆ y I γ +1 , (cid:32) ˆ y I γ ˆ y I γ (cid:33) → (cid:32) ˆ y I γ (cid:0) ˆ y I γ + 1 (cid:1) ˆ y I γ (cid:33) → ˆ y I γ (cid:0) ˆ y I γ + 1 (cid:1) ˆ y I γ ˆ y I γ ( ˆ y I γ +1 ) +1 , (2.12)where ˆ y I γ i again corresponds to the ˆ y γ i of the initial cone, C . The scattering diagram infig. 4a is inconsistent as the ˆ y γ i associated with C are not path independent. To make thescattering diagram self-consistent, we must include the additional wall γ ⊥ = (1 , (cid:32) ˆ y I γ ˆ y I γ (cid:33) → ˆ y I γ ˆ y I γ ˆ y I γ +1 → ˆ y I γ (cid:18) ˆ y I γ ˆ y I γ ˆ y I γ +1 + 1 (cid:19) ˆ y I γ ˆ y I γ ( ˆ y I γ +1 ) +1 → ˆ y I γ (cid:0) ˆ y I γ + 1 (cid:1) ˆ y I γ ˆ y I γ ( ˆ y I γ +1 ) +1 , (2.13)which now matches the second line of eq. (2.12).We now describe the connection between cluster algebras and scattering diagrams.The relation between scattering diagrams and cluster algebra is that the g -vector fan ofthe cluster algebras defines a canonical scattering diagram where each quiver is dual toa cone in the scattering diagram. The B i,j that defines the skew-symmetric product ineq. (2.5) corresponds to the exchange matrix of the principal quiver. For a finite clusteralgebra number, each wall corresponds to a cluster mutation and we fix f (ˆ y γ ⊥ ) = 1 + ˆ y γ ⊥ , (2.14)for all walls. We call walls that correspond to a cluster mutation, cluster walls . Theˆ y -variables of a given cone are the ˆ y γ ⊥ associated with each wall that bound the cone:ˆ y j = (cid:89) ˆ y a ji γ i , γ ⊥ j = a ji γ i , (2.15)where γ ⊥ i is the γ ⊥ associated with ˆ y i . Note that γ ⊥ j points inward from the cone in thisconvention. Furthermore, the exchange matrix associated with a cone is, B i,j = (cid:104) γ ⊥ i , γ ⊥ j (cid:105) . (2.16)For example, for the cone associated with the principle quiver, the principle cone, we find γ ⊥ i | Principle Cone = γ i (2.17)– 14 –o ˆ y γ i | Principle Cone = ˆ y i , B i,j | Principle Cone = γ i · B · γ j = B i,j . (2.18)The cluster mutation in eq. (A.11) corresponds to both a wall crossing transform, eq. (2.9),and a mutation in the γ ⊥ i . To see this, again consider the cluster algebra associated withthe quiver y y x x . The explicit computation of the g -vector lattice in fig. 1 reveals that it is the same as fig.4. Consider a mutation from cone C to C . The ˆ y γ i mutation is given in eq. (2.11) andthe γ ⊥ i mutate as: γ ⊥ = (1 , → γ ⊥ = (1 , ,γ ⊥ = (0 , → γ ⊥ = (0 , − . (2.19)Combining eqs. (2.11) and (2.19), the mutation relation for ˆ y i is (cid:32) ˆ y ˆ y (cid:33) → (cid:32) ˆ y (1 + ˆ y ) y (cid:33) , (2.20)which exactly matches the mutation relation for ˆ y -variables. Again, note that it was acombination of mutating γ ⊥ i and ˆ y γ i that gave the cluster mutation relation for the ˆ y -variables.We can also consider the scattering diagrams of more complex cluster algebras, suchas the A cluster algebra: y y y x x x , which is associated with Gr (2 , /T . Since the cluster algebra is rank 3, the associatedscattering diagram is 3 dimensional. From direct calculation, we find the scattering diagramgiven in fig. 2 in the Introduction. The walls are now 2-dimensional and defined by thespan of two g -vectors. To find the wall associated with the cluster variables x of a givencone/quiver, consider the span of all g -vectors bounding the cone except the g -vectorassociated with x .In summary, scattering diagrams are very useful framework that provide a nice wayto study canonical coordinate transformations for X . Finding self-consistent scatteringdiagrams is naively quite hard since you need to check that relations between the ˆ y γ i ofany two cones are path independent. The g -vector fans of finite cluster algebras providea class of self-consistent scattering diagrams where cluster mutations correspond to a veryspecific type of wall crossing. – 15 – − x − x x − . . . x x x . . . x − x − x − Figure 5 . g -vector fan associated with the A , cluster algebra. There are an infinite number ofcluster variables whose g -vectors approach a limiting ray, (cid:126)g lim = ( − , g -vectors is provided in eq. (2.23). We now turn to infinite cluster algebras. We will show how the scattering diagram frame-work provides a systematic way to study the multiplicative basis of ˆ y i even when the ˆ y i themselves go to infinity. Although γ ⊥ i → ∞ in certain limits, so ˆ y i → {∞ , } , the ˆ y γ -variables remain finite.We first show that the g -vector fans of infinite cluster algebras need to include addi-tional walls that do not correspond to cluster mutations. Furthermore, we will find thefunctions attached to these walls are not elements of the cluster algebra and can be iden-tified with the mysterious cluster algebraic functions of ref. [63]. We will study the clusteralgebra defined by the principal quiver y − y x − x , as our motivating example. A review of relevant derivations and formulas for this clusteralgebra are provided in appendix C. The key results are a closed form solution for any x n : x n = 12 n +2 [( x − + B + (cid:112) (cid:52) )( P + (cid:112) (cid:52) ) n +1 + ( x − − B + (cid:112) (cid:52) )( P − (cid:112) (cid:52) ) n +1 ] ,B + = 2 x − x − P(cid:52) , (cid:52) = P − F , F = y − y , (2.21)and an equation for ˆ y n in terms of x -variables,ˆ y n − = y n y n +1 − x − n , ˆ y n = y − n y − n − x n − , (2.22)– 16 –hich can be used to derive a closed form expression for the g -vectors, (cid:126)g n = (cid:40) ( − n, n + 1) n ≥ − n, − n − n ≤ − . (2.23)The corresponding scattering diagram is illustrated in fig. 5. We will now show that self-consistency of the scattering diagram imposes the existence of a new wall associated withthe limiting ray that does not correspond to a standard cluster mutation.Consider cones that are asymptotically close to the limiting ray. Importantly, the ˆ y i variables go to 0 or ∞ as we approach the limiting ray, which can be seen from eqs. (2.21)and (2.22). To calculate the ˆ y γ i variables in this limit, we first express ˆ y γ i in terms ofmonomials of ˆ y -variables. From the scattering diagram in fig. 5, the γ ⊥ i associated with ˆ y i in eq. (2.22) are [ γ ⊥− ] n = (1 + 2 n, n ) , [ γ ⊥ ] n = − (2 n, n − . (2.24)where [ γ ⊥ i ] n is the perpendicular vector to the wall associated with node x i in the principalquiver after 2 n mutations. We subsequently find that γ = (1 − n )[ γ ⊥− ] n − n [ γ ⊥ ] n ,γ = 2 n [ γ ⊥− ] n + (2 n + 1)[ γ ⊥ ] n . (2.25)Combining eqs. (2.15) and (2.25) gives a formula for ˆ y γ i in the asymptotic limit, denotedas ˆ y + γ i :ˆ y + γ = lim n →∞ (ˆ y n ) − n (ˆ y n − ) − n = 16 y − ( x − + B + (cid:112) (cid:52) ) − ( P + (cid:112) (cid:52) ) − , ˆ y + γ = lim n →∞ (ˆ y n ) n +1 (ˆ y n − ) n = y x − + B + (cid:112) (cid:52) ) . (2.26)These expressions are finite and provide a multiplicative basis for ˆ y -variables asymptoticallyclose to the limiting wall from the right, as shown in fig. 6. We now repeat the samecalculation going in the opposite direction. The new limiting expressions of ˆ y γ i , denotedas ˆ y − γ i , are ˆ y − γ = ˆ y + γ (1 − P − √P − FP + √P − F ) , ˆ y − γ = ˆ y + γ (1 − P − √P − FP + √P − F ) − . (2.27)The fact that eqs. (2.26) and (2.27) are not equal indicates that the scattering diagrammust include another wall to be self-consistent. However, eq. (2.27) can be re-written intothe suggestive formˆ y + γ i = ˆ y − γ i (1 − ˆ y γ ⊥ ) − (cid:104) γ i ,γ ⊥ (cid:105) , ˆ y γ ⊥ = ˆ y + γ ˆ y + γ = ˆ y − γ ˆ y − γ , γ ⊥ = (1 , , (2.28)– 17 – − x − x x − . . . x x x . . . x − x − x − ˆ y + γ i ˆ y − γ i Figure 6 . g -vector fan associated with the A , cluster algebra. There are two paths to conesasymptotically close to the limiting ray (red), which are green and blue respectively. The greenpath leads to the ˆ y + γ i expressions while taking the blue path leads to the ˆ y − γ i expressions. which can be matched to eq. (2.9) by requiring f (ˆ y ) = (1 − ˆ y ) − . Eq. (2.28) showsthat we must include a limiting wall with γ ⊥ = (1 ,
1) for the scattering diagram to beself-consistent. The ˆ y γ ⊥ associated with the limiting wall,ˆ y γ ⊥ = ˆ y + γ ˆ y + γ , = P − √P − FP + √P − F , (2.29)takes exactly the right form for eq. (2.28) to be matched with eq. (2.9). The limiting wallcorresponds to the red line in figs. 5 and 6. Performing a mutation across this limiting wallcannot be identified with a cluster mutation like other walls in the A , scattering diagram.From the perspective of the cluster algebra, these cones are separated by an infinite numberof cluster mutations. Finally, the ˆ y γ ⊥ of the limiting wall obeys the bound 0 < ˆ y γ ⊥ < y -variables which are just positive definite.We now briefly compare our result to previous computations in the literature. Notably,one multiplicative basis of ˆ y ± γ i is the three algebraic functions identified in ref. [88] for agiven A , cluster algebra. Furthermore, due to bound 0 < ˆ y γ ⊥ <
1, the cluster algebriacfunction attached to the limiting walls seem more like the u -variables identified in ref.[66, 67], which obey similar bounds, than standard ˆ y i variables. Finally, note that the ˆ y γ ⊥ attached to the limiting wall is a ratio of the cluster algebraic functions defined in ref. [63].Moving beyond A , , we now turn to a more general discussion. We define asymptoticchambers as cones asymptotically close to the limiting ray that are separated by walls inter-secting the limiting ray. In the higher dimension case, we find that both limiting walls andcluster walls intersect the limiting ray. Furthermore, there are always cluster walls asymp-totically close to the limiting ray that do not intersect the limiting ray and become more– 18 – C C C C C Figure 7 . A schematic representation of the cone structure near the limiting ray of A , a rank 3cluster algebra. The full scattering diagram is three dimensional and we are looking down on thelimiting ray, which is indicated by the green dot. The red line corresponds to the limiting wall. Theblack lines correspond to cluster walls that intersect the limiting ray. The blue lines correspondto asymptotic walls, cluster walls that do not intersect the limiting wall and become more parallelwith a limiting wall as one approaches the limiting ray. There are 6 asymptotic chambers, eachlabeled by C i . parallel to the limiting walls as one approaches the limiting ray. These walls are asymp-totic walls . An example is sketched in fig. 7. For our definition of asymptotic chambersto be self-consistent, we must be able to ignore asymptotic walls if we are infinitesimallyclose to the limiting ray. If the ˆ y γ i associated with asymptotic chambers transformed non-trivially when crossing an asymptotic wall, the ˆ y γ i of asymptotic chambers would not bewell defined. For example, consider the asymptotic chambers C and C in fig. 7. If ˆ y γ i transformed non-trivially across the asymptotic walls, it would be ambiguous which ˆ y γ i was associated with the asymptotic chamber.To see whether asymptotic walls are relevant when infinitesimally close to the limitingray, lets consider crossing one of these asymptotic walls. By the definition of asymptoticwalls, we find that the γ ⊥ of the asymptotic wall asymptotes to: γ ⊥ → lim n →∞ n × γ ⊥ lim , (2.30)where γ ⊥ lim is the γ ⊥ of the limiting wall that the asymptotic walls approaches. Therefore,the wall crossing formula for the asymptotic wall reduces toˆ y γ i → lim n →∞ ˆ y γ i (1 + ˆ y nγ ⊥ lim ) n (cid:104) γ i ,γ ⊥ lim (cid:105) , (2.31)– 19 –hich naively diverges. However, in the previous example, we found that 0 < ˆ y γ ⊥ lim < A , . If this bound holds for general ˆ y γ ⊥ lim , then eq. (2.31) becomes trivial and asymptoticwalls can be ignored when asymptotically close to the limiting ray. We are therefore led toconjecture the bound 0 < ˆ y γ ⊥ lim < , (2.32)in X + for all asymptotic chambers, not just those adjacent to the limiting wall. Eq. (2.32)is a very remarkable bound and a key conjecture of this paper. We explicitly checked thateq. (2.32) held for all examples studied in section 3.In summary, the key insight is that the ˆ y γ i associated with asymptotic chambers arefinite and can be algebraic functions of our initial coordinates. Furthermore, these ˆ y γ i obeywall crossing formula as we mutate around the limiting ray. To find all the ˆ y γ i associatedwith a limiting ray, we simply need to find all the walls in the g -vector fan that intersectthe limiting ray and then use the wall crossing formula in eq. (2.9). The primary difficultyis finding all the walls that intersect the limiting ray. In this section, we describe an algorithm for finding the asymptotic symbol alphabet asso-ciated with a limiting ray. A brute force search algorithm for finding all the asymptoticchambers is given in appendix D. Once we have all the asymptotic chambers associated witha given limiting ray, the calculation for finding the associated symbol alphabet proceeds asfollows:1. Starting from the initial quiver, we mutate until we find a quiver with a A , sub-algebra. We define the quiver with the A , sub-algebra as the principal quiver forthe purposes of defining the scattering diagram and g -vectors2. Repeating the computation in section 2.3 for the A , sub-algebra, we calculate theˆ y γ i of some initial asymptotic chamber, denoted as ˆ y γ i , in terms of ˆ y i of the chosenprincipal quiver.3. Calculate the ˆ y γ i of all other asymptotic chambers in terms of ˆ y γ i using wall crossing.4. Find a complete multiplicative basis of the ˆ y γ i associated with asymptotic chambersin terms of ˆ y γ i .The multiplicative basis calculated in the final step will be the asymptotic symbol alphabetassociated with the limiting ray. Although each element of the multiplicative basis will bea rational function of ˆ y γ i , the ˆ y γ i will themselves often be algebraic functions of ˆ y i .We first apply the above algorithm to a number of lower rank cluster algebras. Afterstudying these lower rank examples, we move onto Gr (4 , /T , calculating the algebraicsymbol alphabet associated with the limiting ray relevant for 8-point MHV amplitudes.Remarkably, we find that there are a finite number of asymptotic chambers associatedwith the limiting ray relevant for the symbol alphabet of 8-point MHV amplitudes. Evenmore remarkably, we find that the symbol alphabet associated with each relevant limiting– 20 –ays consists of only 36 letters.We conclude this section by commenting on how we may need to modify the abovestrategy when faced with more general types of limiting rays. Specifically, the abovestrategy is focused on studying the asymptotic chambers of a limiting ray that can beapproached by repeated mutations on a A , sub-algebra. We give some brief commentson why this may not always be possible for n > We now consider the asymptotic chambers of some lower rank cluster algebras, finding anumber of interesting phenomena: • A , : Both cluster walls and limiting walls can intersect the limiting ray, leading tonon-trivial cluster algebraic functions. • A , : The scattering diagram associated with the limiting ray is not simple. A simplefan is a N -dimensional fan whose cones are each bounded by N walls. • A , , : There can be multiple limiting rays and each limiting ray is associated withits own discriminant. A , We will examine the A , cluster algebra in significant detail, so the algorithm is clear. The A , cluster algebra corresponds to the initial quiver, bz − z , where b , z − and z are x -variables and frozen variables have been suppressed. To find thelimiting ray, we perform a mutation on b , finding the new quiver x : b (cid:48) x : z − x : z , which we choose to be the principal quiver of our cluster algebra whose correspondingexchange matrix is B i,j = − − − . (3.1)Identifying the A , sub-algebra, we perform repeated mutations on nodes x and x , just asin section 2.3, to approach the limiting ray. The ˆ y γ i associated with our initial asymptotic Alternatively, a simple fan is a fan whose dual polytope is simple. – 21 –hamber can be calculated using the same techniques as section 2.3. The γ ⊥ i of the coneassociated with the 2 n -th quiver in the sequence: b (cid:48) b (cid:48) b (cid:48) b (cid:48) z − z , z z , z z , z z , . . . , are [ γ ⊥ ] n = (1 , , , [ γ ⊥ ] n = (0 , n + 1 , n ) , [ γ ⊥ ] n = (0 , − n, − n ) , (3.2)where [ γ ⊥ i ] n is the perpendicular vector of the wall associated with the node x i after 2 n mutations. Setting n = 0 yields eq. (3.1). The limit n → ∞ corresponds to our initialasymptotic chambers, whose ˆ y γ i are denoted as ˆ y γ i . Eq. (3.2) implies γ = [ γ ⊥ ] n ,γ = (1 − n )[ γ ⊥ ] n − n [ γ ⊥ ] n ,γ = (2 n )[ γ ⊥ ] n + (2 n + 1)[ γ ⊥ ] n , (3.3)so ˆ y γ = lim n →∞ [ˆ y ] n = ˆ y (cid:16) y ˆ y + ˆ y + (cid:112) (cid:52) (cid:48) (cid:17) , ˆ y γ = lim n →∞ ([ˆ y ] n ) − n ([ˆ y ] n ) − n = 4ˆ y (cid:52) (cid:48) (cid:0) − ˆ y ˆ y + ˆ y + √(cid:52) (cid:48) (cid:1) , ˆ y γ = lim n →∞ ([ˆ y ] n ) n +1 ([ˆ y ] n ) n = 14 ˆ y (cid:18) − ˆ y (ˆ y + 1) √(cid:52) (cid:48) (cid:19) , (cid:52) (cid:48) = (ˆ y ˆ y + ˆ y + 1) − y ˆ y , (3.4)where [ˆ y i ] n is the ˆ y -variable associated with the x i node in the 2 n -th cone. Eq. (3.12)relates the ˆ y γ i of our initial asymptotic chamber to ˆ y i of our principal quiver. The formulafor ˆ y γ and ˆ y γ are exactly the same as eq. (2.26), except that we wrote the expression interms of ˆ y -variables of the principle quiver instead of x -variables.We now apply the algorithm in appendix D to find the symbol alphabet associatedwith asymptotic chambers of the limiting ray. Repeatedly mutating the x and x nodes,their g -vectors asymptotically approach g lim = (0 , − , , (3.5)– 22 – ⊥ c γ ⊥ a γ ⊥ b C C C C C C Figure 8 . The scattering diagram of asymptotic chambers near the limiting ray in the A , clusteralgebra. We projected down onto the plane perpendicular to the limiting ray, g lim = (0 , − , which we identify as the limiting ray. After applying the algorithm in appendix D, we findall the walls that intersect the limiting ray: γ ⊥ a = (1 , , ,γ ⊥ b = (1 , , ,γ ⊥ c = (0 , , , (3.6)where γ c corresponds to a limiting wall. A visualization of these walls is provided in fig. 8,where we have taken a projection of the scattering diagram onto the plane perpendicularto the limiting ray.With eqs. (3.4) and (3.6), we can mutate around the scattering diagram to find allthe cluster algebraic functions associated with the limiting ray. For example, going alongthe path given in fig. 8, we find the mutations ˆ y γ ˆ y γ ˆ y γ → ˆ y γ ˆ y γ ˆ y γ +1 (cid:0) ˆ y γ + 1 (cid:1) ˆ y γ → ˆ y γ ˆ y γ ( ˆ y γ ˆ y γ ˆ y γ +1 ) ˆ y γ +1 ( ˆ y γ +1 ) ˆ y γ ˆ y γ ˆ y γ ˆ y γ +1 → ˆ y γ ˆ y γ ( ˆ y γ ˆ y γ ˆ y γ +1 )( − ˆ y γ ˆ y γ ) ˆ y γ +1 ( ˆ y γ +1 )( − ˆ y γ ˆ y γ ) ˆ y γ ( ˆ y γ ˆ y γ ˆ y γ +1 ) (3.7)Again, the jump across the limiting wall corresponds to a generalized cluster mutation.Going through all the polynomials, we eventually find the multiplicative basis:ˆ y γ , ˆ y γ , ˆ y γ , (1 + ˆ y γ ) , (1 − ˆ y γ ˆ y γ ) , (1 + ˆ y γ ˆ y γ ˆ y γ ) (3.8)Eq. (3.8) corresponds to all the algebraic functions associated with the limiting ray in the A , cluster algebra. Although the expressions in eq. (3.8) look rational, remember thatˆ y γ i are algebraic functions of ˆ y i . They are all algebraic due to the presence of the quadraticroot, √(cid:52) (cid:48) . – 23 – igure 9 . The scattering diagram associated with the asymptotic chambers of the A , clusteralgebra. A , We now continue to the A , cluster algebra, which corresponds to the quiver x x x x , which is chosen as our principal quiver. The mutation pattern is slightly more complexthan the A , cluster algebra, but the algorithm is the same. The limiting ray is g lim = ( − , , , . (3.9)The scattering walls are γ ⊥ ∈ { (0 , , , , (0 , , , , (1 , , , , (1 , , , , (1 , , , } , (3.10)where the last element corresponds to the limiting wall. To visualize this scattering dia-gram, we project down to three dimensions using the basis,ˆ e (cid:48) = (1 , , , , ˆ e (cid:48) = (0 , , , , ˆ e (cid:48) = (0 , , , , (3.11)– 24 –iving the projected scattering diagram in fig. 9. We call this projection of the scatteringdiagram the asymptotic scattering diagram. Note that the asymptotic scattering diagramassociated with A , is not simple as there are cones bounded by 4 walls instead of 3.We now calculate ˆ y γ i variables of the initial asymptotic chamber in terms of ˆ y -variablesof the principle quiver. The derivation is almost exactly as in section 3.1.1, so we will notwrite it out here. The final result is: i ∈ { , } : ˆ y γ i = ˆ y i (cid:16) y ˆ y + ˆ y + (cid:112) (cid:52) (cid:48) (cid:17) , ˆ y γ = ˆ y (cid:52) (cid:48) (cid:0) − ˆ y ˆ y + ˆ y + √(cid:52) (cid:48) (cid:1) , ˆ y γ = ˆ y (cid:18) − ˆ y (ˆ y + 1) √(cid:52) (cid:48) (cid:19) , (cid:52) (cid:48) = (ˆ y ˆ y + ˆ y + 1) − y ˆ y . (3.12)Due to the number of cones, we will not show all the values of ˆ y γ i in each cone. A completemultiplicative basis in terms of ˆ y γ i isˆ y γ , ˆ y γ , ˆ y γ , ˆ y γ , (ˆ y γ ˆ y γ ˆ y γ +1) , (ˆ y γ ˆ y γ ˆ y γ +1) , (ˆ y γ +1) , (ˆ y γ +1) , (1 − ˆ y γ ˆ y γ ) . (3.13)Again, ˆ y γ i are the ˆ y γ i associated with the initial asymptotic chamber approached by re-peated mutations on x and x in the initial quiver. A , , Our final example before moving onto Gr (4 ,
8) is A , , , which corresponds to the quiver x x x x x x . Unlike the previous examples, there are actually two limiting rays: g lim = ( − , , , , ,g lim = (0 , , , , − , . (3.14)First consider the limiting ray g lim , which is approached by performing repeated mutationson the x and x nodes. The formulas for ˆ y γ i are then:– 25 – ∈ { , , } : ˆ y γ i = ˆ y i , ˆ y γ = ˆ y (cid:16) y ˆ y + ˆ y + (cid:112) (cid:52) (cid:48) (cid:17) , ˆ y γ = ˆ y (cid:52) (cid:48) (cid:0) − ˆ y ˆ y + ˆ y + √(cid:52) (cid:48) (cid:1) , (3.15)ˆ y γ = ˆ y (cid:18) − ˆ y (ˆ y + 1) √(cid:52) (cid:48) (cid:19) , (cid:52) (cid:48) = (ˆ y ˆ y + ˆ y + 1) − y ˆ y . However, upon applying the algorithm in appendix D, we find there are actually an infinitenumber of asymptotic chambers. Rather, an infinite number cluster walls intersect thelimiting ray and the scattering diagram associated with the limiting ray itself containsa limiting ray. It is unsurprising that this phenomena eventually occurs as an infinitenumber of walls intersect a single ray even in the A , cluster algebra. None of these raysare limiting rays so we ignored the phenomena in section 3.1.1.Now consider the second limiting ray in eq. (3.14) by performing repeated mutationson the x and x nodes. As we approach the second limiting ray, the limits of ˆ y γ i , denotedas ˆ y (cid:48) γ i , are: i ∈ { , , } : ˆ y (cid:48) γ i = ˆ y i , ˆ y (cid:48) γ = ˆ y (cid:16) y ˆ y + ˆ y + (cid:112) (cid:52) (cid:48) (cid:17) , ˆ y (cid:48) γ = ˆ y (cid:52) (cid:48) (cid:0) − ˆ y ˆ y + ˆ y + √(cid:52) (cid:48) (cid:1) , ˆ y (cid:48) γ = ˆ y (cid:18) − ˆ y (ˆ y + 1) √(cid:52) (cid:48) (cid:19) , (cid:52) (cid:48) = (ˆ y ˆ y + ˆ y + 1) − y ˆ y . (3.16)We again find an infinite number of asymptotic chambers. Note that the discriminant, (cid:52) (cid:48) ,of the ˆ y (cid:48) γ i variables is different than the ˆ y γ i variables. Rather, the discriminant of the al-gebraic letters associated with a given set of asymptotic chambers seems to be determinedby the associated limiting ray.It is not particularly interesting for us to further study the cluster algebraic functionsassociated with A , , as it is infinite. Furthermore, the scattering diagram associated withthe limiting rays is 6-dimensional and cannot be easily visualized. However, one could takea doubly asymptotic limit to find a finite scattering diagram. More concretely, one couldfirst find the 5-D asymptotic scattering diagram associated with limiting ray g lim and thenfind the 4-D asymptotic scattering diagram associated with the limiting ray of this 5-Dasymptotic scattering diagram. This 4-D asymptotic scattering diagram could be finite.We leave studying such doubly asymptotic limits to future work.– 26 – .2 Gr (4 , /T and algebraic letters We now analyze the algebraic letters associated with 8-point MHV amplitudes in N = 4pSYM. Two classes of known algebraic letters emerge in the N = 4 pSYM symbol alphabetat 8-point, which are related by the cyclic shift: (cid:104) i, j, k, l (cid:105) → (cid:104) i +1 , j +1 , k +1 , l +1 (cid:105) [34, 80].Notably, each class of algebraic letters is associated with a unique discriminant. Since eachlimiting ray seems to be associated with a unique discriminant, (cid:52) (cid:48) , an reasonable conjec-ture is that the asymptotic chambers of only two limiting rays are relevant for the 8-pointMHV amplitude. Furthermore, we only need to analyze the asymptotic chambers of oneof these limiting rays since we can derive the algebraic letters associated with the otherlimiting ray by applying a cyclic shift. We first briefly review the positive kinematic region before summarizing the computa-tion of the algebraic letters. We parameterize kinematic space using momentum twistors, Z Ai . Due to dual conformal symmetry, we can identify Z Ai ∈ Gr (4 , n ). Furthermore,since the Z Ai are projective under a “little group” transform, Z Ai → t i Z Ai , we can identify Z Ai ∈ Gr (4 , n ) /T . The positive kinematic region, K + , corresponds to a compactificationof the positive Grassmannian, Gr + (4 , n ) /T . The cluster algebra structure of Gr + ( k, n ) /T is well known. In particular, there is a famous initial parameterization that at 8-pointcorresponds to the quiver (cid:104) , , , (cid:105) x : (cid:104) , , , (cid:105) x : (cid:104) , , , (cid:105) x , (cid:104) , , , (cid:105) (cid:104) , , , (cid:105) x : (cid:104) , , , (cid:105) x : (cid:104) , , , (cid:105) x : (cid:104) , , , (cid:105) (cid:104) , , , (cid:105) x : (cid:104) , , , (cid:105) x : (cid:104) , , , (cid:105) x : (cid:104) , , , (cid:105) (cid:104) , , , (cid:105)(cid:104) , , , (cid:105) (cid:104) , , , (cid:105) (cid:104) , , , (cid:105) (cid:104) , , , (cid:105) , Ref. [63, 87, 88] have also pointed out that additional types of limiting rays might be relevant forstudying the symbol alphabet at higher loop. However, ref. [63] also pointed out at least some of theseadditional limiting rays are related by a braid group [102] to the limiting rays we study in this section.Therefore, even if the symbol alphabet of associated with these other limiting rays appear in the 8-pointMHV symbol alphabet, it seems plausible they could be derived through braid transformations of thesymbol alphabet derived in this section. – 27 –here boxed elements in the quiver correspond to frozen variables. The ˆ y -variables associ-ated with the quiver areˆ y I1 = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) ˆ y I2 = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) ˆ y I3 = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) ˆ y I4 = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) ˆ y I5 = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) ˆ y I6 = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) ˆ y I7 = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) ˆ y I8 = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) ˆ y I9 = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) , (3.17)where the “I” super-script denotes how these ˆ y -variables are associated with the initialquiver. Note that the above quiver is not chosen as the principal quiver for our scatteringdiagram. Instead, we mutate to the quiver: ˆ y ˆ y ˆ y ˆ y ˆ y ˆ y ˆ y ˆ y , ˆ y which is chosen to be the principal quiver for our scattering diagram, by mutating nodes { , , , , , } of the initial quiver from left to right. As argued in section 2.1, no informa-tion is lost by doing so. Using the ˆ y -variable mutation equation, we find an explicit mapfrom the ˆ y -variables of our initial quiver to those of the chosen principal quiver:ˆ y = (cid:0) ˆ y I6 + ˆ y I1 (cid:0) ˆ y I4 + 1 (cid:1) (cid:0)(cid:0) ˆ y I2 + 1 (cid:1) ˆ y I6 + 1 (cid:1) + 1 (cid:1) ˆ y I1 ˆ y I2 ˆ y I4 × (cid:0) ˆ y I8 + ˆ y I1 (cid:0) ˆ y I2 + 1 (cid:1) (cid:0)(cid:0) ˆ y I4 + 1 (cid:1) ˆ y I8 + 1 (cid:1) + 1 (cid:1) , ˆ y = ˆ y I4 ˆ y I8 ˆ y I8 + ˆ y I1 (cid:0) ˆ y I2 + 1 (cid:1) (cid:0)(cid:0) ˆ y I4 + 1 (cid:1) ˆ y I8 + 1 (cid:1) + 1 , ˆ y = ˆ y I1 ˆ y I2 ˆ y I3 ˆ y I1 (cid:0) ˆ y I2 + 1 (cid:1) + 1 , ˆ y = ˆ y I2 ˆ y I6 ˆ y I6 + ˆ y I1 (cid:0) ˆ y I4 + 1 (cid:1) (cid:0)(cid:0) ˆ y I2 + 1 (cid:1) ˆ y I6 + 1 (cid:1) + 1 , ˆ y = ˆ y I1 ˆ y I2 ˆ y I4 ˆ y I5 ˆ y I1 (cid:0) ˆ y I2 + 1 (cid:1) (cid:0) ˆ y I4 + 1 (cid:1) + 1 , (3.18)ˆ y = ˆ y I1 (cid:0) ˆ y I4 + 1 (cid:1) + 1 (cid:0) ˆ y I1 (cid:0) ˆ y I2 + 1 (cid:1) (cid:0) ˆ y I4 + 1 (cid:1) + 1 (cid:1) ˆ y I6 , ˆ y = ˆ y I1 ˆ y I4 ˆ y I7 ˆ y I1 (cid:0) ˆ y I4 + 1 (cid:1) + 1 , – 28 – y = ˆ y I1 (cid:0) ˆ y I2 + 1 (cid:1) + 1 (cid:0) ˆ y I1 (cid:0) ˆ y I2 + 1 (cid:1) (cid:0) ˆ y I4 + 1 (cid:1) + 1 (cid:1) ˆ y I8 , ˆ y = (cid:0) ˆ y I1 (cid:0) ˆ y I2 + 1 (cid:1) (cid:0) ˆ y I4 + 1 (cid:1) + 1 (cid:1) ˆ y I6 ˆ y I8 ˆ y I9 (cid:0) ˆ y I6 + ˆ y I1 (cid:0) ˆ y I4 + 1 (cid:1) (cid:0)(cid:0) ˆ y I2 + 1 (cid:1) ˆ y I6 + 1 (cid:1) + 1 (cid:1) × (cid:0) ˆ y I8 + ˆ y I1 (cid:0) ˆ y I2 + 1 (cid:1) (cid:0)(cid:0) ˆ y I4 + 1 (cid:1) ˆ y I8 + 1 (cid:1) + 1 (cid:1) . Combining eqs. (3.17) and (3.18) gives explicit expressions of the principal quivers’ ˆ y -variables in terms of external kinematic data.We now turn to analyzing the asymptotic chamber using the A , sub-algebra in theprincipal quiver. By performing an infinite number of mutations on nodes 1 and 9, wemutate to the initial asymptotic chamber, finding i ∈ { , , , } : ˆ y γ i = ˆ y γ i ,i ∈ { , , } : ˆ y γ i = ˆ y γ i f (ˆ y , ˆ y ) , ˆ y γ = 4ˆ y (cid:52) (cid:48) (cid:0) − ˆ y ˆ y + ˆ y + √(cid:52) (cid:48) (cid:1) , ˆ y γ = ˆ y (cid:18) − ˆ y (ˆ y + 1) √(cid:52) (cid:48) (cid:19) , (3.19)where f (ˆ y , ˆ y ) = 12 (cid:16) y ˆ y + ˆ y + (cid:112) (cid:52) (cid:48) (cid:17) , (cid:52) (cid:48) = (ˆ y ˆ y + ˆ y + 1) − y ˆ y . (3.20)We therefore have closed form expressions for the ˆ y γ i of the initial asymptotic chamber.Unlike the previous examples, not all the ˆ y γ i are algebraic function of the ˆ y i -variables. Al-though difficult to see immediately, one can show that the discriminant, (cid:52) (cid:48) , is proportionalto (cid:112) (cid:52) (cid:48) ∝ (cid:112) A − B ,A = (cid:104) (cid:105)(cid:104) (cid:105) − (cid:104) (cid:105)(cid:104) (cid:105) − (cid:104) (cid:105)(cid:104) (cid:105) ,B = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) , (3.21)which corresponds to the limiting ray g in ref. [63].We now employ the algorithm in appendix D to find all the walls that intersect thelimiting ray. After repeated mutations of nodes 1 and 9, the g -vectors approach the limitingray g lim = ( − , , , , , , , , . (3.22)Performing a large number of mutations asymptotically close to the limiting ray, we even-tually found 26 cluster walls and a single limiting wall: γ ⊥ ∈ { (0, 1, 0, 0, 0, 0, 0, 0, 0) , (0, 0, 1, 0, 0, 0, 0, 0, 0) , – 29 –0, 0, 0, 1, 0, 0, 0, 0, 0) , (0, 0, 0, 0, 1, 0, 0, 0, 0) , (0, 0, 0, 0, 0, 1, 0, 0, 0) , (0, 0, 0, 0, 0, 0, 1, 0, 0) , (0, 1, 0, 0, 0, 0, 0, 1, 0) , (0, 1, 1, 0, 0, 0, 0, 0, 0) , (0, 0, 0, 0, 0, 0, 0, 1, 0) , (0, 0, 0, 1, 0, 1, 0, 0, 0) , (0, 0, 0, 1, 0, 0, 1, 0, 0) , (0, 1, 1, 0, 0, 0, 0, 1, 0) , (3.23)(0, 0, 0, 1, 0, 1, 1, 0, 0) , (1, 0, 0, 0, 1, 0, 0, 0, 1) , (1, 0, 0, 0, 0, 1, 0, 0, 1) , (1, 0, 0, 0, 0, 0, 0, 1, 1) , (1, 1, 0, 0, 0, 0, 0, 1, 1) , (1, 0, 0, 1, 0, 1, 0, 0, 1) , (1, 1, 1, 0, 0, 0, 0, 1, 1) , (1, 1, 0, 0, 0, 0, 0, 2, 1) , (1, 0, 0, 1, 0, 1, 1, 0, 1) , (1, 0, 0, 1, 0, 2, 0, 0, 1) , (1, 2, 1, 0, 0, 0, 0, 2, 1) , (1, 1, 1, 0, 0, 0, 0, 2, 1) , (1, 0, 0, 1, 0, 2, 1, 0, 1) , (1, 0, 0, 2, 0, 2, 1, 0, 1) , (1, 0, 0, 0, 0, 0, 0, 0, 1) } , where the last element corresponds to the limiting wall. Since the scattering diagram is 9-dimensional, the number of asymptotic chambers is still large even though there are only 27walls intersecting the limiting ray. Such an analysis is not beyond computers. An extensivecomputer search found a complete multiplicative basis of 27 non-trivial polynomials of ˆ y γ i : f = ˆ y γ + 1 ,f = ˆ y γ + 1 ,f = ˆ y γ + 1 ,f = ˆ y γ + 1 ,f = ˆ y γ + 1 ,f = ˆ y γ + 1 ,f = ˆ y γ + 1 ,f = ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ ˆ y γ + 1 ,f = ˆ y γ ˆ y γ ˆ y γ + 1 , (3.24) f = ˆ y γ ˆ y γ ˆ y γ + 1 ,f = ˆ y γ ˆ y γ ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ ˆ y γ ˆ y γ + ˆ y γ + 1 , – 30 – = ˆ y γ ˆ y γ ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ ˆ y γ (cid:0) ˆ y γ (cid:1) ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ ˆ y γ ˆ y γ ˆ y γ (cid:0) ˆ y γ (cid:1) + 2ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ ˆ y γ ˆ y γ + ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ ˆ y γ ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ ˆ y γ ˆ y γ + ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ (cid:0) ˆ y (cid:1) γ ˆ y γ ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ ˆ y γ ˆ y γ ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ ˆ y γ ˆ y γ + 2ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ ˆ y γ ˆ y γ + ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = ˆ y γ ˆ y γ ˆ y γ ˆ y γ (cid:0) ˆ y γ (cid:1) + ˆ y γ ˆ y γ ˆ y γ + ˆ y γ ˆ y γ ˆ y γ ˆ y γ ˆ y γ + ˆ y γ ˆ y γ + ˆ y γ + 1 ,f = 1 − ˆ y γ ˆ y γ , and the 9 ˆ y γ i , giving a symbol alphabet of 36 independent letters. Combining eqs. (3.17),(3.18), (3.19) and (3.24) gives an explicit expression for the algebraic letters in terms ofmomentum twistors. We have explicitly checked that the algebraic letters of ref. [80] aremonomials of ˆ y γ i and f i . Interestingly, note that many of the letters are obviously notalgebraic. From eq. (3.19), ˆ y γ , ˆ y γ , ˆ y γ , and ˆ y γ are rational, so any f i that is solelya function of these variables will also be rational. From this criteria alone, the algebraicalphabet is reduced from 36 to 25 letters. Further numerical checks show that some ofthese algebraic letters can further simplify to rational function for certain momentumconfigurations.This result is remarkable. There is no reason to expect that there are a finite number ofasymptotic chambers associated with any limiting ray of the Gr + (4 , /T cluster algebra.In section 3.1.3, we saw an example of a limiting ray with an infinite number of asymptoticchambers. Furthermore, although the number of asymptotic chambers is extremely large,the multiplicative basis has rank 36 for the relevant limiting rays! We can further discardletters that are clearly not algebraic, reducing the rank of the alphabet from 72 to 50.We can now conjecture that we have found ALL algebraic letters that could possiblyappear in the N = 4 pSYM 8-point amplitude. The basis shows how the relations betweenalgebraic letters are inherently rational, although the ˆ y γ are algebraic functions of our initialcoordinates, ˆ y I i . Finally, in all examples studied in this paper, the rank of the asymptoticsymbol alphabet has been equal to the number of cluster walls plus the rank of the clusteralgebra. More precisely, there seems to be a correspondence between walls in the clusteralgebra, γ ⊥ in eq. (3.23), and polynomial letters, f i in eq. (3.24). At present, it is unclearto us whether this relation holds for more general cluster algebras or is a red herring. A , sub-algebras Although this paper focuses on limiting rays associated with quadratic cluster algebraicfunctions, we expect that cubic cluster algebraic functions will also be relevant for studyingthe symbol alphabet of N = 4 pSYM beyond 8-point. Quadratic (cubic) algebraic func-– 31 –ions are algebraic functions that are products of roots of quadratic (cubic) polynomials.To see this, note that algebraic letters can at least partially be derived from irrational Yan-gian invariants, as shown in ref. [81, 82]. Using the duality between on-shell super-spacevariables and differentials on kinematic space, η Ai ↔ dZ Ai , (3.25)where η Ai are the on-shell super space variable associated with state i [16, 44], Yangianinvariants in N = 4 pSYM can be written in a manifestly dlog form:Yangian Invariant → (cid:89) i d log( α i ) , (3.26)where α i correspond to functions of external data, Z Ai , that are not necessarily rational.The α i can be interpreted as “letters” of the Yangian-invariant and correspond to singu-larities of the associated component of the super-amplitude. Since we expect the branchpoints of N k MHV amplitudes to match onto branch points of MHV amplitudes, we cantherefore probe the symbol alphabet of MHV amplitudes by studying the α i that appearin Yangian-invariants associated with N k MHV amplitude. Starting at 11-point, we start tosee irrational Yangian invariants that include cubic algebraic letters. Therefore, we expectto find cluster algebraic functions that are cubic at 11-point.The problem with cubic cluster algebraic functions is that it may not be possible toprobe their associated asymptotic chambers using the A , sub-algebra as in section 3.1.Specifically, we will not be able to approach the limiting ray associated with these cubiccluster algebraic functions by performing repeated mutations on a A , sub-algebra. Anylimiting ray that corresponds to a A , sub-algebra must correspond to quadratic clusteralgebraic letters. To see this, note that the generating function for cluster variables in the A , sub-algebra is G n> ( t ) = x − x − F t − P t + F t = ∞ (cid:88) n =0 x n t n . (3.27)Taking limits of x i generated by the above relation, such aslim i →∞ x i /x i − , (3.28)will always generate a function that is either rational or quadratic, but not cubic. Therefore,we must instead identify a more general mutation sequence that allows us to approach anasymptotic chamber associated with cubic algebraic functions. Such a mutation sequencemight correspond to a generating function, G ( t ), with a higher order polynomial in thedenominator, such that specific limits of x i will generate cubic cluster algebraic functions.We expect the methods and results in ref. [103] may be useful for pursuing this direction.– 32 – Degenerate scattering diagrams and tropicalization
We now turn to finding a preferred truncation of ˆ y -variables from the perspective of thescattering diagram. We will first motivate and define the notion of a degenerate scatteringdiagram, commenting on the specific connection to N = 4 pSYM. Although we do not finda definite algorithm for truncating ˆ y -variables, we do find that the notion of asymptoticchambers naturally emerges from degenerate scattering diagrams. In this section, we relate the g -vector fan to tropicalization of the dual cluster algebra. Wethen motivate degenerate fans using tropicalization arguments.We now give a brief review of tropicalization. Since all elements of O ( X ) are positiveLaurent polynomials where the minus operation never appears, we can consider the tropi-calization of such functions. Tropicalization naturally emerges from studying the behaviorof geometric spaces at small (or large) values of their coordinates. For example, given afunction f ( a , a , , . . . , a n ), the tropical function is defined asTrop[ f ( a , a , , . . . , a n )] = lim (cid:15) →∞ (cid:15) log[ f ( e − (cid:15)a , e − (cid:15)a , . . . , e − (cid:15)a n )] . (4.1)The tropicalization of a function effectively amounts to the replacements a × b → a + b ,a + b → min( a, b ) , → , (4.2)where a and b now take values on a semi-field. For example, we find thatTrop[1 + x ] = min(0 , x ) , Trop[1 + x + xy ] = min(0 , x, x + y ) . (4.3)Tropicalization has a number of applications, ranging from mirror symmetry to intersectiontheory. We will now review one aspect of the connection with cluster algebras.In our tropicalization arguments, we do not consider the O ( X ) associated with ourinitial principal quiver. Instead, we consider the dual principal quiver and the associateddual cluster algebra, X ∨ . The dual principal quiver is given by the initial quiver, exceptthat we flip all arrows between mutable nodes. As an example, given the initial quiver, y y x x , – 33 – ˆ y ∨ (a) Trop[1 + ˆ y ∨ ] y ∨ (b) Trop[1 + ˆ y ∨ ] y ∨ ˆ y ∨ γ + ˆ y ∨ (c) Trop[1 + ˆ y ∨ + ˆ y ∨ ˆ y ∨ ] Figure 10 . The fan associated with the tropicalization of functions in eq. (4.5). the dual quiver is y ∨ y ∨ x ∨ x ∨ , We now consider the O ( X ∨ ). For example, in the case the A , O ( X ∨ ) is generated by f = 1 + ˆ y ∨ ,f = 1 + ˆ y ∨ ,f = 1 + ˆ y ∨ + ˆ y ∨ ˆ y ∨ . (4.4)Any ˆ y -variable in X ∨ can be written as a product of functions in eq. (4.4) and ˆ y ∨ i .The tropicalization of each f ∈ O ( X ∨ ) defines a fan that splits R N into regions whereTrop[ f (ˆ y )] is constant. We simply state without proof that all such fans together give thescattering diagram in the finite case [39, 40, 67]. For example, again consider the A clusteralgebra and the tropicalization of functions in eq. (4.4): f = 1 + ˆ y ∨ → Trop[ f ] = min(0 , ˆ y ∨ ) ,f = 1 + ˆ y ∨ → Trop[ f ] = min(0 , ˆ y ∨ ) ,f = 1 + ˆ y ∨ + ˆ y ∨ ˆ y ∨ → Trop[ f ] = min(0 , ˆ y ∨ , ˆ y ∨ + ˆ y ∨ ) . (4.5)The tropicalization of each f i defines a fan in R , which are given in fig. 11. In this example,one can immediately see that the combination of all fans defined by tropicalization of O ( X ∨ )is equivalent to the scattering diagram for X . Rather, cones in the scattering diagram areregions where all tropical f i functions are constant. We now motivate degenerate scattering diagrams. Suppose we do not tropicalize allregular functions in O ( X ∨ ), but only a subset. For example, suppose we only consideredthe tropicalization of f and f in eq. (4.5). We would find only 4 walls in the scattering The relation between the scattering diagram of X and O ( X ∨ ) is easier to understand from a mirrorsymmetry perspective. A ∨ is dual to X under mirror symmetry [41]. – 34 – w w w w (a) Non-degenerate scatteringdiagram corresponding to A . w w w w + w (b) A degenerate scatteringdiagram Figure 11 . A demonstration of how to derive a degenerate scattering diagram from the non-degenerate scattering diagram for the A cluster algebra. diagram. Naively, this does not correspond to a well-defined scattering diagram if weassume the walls are single cluster walls. However, one might conjecture that it correspondsto a degenerate scattering diagram , where certain walls are combined so certain chambersare inaccessible. We now introduce the notion of degenerate scattering diagrams to motivate this truncation.Suppose that we want to truncate some cones from the scattering diagram while keepingothers. Rather we want to enforce certain conditions of the form: “If you cross wall A, youmust also cross wall B and vice-versa.” This is a well defined procedure if we combine wallsin the scattering diagram. For example, again consider the scattering diagram associatedwith A . Suppose we consider the fan derived by only tropicalizing f and f in eq.(4.5), leading to the degenerate fan in fig. 11b. We can derive this fan from a wallcombination procedure by combining walls w and w in the full scattering diagram in fig.11a. However, by combining walls, we lose a number of nice properties associated with theoriginal scattering diagram. First, we find multiple functions associated with a degeneratewall, so wall crossing across a degenerate wall takes the form: µ γ ⊥ ˆ y γ i = ˆ y γ i (cid:89) a f a (ˆ y γ ⊥ ) (cid:104) γ i ,γ ⊥ a (cid:105) . (4.6)Second, we find that the functions, f a , in eq. (4.6) change depending on whether youare mutating forward or backward across a degenerate walls. For instance, the functionsassociated with the degenerate wall in fig. 11b take the form: µ + γ ⊥ =(1 , ˆ y γ i = (1 + ˆ y γ ˆ y γ y γ ) (cid:104) γ i ,γ (cid:105) (1 + ˆ y γ + ˆ y γ ˆ y γ ) (cid:104) γ i ,γ (cid:105) ,µ − γ ⊥ =(1 , ˆ y γ i = (1 + ˆ y γ ˆ y γ ) (cid:104) γ i ,γ (cid:105) (1 + ˆ y γ + ˆ y γ ˆ y γ ) (cid:104) γ i ,γ (cid:105) , (4.7)– 35 –here the +( − ) indicates if you going counter-clockwise (clockwise) around the scatteringdiagram respectively. We can view this procedure as a “wall combination procedure” or“cone truncation” procedure.Although the degenerate walls are useful for motivating asymptotic chambers, thereis significant ambiguity in their construction. Primarily, given an arbitrary fan, we do nothave a procedure for associating a unique degenerate scattering diagram to this fan. Forexample, again consider the fan in fig. 11. We could construct this fan by combining wall w with walls w or w . Given only the fan, there is no canonical choice without additionalinput. We now consider the above procedure in the infinite case, working with the associateddegenerate cluster polytope instead of the degenerate scattering diagram. We note thatif we only tropicalize a finite subset of O ( X ∨ ), we will find that the associated degeneratecluster polytope includes a facet corresponding to the limiting ray. This facet does notappear if we tropicalize all functions in O ( X ∨ ) as this facet is then pushed to infinity. Forexample, consider the following principle quiver: y x y x x y , which corresponds to the A , cluster algebra, and its dual quiver, y ∨ x ∨ y ∨ x ∨ x ∨ y ∨ . Now consider the tropicalization of the following subset of regular functions of A ∨ , : f = ˆ y ∨ + 1 ,f = ˆ y ∨ + 1 ,f = ˆ y ∨ + 1 ,f = ˆ y ∨ ˆ y ∨ + ˆ y ∨ + 1 ,f = ˆ y ∨ ˆ y ∨ + ˆ y ∨ + 1 .f = ˆ y ∨ ˆ y ∨ + ˆ y ∨ + 1 , (4.8)The corresponding polytope is given in fig. 12, where the facet corresponding to the lim-iting ray is highlighted red. We argue that the vertices containing this facet correspond Working with the degenerate cluster polytope is purely for visualization purposes and contains equiv-alent combinatorial information to the degenerate scattering diagram. A review of the map is provided inappendix E. – 36 – rinted by Wolfram Mathematica Student Edition
Figure 12 . A degenerate cluster polytope of A , corresponding to the tropicalization of polyno-mials in eq. (4.8). The red facet corresponds the limiting ray. Figure 13 . The dual polytope of the asymptotic scattering diagram in fig. 8. Each asymptoticchamber corresponds to a vertex and walls between asymptotic chambers correspond to 1-dim edges. with asymptotic chambers in the degenerate scattering diagram. Such a conjecture nat-urally explains the appearance of algebraic letters found at 8-point. Note that the facetassociated with the limiting ray is not dual to the asymptotic scattering diagram given insection 3.1.1, whose associated cluster polytope is given in fig. 13. This is because eventhe asymptotic scattering diagram is degenerate. There have been a number of proposals for deriving degenerate scattering diagrams,which amount to choosing finite subsets of O ( X ∨ ) to tropicalize. For example, the authorsof ref. [63] proposed that the desirable subset of O ( X ∨ ) corresponds to the smallest subsetof minors closed under parity: (cid:104) i, i + 1 , j, j + 1 (cid:105) and (cid:104) i, j − , j, j + 1 (cid:105) . These functions canbe identified with some subset of O ( X ∨ ) using the “web-variables” originally given in ref.[104]. A number of alternate subsets have been proposed [63, 87, 88, 105]. However, in con-trast to our conjecture, which motivates a truncation of the clusters, these proposals arguefor a truncation of the x -variables. In the finite case, the authors identify a subset of x -variables whose g -vectors are in bijection with facets of the degenerate cluster polytope andconjecture that this subset acts as a complete multiplicative basis for desired ˆ y -variables.In the infinite case, where facets corresponding to limiting rays appear, they conjecture thelimiting rays correspond to cluster algebraic functions. It may turn out these conjecturesare equivalent to our proposal. To make any definite conclusion, one would have to find It seems that the facet associated with the limiting ray will always be degenerate unless you include F -polynomials associated with elements on the limiting ray in the tropicalization. These polynomials arenot elements of O ( X ∨ ) as they are not critically positive [63]. In our example, you need to include the F -polynomial associated with the generalization of P in appendix C, even though P is not an x -variable ofthe dual cluster algebra. – 37 – more concrete procedure for isolating the correct degenerate scattering diagram, as theprocedure provided here is still ambiguous. The structure of scattering amplitudes beyond Feynman diagrams has undergone intensestudy in a number of contexts over the past 60 years. This program has been very successfulat tree-level, where numerous bottom-up approaches have almost completely circumventedthe Lagrangian approach [11, 12, 14, 106, 107]. However, a systematic understanding ofhow locality, causality and unitarity are precisely encoded at all orders in scattering ampli-tudes remains surprisingly elusive. Many approaches, ranging from topological strings ontwistor space [108] to flat space holography [109], have given partial answers to this prob-lem. For example, the infrared structure of scattering amplitudes is famously connected tothe vacuum structure of the theory and asymptotic symmetries [110–113]. Recent researchsuggests that the underlying structure of scattering amplitudes is deeply connected to ge-ometric and combinatorial notations such as total positivity and motives [4, 6, 31, 44, 46].The amplituhedron provides a precise geometric description of integrands at all-loop or-ders in N = 4 pSYM. However, although the amplituhedron has led to many interestingresults in the study of scattering amplitudes, it is a fundamentally perturbative descriptionof the underlying physics. The ultimate goal of this program is a geometric descriptionof the integrated, all-loop amplitude independent of the chosen perturbation method, a“non-perturbative geometry” [63].One possible manifestation of this non-perturbative geometry is the connection be-tween boundaries of the positive kinematic region and logarithmic branch points of inte-grated MHV amplitudes in N = 4 pSYM. This conjecture is more subtle than it initiallyappears due to ambiguities in the precise definition of the positive kinematic region, such asthe chosen compactification. In this paper, we focused on studying the positive kinematicregion of the MHV sector and proposed that scattering diagrams are a useful mathematicalframework to study the boundary structure of the positive region. Furthermore, we devel-oped the notion of asymptotic chambers to explain the appearance of algebraic letters inthe symbol alphabet of MHV amplitudes. Interestingly, the asymptotic diagram approachprovides manifestly rational relations for the asymptotic ˆ y γ -variables.As a proof of concept, we used scattering diagrams to study the branch point structureof 8-point MHV amplitudes. Using the scattering diagram framework, we made a conjec-ture for all possible algebraic letters that could appear in the 8-point symbol alphabet.We confirmed that the algebraic letters found in explicit computations could be written asmonomials of letters in our alphabet. We also developed the notion of degenerate scatter-ing diagrams and commented on a possible truncation procedure for ˆ y -variables, followingthe philosophy of ref. [63, 87, 88].Our results are especially interesting in the context of the Landau equations. The Lan-dau equations provide a direct link between the structure of the integrand and singularitiesof the integrated amplitude. In particular, branch points of amplitudes at high multiplicity– 38 –nd loop order have been calculated by applying Landau equations to the amplituhedron[30, 35, 84]. However, although the Landau equations provide an non-trivial probe of theintegrated amplitudes’ branch points, knowledge of the branch points is not enough touniquely determine the symbol alphabet (see section 7 of ref. [30]). For example, althoughsome letters in the alphabet may take the schematic form φ ∼ f − √(cid:52) (cid:48) f + √(cid:52) (cid:48) (5.1)where f and (cid:52) (cid:48) are rational functions of external kinematic data, the Landau equationsonly predict branch points of the form (cid:52) (cid:48) = 0. This mismatch results from how thesolution to the Landau equations corresponds to the algebraic branch cut from the square-root in φ i instead of the full logarithmic branch point. A related mismatch also occurs forrational branch points. Similar to how cluster algebras provide the missing link betweenLandau singularities and the symbol alphabet at 6-point and 7-point, asymptotic chambersprovide the missing link between the algebraic symbol alphabet and specific solutions tothe Landau equations at 8-point. It has been argued that the branch points of N = 4pSYM associated with solutions to Landau equations are universal to all gauge theories. Itwould be interesting to understand whether the logarithmic branch points studied in thispaper, which contain more information than the solutions to the Landau equations, retainany degree of universality.The notion of degenerate scattering diagrams has applications beyond planar gaugetheories, specifically higher loop integrands of φ . However, it is instead the cluster polytopepicture that is more interesting for studying higher loop integrands of φ [114, 115] andgeneralized scattering amplitudes [116–122]. Both the higher loop integrands of φ andgeneralized scattering amplitudes can be identified with the canonical rational function ofthe (degenerate) cluster polytopes discussed in section 4 [105]. For example, the degeneratecluster polytope associated with A , , fig. 14, is associated with the multi-trace, 1-loop3-point integrand of φ theory [115]. This degenerate polytope can be derived from thetropicalization of f and f in eq. (4.8) along with the polynomial: f P = 1 + ˆ y ∨ ˆ y ∨ + ˆ y ∨ ˆ y ∨ ˆ y ∨ , (5.2)which is the F -polynomial of the generalization of P in appendix C to A , . Each ver-tex in the (degenerate) cluster polytope can be uniquely mapped to a specific Feynmandiagram contribution. However, considering all vertices in the full cluster polytope over-counts Feynman diagrams with specific topologies. Therefore, it is instead more naturalto consider degenerate cluster polytopes, where redundant vertices have been truncated.Although the motivation for truncating the unwanted vertices is very different, the notionof truncating undesirable vertices (cones) from the degenerate cluster polytope (scatteringdiagram) is exactly the same as section 4.Finally, the notion of asymptotic chambers has applications outside of scattering am-plitudes, such as studying coordinate systems of (higher) Teichmuller spaces. Specifically,when the cluster algebra corresponds to the (higher) Teichmuller space of a Riemann– 39 – rinted by Wolfram Mathematica Student Edition Figure 14 . A degenerate cluster polytope of A , corresponding to the tropicalization of f and f in eq. (4.8) along with eq. (5.2). surface, one can identify cluster algebraic functions with Fenchel-Nielsen coordinates [96].Non-trivial relations obeyed by Fenchel-Nielsen coordinates, and their generalizations, havebeen studied in the context of spectral networks [97, 98]. However, to our knowledge, noone has systematically studied relations between Fenchel-Nielsen coordinates in the contextof scattering diagrams and cluster mutations.Since our core result is a general framework for approaching these questions ratherthan individual results, there are a number of future directions. The most obvious direction for future work is developing a more systematic understandingof limiting walls and their connection to algebraic letters. For example, proving the bound:0 < ˆ y γ ⊥ lim < , (5.3)holds in any asymptotic chamber would certainly be very interesting. However, even witheq. (5.3), it is unclear the notion of asymptotic chambers is always well defined. Alter-natively, another interesting problem could be proving, or disproving, that the number ofmultiplicatively independent asymptotic letters associated with a limiting ray always is thenumber of intersecting walls plus the rank of the cluster algebra. Algorithm for Finding All Asymptotic Chambers
A very practical direction for future work is the development of new algorithms for findingˆ y γ i of more general asymptotic chambers. For instance, if one cannot identify a A , sub-algebra with the limiting ray, one would instead need to generalize the generating functionmethod of appendix C to more general sequences of quivers. In addition to finding explicitexpressions for ˆ y γ i , one also needs to find all the walls that intersect the limiting ray. Thealgorithm in appendix D becomes highly inefficient for limiting rays in higher rank clusteralgebras. Some possible approaches are: (1) calculating the walls using the tropicalization– 40 –rocedure in section 4.1, (2) developing some generalization of eq. (D.1) for asymptoticchambers, or (3) generalizing the causal diamond picture of ref. [114, 115]. Degenerate Scattering Diagrams
The discussion regarding degenerate scattering diagrams and truncated cones in section4.3 was largely qualitative. Although the construction naturally motivated the notion ofasymptotic chambers, it did not provide a concrete procedure for identifying the exactdegenerate scattering diagram given the corresponding fan. A primary goal of future workwould be developing the notion of canonical degenerate scattering diagrams, perhaps mod-ifying the somewhat ad-hoc construction in section 4.2. Motivated by related results inref. [116, 123], we note that understanding f i ∈ O ( X ∨ ), and not just their tropicalization,could be important for developing such a notion. One path is investigating the explicitcalculations in ref. [85] that connect boundary points of Gr ( k, n ) /T to f i using generalizedscattering equations. Finding Critically Positive Coordinates for Gr k (4 , n ) /T We focus on the positive kinematic region defined by Gr + (4 , n ) /T due to its connectionto logarithmic branch cuts of MHV amplitudes. However, the amplituhedron story nat-urally suggests the existence of alternate positive regions, Gr k (4 , n ), that are relevant forN k MHV amplitudes [44]. These positive spaces are much more non-trivial than Gr + (4 , n )and could be tied to the appearance of more general functions beyond the MHV sector,such as elliptic polylogarithms [46, 124–128]. It seems highly plausible that some notion ofcritically positive coordinates does generalize to these positive kinematic regions. Positive Kinematic Region of More General Theories
One can also attempt to understand the significance of the positive kinematic region formore general theories. A prime target would be understanding the positive kinematicregion associated with N = 4 pSYM amplitudes on the Coulomb branch [129, 130], ormore general massive theories [14, 131]. The kinematic space of these theories does notcorrespond to Gr (4 , n ) /T due external states being massive. Alternatively, one could tryto understand the positive kinematic region of more general scalar theories, such as bi-fundamental φ theory [132, 133] and φ p theory [134, 135]. The Positive Kinematic Region and (2 , Signature
Although the positive kinematic region exhibits remarkable properties and is intimatelyconnected with the analytic structure of amplitudes, there is currently no physical expla-nation of its importance. However, instead of directly addressing the importance of thepositive region, perhaps an easier preliminary question is the importance of (2 ,
2) signature.Using (2 ,
2) signature has often been viewed as a simple trick for removing ambiguities incertain computations due to the plethora of subtleties that emerge from trying to actuallyunderstand field theory dynamics in (2 ,
2) signature. However, there has been a recentrevival in trying to systematically understand the behavior of theories in (3 ,
1) signature,most notably the development of light ray operators, which has led to a number of non-– 41 – jk l → ijk l Figure 15 . A visual representation of the plucker relations for Gr (2 , n ). trivial results [136–142]. It would be interesting if such an analysis in (2 ,
2) signature couldlead to similarly novel results.
Acknowledgments
AH would like to thank Nima Arkani-Hamed, Will Dana, Song He, Thomas Lam, ZhenjieLi, Lecheng Ren, Chi Zhang, and Peng Zhao for stimulating discussion. AH would like toespecially thank Henriette Elvang for continued support and comments. AH is supportedin part by the US Department of Energy under Grant No. DESC0007859 and in part bya Leinweber Center for Theoretical Physics Graduate Fellowship.
A Introduction to cluster algebras
In this Appendix, we give a brief introduction to cluster algebras [99, 143–145]. Thoroughintroductions are ref. [146–149], while those looking for a review that focuses on theconnection with scattering amplitudes are referred to ref. [31]. Cluster algebras wereinitially motivated by the notion of total positivity. For example, one major motivatingquestion was how much information is generically needed to prove that minors of a givenmatrix are totally positive. Due to non-linear relations between minors, this questionquickly becomes very hard from a brute force approach of writing out all relations betweenminors and solving these polynomials. The advent of cluster algebras gave a fundamentallydifferent approach.Suppose you are given a 2 × n matrix and asked to find the minimal informationneeded to determine whether all 2 × (cid:104) i, k (cid:105) ( (cid:104) i, j (cid:105)(cid:104) k, l (cid:105) + (cid:104) i, l (cid:105)(cid:104) j, k (cid:105) ) = (cid:104) j, l (cid:105) i < j < k < l . (A.1)A brute force approach would be to calculate all quadratic relations of the form eq. (A.1)at once and find some minimal subset directly. This computation would be problematic foreven the best computers. We instead take a cluster algebra approach and find a preferredset of coordinates on the space of minors. To do so, we note that eq. (A.1) can be visuallyinterpreted as a mutation on the triangulation of the 4-gon with edges, i, j, k, l , as visualized– 42 – Figure 16 . The first triangulation of the 5-gon corresponds to the parameterization in eq. (A.3).Each minor in eq. (A.3) corresponds to an edge. The remaining n -gons correspond to the mutationpattern that leads to (cid:104) , (cid:105) . in fig. 15. Therefore, at n = 4, a natural set of preferred minors is (cid:104) , (cid:105) , (cid:104) , (cid:105) , (cid:104) , (cid:105) , (cid:104) , (cid:105) , (cid:104) , (cid:105) . (A.2)We can calculate the remaining coordinate, (cid:104) , (cid:105) using eq. (A.1), interpreting eq. (A.1)as a mutation on the 4-gon. Going beyond n = 4, it is natural to start with coordinatesthat can be associated with the triangulation of an n -gon and interpret eq. (A.1) as amutation on this n -gon, just as we did for the 4-gon. For example, consider n = 5 andinitial coordinates: (cid:104) , (cid:105) , (cid:104) , (cid:105) , (cid:104) , (cid:105) , (cid:104) , (cid:105) , (cid:104) , (cid:105) , (cid:104) , (cid:105) , (cid:104) , (cid:105) (A.3)which are associated with the first triangulation in fig. 16. Suppose we want to write (cid:104) , (cid:105) in terms of our initial coordinates. We first perform a “mutation” on (cid:104) , (cid:105) , finding (cid:104) , (cid:105) = 1 (cid:104) , (cid:105) ( (cid:104) , (cid:105)(cid:104) , (cid:105) + (cid:104) , (cid:105)(cid:104) , (cid:105) ) (A.4)and a new triangulation where (cid:104) , (cid:105) is replaced with (cid:104) , (cid:105) . We then perform a mutationon (cid:104) , (cid:105) , finding (cid:104) , (cid:105) = 1 (cid:104) , (cid:105) ( (cid:104) , (cid:105)(cid:104) , (cid:105) + (cid:104) , (cid:105)(cid:104) , (cid:105) ) (A.5)Therefore, assuming that all our initial minors in eq. (A.3) are positive, then (cid:104) , (cid:105) mustbe positive as well. One can repeat the above calculation for any minor not in eq. (A.3),showing that all minors are positive if our initial minors in eq. (A.3) are positive. Notethat we never mutate the edges that define the boundary of the n -gon. These are calledfrozen variables as they appear in the plucker relations, but do not themselves mutate.The above discussion focuses on the positivity of a Gr (2 , n ) matrix. However, we willultimately be interested in Gr (4 , n ) /T , where T acts on individual columns by a re-scaling: Z Ai → t i Z Ai . (A.6)– 43 – (cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) , (cid:105)(cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) , (cid:105) (cid:104) , (cid:105) Figure 17 . A triangulation of a 5-gon and its dual quiver representation. The boxed elements inthe quiver correspond to frozen nodes. where i and A index the columns and rows respectively. Therefore, it is natural to considerthe same question as above, except now for Gr (2 , n ) /T . Our minors, (cid:104) i, j (cid:105) , are no longersuitable coordinates as they are not invariant under T . Instead, we must develop a newset of coordinates, ˆ y -variables, for a given triangulation that are invariant under T trans-formation. Again consider the coordinates in eq. (A.3). A natural combination of minorsthat are invariant under T is,ˆ y = (cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105) , ˆ y = (cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105)(cid:104) , (cid:105) . (A.7)These variables form a natural set of coordinates on the compactified space Gr + (2 , n ) /T .To see their importance, lets interpret Gr (2 , n ) /T as a manifold and Gr + (2 , n ) /T a positiveregion. Each triangulation, with its own ˆ y i variables, corresponds to a different “corner”of Gr + (2 , n ) /T , as visualized for n = 5 in fig. 1.The above strategy of finding an initial “cluster” of coordinates and developing asequence of coordinate transforms turns out to be very versatile. Generalizing beyond Gr (2 , n ), one can systematically develop the notation of a cluster algebra. Instead of atriangulation, we associate to each cluster a quiver, Q , and exchange matrix, B i,j : B i,j = n if there are n arrows from i to j − n if there are n arrows from j to i j to i (A.8)– 44 –ach triangulation of an n -gon for Gr (2 , n ) maps onto a triangulation in the following way • Each edge in the n -gon triangulation corresponds to a node in the quiver. The edgescorresponding to the boundary of the n -gon are frozen nodes that never mutate • For each triangle in the n -gon triangulation, we draw a clock-wise orientated cycle in Q connecting the vertices associated to the bounding edges.For example, a visualization of the quiver associated with the first triangulation in fig. 16 isgiven in fig. 17. Given a mutation, the minors generalizes to cluster variables that mutateas µ k x i = (cid:40) x i ( (cid:81) j → i x j + (cid:81) j ← i x j ) i = kx i i (cid:54) = k (A.9)If we perform a mutation on node k , the quiver, and corresponding exchange matrix,mutate according to the rules • Reverse all arrows going in or out of k , • For each sub-path of the form i → k → j , add the arrow i → j , • Remove any two cycles that have formed.One can explicitly check that eqs. (A.9-A.11) are a self consistent generalization of the Gr (2 , n ). Finally, the ˆ y coordinates also have a natural generalization as,ˆ y i = (cid:89) j x − B i,j j , (A.10)and mutate as µ k ˆ y i = (cid:40) y i i = k ˆ y i (1 + ˆ y Sign( B i,j ) k ) B i,j i (cid:54) = j . (A.11)We will denote the positive space parameterized by x i coordinates as A and the spaceparameterized by ˆ y i coordinates as X . The relation between A and X is still under activeresearch and not completely understood.Cluster algebras have a number of remarkable properties, such as the Laurent phe-nomena. A cluster variable of any quiver can be written as a Laurent polynomial of x i of some initial cluster. For example, consider the cluster algebra associated with Gr (2 , (cid:104) i (cid:48) , j (cid:48) (cid:105) can be written as a Laurent polynomial of (cid:104) i, j (cid:105) in eq. (A.3).To learn about other amazing properties of cluster algebras, the reader is referred to ref.[146, 147]. B Differentiating x -variables with frozen nodes Like ˆ y -variables, x -variables will also obey additional relations when there are fewer frozennodes. To see this, again consider the A cluster algebra with initial quiver: x x x . – 45 –ithout any frozen nodes, all x -variables in the cluster algebra are { x , x , x x + 1 x , x x + x + 2 x + 1 x x x x + 1 x , x x + x + 1 x x x x + x + 1 x x , x x + 1 x } . (B.1)The minimal multiplicative basis of the x -variables is rank 7: { x , x , x x + 1 , x x + x + 2 x + 1 ,x x + 1 , x x + x + 1 } . (B.2)However, suppose we include the additional frozen node, z , so the initial quiver is now: zx x x . All x -variables in the cluster algebra are now { z, x , x , x x + zx , x x z + zx + (1 + z ) x + zx x x x z + 1 x , x x z + x + zx x x x + x z + 1 x x , x x + 1 x } , (B.3)so the multiplicative basis for x -variables is now rank 10: { z, x , x , x ,x + z, x x z + zx + (1 + z ) x + z, (B.4) x x + 1 , x x z + x + z,x z + 1 , x x + x z + 1 } . Comparing eqs. (B.1) and (B.3), one can clearly see that adding the frozen node, z , removesrelations between the x -variables. Adding more frozen nodes would further disentangle the x -variables.Remarkably, the frozen nodes of a principle quiver is enough to ensure that all the x -variables are maximally disentangled. To see this, note that the x -variables of a cluster– 46 –lgebra with completely arbitrary frozen nodes can be always be written in the form: x = x (cid:126)g F (ˆ y i ) × (monomial of frozen variables) , (B.5)where (cid:126)g and F (ˆ y i ) are the same (cid:126)g and F (ˆ y i ) that appear in eq. (2.2). The exact formulafor computing the monomial of frozen x -variables is unimportant our purposes and thereader is referred to Appendix B of ref. [87] for details. From eq. (B.5), we see that the x -variable of a cluster algebra with arbitrary frozen nodes is the same as the x -variable ofcluster algebra with a principal quiver up to a monomial of frozen x -variables. Therefore,one multiplicative basis of the ˆ y -variables of a cluster algebra with arbitrary frozen nodesis the multiplicative basis of a ˆ y -variables of a cluster algebra with a principal quiver inaddition to the frozen variables themselves. C Review: A , cluster algebra In this appendix, we consider the cluster algebra and scattering diagram associated withthe principal quiver, y − y x − x , reviewing the results in ref. [63, 87, 88]. After an arbitrary sequence of mutations towards x n where n >
0, we find the quiver y x n − x n y − . n n − nn +1 Using cluster mutations and the above represetion of the quiver after n mutations, wedefine a recursive solution for x n in this model, finding for even n : x n − x n − = ( y n − − y n − + x n − ) ,x n x n − = ( y n − y n − + x n − ) . (C.1)This form of the mutation relations is still too complicated to solve analytically due tobeing inherently non-linear. Instead, we identify a new variable, P = y − x − x + x x − + x − y − y x , (C.2)such that x n − = x n − P − x n − F , F = y − y . (C.3)– 47 – is not an element of the cluster algebra, but a cluster-like variable associated with thelimiting ray. For further discussion of P , the reader is referred to ref. [63]. Only eq. (C.3)is important for our purposes, which one can explicitly check.We can solve eq. (C.3) by first writing down the associated generating function: G n> ( t ) = x − x − F t − P t + F t = ∞ (cid:88) n =0 x n t n . (C.4)and then finding a closed form expression for the derivatives of G n> ( t ): x n = 12 n +2 [( x − + B + (cid:112) (cid:52) )( P + (cid:112) (cid:52) ) n +1 + ( x − − B + (cid:112) (cid:52) )( P − (cid:112) (cid:52) ) n +1 ] B + = 2 x − x − P(cid:52)(cid:52) = P − F (C.5)Using our closed form expressions for x n in eq. (C.5), it is trivial to calculate closed formexpressions for ˆ y i : ˆ y n − = y n y n +1 − x − n , ˆ y n = y − n y − n − x n − (C.6) D Algorithm for finding asymptotic chambers from A , sub-algebra We now outline a search algorithm we used for finding the asymptotic chambers associatedwith a given limiting cluster. In a normal search algorithm for a finite cluster algebra, onewould perform sequences of mutations until one found all the cones in the fan, defining acone by its associated g -vectors. This method does not work for infinite cluster algebrasas there are an infinite number of cones, even asymptotically close to the limiting ray. Wecircumvented this issue by defining a new equivalence class of cones arbitrarily close to thelimiting ray called pre-asymptotic cones.We first mutate the initial quiver until we find a A , sub-algebra: x i x j , which we define as the principal quiver. To calculate the g -vectors and walls of adjacentcones, we use the g -vector mutation formula originally derived in ref. [99]: µ k (cid:126)g i = (cid:40) (cid:126)g i if i (cid:54) = k − (cid:126)g i + (cid:80) Nm =1 [ B m,k ] + (cid:126)g a − (cid:80) Nl =1 [ B N + l,k ] + (cid:126)b j if i = k , (D.1)where [ x ] + = max( x,
0) and (cid:126)b j is column j of the initial B i,j matrix. This formula can bederived by combining eq. (A.9) in appendix A, µ k x i = (cid:40) x i ( (cid:81) j → i x j + (cid:81) j ← i x j ) i = kx i i (cid:54) = k , (D.2)– 48 –ith eq. (2.2) in section 2.1: x = x (cid:126)g F (ˆ y i ) , x (cid:126)g = (cid:89) i x g i i . (D.3)This allows us to compute the g -vectors of adjacent cones in the g -vector fan very efficiently.Using eq. (D.1), we find the g -vectors of cones associated with repeated mutations on nodes x i and x j . We find that the rays asymptotically approached some limiting ray, (cid:126)g lim . Inprinciple, we could now perform a brute force search of all walls, just performing randommutations asymptotically close to the limiting ray until we found no new walls intersectingthe limiting ray. This approach would be highly inefficient however as there are always aninfinite number of cones asymptotically close to the limiting ray. Although we do performa brute-force search, we partially streamline the algorithm by defining a new equivalenceclass of cones: pre-asymptotic chambers.Consider the schematic scattering diagram in fig. 7, which corresponds to the A , cluster algebra. Suppose we mutate to one of cones in the sequence that approaches theasymptotic chamber C . Mutating across an asymptotic wall toward or away from thelimiting ray does not give us any new information. In some sense, the sequence of conesapproaching C are equivalent, and therefore redundant, for the purposes of trying to findwalls intersecting the limiting ray. We wish to find some criterion that allows us to avoidmutating into these redundant cones. To see what this criterion should be, we first notethat the sequence of g -vectors along the black cluster wall generically obey the relation: (cid:126)g ( x n ) − (cid:126)g ( x n − ) = (cid:126)g lim . (D.4)Therefore, it is natural to consider the projection of the g -vectors onto the hyperplaneperpendicular to the limiting ray, such that P ⊥ ( (cid:126)g ( x n ) − (cid:126)g ( x n − )) = P ⊥ (cid:126)g lim = (cid:126) . (D.5)If we define equivalence classes of cones by considering the projection of their g -vectors,not the g -vectors themselves, the sequence of cones approaching C correspond to thesame cone under this projection. This is true for the sequences of cones approaching C , C , and C as well. For C and C , we find two classes of cone upon taking theprojection. To see why, lets focus on C . Denoting the projection of the g -vectors onthe two bordering cluster walls as g a and g b , the two classes of cones are defined by thesets { g a , g a , g b } and { g a , g b , g b } . Turning to more general cluster algebras, we define pre-asymptotic chambers as the equivalence classes of cones defined by the projection of g -vectors onto the hyperplane perpendicular to the limiting ray. Infinite sequences of conesapproaching some asymptotic chamber correspond to the same equivalence class under thisprojection. Again, pre-asymptotic chambers do not correspond to asymptotic chambers,as a single asymptotic chamber can correspond to multiple pre-asymptotic chambers.The final subtlety to consider is that we can only find some subset of the pre-asymptoticchambers when we perform a search of pre-asymptotic chambers asymptotically close to– 49 – igure 18 . Cluster polytopes corresponding to A (left) and A (right). the limiting ray. We cannot find all pre-asymptotic chambers from a single search becausewe cannot use the mutation rule in eq. (D.1) to mutate across the limiting wall. Therefore,we generically perform two searches for pre-asymptotic chambers, one on each side of thelimiting wall. E Review: cluster polytopes
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