Scattering Equations: From Projective Spaces to Tropical Grassmannians

We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ${\mathbb{CP}^1}$, to higher-dimensional projective spaces $\mathbb{CP}^{k-1}$. The standard, $k=2$ Mandelstam invariants, $s_{ab}$, are generalized to completely symmetric tensors $\textsf{s}_{a_1a_2\ldots a_k}$ subject to a `massless' condition $\textsf{s}_{a_1a_2\cdots a_{k-2}\,b\,b}=0$ and to `momentum conservation'. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the $k=3$ case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all `biadjoint amplitudes' for $(k,n)=(3,6)$ and find a direct connection to the tropical Grassmannian. This leads to the notion of $k=3$ Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for $k=2$, and provides analytic solutions analogous to the MHV ones.


Introduction: Generalizing the Potential Function
Scattering equations connect the space of Mandelstam invariants of n massless particles to that of n points on CP 1 [1]. They can be obtained as the conditions for finding the critical points of the "potential" function S := For S to be a function of points on CP 1 , it has to be invariant under the equivalence relation of projective space (σ a,1 , σ a,2 ) ∼ t a (σ a,1 , σ a,2 ) for every a. This is achieved provided These conditions have the physical interpretation of momentum conservation. Note that we have not used the diagonal components s aa , which in physical applications are taken to be zero for massless particles. Working in inhomogeneous coordinates x a := σ a,2 /σ a,1 critical points of S are found by requiring ∂S ∂x a = n b=1 b =a s ab x a − x b = 0 ∀a, (1.4) which are known as the scattering equations [1]. The scattering equations are at the heart of the Cachazo-He-Yuan (CHY) formulation [2,3] of scattering amplitudes of a large variety of theories and have made manifest properties such as the Kawai-Lewellen-Tye (KLT) [4] and Bern-Carrasco-Johansson (BCJ) relations [5]. This is one of the motivations for finding natural generalizations. Moreover, generalizations to spaces other than CP 1 can help in better understanding the original equations and in finding new physical applications.
For instance, a large generalization of scattering equations was introduced by one of the authors in [6] and later used to study algebraic properties of multi-loop Feynman integrals [7,8]. Earlier exploration into constructing a higher "Möbius spin" extension of scattering equations was taken in [9].
In this work we consider another natural generalization: from points on CP 1 to points on CP k−1 . The standard case has been chosen to correspond to k = 2 for historical reasons. In physical applications the scattering equations are used to integrate functions on the moduli space known as Parke-Taylor functions [10], which are the simplest examples of what are known as leading singularities [11]. In recent work [12], we considered generalization of leading singularities from CP 1 to higher-dimensional projective spaces in terms of so-called ∆-algebras. Such higher-k leading singularities were first introduced by Franco et al. [13] and can be used to construct general non-planar on-shell diagrams in N = 4 super Yang-Mills [14]. On CP k−1 the corresponding SL(k, C) invariants are determinants of the homogeneous coordinates of k points (a 1 , a 2 , . . . , a k ). It is then natural to introduce the potential function S k := 1≤a 1 <a 2 ···<a k ≤n s a 1 a 2 ···a k log (a 1 , a 2 , . . . , a k ). (1.5) Once again, requiring S k to be independent of the scaling of each point imposes conditions on the generalized Mandelstam invariants s a 1 a 2 ···a k . It is easiest to express the condition by completing s a 1 a 2 ···a k into a symmetric rank k tensor, n a 2 ,a 3 ,...,a k =1 a i =a j s a 1 a 2 ···a k = 0 ∀a 1 . (1.6) It is tempting to think about the tensor s a 1 a 2 ···a k as the multiparticle generalization of s ab given by the norm of the sum over the corresponding momentum vectors. As it turns out, while multiparticle invariants satisfy (1.6), they are not the most general solution and therefore we do not specialize to them as doing so would lead to singular configurations. In order to make this manifest we use a different font. Of course when k = 2 we can write s a 1 a 2 = s a 1 a 2 .
The CP k−1 scattering equations are then given by the conditions for finding critical points of S k , where x (i) a represent inhomogeneous coordinates of the puncture a on CP k−1 . In this work we initiate the study of these equations.
We denote the moduli space on which the scattering equations (1.7) are defined by X(k, n). It can be written as a quotient of the Grassmannian G(k, n) by the n-torus action, X(k, n) := G(k, n)/(C * ) n . (1.8) Since the diagonal torus action is redundant, the complex dimension of X(k, n) turns out to be (k−1)(n−k−1). Understanding the boundary structure of X(k, n) in general proves to be a difficult mathematical problem [15][16][17][18][19][20][21][22]. Hence we focus mainly on the case k = 3 with n ≤ 6. We start in Section 2 by defining the analogues of the CHY formulae for k = 3 and computing the associated "amplitudes" based on a natural generalization of Parke-Taylor factors. We find that they are rational functions with poles in the kinematic invariants s ijk , as well as more complicated linear combinations of s ijk . We then proceed by identifying singular configurations of points on CP 2 associated to these kinematic poles, for example when multiple points become simultaneously collinear. In Section 3 we show that there exists a duality between scattering equations on X(k, n) and X(n−k, n), which follows from the corresponding duality on the Grassmannian.
In the case of k = 3 we find a surprising relation to the so-called tropical Grassmannian [23], which seems to govern the space of kinematic singularities allowed by the CHY formalism on X (3,6). This allows us to associate a set of "Feynman diagrams" to each planar ordering α of 6 labels, and understand generalizations of biadjoint scalar amplitudes m (3) 6 (α|β) as a sum over the diagrams compatible with both permutations α and β at the same time. This is described in Section 4.
In Section 5 we provide an interpretation of the generalized kinematic invariants s a 1 a 2 ···a k as coming from variables similar to the standard spinor-helicity formalism for k = 2 in four dimensions. Using this kinematics we are able to prove that for any k and n there exist special classes of analytic solutions to the scattering equations akin to the MHV and MHV sectors in four dimensions.
In [24] it was shown that there exist large kinematic regions in which all the solutions of the standard scattering equations are real and bounded. We generalize this construction to k = 3 cases by interpreting (the real part of) S 3 from (1.5) as a potential for interacting particles on RP 2 . We discuss limitations of this procedure in predicting the number of solutions of the general-k scattering equations due to the fact that different soft kinematic regimes are separated by new singularities appearing at k = 3.
We conclude with a discussion of results and future directions in Section 7. We also include two appendices. In Appendix A we prove the number of solutions to the scattering equation by computing the Euler characteristic of X (3,6), while in Appendix B we give a lower bound for the number of solutions for general n using soft limits.
2 Scattering Equations on CP 2 : Jacobians and Amplitudes In this section we specialize to k = 3 in order to carry out explicit computations and build intuition by developing some of the same tools already known for k = 2. Let us simplify the notation by denoting inhomogeneous coordinates on CP 2 by (x, y). This means that the potential function is (2.1) The scattering equations are then Here we have introduced the shorthand notation |abc| for the determinant in S 3 . At first sight, these are 2n equations for 2n variables. However, as it is familiar in the k = 2 case, these equations are covariant under SL(3, C) transformations which is the automorphism group of CP 2 . This means that 8 equations are redundant and that we can use the group to fix the positions of 4 points (each having two coordinates) to generic, i.e., non-collinear, positions.
This makes it clear that n ≥ 4 in order to have a stable CP 2 , i.e., one in which all the automorphism group is fixed. Recall that when k = 2, this is equivalent to the statement that at least three points are needed in order to have a well-defined set of equations and develop the CHY formalism.

Jacobian Matrix
The next key ingredient in the CHY formalism of scattering amplitudes is the Jacobian matrix associated with the system of scattering equations. Once again, let us review the k = 2 case before discussing the k = 3 case.
The Jacobian matrix is a n × n symmetric matrix with components Φ (2) ab := ∂ 2 S 2 /∂x a ∂x b . It is well-known that this matrix has rank n − 3 due to the SL(2, C) action, and therefore one has to defined a reduced determinant where the Vandermonde determinants are defined as (2)abc pqr is a (n−3) × (n−3) matrix obtained by deleting rows i, j, k and columns p, q, r of Φ. It is easy to show that the right-hand side is independent of the choice of rows and columns to delete and this is why det Φ (2) is well-defined.
Moving on to k = 3, one has that the Jacobian matrix is a 2n × 2n matrix with a block structure. The n×n blocks have components Φ Once again the matrix is singular and it has rank 2n−8 due to the SL(3, C) action. The reduced determinant in this case is obtained in a completely analogous manner, pqrs is the matrix obtained from Φ (3) by deleting rows {i, i+n, j, j+n, k, k+n, l, l+n} and columns {p, p+n, q, q+n, r, r+n, s, s+n}. The generalized Vandermonde factors are defined as V ijkl := |ijk||jkl||kli||lij|. (2.5)

Generalized Biadjoint Amplitudes
The simplest scattering amplitudes that admit a CHY representation are those of the biadjoint scalar theory [3]. In this k = 2 construction the amplitudes depend on the choice of two cyclic orderings α and β and are computed as where Parke-Taylor functions are defined as and the pre-factor containing vol(SL(2, C)) is there to indicate that the integral as such is not well-defined and the SL(2, C) redundancy has to be fixed. This is done by fixing three coordinates, say x i , x j , x k and removing three delta functions, say p, q, r. There is a Fadeev-Popov factor that is generated and it is given by V ijk V pqr .
Explicitly evaluating the formula on the solutions x (I) a to the scattering equations gives (2.8) The simplest amplitude is for n = 3. In that case det Φ (2) = 1/V 2 123 and m 3 (α|β) = 1. The first non-trivial amplitude is for n = 4. A simple computation reveals that, e.g., Let us also give a result for n = 5 as this will be useful in the k = 3 discussion, m  Once again, the factor vol(SL(3, C)) is there to indicate that the corresponding redundancy must be fixed before attempting the integration. After fixing the redundancy one finds (2.14) Here N 4 (α|β) = 1. The first nontrivial amplitudes are for n = 5. We start with this case and then move on to n = 6. This correspondence between the k=3, n=5 case and the k=2, n=5 is not an accident. In Section 3, we prove that solutions and amplitudes for (k, n) map to those for (n−k, n) as a consequence of the isomorphism between the Grassmannians G(k, n) and G(n−k, n).

Case II: k = 3 and n = 6
Considering n = 6 and k = 3 produces new objects but still with similar features to those of n = 6 and k = 2. We have computed several explicit examples.
The first class is the set of orderings orthogonal to the identity or canonical order, i.e., those for which the k = 2 amplitudes vanish. As it turns out the same amplitudes vanish for k = 3. Explicitly, introducing the notatation I = 123456 for the canonical ordering, we have:   There are two interesting features that deserve attention. The first is the appearance of both s 612 and s 345 . If they were standard k = 2 three-particle kinematic invariants momentum conservation would have equated both of them. Here however they are independent and as discussed in the next section, the presence of both poles corresponds to two different geometric configurations; one in which 6, 1, 2 are collinear and the other where 3, 4, 5 are collinear. The second feature is the structure of the other two poles. Note that they are the straightforward generalization of (k+1)-particle invariants when k particle invariants are taken as fundamental. Motivated by this we define t a 1 a 2 ...a k a k+1 := For example, when k = 2 one has t abc = s bc + s ac + s ab , recall that we use the standard font for both t and s when k = 2. Using this notation one has There is one more topology of k = 2 Feynman diagrams associated with the orderings (I|125634). This turns out to introduce a new class of poles with no k = 2 analog. This is why we first discuss other amplitudes which contain two Feynman diagrams in k = 2 before returning to (I|125634), m using momentum conservation. The form of (2.26) suggest that the expression should be expanded in terms of three objects; each containing poles of the form ttRR. However, it turns out that (2.26) proves to be a single object. In order to motivate this interpretation it is important to note the identity R +R = t 1234 + t 3456 + t 5612 . (2.29) This means that (2.26) can also be written as the sum over only two terms, each with a pole structure tttR. This 3 equals 2 identity is reminiscent of identities found by computing the "volume" of a bipyramid. The volume can be computed either by summing over the top and bottom tetrahedra or by slicing through the middle line and produce three tetrahedra, each containing the top and bottom vertices. This means that none of the two ways of computing the volume are fundamental and that the object of interest is the whole bipyramid. In Section 4 we will see that this intuition is correct, which is why we introduce the notation Θ := (2.26). This object completes the lists of possible singularities appearing in m 6 (α|β). We can now construct other amplitudes containing two or more k = 2 Feynman diagrams: There is only one more amplitude missing from the set of all possible 'biadjoint' amplitudes. The one missing is m (3) 6 (I|I). As it is familiar from k = 2 amplitudes, this case is the one with the largest number of terms. We postpone its computation to Section 4 where we present the analog of a Feynman diagram computation, i.e., a purely combinatorial argument which turns out to be based on a interesting connection to tropical Grassmannians.
It can be checked that all the amplitudes we have presented are invariant under the exchange s abc → s def , where {a, b, c, d, e, f } = {1, 2, 3, 4, 5, 6}. This exchanges R withR and leaves t's invariant. This property is the n = 6 analog of the property (2.15) and, as we will see in Section 3, follows from self-duality of G(3, 6).
Finally, the presence of the different types of singularities in the above amplitudes hints at a richer boundary structure of X(3, n) as compared to X(2, n). Therefore we end this section by studying configurations of points on CP 2 that give rise to these singularities.

Geometric Interpretation
Singularity structure of the amplitudes computed with the new scattering equations is governed by the potential function S k . To be more precise, the pole locus of the logarithmic 1-form dS k specifies the boundaries of X(k, n). The only difficulty in studying these boundaries is that not all of them are accessible in the same chart of X(k, n) as blow-ups might be necessary. Therefore the strategy for checking whether a given configuration of points on CP k−1 is a codimension-1 boundary is to first change coordinates such that it is approached as ε → 0 from a generic configuration, and then compute which gives the corresponding factorization channel. If the above residue is zero or the change of variables is not valid then the configuration is not a codimension-1 boundary of X(k, n).
Higher-codimension boundaries are easily obtained by intersecting multiple codimension-1 boundaries. Before discussing the singularity structure of X(3, 6) let us consider that of X(3, 5), which is more familiar because of the duality to M 0,5 ∼ = X(2, 5) (as discussed in Section 3). Naively, there are two types of singularities that are allowed. The first one is when two points, say a and b, collide with each other. It can be parametrized by Using momentum conservation we find c =a,b s abc = s def , where {d, e, f } are the three points in the complement of {a, b}. Therefore s def is the factorization channel associated to x a and x b colliding. More points cannot collide since there is no change of variables allowing for such a scenario. Another option is that three points, say d, e and f , become collinear. Clearly the only vanishing angle bracket is |def | ∼ ε, which means that 36) and hence this yields the same type of singularity as two complementary points colliding with each other. It is straightforward to check that two codimension-1 boundaries intersect only if their channels s def share exactly one label, e.g., s 123 and s 145 are compatible, but s 123 and s 124 are not. This concludes the description of the boundary structure of X(3, 5). Given the above discussion we move on to studying boundaries of X(3, 6), where we have the following classes of singularities.

Two or Three Punctures Colliding
For n=6 we can have two or three punctures colliding with each other (four or more is not allowed as it would be inconsistent with SL(3, C) invariance). In the first case, when a and b collide, we have |abc| ∼ ε for all other c and hence find the channel where {d, e, f, g} is the complement of {a, b}. In the second case, say a, b and c colliding at the same speed we find for all d, e = a, b, c. Hence the corresponding factorization channel is Note the factor of 2 in front of s abc in the first term due to faster vanishing of |abc|. The resulting channel is simply s f gh for the complementary set {f, g, h} to {a, b, c}.

Three or Four Punctures Becoming Collinear
Next we consider the singularity in which three or four points become simultaneously collinear.
In the first case we only have |abc| ∼ ε when a,b,c become collinear and hence the singularity is simply s abc . Hence it is the same as the complementary three points colliding. In the second case, when a, b, c, d become simultaneously collinear, we have while other brackets stay finite. Hence the corresponding singularity is s abc +s abd +s acd +s bcd = t abcd , which is the same as in the case of the complementary two punctures colliding. Indeed, by applying SL(3, C) transformations one can show equivalence of the two sets of singularities.

Two Punctures Colliding on a Line
Finally we have a codimension-1 singularity in which two points, say a, b collide with each other and at the same rate become collinear with two other punctures c, d. In this case we have the vanishing brackets: for any e = a, b, c. Hence we obtain the associated channel e =a,b which is the R-type singularity: for example when (a, b, c, d) = (1, 2, 3, 4) the sum in (2.42) equals to R from (2.27). By a change of the SL(3, C) frame this singularity is the same as a colliding with b, while c simultaneously becomes collinear with a and d. Using SL(3, C) transformations, we can check the invariance of the R-type singularity under cyclic shifts of the pairs (a, b), (c, d), (e, f ).
3 Duality Between X(k, n) and X(n − k, n) The scattering equations inherit a duality from that of Grassmannians G(k, n) and G(n−k, n). In order to show this it is enough to study the potential function.
Let us start with the potential function (1.5) for n points on CP k−1 , Combining the invariance of this function under SL(k, C) transformations and rescaling of individual points one has GL(k, C) as a subgroup. This means that we can go to a gauge fixing or frame in which the first k × k submatrix for the k × n matrix defined by the columns of the points is set to the identity. Once this is done, all the maximal minors of the matrix can be interpreted as the minors of a (n−k) × n matrix in which GL(n − k, C) has been used to set the maximal minor of the last n−k columns to the identity. This map identifies the k × k minor (a 1 , a 2 , . . . , a k ) with the (n−k) × (n−k) minor (b 1 , b 2 , . . . , b n−k ) where the set {b 1 , b 2 , . . . , b n−k } is the complement of {a 1 , a 2 , . . . , a k } in {1, 2, . . . , n} which can be denoted as a 1 , a 2 , . . . , a k .
Applying this to S k one finds The goal is to show that this function can also be thought of as a general potential function for n points on CP n−k−1 . Given that the number of terms, n k also happens to be equal to n n−k , the sum can be transformed into a sum over the complement sets by writing All that remains to be shown is that the set of kinematics invariants for CP n−k−1 defined by the identification is generic and satisfies the conditions Once this is done the full SL(n−k, C) invariance can be restored. Without loss of generality, let us choose b 1 =1. Using the identification (3.4) one has to prove that Note that the sum does not include the label 1. In order to prove this property, let us start with the conditions s a 1 a 2 ...a k are known to satisfy, (1.6), as valid kinematic variables for n points on CP k−1 , i.e., Consider the linear combination C 2 + C 3 + . . . + C n and collect terms into two groups. The first contains all terms that involve the label 1 and the second is the rest. This gives The first group of terms on the RHS is nothing but C 1 while the second is (3.6). Since all C i = 0 this concludes the argument that (3.3) with the identification (3.4) defines a valid potential function S n−k . Before closing this section note that this duality already shows that the standard scattering equations which are defined on the space of n point in CP 1 are also equations for n points on CP n−3 . This is yet another indication of the importance of filling in the gap for other projective spaces in between.

Tropical Grassmannians and Higher-k Feynman Diagrams
In this section we use a surprising connection between tropical Grassmannians, the space of kinematic invariants, and the singularities that can arise in computing amplitudes using the scattering equations.
The connection starts with the tropical G(2, n) Grassmannians and the standard space of kinematic invariants, i.e., the k = 2 case. In this section we follow the construction of Speyer and Sturmfels [23] and use their notation. The map connects the vertices of the tropical Grassmannians with all possible kinematics invariant that can be poles of a Feynman diagram in a φ 3 theory. For example, G 2,4 is given by three points. These correspond to s, t, u. For G 2,5 one has 10 vertices. Each vertex is associated with a given s ij . One more case is needed to reach some generality. Consider G 2,6 which has 25 vertices and are in correspondence with s ij and t ijk . In general G 2,n is known to have 2 n−1 −n−1 vertices labeled by all ways of partitioning the set [n] := {1, 2, . . . , n} into two sets A, B with |A| > 1 and |B| > 1.
The analogy goes even beyond the structure of vertices. The edges of G 2,n can be mapped to all possible pairs of consistent poles or factorization of an amplitude. In other words, there is an edge between the vertices {A, Likewise the correspondence continues all the way to the facets for G 2,n . As it turns out, the tropical Grassmannian G 2,n has (2n − 5)!! facets which is precisely the number of all possible Feynman diagrams in φ 3 theory. 1 Motivated by this and by the careful study of G 3,6 done in [23] we propose to extend the analogy to k = 3 kinematics and the possible biadjoint scalar amplitudes and the corresponding generalization of Feynman diagrams.
Let us review the G 3,6 results of [23] to exhibit the surprising connection with the object found up to now and then use it to give a prescription for the computation of the most complicated of the biadjoint scalar amplitudes, i.e., m Setting the image of this function to zero then amounts to imposing momentum conservation, thus we can identify the set E with {s ijk } in this work. Moreover, for each 4-subset of labels one defines [23] f ijkl = e ijk + e jkl + e kli + e lij . (4. 2) The set of such 15 vectors is denoted by F . Clearly, these correspond to t ijkl . Even more surprising is the fact that Speyer and Sturmfels consider objects with six labels which map to our R andR. The set of these 30 vectors is G. The collection E ∪ F ∪ G has 65 vectors and represent the vertices. From now on, we will not distinguish between the vertices and their corresponding kinematic invariants. The next step is to define edges. There are six classes of edges whose definition can be read off from [23]. For instance, the first class is "EE" and is given by pairs of the type {s abc , s ade } or complementary labels as in {s abc , s def }. Inspection of the previous amplitudes reveals that only these combinations appear in m  This is noting but the clover diagram Θ. As explained by the authors, tetrahedra contained in these do not correspond to facets. These tetrahedra precisely correspond to the individual terms after splitting Θ using (2.29). This is the reason why they always appear combined in our previous examples. Now we are ready to give a prescription for a combinatorial computation of m of twelve elements (by . . . we denote cyclic shifts). Among the 12 4 four-element sets of L(I), we simply pick the ones that correspond to facets of G 3,6 ! There are 46 such matches, each of these corresponds to a Feynman diagram of our amplitude. Finally, we append to the list the contribution from the clover diagram Θ and its cyclic shiftΘ. This gives a list of 48 elements we denote by J(I). The sum of all the elements of J(I) corresponds to the m  We leave the complete study of these correspondence to future work.

Matrix Kinematics and MHV Sectors
The generalized kinematic invariants s abc have been treated as abstract objects so far. In this section we explore a generalization of the notion of momentum vectors that gives rise to a special class of kinematic invariants s abc analogous to four-dimensional kinematics when k = 2. Moreover, on the support of these kinematics one can identify at least four analytic solutions of the scattering equations for all multiplicity, which are analogous to the four-dimensional MHV and MHV solutions [26][27][28]. Let us start discussing the case of generic k and then specialize to k = 3. To each particle we associate two k-dimensional complex vectors λ  αα has rank one and therefore its determinant vanishes. Moreover, any linear combination of fewer than k such matrices has rank smaller than k and therefore vanishing determinant. This motivates the following definition s a 1 a 2 ···a k := det K (a 1 ) + K (a 2 ) + · · · + K (a k ) . (5.1) With this definition it is clear that s a 1 a 2 ···a k is completely symmetric in its indices. Moreover, if any label repeats the invariant vanishes, i.e., s a 1 a 2 ···a k−2 b b = 0. Furthermore, for B = {b 1 , . . . , b j } with j > k, the Cauchy-Binet theorem states that the determinant decomposes Hence for k = 3 the object t abcd can be identified as the LHS. The k = 2 analogy can actually be taken further as one can write α . This can be seen by writing the argument of (5.1) as ΛΛ T . Formula (5.3) also makes explicit the fact that each s a 1 ···a k is linear in the K (a) 's.
So far the invariants built from (5.1) satisfy properties analogous to the k = 2 case. Let us then impose the generalized momentum conservation constraints (1.6). First note that adding all the n conditions gives 0 = a 1 <···<a k s a 1 ···a k = det K (1) + · · · + K (n) (5.4) by (5.2). Denoting the sum of all momenta by Q = n i=1 K (i) , this condition is the fact that Q is of rank at most k−1, or, that Q is the sum of at most k−1 rank-1 matrices: We can write this as n a=1 λ (a)λ(a) = − k−1 i=1 q (i)q(i) and insert it into the conditions (1.6) to obtain 0 = a 2 <···<a k a 1 · · · a k [a 1 · · · a k ] = a 1 q (1) · · · q (k−1) [a 1 q (1) · · · q (k−1) ] , ∀a 1 . (5.6) To solve these conditions one can assume without loss of generality that 2 A proof of this is obtained by regarding each K (a) = λ (a) ∧λ (a) as a two-form in k + k dimensions. The determinant of any two-form ω is the coefficient of the top form ω ∧k := 1 k! ω ∧ . . . ∧ ω. Hence we can identify sa 1 ···a k with ( i K (a i ) ) ∧k = K (a 1 ) ∧ . . . ∧ K (a k ) and tB with ( i K (b i ) ) ∧k . The result (5.2) follows after expanding tB. This also motivates the formula (5.3) and the subsequent manipulations.
i.e., we have n 2 vectors living in a k−1 plane, where n 2 is the ceiling function. But in the cases we are studying (n ≥ 3 for k = 2 and n ≥ 5 for k = 3) we have n 2 > k−1. This means that either the n 2 vectors λ (b) are degenerate, which would make the kinematic invariants s b 1 ···b k vanish, or the k−1 vectors q (i) are degenerate, which would decrease the rank of Q to k−2. As the first case corresponds to singular kinematics we focus on the second setup, which without loss of generality is obtained by writing (5.5) with For k = 2 this sets Q = 0 and recovers the standard momentum conservation condition. Specializing to k = 3, generalized momentum conservation now reads where we call q = q (1) andq =q (1) . At the level of the invariants we can use the familiar manipulations of the k = 2 case, for instance, for k = 3, n = 6 we can write which is (2.37).

Analytic Solutions to the Scattering Equations
In the k = 2 case it is well known that solutions of the scattering equations split into n−3 sectors [29,30]. The d-th sector can be associated with two maps ρ,ρ : CP 1 → CP 1 of degrees d and n−2−d respectively. For d = 1 and d = n−3 we find linear maps and the corresponding solutions are said to be in the MHV and MHV sectors.
Here we provide evidence of the existence of such sector decomposition for higher k. We do this by constructing four analytic solutions which are present at any multiplicity for k = 3. In particular this proves the existence of solutions to our scattering equations. The first two solutions lie in the direct analog of MHV and MHV sectors and are easy to construct for any k. The other two are particular for k = 3 as all the points in CP 2 are found to lie on a conic. As they can also be identified as MHV-like solutions under a Veronese action, we will refer to them as MHV q solutions. For X(3, 5) the MHV q sector coincides with the MHV ones, which in particular gives the two solutions to the scattering equations at five points.
Let us first discuss the MHV solutions, the MHV case being obtained by exchanging λ α ↔λα. A degree-one map ρ : CP k−1 → CP k−1 can be written as ρ(σ) = Gσ, where G can be set to the identity by means of a GL(k, C) transformation in X(k, n). Setting k = 3, we adopt inhomogeneous coordinates by putting σ = t (1, x, y). This means we can obtain our MHV solution simply as λ (a) = σ a , i.e., This feature will have a nice analog for the MHV q solutions, which we now introduce. For k = 3 we define such solutions as the ones lying on a conic in CP 2 . As there is always a conic passing through five points, this explains why the MHV q sector is contained in the MHV ones for X (3,5). For n > 5 one needs to set additional conditions so that the n points lie on the conic defined by any five of them. Such conditions, i.e., the conic equations, can be stated as (abf )(cdf ) (adf )(cbf ) = (abg)(cdg) (adg)(cbg) , f, g = a, b, c, d (5.14) (this is an equation for σ g if {σ a , σ b , σ c , σ d , σ f } are considered fixed). It can be shown that these conditions are enough to arrange, by means of a GL(k, C) × (C * ) n transformation, any element of X(3, n) into the Veronese form [31]. Such form is defined as y a = x 2 a in inhomogeneous coordinates. Under this parametrization (abc) = x ab x bc x ca and the scattering equations become As the objects abq and [abq] can be identified with the standard k = 2 brackets ab and [ab] (for instance by choosing a frame where q = (0, 0, 1)), we recognize in the second condition the standard scattering equations over CP 1 . As before, we know it admits two MHV-type solutions, which can be stated in a covariant form as where X, Y are two reference vectors parametrizing the SL(2, C) redundancy of (5.15). As it turns out only these two solutions of the scattering equations are also solutions to the first condition. To see that such condition holds, let us take the MHV q solution and use the Schouten identity to write x ab = abq XY q aY q bY q so that where the sum over b again vanishes due to momentum conservation. Thus we have found two new solutions which lie on a conic given by (5.14). In fact, the cross-ratio in (5.13) now becomes x ab x cd x ad x cb so that we can write abq cdq adq cbq = (abf )(cdf ) (adf )(cbf ) ∀f , (5.18) which is a GL(k, C)-invariant statement.

Positive Kinematics
Motivated by the work of Kalousios [32], Zhang and two of the authors found a subregion of the n(n−3)/2-dimensional space of kinematic invariants s ab where all (n−3)! solutions to the standard scattering equations, i.e., k = 2, are real [24]. Moreover, the equations had the interpretation of the equilibrium points of a potential describing interacting particles on the interval [0, 1]. This is easy to see by singling out three particles, A, B, C and using SL(2, C) to set x A = 0, x B = 1 and x C = ∞. The potential function is then 3 Letting all s Aa , s Ba , s ab be positive gives rise to a system of n−3 particles on an interval, where all particles repel each other and are also repelled from the boundaries of the interval at x=0 and x=1. This region is called K + n . Note that the choice s Aa , s Ba , s ab > 0 is possible because they form a basis of the n(n−3)/2-dimensional kinematic space. 4 In this section we generalize the notion of positive kinematics to k=3 and discuss some of the new features that appear. In the same way as for k=2, we find for n<7 that all solutions are real and give rise to a very elegant and pictorial derivation of the number of solutions.
We start by selecting four particles A, B, C, D to be fixed by the action of SL(3, C). This time two particles, say C, D, can be sent to infinity by setting their homogeneous coordinates to (0, 1, 0) and (0, 0, 1) respectively. The other two are chosen to be, in inhomogeneous coordinates, at the origin and at (1, 1) on the plane (x, y) ∈ R 2 , i.e., A has homogeneous coordinates (1, 0, 0) while B has (1, 1, 1).
Clearly the n = 4 case is trivial as all four particles are gauge fixed. Since interactions in the potential function are controlled by the determinants |abc| a given particle is not directly sensitive to the location of any other particle but only to the lines defined by any pair of particles. In order to find the analog of the positive region in this case let us again consider the potential function Our first approach to the k = 3 positive region K + 3,n is then to ask all invariants that explicitly appear in (6.2) to be positive. This is possible because once again they form a basis for the space of k = 3 kinematic invariants. More explicitly, the only constrains on kinematic invariants are determined by the n conditions which generalize k = 2 momentum conservation. These are linear equations for the n variables s ABC , s BCD , s ACD , s ABD and s CDa for 1 ≤ a ≤ n−4.
Now we are ready to study the dynamics generated by the potential S 3 . Consider n = 5. Since only one point is free to move we use x, y for its coordinates. Here the potential is Since all coefficients are positive it is possible to understand the dynamics as that of a particle in R 2 which is repelled from five lines. The lines are, in the order in which they appear in the potential, x = y, y = 0, x = 0, y = 1, and x = 1. This is shown in Figure 1 (left). Since critical points of the potential correspond to equilibrium points, it is clear that they can only lie in the bounded chambers of the space, i.e., those with finite area. Note that the five lines divide the plane into 12 chambers. Ten of them are unbounded and two are bounded. Therefore there are only two possible places for particle 1 to be located, and since we know from Section 2.2 that there are exactly two solutions to the scattering equations, they have to lie in these two chambers. Much more interesting is the n = 6 case. Here we use the soft limit approach in order to more clearly understand the dynamics. We assume that |s ij2 | |s klm |. This means that one can effectively solve the scattering equations in steps. First we find solutions for particle 1. As before, we find two of them in the two bounded chambers. Once a solution is picked, we study the dynamics of the second particle, assumed to be soft. Let us choose the lower chamber (chamber 2 in Figure 1). Particle 2 now interacts with the same five lines given above and also with four new lines. The new lines all pass through the point 1. One is parallel to the x axis, another to the y axis, while the remaining two pass one through A at (0, 0) and the other through B at (1, 1). These lines are depicted in Figure 1 (right).
Clearly, particle 2 can only find equilibrium points on bounded chambers. It is easy to count 31 chambers in total with 13 bounded and 18 unbounded. On the figure we have explicitly labeled all 13 bounded chambers. This can be repeated again for the second equilibrium point of particle 1 thus obtaining another set of 13 critical points, summing to 26. This saturates the total number of solutions (see Appendix A) and hence the above argument describes all of them.
These pictures also reveal that our first attempt at defining a positive region cannot be a completely connected one. The reason is that according to our definition, one can smoothly change the kinematics invariants from the soft region |s ij2 | |s klm | to a new one where the roles of 1 and 2 are reversed, i.e., |s ij1 | |s klm |. This leads to a problem as the solutions where particle 2 is placed on region 7 or region 13 in the figure disappear as soon as particle 1 becomes soft and 2 hard.
This means that there must be new singularities that separate the two soft regions. These are not of the form s ijk = 0 or t ijkl = 0 since they all have definite sign in the positive region. This puzzle is resolved by the novel k = 3 singularities found in the biadjoint amplitudes in Section 2.2 and denoted by R andR.
It is possible to see that some of the R's andR's do not have definite sign and hence can become zero as we move from one soft region to the next. In fact, on kinematics that is positive but generic, i.e., not near any soft region, we find that the 26 solutions split in classes shown in Figure 2. In 16 of them points 1 and 2 are located on the same n = 4 bounded chamber while 6 have 1 and 2 on two distinct bounded chambers. The remaining four have both 1 and 2 in unbounded chambers. This is possible because one particle generates a new bounded chamber for the other! In the examples given in Figure 2 (right), point 2 generates a bounded chamber for 1 with the line 2−B, while particle 1 generates a bounded chamber for 2 with the line 1−B, both indicated by dashed lines.
Before ending this section let us mention that one can be tempted to continue the construction of chambers for more particles and count the number of solutions to the k = 3 scattering equations in this way. More explicitly, placing particle 2 in any of the 13 bounded chambers generated by particle 1 gives rise to a diagram with 42 bounded chambers. This implies that there must be at least 2 × 13 × 42 solutions for n = 7. In Appendix B we carefully study soft limits directly from the scattering equations and find a closed formula for the bounded regions and also explain why for n > 6 and k = 3 these numbers are strict lower bounds.

Discussion
In this work we have initiated the study of a natural generalization of scattering equations to moduli spaces of points on CP k−1 . We have only scratched the surface of what seems to be a vast subject. The duality explained in Section 3 between the space of n points on CP k−1 and that of n points on CP n−k−1 shows that the standard k = 2 scattering equations, which are at the heart of the CHY formalism, already imply the existence of the k = n−2 scattering equations. Moreover, the standard Mandelstam invariants s ab give rise to k = n−2 kinematic invariants. In this work we have filled the gap between these two end points by studying intermediate values of k.
We found that the all the elements needed to define generalized CHY amplitudes are also present for any k and gave the explicit formulation of "biadjoint amplitudes". The first nontrivial case is six points on CP 2 , i.e., the scattering equations relating the space of kinematic invariants s abc to X (3,6). Explicit computations of biajoint amplitudes led to the discovery of new kinematic poles R andR in addition to the expected s and t poles. A very pressing question is the explicit computation for n > 6 and k = 3. A direct computation for seven particles seems technically very challenging as the number of solutions jumps from 26 for n = 6 to more than a thousand for n = 7. However, it is well known that in the k = 2 case, a variety of techniques have been developed that allow the evaluation of CHY formulae without actually solving the scattering equations, see, e.g., [34][35][36].
One of the early successes of the CHY formalism was the direct connection between biadjoint scalar amplitudes and the Kawai-Lewellen-Tye construction and the simple derivation of the Bern-Carrasco-Johansson basis of color ordered amplitudes. These developments rely on the fact that the basis of Parke-Taylor functions on X(2, n) has size (n−3)! when evaluated on the solutions of the scattering equations which happens to be the same as the number of solutions. It is clear that a generalization of these construction to higher k is very desirable. At this point there is evidence that Parke-Taylor functions might not provide the most general basis of functions needed for 2 < k < n − 2.
Another natural question is the generalization of the notion of Feynman diagrams for k > 2. When n = 6 and k = 3 we have made a proposal based on the computations of Section 2 and on the surprising connection to tropical Grassmannians explored and used in Section 4. Tropical Grassmannians are polyhedral complexes with what seems to be a very direct connection to the space of kinematic invariants. The vertices of the tropical Grassmannian are in bijection with all possible poles in a φ 3 theory. Moreover, the facets are in bijection with all possible individual Feynman diagrams of the theory. We have found that the same is true for k = 3 and n = 6 for the vertices and kinematic poles. Using the natural assumptions that facets must also correspond to Feynman diagrams we developed a combinatorial computation of all biadjoint amplitudes by computing the most fundamental one, m 6 (I|I). We noted that one class, the EEEE class, of k = 3 Feynman diagrams did not contribute to m (3) 6 (I|I) and hence to any of the biadjoint amplitudes. This is strong evidence that for k = 3 one has to go beyond Parke-Taylor factors as building blocks for the integrand.
Another piece of evidence for the need of more general integrands comes from the k = 3 Schouten identity. In k = 2 the Schouten identity (13) None of these four terms are Parke-Taylor functions. However, it is possible to multiply (7.4) by a product of six factors, e.g., (236) 2 (145) 2 (612)(345), such that each of the four terms are turned into the product of two Parke-Taylor functions but we take this still as an indication that more general functions are needed. We also found an explicit realization of the k space of kinematic invariants in terms of a generalization of the spinor-helicity formalism. Rank one 2 × 2 matrices are replaced by rank one k × k matrices and invariants are computed as determinants of sums of such matrices. Since the factorization, s ab = ab [ab], that led to miraculous simplifications for k = 2 also happens for any k > 2, it is tempting to suggest that there is a generalization of constructions such as BCFW recursion, superamplitudes, etc. One immediate challenge is the definition of the notion of helicity and polarization vectors. We leave these fascinating questions for future research.
Another interesting connection made for k = 2 is that between the CHY formalism, string theory [6,[37][38][39][40][41], Z-theory [42][43][44], and ambitwistor strings [45,46]. It is clear that a generalization of Z-theory integrals is possible for k > 2 and natural to expect that on a subregion of the positive kinematics defined in Section 6, where appropriate R's are positive, such integrals should exists.
Finally, there are two more very useful constructions known for k = 2 which can greatly impact the ability of solving k > 2 scattering equations. The first is obtaining what is known as the polynomial form of the scattering equations first constructed by Dolan and Goddard in [9]. These polynomial form allows a faster and more stable numerical search as well as a direct way of counting the number of solutions using Bezout's theorem since the equations increase their degree in steps of one. The second development is the identification of an integrable system with the scattering equations for a particular set of kinematics invariant. This was done by Kalousios and led to a connection to Jacobi polynomials [32]. We started the exploration of such kind of kinematics in Section 6 but leave the search for an integrable system for future work. Either one of these construction would help in completing the counting of solutions started in Section 6 and in Appendix B where an attempt to follow the soft limit approach led only to lower bounds. of the scattering equations (2.2), purely topologically as the Euler characteristic of X(3, 6). More precisely, under the assumption that the kinematics is generic (so that all other twisted cohomology groups vanish), one can show that using Morse theory, see, e.g., [6,33]. We can use a chart in which punctures {1, 2, 3, 4} are held fixed, such that X(3, 6) can be written as the complement of the projective variety in (CP 2 ) 2 . Using the inclusion-exclusion principle we can write the Euler characteristic as where we used χ(CP 2 ) = 3. The remaining contribution χ(V ) can be evaluated using the algebraic geometry system Macaulay2 [47] with the package CharacteristicClasses.m2 [48,49] as follows. Let R be the coordinate ring of (CP 2 ) 2 (say over Z/pZ for p=32749) with coordinates r_i for i = 0, 1, . . . , 5, and I be the ideal generated by vanishing of all the relevant maximal minors of the matrix    1 0 0 1 r_0 r_3 Then the Euler characteristic in (A.4) can be computed using the following script: load "CharacteristicClasses.m2"; R = MultiProjCoordRing(ZZ/32749, symbol r, {2,2}); I = ideal(r_0*r_1*r_2*r_3*r_4*r_5*(r_0-r_1)*(r_0-r_2)*(r_1-r_2)*(r_3-r_4) *(r_3-r_5)*(r_4-r_5)*(r_0*r_4-r_1*r_3)*(r_0*r_5-r_2*r_3) *(r_1*r_5-r_2*r_4)*(r_0*(r_4-r_5)+r_1*(r_5-r_3)+r_2*(r_3-r_4)));

3*3-Euler(I)
The output of this computation is 26, which by (A.2) gives the number of solutions of scattering equations, N

B Soft Limits and Numbers of Solutions
One of the most basic questions about the scattering equations is the number of solutions. It is well-known that the k = 2 scattering equations for n particles have (n − 3)! solutions [1]. Let us review one technique for proving it and then try and apply it for k = 3. The idea is to approach what is known as the soft-limit region for the n th particle. This is done by writing all invariants of the form s an as τŝ an in the limit when τ → 0. In the limit the n th particle drops from the first n − 1 scattering equations which become those for a system of n − 1 particles. The n th equation is proportional to τ even for finite τ and therefore it fixes the location of x n . Assuming that the system for n − 1 has been solved and N It is easy to show that this leads to n − 3 solutions for x n . Therefore N Strictly speaking, the soft argument only leads to a lower bound on the number of solutions as one has to prove that when taking τ = 0 in the first n − 1 equations one is allowed to drop terms that depend on particle n. For instance, it could happen that there are solutions where x n − x a ∼ O(τ ) and hence the term s an /(x n − x a ) cannot be dropped. Indeed this happens when collinear limits, s an → 0, are taken and it is well-known that solutions split into two classes: singular and regular. Regular solutions are those for which the term s an /(x n − x a ) can be dropped. This means that the soft limit argument is only guarantee to count the regular solutions. It turns out that when k = 2 there are no singular solutions in the soft limit. Unfortunately, this is not the case when k > 3.
Here we repeat the same argument for k = 3, one takes the "soft" limit s abn = τŝ abn with τ → 0. We will remove all dependence on particle n in the equations that defined the (n − 1) system, i.e., we will only compute the number of regular solutions. Assuming that the system for (n−1) particles has been solved, one finds for each solution x a,(I) , y a,(I) two equations We are only interested in counting the number of solutions to these "soft" equations. In order to analyze these equations one has to rewrite them as the ratio of two polynomials P A (x n , y n ) Counting the number of solutions to the system P A (x n , y n ) = P B (x n , y n ) = 0 is harder than in the k = 2 case for two reasons.
In order to understand the first, recall that when dealing with a single polynomial in one variable, as in k = 2, the degree of the polynomial equation coming from the numerator of (B.1) directly gives the number of solutions. The first complication for k = 3 arises from the fact that for two polynomials in two variables, (x n , y n ), Bezout's theorem only gives an upper bound for the number of solutions at finite locations. The bound is the product of the degrees of the two polynomials.
The second difficulty comes from the fact that the system P A (x n , y n ) = P B (x n , y n ) = 0 has solutions which are not solutions to the original equations (B.2). Such solutions come from where two factors in the denominator of (B.3) vanish. To see this note that if the poles are approached as → 0, then the original equations (B.2) diverge as 1/ while the denominators in (B.3) diverge as 1/ 2 . This shows that both P A and P B must vanish as → 0. This means that we have to remove these spurious solutions from Bezout's bound.
Very nicely, the final formula turns out to be simple Soft However, as discussed above, the soft limit computation only captures regular solutions. We will now show why when k = 3 there must also be singular solutions as well and leave their enumeration for future work. Note that N To see this more explicitly, note that when n = 7 one can repeat the same soft argument for k = 4. Starting from n = 6, k = 2 which has six solutions one finds that so does n = 6, k = 4 by the duality in Section 3. An explicit computation reveals that the soft equations for the seventh particle have 192 solutions. This means that the number of solutions must be at least 6 × 192 = 1152. This is 60 more than N