$\Delta$-Algebra and Scattering Amplitudes

In this paper we study an algebra that naturally combines two familiar operations in scattering amplitudes: computations of volumes of polytopes using triangulations and constructions of canonical forms from products of smaller ones. We mainly concentrate on the case of $G(2,n)$ as it controls both general MHV leading singularities and CHY integrands for a variety of theories. This commutative algebra has also appeared in the study of configuration spaces and we called it the $\Delta$-algebra. As a natural application, we generalize the well-known square move. This allows us to generate infinite families of new moves between non-planar on-shell diagrams. We call them sphere moves. Using the $\Delta$-algebra we derive familiar results, such as the KK and BCJ relations, and prove novel formulas for higher-order relations. Finally, we comment on generalizations to $G(k,n)$.


Contents
Scattering amplitudes in N =4 super Yang-Mills in four dimensions are known to be constructible in terms of two classes of three-particle amplitudes, usually represented as black and white trivalent vertices [1]. Gluing such vertices so that each edge is on-shell gives rise to on-shell diagrams. Choosing an ordering for n external particles, planar on-shell diagrams become plabic graphs [2] and are deeply connected to the positive Grassmannians G ≥0 (k, n) [1,2]. In physics, k represents the N k−2 MHV sector the diagram belongs to. It is known that the simplest sector, i.e., k = 2 or MHV, is special in many ways. In particular, any on-shell diagram, planar or not, associated with a top dimensional region of G(2, n) can be characterized by a set of n−2 triples of labels [3]. This is because each such on-shell diagram contains exactly n−2 black trivalent vertices even after contracting like-colored vertices to make the diagram bipartite. Moreover, in the bipartite diagram each white vertex has exactly one external leg attached to it and each black vertex is connected to three distinct white vertices. As explained in [3], and recently explored in [4] from a novel view point, one can associate a 2-form to each black vertex Ω abc = d log a, b a, c ∧ d log b, c a, c (1.1) -2 -so that the top form, i.e., a 2(n−2)-form associated with a given on-shell graph is simply given by where T is a list of n − 2 triples τ defining the diagram. Also in [3], an alternative formula for a rational function associated with the on-shell diagram T was given in term of the reduced determinant of a matrix M constructed from T Here we rewrite the determinants in (1.3) in terms of integrals over Grassmann variables θ a , χ a motivating the definition of the following object: This object naturally maps to Ω abc under the map introduced by He and Zhang [4] that takes Grassmann variables to differential forms. The formula for F T can then be written as Grassmann integrations over the product of n−2 ∆'s; one for each triple of labels in T . Moreover, we identify the integrand as the physically relevant object by making the following definition LS T := τ ∈T ∆ τ 1 ,τ 2 ,τ 3 . (1.5) LS stands for "leading singularity" which is the terminology for the physical meaning of the quantity [5][6][7]. Note that LS T has the same structural form as (1.2). Two crucial properties of ∆'s are that they are nilpotent, ∆ 2 = 0, and they commute, ∆∆ = ∆ ∆. Using these two properties, LS T can be rewritten as (1.6) The simplest leading singularity is known as the Parke- Taylor  (1.7) In a different line of developments, the study of positive geometries [8], and the amplituhedron [9] has led to the development of volume functions usually denoted by [i 1 . . . i m+1 ], see, e.g., [10,11]. Here m denotes the CP m where the object lives. These functions are as--3 - sociated with simplices that can be put together to form more complicated geometries. The corresponding volume is the sum of the volume functions. Letting m = 2 gives rise to volume (or area) functions with three labels. The simplest example is an n-sided polygon whose area is computed, e.g., as (1.8) These two lines of development naturally motivate the study of the algebra generated by ∆'s which we call the ∆-algebra.
Interestingly, the ∆-algebra has also appeared in the study of configuration spaces, in particular the configuration space of n distinct points in SU (2). More precisely, in [12], the cohomology ring of the configuration space, H * (Conf n (SU (2))/SU (2), C), of n distinct points on SU (2) modulo the diagonal action, was constructed as a subalgebra, denoted V n , of the cohomology ring H * (Conf n (R 3 ), C). 1 By comparing generators and relations, one finds that in fact the generators ∆ abc satisfy the same relations as the generators v abc for V n . Therefore the ∆-algebra is a representation of V n , with v abc → ∆ abc . For background on configuration spaces, see, e.g., the review [14] and the classic papers [15][16][17][18]. For related motivating work on permutohedral tessellations and blades, see [19,20].
In this paper we initiate the study of the ∆-algebra applications in scattering amplitudes. In particular, MHV on-shell diagrams that produce leading singularities. A crucial property of on-shell diagrams is that any two diagrams related by an operation known as the "square move" give rise to the same rational function (1.3), i.e., the same physical object. Moreover, if the diagrams are planar (and reduced) it was proven by Postnikov [2] that the square move, combined with expansion and contraction moves of like-colored vertices, is enough to define equivalence classes of diagrams, known as plabic graphs, each encoding a different cell in 1 See also [13] for a related conjectural description of the cohomology ring H * (Confn(SU (2))/SU (2), C).
In the non-planar case, already at six points this is not the case as found by the second author in the study of permutohedral tessellations and blades [20]. There are two MHV onshell diagrams not connected via square moves that give rise to the same rational function via something that can be called the octahedral move. As anticipated in [21], finding a structure such as the ∆-algebra associated with on-shell diagrams makes these new identifications natural and easy to prove. In fact, we show that the square and octahedral moves are the simplest cases of an infinite family of moves we call sphere moves. Moreover, the very fact that one needs n−2 triples (to which we associate oriented triangles) to define a leading singularity and therefore the equivalence of two such objects requires 2(n−2) triangles gives the triangulation of the surface of a sphere with n vertices and hence the name. We illustrate how a given on-shell diagram, that admits a sphere move, induces a triangulation in Figure 1.1 that shows the emergence of a sphere.
The sphere move is performed by subdividing the triangle corresponding to ∆ abc into three new ones with a reference point r, such that: Moving all reference points r into a common one, say R, it is possible to open them up in a different order, such that the resulting triangles are the complement of the ones we started with. See Figure 1.2 for an example. In fact, it is most convenient to send the reference point R to infinity, which gives variables associated to (oriented) edges: with u ab = −u ba , so that ∆ abc = u ab + u bc + u ca . In this language, two sets of triangles are connected by a sphere move if their boundaries (i.e., ∆ abc 's expanded as sums of u ab 's) are -5 -  (13) and (24)  the same. See Figure 1.3 for an example of a square move as the simplest case of a sphere move.
Many of the applications we find of the ∆-algebra follow from the fact that one can use a triangulation independent formulation as the one introduced by Enciso [22,23] and used in the mathematical construction of configuration spaces [12].
On-shell diagrams have been the subject of many studies and there are many properties which are understood with very elegant proofs [24], for reviews see [25,26]. Here we present the ∆-algebra formulation some of these properties, e.g., the U (1) decoupling identity and the fundamental Bern-Carrasco-Johansson (BCJ) relation [27]. Moreover, the construction naturally leads to higher order identities such as those expected from the string theory formulation in terms of amplitudes involving TrF 4 amplitudes [28,29]. In our derivation, the objects that appear are double-trace-like instead.
The fact that the determinant formula (1.3), as shown by Franco, Galloni, Penante, and Wen [30], and Enciso's construction can both be generalized to higher k, motivated us to also generalize the ∆-algebra to higher k. Here we only start the study of the object and explain some of the most basic properties. This paper is organized as follows. In Section 2 we introduce the ∆-algebra and explain its connection to leading singularities. In Section 3 we use it to define sphere moves between classes of on-shell diagrams. In Section 4 we show identities satisfied by the ∆-algebra, which allow us to prove known and new relations among leading singularities. Longer proofs needed in this section are relegated to the appendices. In Section 5 we discuss higher-k generalizations of the ∆-algebra. We end with a discussion of future directions in Section 6.

Leading Singularities: Introducing ∆ abc
Leading singularities in N =4 super Yang-Mills are the most basic IR finite quantities in the theory [31]. They are computed using on-shell diagrams and are classified by their R-charge in sectors [1]. The simplest and most well-understood sector is the maximal helicity violating (MHV) one. All n-particle MHV leading singularities can be constructed from a set of n−2 cyclically-ordered triples of distinct labels T = {(a 1 , b 1 , c 1 ), . . . , (a n−2 , b n−2 , c n−2 )}. The explicit form of a given leading singularity can be compactly encoded in the reduced determinant of a (n−2) × n matrix M [3]. Rows are labeled by triples while columns by particles. A row corresponding to triple (a i , b i , c i ) has only three non-zero components at columns a i , b i and c i with values b i , c i , c i , a i and a i , b i respectively. MHV leading singularities are known to depend only on holomorphic spinors λ a := {λ a,1 , λ a,2 } through SL(2, C) invariant combinations a, b , which can be also thought of as Plücker coordinates of G(2, n).
The matrix M has two column null vectors with components v a,2 . Using this it is possible to remove two columns, say the a th and b th columns, of M to obtain a (n−2) × (n−2) submatrix M (ab) and show that is independent of the choice of deleted columns. The rational function associated to a list of triples T is then obtained as (1.3)

Product of Triples
In order to obtain a reformulation of leading singularities that leads to the ∆-algebra, the first step is to use the well-known formulation of determinants in terms of Grassmann variables. 2 Given any m×m matrix R, the determinant of R can be expresses as an integral over two sets of Grassmann variables θ a andθ a as follows 3) It will be useful to carry out the integration over either set of Grassmann variables, saȳ θ to obtain Let us apply (2.4) to the determinant entering in the formula for rational function associated to a leading singularity (2.2). One can write Note that the choice of columns that are removed is done by the choice of Grassmann 2 Recall that any pair of Grassmann variables anticommutes, {θa, θ b } = 0, {θa,θ b } = 0, {θa, θ b } = 0, and the integration rule is dθa(α + βθ b ) = βδ ab for constants α, β ∈ C, see, e.g., [32] for a review.
-7 -variables in the last factor. Indeed, since dθ a θ a = 1, it is useful to think about θ = δ(θ) and therefore the last factor can be written as δ(θ d )δ(θ e ) which then imply that θ e and θ d must be set to zero in the rest of the integrand thus removing the corresponding columns.
Finally, since the leading singularity formula (2.2) has two powers of the determinant, one can introduce a different set of Grassmann variables, say χ a , to write it. This shows that a leading singularity associated to a set of triples T is given by The rest of the paper is devoted to studying properties of ∆ abc 's as generators of a commutative algebra and its applications to physical quantities.

From Homogeneous to Inhomogeneous Variables
The variables λ a can be thought of as homogeneous variables of points on CP 1 . In physics applications, such as Cachazo-He-Yuan (CHY) [33] and Witten-Roiban-Spradlin-Volovich formulas [34,35] , it is more convenient to work with inhomogeneous variables. In order to go from one to the other it is enough to write and (θ a , χ a ) → t a (θ a , χ a ). Under these operations one finds that ∆ abc has a much more compact form where we have introduced the shorthand notation x ab := x a − x b . Given that the expression for a given leading singularity is independent of the choice of columns deleted, i.e., the choice of d, e, f and g in the factor we choose to refer to a leading singularity as simply the product of n−2 ∆ abc It is straightforward to see that the map (2.6) between LS T and F T is an isomorphism. This -8 -is the formula presented in the introduction in (1.5). Of course, the physically relevant object is the rational function F T associated with it which is obtained after integrating out the Grassmann variables.

Properties of ∆ abc
The building block of leading singularities has several crucial properties. First, ∆ abc is completely antisymmetric in its indices. Second, once again one can use that for any Grassmann variable θ = δ(θ) to conclude that x ab x bc x ca (2.12) and therefore ∆ 2 abc = 0. Moreover, since the Grassmann degree is two, they commute ∆ abc ∆ ef g = ∆ ef g ∆ abc .
Finally, a small amount of algebra reveals that (2.14) Here θ ab := θ a − θ b and χ ab := χ a − χ b . Note that u ab = −u ba while, once again, the nature of Grassmann variables implies that u 2 ab = 0 while ∆ 2 abc = 0 shows that u ab u bc + u bc u ca + u ca u ab = 0.
(2.15) These relations for u ab are exactly those defining the commutative algebra governing the cohomology ring of the configuration space, H * (Conf n (SU (2))/SU (2), C), of n distinct points in SU (2) modulo the diagonal action was constructed as a subalgebra of the cohomology ring H * (Conf n (R 3 ), C) as found in [12]. In fact, our main motivation was to find a physical representation of this algebra. It is also possible to derive the expression (2.13) using the intuition developed by Enciso [22,23] in his construction of a triangulation independent version of amplituhedron formulas. In order to see this, one starts by showing that ∆ abc satisfies properties of the boundary of an oriented triangle. Without assuming (2.13) one can easily show from (2.9) that ∆ abc = ∆ abr + ∆ bcr + ∆ car . (2.16) Note that since the LHS does not depend on x r , it must be that neither does the RHS and therefore one can set x r to any value. Taking the limit x r → ∞ gives (2.13) and we discover that This point of view will be very useful in generalizations to higher values of k, as discussed in Section 5. This concludes our presentation of the ∆-algebra and we turn to physical applications.

Sphere Moves for Non-Planar On-Shell Diagrams
The first application of the ∆-algebra formulation of leading singularities has to do with their graphical representation as on-shell diagrams. Associated with a given leading singularity function there can be many different lists of triples, each representing an on-shell diagram. It is well-known that any two on-shell diagrams related by a square move give rise to the same physical object [1,2]. In the planar case, this move, combined with expansion and contraction moves of like-colored vertices, is enough to define equivalence classes of physical objects. Here we find that the square move is the simplest example of an infinite class of moves that become available once non-planarity is allowed.
We start by recalling that a leading singularity is represented by a set of triples is computed as This object can be rewritten in a variety of ways. One possibility is by choosing complex numbers α i such that α 1 α 2 · · · α n−2 = 1 to write the identity where we used that ∆'s commute and square to zero. Motivated by the expression of ∆ abc in terms of u ab given in (2.13), it is natural to choose α i 's such that as many u ab 's can be canceled as possible.
Consider the simplest non-trivial example at n=4 for which one choice of triples is T = {(1, 2, 3), (1, 3, 4)} so that Since none of the α's can vanish, the only u ab that can be eliminated is u 13 whose coefficient is α 2 − α 1 (recall that u 31 = −u 13 ). Therefore asking this to cancel leads to α 1 = α 2 = ±1 and to a very illuminating formula This form has a cyclic symmetry that the triples obscured. In particular, it means that which is known as the square move in the on-shell diagram representation of MHV leading singularities. Given these properties it is clear that u ab should be associated with an oriented edge E ab of a graph connecting vertices a and b, and hence ∆ abc should be associated to the boundary ∂T abc = E ab + E bc + E ca of an oriented triangle (2-simplex) T abc , Generalizing the LHS of the four-point formula (3.5) to any n is trivial if we want to preserve the cyclic symmetry by replacing a square by an n-gon It is clear that there are as many representation of this object in term of products of n−2 ∆'s as the number of triangulations of an n-gon, the Catalan number C n−2 .
The rational functions associated with these polygons are the most basic examples of MHV leading singularities and are known as Parke-Taylor factors [36]. Once all integrations over the Grassmann variables are done the final result is [3]: PT(1, 2, . . . , n) := 1 Here we used the opportunity of introducing the notation for a Parke-Taylor factor. Motivated by the above discussion we make the following definition.
Definition 3.1. Two distinct sets of n−2 triples T 1 and T 2 are said to be equivalent if they give rise to the same leading singularity, up to a sign, By equation (3.3), a sufficient condition for this to happen is that there exist non-zero coefficients α τ , β τ such that In the sequel we explore a particular infinite family of equivalences between non-planar on-shell diagrams which have a nice geometric interpretation.

The Sphere Move
The first example of a new move was found by one of the authors in [20], where the 6-point non-planar leading singularity associated with triples T = {(1, 2, 3), (3,4,5), (5, 6, 1), (6,4,2)} was found to be invariant under a cyclic shift of labels, i → i + 1. This property does not follow from square moves as none are possible for this set of triples. Using the results of the previous discussions it is easy to see the cyclic property by writing In other words, the graph given by the u ab 's is the oriented circulant graph C 6 (1, 2) which is clearly cyclic invariant. Let us explain how the square move and the one just discussed are the first two examples of a family of moves.
Consider a triangulation of a sphere. The simplest one gives rise to a tetrahedron. Each triangle, T abc , has an orientation induced by that of the sphere. Let the vertices of the tetrahedron be labelled {1, 2, 3, 4}. The oriented triangles are {T 123 , T 134 , T 142 , T 243 }. Clearly, the boundary of the surface is empty. Using the boundary map ∂T abc = E ab + E bc + E ca together with the identification (3.7) one finds that (3.14) This is nothing but the square move, see  A general triangulation of the sphere has F triangular faces, E = 3F/2 edges and V = n vertices satisfying V − E + F = 2. This gives exactly F = 2(n−2) triangles. One can then split the set of triangles into two disjoints sets of n−2 triangles. Each set gives rise to a list of n−2 triples and therefore to a leading singularity or to something that vanishes. In either case, one finds that both must give rise to the same object, up to a sign, as a consequence of the boundary of the surface of the sphere being empty. This is why we call these families of identities sphere moves. Sphere moves can also be performed locally on a subset of triples from the two on-shell diagrams. This motivates the following definition.
Definition 3.2. Two sets of triples T 1 and T 2 are said to be related by a sphere move of type S, for a triangulation S of a 2-sphere, if where T abc is a triangle corresponding to the triple (a, b, c). In particular, by the identification (3.7) it implies that since the boundary of a sphere is empty. Note that here the number of triangles in S can be smaller than that of T 1 ∪ T 2 , i.e., sphere moves can be performed locally.
We will often say that two sets of triples are related by a sphere move if there exists any S satisfying the above criteria.
At four points the only sphere move is the square move, see . Of course, this can be obtained by a sequence of two square moves and therefore it is not a new identity. Examples of sphere moves beyond the square one start at six points, which is what we will discuss next.

Further Examples of Sphere Moves
We start by giving an infinite family of examples that generalize the sphere move on the octahedral leading singularity in a natural way into bipyramids with more faces. Let us consider the following set of triples for n even: It is related by a sphere move to the following set of triples obtained by exchanging n−1 ↔ n: We illustrate this sphere move in Figure 3.2.
Clearly, one can generate an infinite number of new sphere moves by starting with an arbitrary triangulation (not necessarily invariant under any relabeling) of a sphere with 2(n−2) triangles and dividing it into two sets of n−2, which yields an equality between two leading singularities. We leave such classification questions to future work.
Consider now another set of triples, given by (8,9,10), (10,6,5), (5,7,8), (9, 7, 6)}. (3.23) The difference between two sets of triangles corresponding to T 2 and T 3 is where once again the labels surviving on the RHS are only those from the set {5, 6, 7, 8, 9, 10} and they can also be understood as a triangulation of a sphere. Hence T 2 and T 3 are connected by a sphere move. As a result of this T 1 and T 3 are in fact equivalent. Note that each of them can be obtained by gluing two octahedra by an edge. Hence the corresponding triangles cannot be embedded on a surface of a sphere. Alternatively, one can arrive at the set of triples T 3 from T 1 by a composition of two different sphere moves. Using the intermediate set (3,4,5), (1,7,6), (6, 4, 2), (7,8,9), (9, 10, 6), (1, 5, 7), (8, 5, 10)} (3.25) we find: where we recognize that S is a tetrahedral triangulation of a sphere, i.e., T 1 and T 2 are -15 -connected by a square move in the labels {1, 5, 7, 6}, similar to the one given in (3.14). Let us consider the difference between T 2 and T 3 : T τ 1 ,τ 2 ,τ 3 = T 1,2,3 + T 3,4,5 + T 1,7,6 + T 6,4,2 + T 7,8,9 + T 9,10,6 + T 1,5,7 + T 8,5,10 Here the RHS involves all the available labels and S is a triangulation of a sphere with 10 vertices, 24 edges, and 16 triangles. Hence T 2 and T 3 are connected by a sphere move, which shows equivalence between T 1 and T 3 by a composition of two sphere moves. We illustrate this procedure in Figure 3.4.  Indeed, it can be confirmed directly from the determinant formula (2.6) that the rational functions F T i for i = 1, 2, 3 are all equal to 10 x 67 x 69 x 6,10 x 78 x 79 x 89 x 8,10 x 9, 10 . (3.28) The numerator factors, which is a consequence of the fact that the sets of triples can be obtained by gluing two leading singularities. In fact, in Appendix A we show that this is a generic property of any leading singularity that contains a smaller one inside of it.
- 16 -In this section we present new identities among sets of triples, when realized in terms of ∆ variables. These identities are the lower order analogues of well-known relations between leading singularities. In fact, we show that these correspond to parent identities in the sense that the U (1) decoupling, Bern-Carrasco-Johansson (BCJ) [27] and Kleiss-Kuijf (KK) relations [37] follow almost directly from the low order relations. The scope of these relations between ∆ objects is however far more general. We end section presenting new extensions of the U (1) decoupling identity involving different types of leading singularities. We will concentrate on the study and applications of expressions of the form We will interpret the x i variables appearing in the ∆'s as the location of punctures on a CP 1 . The power j = n−2 of this expression can be taken as an integrand for the CHY construction [33]. Relations among integrands are then mapped to relations among amplitudes in theories which admit a CHY representation.

Decoupling of an Internal Label
We have discussed how different on-shell diagrams can give rise to the same leading singularity function. It is also known that leading singularity functions are not linearly independent. The simplest relation is the so-called U (1) decoupling identity for n = 4. This is the following three-term identity PT(1234) + PT(2314) + PT(3124) = 0.
It is natural to ask for a ∆-algebra interpretation of these kind of relations. Using the following representation of the left hand side in terms of ∆'s reveals the general structure Now it is obvious why this has to vanish. Using the triangulation property (2.16), i.e., we conclude that its square vanishes. In addition to making the identity a consequence of a simple cancellation, this discussion also reveals an important structure. Given the one-element list of triples T = {(123)} its leading singularity is simply ∆ 123 . In (4.3) we found that ∆ 2 123 /2 is also useful for deriving identities. This suggests that one can talk about leading singularities and identities in a -17 -unified way by introducing the following exponential formula (4.5) Indeed, following [20], we propose that associated with any list of triples one should construct an exponential representation so that While it is clear that the coefficient of κ n−2 computes the leading singularity function, the previous example shows that other orders of the expansion contain valuable information as well. We investigate these orders in the next section.

U (1) Decoupling
Let us start with the U (1) decoupling identity. The name of the identity refers to the fact that it is a property of a U (N ) Yang-Mills theory where one particle is taken to be in U (1) ⊂ U (N ). The Lagrangian of a U (N ) theory splits into two parts, one for the U (1) "photon" and one for the SU (N ) "gluons". There is no cross term and therefore no coupling among the photon and the gluons, i.e., the photon is decoupled. This means that the scattering amplitude of n gluons and a photon must vanish, see, e.g., [38] for a review. Let us start with the usual Parke-Taylor interaction of n gluons described at the linear order by the n-gon d 1,2,...,n := u 12 + u 23 + · · · + u n−1,n + u n1 . (4.7) Clearly, the power d n−2 1,2,...,n , integrated over the Grassmann variables, computes the Parke-Taylor factor PT(1, 2, . . . , n) as in (3.9). This combination of signed edges, u ab , can be written in terms of the triangle boundaries ∆ abc in many ways, in particular, one can choose any triangulation of the n-gon.
Now we want to include the photon as the particle n+1 in a way that manifestly decouples. This is done by realizing that any point in the interior of the n-gon can be used to define a triangulation. Since the point is in the interior nothing depends on it and it is "decoupled". Denoting the interior point as n + 1 one finds The (n−1) th power vanishes as the left hand side can be written in terms of n−2 ∆'s, e.g., u 12 + u 23 + . . . + u n−1,n + u n1 = ∆ 1,2,n + ∆ 2,3,n + · · · + ∆ n−2,n−1,n . (4.9) and therefore saturates at the power n−2. Expanding the right hand side of (4.8) gives the identity we are looking for. Note that the right hand side has n ∆'s and therefore the (n−1) th -18 -power is a sum over n terms in which exactly one of the ∆'s is missing. This can be written as d n−1 1,2,...,n,n+1 + d n−1 1,2,...,n+1,n + · · · + d n−1 1,2,n+1,3...,n + d n−1 1,n+1,2,...,n = 0. (4.10) After Grassmann integration each term gives rise to a (n+1)-particle Parke-Taylor factor leading to PT(1, 2, . . . , n, n+1) + PT(1, 2, . . . , n+1, n)+ Given the linear realization of the Parke-Taylor factors defined in (4.7) it is natural to ask if there is an analogue of (4.10) for lower orders. The answer is yes. Indeed, let us now show how to generalize (4.10) to all powers of d's. Denoting d i := d 1,...i,n+1,i+1,...n and using (4.8) it is easy to see that we can write Now consider the sum In the second line we used e κ∆ = 1 + κ ∆. Now, using the fact that d n−1 1,2,...,n = 0 we conclude that the right hand side is at most of order κ n−2 . Extracting the order κ n−1 in the left hand side then recovers the standard U (1) decoupling (4.10). However, we now see that this is just the first of a family of identities that follow from extracting the lower orders in κ. At order κ j we have d j 1,2,...,n,n+1 + d j 1,2,...,n+1,n + · · · + d j 1,2,n+1,3...,n + d j 1,n+1,2,...,n = (n − j) d j 1,2,...,n . (4.14) Because these relations follow from an straightforward application of (4.8) we refer to the latter as the "parent identity" for U (1) decoupling. Next we study similar identities for BCJ and KK relations.

Fundamental BCJ Identity
Here we present the derivation of the fundamental Bern-Carrasco-Johansson relations [27]. In theories which admit a BCJ representation, these relations reduce the number of independent partial amplitudes, corresponding to Parke-Taylor factors, to (n−3)!. Here these identities arise naturally from a generalization of the decoupling procedure at the linear level in ∆.
We are now in position to derive BCJ relations from a generalization of the decoupling identity (4.8). We will need to relate the coefficients α i in the general form (4.1) to kinematic -19 -invariants. The starting point is to note that the decoupling condition of particle n+1 does not require the coefficients of ∆ 1,2,n+1 + ∆ 2,3,n+1 + · · · + ∆ n−1,n,n+1 + ∆ n,1,n+1 (4.15) to be necessarily one. In order to find the most general setup for this to happen, let us consider the linear combination A 12 ∆ 1,2,n+1 + A 23 ∆ 2,3,n+1 + · · · + A n−1,n ∆ n−1,n,n+1 + A n1 ∆ n,1,n+1 = I n+1 + Γ n+1 , (4. 16) where have decomposed the LHS into two parts. In the term Γ n+1 we have collected the dependence on particle n+1, i.e., whereas the rest reads I n+1 := A 12 u 12 + A 23 u 23 + · · · + A n−1,n u n−1,n + A n1 u n1 . (4.18) The linear expression (4.16) will be the parent identity generating BCJ relations (and other new relations among CHY integrands), provided the coefficients A i,i+1 satisfy a decoupling condition. Here we will be mainly interested in the combination (4.16) raised to its (n−1) th power. When computing the leading singularity the only non-trivially zero terms are those for which the Grassmann variables saturate after the gauge fixing term is included. In other words, in the expression (A 12 ∆ 1,2,n+1 + A 23 ∆ 2,3,n+1 + · · · + A n−1,n ∆ n−1,n,n+1 + A n1 ∆ n,1,n+1 ) n−1 θ d θ e x de the only potentially non-zero terms are those proportional to n+1 i=1 θ i χ i . If one chooses d, e, f, g to be different from n+1, then the only source of θ n+1 χ n+1 is the combination Γ n+1 defined in (4.17). In the following we will thus consider the expansion: Clearly, the first term in the left hand side vanishes after Grassmann integration as it does not have any factors of θ n+1 χ n+1 . We anticipate that each order in Γ n+1 will give us a relation among Parke-Taylor factors and other CHY integrands. Let us start by considering the linear part, i.e., the second term in (4.20). The combination Γ n+1 must provide all the dependence in θ n+1 χ n+1 and therefore we can drop all other Grassmann variables inside it. Using that u ab = θ ab χ ab /x ab one finds -20 -that the relevant piece of Γ n+1 is This means that in order to have an identity among Parke-Taylor functions it is necessary for this to vanish. More explicitly, if Clearly, A ab = 1 is a solution to (4.22) and gives rise to the standard U (1) decoupling identity since Γ n+1 vanishes identically. This system can be written by performing the change of variables A a,i+1 − A a−1,i = s n+1,i , in which case it reads One can easily express the A a,a+1 variables in terms of s n+1,a as follows Note that this requires n i=1 s n+1,i = 0. In the context of CHY formalism [33] the condition (4.24) is precisely the scattering equation for label n + 1. For m = n + 1 massless particles, the scattering equations relate punctures in CP 1 , located at x a , a = 1, 2, . . . , m, to kinematic invariants. The latter are nicely encoded in a symmetric matrix with components s ab , whose diagonal elements are set to zero by the on-shell conditions. The space is m(m−3)/2 dimensional as one further imposes m momentum conservation conditions m b=1 s ab = 0, removing in total 2m variables from the m(m+1)/2 elements in a generic symmetric matrix. The scattering equations then fix the positions of the punctures x a by imposing We have found that only the momentum conservation condition and the scattering equation corresponding to the (n+1) th particle is needed for the decoupling of such label. Given -21 -such conditions we would like to derive relations among Parke-Taylor factors from (4.23). This requires us to find a way to ignore O(Γ 2 n+1 ). In order to achieve this it is necessary to introduce an expansion parameter. This is done by the replacement s ab → s ab and declaring that A n1 = 1. Clearly Γ n+1 = O( ) and an expansion around = 0 of (4.23) leads to relations for the leading and next-to-leading order terms.
The leading order term gives A ab = 1 and becomes the U (1) decoupling identity. The next to leading order term becomes n−1 i=1 s n+1,i PT(1, 2, . . . , n−1, n+1, n) + · · · + s n+1,1 PT(1, n+1, 2, . . . , n) = 0, (4.27) which is the fundamental BCJ relation [27] at the level of Parke-Taylor factors. In [39] this relation was in fact shown to hold on the support of the scattering equations. 3 As we extract higher orders in , we will see that the higher powers of Γ n+1 in the expansion (4.20) do not vanish. Instead, it turns out that only the vanishing of the linear term, imposed through the scattering equation (4.24), is enough to obtain a new family of relations between the Parke-Taylor factors in (4.23) and multi-trace-like objects.

Beyond BCJ Relations
The derivation of the U (1) decoupling identity and the fundamental BCJ relations presented in the previous section motivates the study of higher orders in the expansion. Here we will illustrate with this the second order identity, i.e., O( 2 ), obtained from the expansion, and we will leave the study of higher orders to future work.
Here instead let us present a five-point example that can be explicitly checked. Taking n=4 produces: Let us now provide an interpretation for these new identities. It is well-known that the fundamental BCJ relations exhausts the relations among Yang-Mills partial amplitudes. However, the string theory derivation of the identities which involves moving a vertex operator around the boundary of a disk [28,29], implies and infinite number of other identities involving higher-derivative terms in the EFT expansion of the open string. In string theory, the monodromy picked up by passing one vertex operator over another is an exponential of s ab , with identified with α . Expanding in gives the U (1) decoupling and BCJ identities as the first two others. The O( 2 ) contains amplitudes involving TrF 4 terms, whose CHY formulation was found by He and Zhang in [43]. Let us write down the monodromy relations for these CHY integrands, which hold on the support of the scattering equations: where the ellipsis on the RHS stands for all the insertions of particle label n+1 in the ordered set (1, 2, . . . , n). Note that the cross ratio on the RHS is in fact a product of the two soft factors (Λ (p) ) −1 qr Λ (r) sp , with respect to labels p and r respectively. We have seen that the ∆-algebra parent formula provides a different representation of the monodromy identity. In particular, (4.28) exhibits the same soft behaviour on the particle n+1 on both sides of the equation. We leave the discussion of higher orders in s ab as well as their explicit equivalence to monodromy relations for future work.

KK Relations and Identities for Shuffle Sums
There is another direction in which the U (1) decoupling identity can be generalized. It is well-known that at the level of Parke-Taylor factors U (1) decoupling is the simplest example of a more general set of relations known as KK identities. In its standard form, the KK  The shuffle product α¡β T corresponds to all the permutations of the labels in {α}∪{β T } such that the respective orderings of α and β T (i.e., the transpose permutation of β) are preserved. One can now ask how the sum over the permutations in α ¡ β T is realized at the level of sums of triples. It is the goal of this section to provide a linear version of (4.32), as well as other useful identities at various orders in ∆'s. The relations (4.32) will then be a direct consequence of such parent identities. In order simplify the notation, let us take α = (2, 3, . . . , m) and β = (m+1, m+2, . . . , n−1). Such orderings correspond to the following object at the linear level: d := d 1,2,...,m,n,m+1,...,n−1 =∆ 123 + . . . + ∆ 1mn + ∆ 1,n,m+1 + . . . + ∆ 1,n−2,n−1 . (4.33) We will also denote by d ω i the shuffles of d, in the case where ω i is a permutation in (2, . . . , m) ¡ (n−1, . . . , m+1). In this setup, the realization of the KK relations (4.32) is We are now in position to ask if there exists any identity at linear order between {d ω i } and d giving rise to the physical KK relation, in an analogous way to the U (1) and BCJ relation. To see this, let us split the RHS of (4.33) as = ∆ 123 + . . .+∆ 1mnˆ = ∆ 1,n,m+1 + . . . + ∆ 1,n−2,n−1 , (4. 35) i.e., d = +ˆ . Note that these objects each takes (m − 1) and (n − m − 1) triangles, respectively. It turns out that for each shuffle ω one can define an induced splitting of d ω , that is We show in Appendix C that each simplicial complex in ω (respectivelyˆ ω ) is composed of m−1 (respectively n−m−1) triangles, and that we have Furthermore, we also show that the following relations between powers of ω hold: -24 -Let us first focus on the linear relation (4.37). Using the decomposition (4.36) we find This is the reflection of the KK identity (4.34) at the linear order in ∆. For instance, for n = 6 and m = 3 this identity reads: Finally, let us close this section by giving a direct derivation of the KK relations at the level of Parke-Taylor factors, as following directly from their parent identities. We can multiply both sides of the first equation in (4.38) byˆ n−m−1 to give: Here we have used thatˆ is composed of n − m − 1 triangles and henceˆ n−m = 0.
Finally, we can use (4.36) together with m ω i = 0 to write which is the identity (4.34) we wanted to prove.
Consider now the octahedral leading singularity and try to obtain a new relation by reversing the previous procedure. The first step is to use the equivalence of its two presentations in triples, The resulting identity expresses the octahedral leading singularity, first term in the sum, as a sum of four Parke-Taylor factors, each of them with a soft factor attached.
These two examples can be further generalized as follows.
Consider any given set of n−2 triples T 1 and find a second set of m ≥ n − 2 triples T 2 Note that when T 1 does not allow any moves m is necessarily larger than n−2.
By taking the (n−2) th power of this parent identity or some rearrangement of it (obtained by moving terms from the RHS to LHS) we obtain new identities. Notice that when the LHS is taken to the (n−2) th power, one of the terms will always correspond to the leading singularity of T 1 .
Let us illustrate these general identities with an example involving a singularity that does not admit any moves. Consider the set of triples depicted in Figure 4.1: (1,4,7), (7,6,3), (4,5,6), (2,7,5)} , (4.51) Since this object is rigid it must be that we require m > 5 to produce relations. It turns out that it is necessary to consider at least m = 7 triples. The choice is not unique. For example, using (1,4,5), (7,5,6), (7, 6, 3), (7,3,2), (4,7,6), (1, 3, 7)} (4.52) it is easy to check that Taking the fifth power of LHS gives F T 1 , whereas the fifth power of RHS leads to 9 2 = 36 terms, each of them being a leading singularity. This 37-term identity can be replaced by a 12-term identity by simply moving ∆ 467 from the RHS to the LHS, i.e. (4.54) This time the fifth power only gives six terms on the LHS and five on the RHS (one of them is zero). It can be checked that each of the new terms simply corresponds to Parke-Taylor -27 -factors with soft factors attached. This gives a presentation of T 1 in terms of 10 simple leading singularities. Note that T 1 is the unique, up to relabeling, irreducible leading singularity at seven points [44]. Let us finally point out that the (unoriented) set of triples in T 1 contains a G S 3 Z 2 symmetry. The  Even though T 1 is invariant under these transformations, T 2 is not. Therefore one finds different representations of the identity (4.54) given by the action of the group on the labels of T 2 . If we were to draw on-shell diagrams for the different sets of triples associated with T 2 and its images we would then find diagrams related via moves. Of course, these on-shell diagrams do not have the correct dimension to be leading singularities. We leave the study of such moves for future work.

N k−2 MHV Leading Singularities and the ∆-Algebra
In this section we turn to the generalization of the ∆-algebra to helicity sectors other than MHV. A general on-shell diagram describing a leading singularity in the N k−2 MHV sector must satisfy the following condition α + β = 2(k−2), where α is the number of white vertices that are not connected to any external label either directly or via a path only involving white vertices, while β is the number of edges that connect two black vertices [3]. It is now clear why the k = 2 sector is special. All MHV diagrams must have α = β = 0 and this leads to the description in terms of lists of n−2 triples of labels. Already the k = 3 sector allows for several possibilities, since α + β = 2. One attempt to simplify the general problem is to restrict to diagrams with α = 0. In this case one can contract all clusters of trivalent black vertices into higher-valent black vertices and associate with them a tuple of labels. For example, if k = 3 then β = 2 and there are two possibilities. The first is to have two 4-valent black vertices and n−5 trivalent black vertices. The second is to have one 5-valent black vertex and n−4 trivalent black vertices.
Having sets of labels of different length seems to prevent the development of a ∆-algebra formulation. Luckily, Franco, Galloni, Penante, and Wen [30] showed that at the level of the integrand in the Grassmannian formulation of scattering amplitudes [1] it is possible to work with only (k+1)-tuples of labels and lists containing exactly n−k of them. The price to pay is to abandon a direct connection to on-shell diagrams. The reason is that one has to work in the top cell of G(k, n) which has dimension k(n−k), while leading singularity on-shell diagrams live in 2(n−2)-dimensional cells. However, after computing with (k+1)-tuples, one can go to boundaries of G(k, n) of dimension 2(n−2) and make connection to leading singularities.
We will not discuss this last step and instead concentrate on the ∆-algebra related to the integrand.
The two-components objects λ a are replaced by k-component ones Z a ∈ P k−1 and Plücker coordinates a, b are replaced by 1, 2, . . . , k := det(Z 1 , Z 2 , . . . , Z k ). Given a list of (k+1)tuples the determinant formula (1.3) can be generalized to [30]: where |a 1 , a 2 , . . . , a k+1 | = a 1 , a 2 , . . . , a k a 2 , a 3 , . . . , a k+1 · · · a k+1 , a 1 , . . . , a k−1 . Let us denote the product of ∆'s in the set T by so that a 2 · · · θ (r) a k a 1 , a 2 , . . . , a k (5.7) for an arbitrary choice of labels a i . Of course, this is an abuse of notation, since in order to obtain a physical leading singularity associated with an on-shell diagram one has to not only perform the integration over θ (r) i 's, but also reduce the dimension of the cell in G(k, n) down to 2(n−2).
Other types of identities can be obtained by moving ∆'s from RHS to LHS of (5.11) and squaring both sides (for even k), or alternatively multiplying the whole equation by one of the sides. It turns out that k=2 is the only case for which this operation relates a single leading singularity to another one. This is because only for k=2 the ∆'s can be partitioned into two on the LHS and two on the RHS, which after squaring gives an equality between two leading singularities: the square move. For instance, (5.11) for k = 4 implies that the ∆'s satisfy the identity (the 3−3 bistellar flip): ∆ 12345 + ∆ 26345 + ∆ 61345 = ∆ 12645 + ∆ 12365 + ∆ 12346 , (5.14) where we also used antisymmetry under exchanging labels, ∆ abcde = −∆ bacde . This identity is very familiar in the context of NMHV amplitudes in momentum twistor space. In fact, identifying our variables Z a with momentum twistors and θ (r) a with the Z a superpartner η I a , one finds that ∆ abcde is exactly what is known as the R-invariant R abcde . Moreover, (5.14) is nothing but the equivalence of the six-point NMHV amplitude when computed using the BCFW technique or its parity conjugated version.
The straightforward analogy with R-invariants somehow breaks down when we square both sides of (5.14) in order to get a relation among k = 4 leading singularities. 5 The identity is then ∆ 12345 ∆ 26345 + ∆ 12345 ∆ 61345 + ∆ 26345 ∆ 61345 = ∆ 12645 ∆ 12365 + ∆ 12645 ∆ 12346 + ∆ 12365 ∆ 12346 (5.15) and it is the analogue of the square move identity ∆ 123 ∆ 134 = ∆ 234 ∆ 241 at k = 2. Similarly, squaring both sides of (5.12) reveals that u's satisfy u 1234 (u 2345 + u 3451 ) + cycl. = 0, (5.16) which is the analogue of the identity u 12 u 23 + cycl. = 0 at k = 2. It is clear that there is much more to be explored in the higher-k extension of the ∆algebra. A particularly interesting direction is the recent use of volumes of k-simplices in the computation of φ 3 biadjoint scalar amplitudes in the context of their corresponding amplituhedron, i.e., the associahedron in the Mandelstam space [46,47]. We leave the exploration of consequences of the above algebras for future work.

Discussions and Future Directions
In this work we introduced the ∆-algebra as a natural structure in the construction of MHV non-planar leading singularities. Physical applications of the algebra structure range from manifesting symmetries, such as cyclic invariance in the Parke-Taylor case, to revealing new equivalences of on-shell diagrams not connected via square moves. Also, identities involving several leading singularities found a natural description in terms of parent (or linear) relations in the algebra. It is clear that we only scratched the surface of the structure. In particular, 5 Recall that by this we mean the integrand of the Grassmannian integral formulation of the object.
-31 -extensions of identities to k > 2 and applications to loop integrands, as well as less supersymmetric [48] and gravity on-shell diagrams [49,50] are some of the most pressing directions. Another direction which is hard to overlook is to find possible connections to the triangulation of the m = 2 amplituhedron for general k as discussed in [8,51] where a structure of the form "(polygon) k " was found.
In addition, there are some directions for future developments for which we can provide more detailed explanations.

Comparison with (Combinatorial) BCFW Relations
It is well-known that it is possible to obtain relations among leading singularities by attaching a BCFW-bridge to one of them and using the residue theorem to get the others [52,53]. A BCFW-bridge is a 4-particle diagram with a single black vertex connected to a single white vertex (see Chapter 17 of [1] for a review). It is not a leading singularity but it is useful when two of the four legs are connected to a complicated leading singularity. In fact, it can be shown that this procedure decomposes any MHV leading singularity into a sum of leading singularities with at least one soft factor attached. Here we compare the identities we obtained by moving ∆'s in Section 4.3 with those from a purely combinatorial version of the BCFW procedure when applied to a list of triples T .
Let us start by explaining the combinatorial BCFW identity related to a BCFW-bridge (1, n).
Given a list of triples T , identify all those where label n enters and make a list T n . For each triple in T n , say (nab), create the following two lists: Note that the orientation of the triples (1na) and (1bn) was chosen to match that of the shared edges with the original triple (nab). Also, when either a=1 or b=1, the corresponding contribution is defined to vanish, i.e., T n1 := 0. The new lists either correspond to new leading singularities or to zero. The combinatorial BCFW identity is then The simplest example is the four-particle U (1) decoupling identity. In our language we start with ∆ 123 = ∆ 124 + ∆ 234 + ∆ 314 . In the BCFW framework, we can attached a BCFW-bridge (2,4) to PT(1234). Following the combinatorial procedure lead us to (6.5). This coincidence of results turns out to be an accidental coincidence. One can check that there is not any single BCFW-bridge that can be attached to the leading singularity {(123), (345), (561), (642)} that gives rise to the identity found in Section 4.3 by moving a triangle, i.e., Here we used that It is clearly a very important question to find out how to connect these two constructions. Moreover, it is hard to overlook the fact that the combinatorial BCFW relation involves a procedure very reminiscent of the circuit elimination axiom of a matroid [54].

Gauge Redundancies in ∆ abc and Differential Form Interpretation
The realization of the ∆-algebra discussed in this paper is based on that for the building block However, this representation is not unique, there are redundancies similar to gauge transformations, for example, u ab → u ab + αθ a θ b (6.11) -33 -for any constant α leaves all properties of the algebra invariant. Under this transformation Now, if both u ab and ∆ abc are not gauge invariant, what is the physical observable? The answer is, of course, the rational function associated to a leading singularity. More precisely, The fact that there are gauge redundancies suggests that an interpretation as differential forms could be the link to an even more geometrical picture. In fact, as mentioned in the introduction, the object ∆ abc defined in homogeneous variables becomes the 2-form Ω abc under the Grassmann variable ↔ (1-form) identification used by He and Zhang [4]. It is interesting that when Ω abc is written in inhomogeneous variables the form that appears is a gauge transform of ∆ abc (x). Let us show this more explicitly. Let us parametrize holomorphic spinors in the following way: Let us apply this to Ω ijk , which we rewrite below for the reader's convenience, Using the above change of variables we find that it decomposes as: Under the identification (θ a , χ a ) → (dx a , dx a ) we find a gauge transform version, with α = 1, of the representation we used. Now we are guaranteed that the 2-forms defined above also furnish a representation of H * Conf n (R 3 ), i.e., Note that if we were to identify (x a ,x a ) → (z a ,z a ), then U ij would be a combination of a (1, 1)-form and a (0, 2)-form. However, the (0, 2) part is pure gauge. It would be interesting to explore this further, especially in view of the following discussion.

det Φ(x, y) and u ij
There is an object that made its appearance for the first time in a twistor formulation of gravity amplitudes introduced by one of the authors and Geyer in [55] and which admits an -34 -interesting formulation in terms of u ij . This is the reduced determinant of It is easy to show that if x's are any solution to the scattering equations then n a=1 x m a Φ ab (x) = 0 for m = {0, 1, 2}. (6.19) This implies that Φ is a n × n symmetric matrix of corank 3. The object of interest for this part is a generalization of Φ introduced in [56] to prove a property known as KLT orthogonality. The object is defined using the matrix In order to obtain a non-zero determinant it is necessary to remove four rows and four columns. However, for our purposes it will be sufficient to consider the submatrix of Φ(x, y) obtained by deleting rows i, j, k and columns p, q, r. Clearly, the determinant vanishes unless x a and y a are the same solution to the scattering equations. Let us denote by I x and I y the label in {1, 2, . . . , (n−3)!} corresponding to the solutions x and y belong to. Therefore one finds that the reduced determinant is det Φ(x, y) = detΦ pqr ijk (x, y) x ij x jk x ki y pq y qr y rp = δ Ix,Iy det Φ(x, x). (6.22) The Grassmann integral formulation is Once again it is possible to rewrite this in term of u ij as follows up to an irrelevant numerical factor. Following the spirit of this work in the way leading singularities were associated with the product of ∆'s it is natural to propose that det Φ(x, y) must likewise be associated with Moreover, as proven by the second author in [20], any linear combination of u ab 's with antisymmetric coefficients c ab that satisfy a c ab = 0 can be written in terms of ∆ abc . Identifying c ab with s ab /y ab and recalling that the y's satisfy the scattering equations one finds n a,b=1 b =a for some set of triples T and coefficients α τ . It would be interesting to explore this connection further.

A Reducible Leading Singularities
Recall from (2.6) that an n-point MHV leading singularity has the associated rational function F T computed as follows: where the labels a, b, c, d can be chosen arbitrarily.
We can ask what happens if T contains a subset of triples T ⊂ T , which itself forms a leading singularity, i.e., Note that here the complementary set T \ T does not necessarily give a leading singularity on its own. Let us say, without loss of generality, that T contains m−2 triples constructed out of the labels {1, 2, . . . , m} for m < n. Plugging the decomposition (A.2) back into the expression for the rational function (A.1), we find: since Grassmann integrals associated to the labels {1, 2, . . . , m} are saturated in the left factor. We also chose the arbitrary labels a, b, c, d to be taken out of the same set. The rational function factors and the left factor is precisely F T . In the right factor we set the Grassmann variables from the set {1, 2, . . . , m} to zero. Since this statement holds on the level of the rational functions, it is clearly independent of a given representation in terms of sets of triples. This extends the notion of reducibility of [44] and motivates the following definition.
Definition A.1. A set T of n−2 triples is called reducible if there exists a proper non-empty subset T T of m−2 triples consisting of exactly m > 3 labels.
Thus, as above, the rational function F T contains F T as a factor. Here we excluded the trivial case where m = 3. Note that by the same logic, if there exists a subset of m−2 triples T ⊂ T , which contains fewer than m labels, then LS T = 0 and hence also LS T vanishes. For example, any leading singularity containing two repeated triples T = {(a, b, c), (a, b, c)} vanishes.
In the following discussion we give examples of how to construct reducible leading singularities by gluing other ones.

A.1 Soft Factor
First, let us consider the simplest case in which m = n−1, i.e., removing one triple from a leading singularity leaves us with another leading singularity. The complementary set is necessarily of the form where y, z ∈ {1, 2, . . . , n−1}.
Hence the right factor in (A.3) becomes: This is nothing but the so-called soft factor [31] glued onto the leading singularity LS T by the legs y and z.

A.2 Gluing by an Edge
The above soft factor can be alternatively understood as gluing of a single triple, say, {(n+1, n+2, n)}, to the original set of triple T by identifying n+1 with y and n+2 with z. Let us use the notation: T (y,z) to define the result of this operation. It is in fact the simplest example of a more general gluing operation, which takes two sets of triples T 1 and T 2 with n 1 −2 and n 2 −2 elements respectively, and identifies a pair of labels (y, z) from the first set with a pair of labels (v, w) from the second. The result is denoted by: (A.7) It consists of n 1 +n 2 −4 triples and involves exactly n 1 +n 2 −2 labels. Therefore it is a valid leading singularity. For the above gluing operation, the right factor in (A.3) simplifies as follows: where we assumed that prior to gluing T 2 consists of labels {n 1 +1, n 1 +2, . . . , n 1 +n 2 }. The equality was obtained by multiplying by Hence the resulting leading singularity T 3 from (A.7) has a rational function factorizing as a product of the two rational functions of T 1 and T 2 : Let us consider an example in which we can take two copies of the octahedral set of triples: (3,4,5), (5, 6, 1), (6, 4, 2)}, (7,8,9), (9,10,11), (11,12,7), (8, 12, 10)} (A.11) and attach them by legs respectively 5 ↔ 12 and 6 ↔ 11. The resulting leading singularity is determined by the data: The corresponding rational function is given in (3.28).

A.3 Gluing by a Triangle
A natural generalization of the above procedure is to glue two sets of triples by a triangle. Let us identify the triangle (y, z, t) from T 1 with a triangle (u, v, s) from T 2 and denote the result with In this case, by using the identity 1 = x yz x zt x ty dθ y dθ z dθ t dχ y dχ z dχ t ∆ yzt θ y θ z χ y χ z x 2 yz . (A.14) it is straightforward to show that the right factor in (A.3) simplifies and the resulting rational function becomes: For instance, we can consider the triples from (A.11) glued by the triangles (5, 6, 1) ↔ (11, 12, 7) yields T 1 (5,6,1) • (11,12,7) T 2 = {(1, 2, 3), (3,4,5), (5, 6, 1), (6,4,2), (1,8,9), (9,10,5), (8, 6,  More generally, one can glue two leading singularities T 1 and T 2 by a common leading singularity, if there exists one. It is straightforward to show that the resulting rational function always factors into a product of rational functions F T 1 F T 2 times a factor that depends on the gluing object.
Here we think of the RHS as Grassmann integrands, i.e., top forms in θ and χ.
We now concentrate on the binomial expansion of the LHS, where the ellipsis represents higher order terms as well as terms independent of θ n+1 χ n+1 .
Proof. The first step is to construct the triangulations of ω i andˆ ω i . For this we note that each ω i represents a path from label 1 to label n in the "two-route" arrangement of Figure C.1. Each path passes through all the labels, and covers the labels in each route in a consecutive order. The shaded region we call the "interior" (exterior) of the path, and we associate the complex ω i (ˆ ω i ). It is easy to check that constructed in this way ω i consists of m − 1 triangles. Let us denote by {∆ We are now in position to prove our main identities. Define the function: where d n = +ˆ . Note that such function can be expanded explicitly in terms of, in principle, all the possible triangles {∆ j } with a side laying on one of the routes, since we can always triangulate ω i using such. Note also that f n,m ({∆ (i) j }) is of uniform Grassmann degree, and thanks to the nilpotency of ∆'s it is at most linear in each ∆ j . Hence, it is sufficient to show that the coefficient of each ∆ j vanishes to argue that f n,m ({∆  any set of triangles. Consider then a given ∆ ( * ) that is contained among a subset of paths P ∆ ( * ) = { i } i∈I . We can single out the contribution from ∆ ( * ) in the sum over paths as Now, it is easy to check that removing ∆ ( * ) from a path i ∈ I defines a new path for n − 1 labels, whose interior contains m − 2 triangles (see Figure C We can now see that the coefficient of the second term precisely cancels the contribution from the path sum, hence f n,m = 0. This concludes the proof.
As a corollary of our construction, we note that, given a triangulation of the region d n , each of the triangles will appear |I| = n−3 m−2 times when summing over all possible paths, that is This proves our formula (4.37). It is interesting to note that the idea of the proof can be extended to a general piece (i.e., subset of triangles) of Ω n , say Ω n = R +R. If R has k triangles then Ω m−1 n = m−1 k R kRm−1−k + . . . (lower orders). The binomial in the LHS of the identity is unique in that it accounts for all the combinatorial factors associated to singling out R k in the path sum.