Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals

We continue the study of positive geometries underlying the {\it Grassmannian string integrals}, which are a class of"stringy canonical forms", or stringy integrals, over the positive Grassmannian mod torus action, $G_+(k,n)/T$. The leading order of any such stringy integral is given by the canonical function of a polytope, which can be obtained using the Minkowski sum of the Newton polytopes for the regulators of the integral, or equivalently given by the so-called scattering-equation map. The canonical function of the polytopes for Grassmannian string integrals, or the volume of their dual polytopes, is also known as the generalized bi-adjoint $\phi^3$ amplitudes. We compute all the linear functions for the facets which cut out the polytope for all cases up to $n=9$, with up to k=4 and their parity conjugate cases. The main novelty of our computation is that we present these facets in a manifestly gauge-invariant and cyclic way, and identify the boundary configurations of $G_+(k,n)/T$ corresponding to these facets, which have nice geometric interpretations in terms of $n$ points in $(k{-}1)$-dimensional space. All the facets and configurations we discovered up to $n=9$ directly generalize to all $n$, although new types are still needed for higher $n$.


Introduction
In [1], N. Arkani-Hamed, T. Lam and one of the authors introduced stringy canonical forms, which are string-like integrals as α ′ -deformation of canonical form [2] for general polytopes. Let's briefly review the idea: we consider an integral over R d >0 , which depends on a collection of polynomials, p I , with positive coefficients: and it is a meromorphic function of the exponents, X i > 0 for x i factor and c I > 0 for polynomial p I , respectively. As explained in [1], such integrals resemble open-string amplitudes in a number of ways, and there is a natural polytope associated with it. The most basic observation of [1] is that the integral converges if X := (X 1 , · · · , X d ) is inside the polytope P that is defined as the Minkowski sum of Newton polytopes of p I , P = I c I N(p I ). Exactly this polytope controls the leading order of I as α ′ → 0, which is given by the canonical function of P (or the volume of the dual polytope) computed at X: lim α ′ →0 I = Ω(P; X) . (

1.2)
Another remarkable property is that the same polytope can be obtained as the image of the so-called scattering-equation (SE) map from the integration domain. The map is given by a rewriting of the saddle-point equations, which in turn are given by taking α ′ → ∞ limit of the regulator R : As shown in [1], this provides a diffeomorphism from x ∈ R d >0 to the interior of the polytope P, thus one can obtain the canonical form of the latter, Ω(P; X) := d d XΩ(P; X), as the pushforward of that of the former by summing over all solutions of (1.3): where equivalently the canonical function can be written as a generalization [3] of the CHY formula for bi-adjoint φ 3 amplitudes [4,5] (see also [6,7]).
As discussed in [1], open-string amplitudes can be thought of as the stringy canonical form for the so-called ABHY associahedron [3]; in the α ′ → 0 limit we obtain the fieldtheory, bi-adjoint φ 3 amplitudes as its canonical function, which can also be obtained using the pushforward/CHY formula from the scattering-equation map for the associahedron [3]. As shown in [1], the two most natural extensions of string amplitudes are the so-called cluster string integrals and the Grassmannian string integrals. The former are stringy integrals for generalized associahedra of finite-type cluster algebra (usual string integrals correspond to type A), which exhibit various remarkable properties relevant for scattering of generalized particles [8,9] and strings [10,11]. The latter are natural integrals over positive Grassmannian mod torus action, G + (k, n)/T (string integrals correspond to k = 2), which were first studied very recently in [12]. At leading order, the Grassmannian string integral is computed by the canonical function of a class of polytopes, which are closely related to the positive tropical Grassmannian [13], as shown in [14][15][16][17]). It has also been shown that the canonical function can also be obtained using a CHY formula which are higher-k generalization of the bi-adjoint φ 3 amplitudes (see also [18][19][20]).
On the other hand, just as the momentum twistor space [21], G(4, n), is crucial for the amplituhedron reformulation [22,23] of all-loop integrand in planar N = 4 SYM [24], the space G + (4, n)/T can play a crucial role for all-loop integrated amplitudes (or even non-perturbatively). Recall that for n = 6, 7, all evidence so far [25][26][27][28][29][30][31] is consistent with the conjecture that the alphabet for symbol letters consists of the 9 and 42 cluster variables associated with G + (4, n)/T , respectively; it is thus highly desirable to study the structure of G + (4, n)/T for n ≥ 8. In [12,16,17], it has been shown that polytopes P(4, n) and the associated tropical positive Grassmannian indeed have intriguing applications to the mathematical structures of multi-loop scattering amplitudes in N = 4 SYM.
In this paper, we continue to investigate the leading order of Grassmannian string integral for G + (k, n)/T as α ′ → 0, which is the canonical function of polytope denoted as P(k, n), and we call it the generalized amplitudes A k,n = Ω(P(k, n)) since these functions generalize bi-adjoint φ 3 amplitudes for k = 2 case [14]: A k,n := lim α ′ →0 (α ′ ) d G + (k,n)/T Ω(G + (k, n)/T ) R k,n (1.5) where d = (k − 1)(n − k − 1), and Ω(G + (k, n)/T ) denotes the canonical form of the domain G + (k, n)/T . The regulator R k,n is defined as R k,n := 1≤i 1 <i 2 <···<i k ≤n (i 1 , i 2 , · · · , i k ) α ′ s i 1 ,i 2 ,··· ,i k , (1.6) where the SL(k, C) invariant (i 1 , i 2 , · · · , i k ) is the minor of the k vectors labelled by i 1 , · · · , i k . The generalized Mandelstam variables are symmetric tensors satisfying "onshell" condition s i,i,··· = 0 and "momentum conservation" condition Note that the dimension of kinematic space for (k, n) is thus n k − n. As we will review shortly, once we recast the Grassmannian string integral in a positive parametrization (1.1), we can directly apply the two general methods mentioned above to compute the polytope P(k, n): (1). by taking the Minkowski sum of the Newton polytopes, and (2). by working out the image of the scattering-equation (SE) map. These two methods are usually complementary: while the Minkowski sum naturally gives the vertices of the polytope, the SE map gives the facets which cut out the polytope. Once the polytope P(k, n) is obtained, it is straightforward to compute its canonical function, A k,n e.g. by any triangulation of the dual polytope. However, we emphasize that a priori our computation for P(k, n) and A k,n are independent of the positive tropical Grassmannian or higher-k generalization of CHY formulas. Note that in the form of (1.1), we automatically have the subspace where the polytope lives: the d-dim subspace of the kinematic space can be specified by choosing all the c's to be constant, and a basis for it is given by the X. Thus it suffices to give all the linear functions for the facets of P(k, n) (or equivalently all the poles of A k,n ), which we denote as F a for a = 1, 2, · · · , N , and requiring all F a ≥ 0 immediately cut out P(k, n). The canonical function A k,n follows from any triangulation of the dual polytope: for a simple vertex of P(k, n), i.e. with d facets touching it, so its contribution to A k,n (the volume) is given by the product of the d factors 1/F ; for non-simple vertex we will need to triangulate its dual which we will discuss. We emphasize that having the polytope is the most important part of the calculation. All the facets (or equivalently the poles) up to k = 3, n = 10 and k = 4, n = 9, and the explicit results for generalized amplitudes up to n = 8, are presented in the supplementary material.
In addition to providing more data and an independent method for computing the polytope and its canonical function, the main novelty of our calculation is that, up to n = 9 (for k = 3, 4 and their parity conjugate k → n−k), we obtain interesting patterns for all the facets of the polytope, or the poles of the generalized amplitudes, which can be written in a completely invariant way using generalized Mandelstam variables (independent of any particular parametrization of G + (k, n)/T ). These facets/poles are direct generalizations of planar variables X a,b = a≤i<j<b s i,j of the k = 2 case to higher k. As we will see, already for k = 3, as n increases there are more new types of facets, and even more so for k = 4 case. An interesting phenomenon is that there are usually more than one way to represent a facet, and their equivalence is guaranteed by momentum conservation.
More importantly, we will see that all the facets of the polytope are associated with certain "boundary" configurations of G + (k, n)/T ; each configuration can be naturally mapped to the corresponding facet via the SE map. These boundary configurations and more generally the stratification of G + (k, n)/T have been studied in [32] in terms of "hypersimplex decomposition". Here we will not try to systematically study the boundary configurations in general, but instead we find the configurations by finding the pre-image of the facets. Such a boundary is controlled by a single parameter ǫ → 0 and it is specified by to which order in ǫ some of all the minors vanish; the facet is then given by the residue at ǫ → 0 of the d log R k,n at the boundary. In this way, we find all facets and boundary configurations for up to n = 9, written in a manifestly gauge-invariant and cyclic manner.
The boundary configurations have nice geometric interpretations which can be described in terms of n points in (k−1)-dim space. The identities between different representations of a given facet reflect equivalence relations between different configurations of G + (k, n)/T , which can be understood systematically [32]. All the facets and associated boundary configurations we found directly generalize to all n, which represent a subset of the facets and configurations, although new types will appear for higher n.

P(k, n) from Minkowski sum and scattering-equation map
Let us first review a canonical way to parametrize G + (k, n)/T using the so-called "web variables" [13]. We illustrate it by showing an example of k = 3 in figure 1.
We first gauge fix the k × n matrix to have an identity matrix I k×k in its first k columns. The remaining entries are determined as follows. There are (k − 1)(n − k − 1) web variables, one in each box (while all the boundary slots are assigned with x = 1). Following the decorated directions, we find all paths from i with 1 ≤ i ≤ k to j with k + 1 ≤ j ≤ n, and for each path we assign the product of all web variables below it, which we denote as Prod path (x). The sum over all paths gives the entry (i, j) of the matrix: Figure 1: A parameterization of G + (k, n)/T With this parameterization, note that entries of the last row are trivial, and we have n trivial minors that equal to 1: k+1 of them from the first k+1 columns and n−k−1 ones, (1, 2, · · · , k−1, i) with k+2 ≤ i ≤ n. The following d = (k − 1)(n − k − 1) minors are monomials of x's, (1, 2, · · · , j, l, l+1, · · · , l+k−j−1) with 0 ≤ j ≤ k−2 and j+3 ≤ l ≤ n−(k−j−1). For example, for (k, n) = (3, 6) we have (3,4,5) The remaining n k −n−d minors are non-trivial polynomials with positive coefficients, and for each of them one can always factor out a monomial of x's which gives a polynomial p I with constant term 1. We can reorganize the regulator into d monomials and n k − n − d polynomials, so the Grassmannian string integral is written in the standard form: where c I = −s I for each polynomial (non-trivial minor) p I , and we collect all contributions to x a thus its exponent X a is a sum of the s I . Note that this simple rewriting already gives a family of d-dim subspaces of the kinematic space for (k, n), where the polytope P(k, n) lives in. Such a subspace is defined by setting all c I to be positive constants, and it can be spanned using the basis X = {X 1 , · · · , X d }. Now the polytope P(k, n) can be cut out by all inequalities F a (X, {c}) ≥ 0 for a = 1, 2, · · · , N , where F a is a linear combination of X's and c's. In the rest of the paper, we will present these facets (which are poles of A k,n ), and by [1] they can be computed by taking the Minkowski sum of Newton polytopes for all p I , P(k, n) = I c I N(p I ). The same polytope P k,n can be obtained as the image of the scattering-equation (SE) map from the integration domain R d >0 . Example (k, n) = (3, 5). Using web variables, the regulator reads with X 1 = s 345 , X 2 = s 123 . The Newton polytopes from the regulator R 3,5 are two intervals  The facets of P (3,5), which correspond to the poles of A 3,5 , are where we have seen that each facet/pole can be written more invariantly as s i,i+1,i+2 for i = 1, 2, · · · 5. The pentagon P (3,5) can also be obtained as the image of SE map Obviously the boundary x i = 0 of the domain, is mapped to the facet X i = 0 of P(3, 5), for i = 1, 2. With x 1 → ∞ and x 2 → ∞, the map gives facets c 235 + c 245 − X 1 = 0 and c 135 + c 235 − X 2 = 0, respectively. The last facet is more non-trivial which can be seen as blowing up the corner with x 1 → ∞ and x 2 → 0. Let's use x 1 =x 1 ǫ −1 , x 2 =x 2 ǫ with ǫ → 0 to parametrize this corner, then eq. (2.6) becomes which implies that X 1 − X 2 → c 245 . Thus under SE map it indeed gives the last facet in (2.5), and we seex 1x2 can be used as a new variable to parameterize this facet. Finally, let's comment on computing the canonical function A k,n once the polytope is obtained, which is in fact the volume of the dual polytope P • (k, n) [2]. It is straightforward to triangulate P • (k, n) by using the reference point (1, 0, · · · , 0) inside it; the total volume is given by the sum of the volume of "cells" formed by connecting the point to each facet of P • (k, n), which corresponds to a vertex of P(k, n). The facet is a (d−1)-dim simplex if and only if the corresponding vertex in P(k, n) is simple, in which case the cell is a d-dim simplex, with volume given by the product of 1/F 's for the d facets adjacent to that vertex in P(k, n). Otherwise, we have to triangulate that facet into simplices. We emphasize that there are numerous other triangulations and ways for evaluating the canonical function, and once the (invariant) polytope is given, it is not very important which particular triangulation we use. For example, the dual polytope P • (3, 5) and its triangulation are shown in figure 2 (b). In this case, all facets of the dual polytope are simplices (intervals) and the contribution to the volume from the i-th facet is 1/(s i−2,i−1,i s i,i+1,i+2 ), and the sum obviously gives the amplitudes m Next, let us study the first non-trivial example with (k, n) = (3, 6).

Example: P(3, 6) and its generalized amplitudes
Using web variables, the regular for G + (3, 6)/T becomes where we have X 2 = s 123 , X 3 = s 456 , and X 1 = S 3456 , X 4 = S 1234 . For the latter we have defined S A for any subset A with |A| ≥ 3 (for |A| = 3, S A := s A ): It is straightforward to compute the polytope P(3, 6) by taking the Minkowski sum of Newton polytopes from R 3,6 . Let's list the result first, namely the 16 linear functions for the facets which cut out the polytope (or the 16 poles of the canonical function). Clearly X 1 to X 4 are 4 of the facets, and by cyclic permutations we expect the following two cyclic classes: S 123 , · · · , S 612 ; S 1234 , · · · , S 6123 . (2.10) Indeed we find these 12 facets, but there are 4 additional ones, which are of a distinct type. Let's first define S (B) = SB whereB denotes the complement set, for any subset B with 2 ≤ |B| ≤ n − 3 (this notation will become clear in the next section), thus e.g. We emphasize that the 16 facets/poles of (2.10) and (2.11) are result from Minkowski sum, which express them as linear combinations of X's and c's, and here are a few examples with rewriting in our new notation For example for the last one we indeed find c 356 + X 1 − X 3 + X 4 = (s 125 + s 256 + s 561 + s 612 ) + (s 123 + s 124 ) = S 12 (34) . (2.14) Remarkably, these facets, and the polytope cut out by them, can be obtained using SE map (1.3). For example, the equation for X 1 is The first two facets in (2.13) can be obtained by taking the limit of x 1 → 0 or ∞ respectively. More generally, a facet is mapped from taking limits for multiple web variables. For example, the limit x 1 =x 1 ǫ, x 4 =x 4 ǫ, x 3 =x 3 ǫ −1 with ǫ → 0 gives the last facet in (2.13). Before proceeding, let us comment on identities satisfied by the last type of facets. On the support of "momentum conservation" (1.7), we have an identity, (2.16) Thus e.g. we have S 12(34) = S 34(56) = S 56 (12) , S 23(45) = S 45(61) = S 61 (23) , so only 2 of these 6 cyclic images are distinct; the other orientation also gives 2 distinct facets, which are the 4 in (2.12). The list of these 16 facets, which can be expanded in X's and c's, is a rather compact and invariant way to describe the polytope. Note that there are three kinds of facets for P (3,6), which are all three-dimensional polytopes and we draw them in figure 3.
Last but not least, let us also describe the 48 vertices which can be computed from the facets. Out of them, 46 of them are simple vertices, and the remaining two are non-simple ones. The former correspond to facets of the dual polytope P • (3, 6) that are simplices, and the latter correspond to non-simplicial ones. Examples for these two kinds are given in figure 3 (b). The f -vector for the polytope P(3, 6) is (1, 48, 98, 66, 16, 1).
As we have mentioned, the canonical function, A 3,6 is given by the volume of the dual polytope P • (3, 6), which can be triangulated using the reference point (1, 0, 0, 0, 0) in CP 4 and written as a sum over its facets. For a simplex such as the one in figure 4 (a), the contribution to the volume is simply the product of inverse facets  For a non-simplicial facet of the dual polytope such as the one in shown in figure 4 (b), one can triangulate it further. Note that it is a bipyramid with 5 vertices, which in the original polytope P(3, 6) correspond to the 5 facets touching the vertex, namely S 1234 , S 3456 , S 5612 , S 12(34) , S (12)34 . One way to triangulate it is to cut it into two simplices by the 2-plane passing the vertices S 1234 , S 3456 , S 5612 in the dual polytope. Therefore, the contribution to the volume is given by . (2.17) By adding up the contribution from 48 facets of P • (3, 6), we obtain its volume, or equivalently the canonical function of P(3, 6), which we also refer to as the generalized amplitudes A 3,6 . We emphasize that there are numerous ways to compute A 3,6 by triangulating the dual (or the polytope itself). The invariant object is the polytope P(3, 6) cut out by the 16 facets as we discussed.

Polytopes, amplitudes and boundary configurations for k = 3
Using the methods of Minkowski sum and SE map, we have computed the polytope P(3, n) up to n = 10; since the subspace is already given in the above section (setting all c I in (2.3) as positive constants), the knowledge of all facets (or equivalently all the poles of the canonical form) suffices to determine P(3, n) completely. As we discussed, a natural basis for the subspace X is given for all multiplicity,  We emphasize that by expanding the facets/poles into the basis X and the constants in (2.3) cuts out the polytope P(3, n).
As we have seen, any facet of P(k, n) is the image of certain boundary configurations of G + (k, n)/T under SE map. Given a parametrization, such a boundary configuration can be described by taking certain limit of the web variables, which is controlled by one parameter ǫ → 0. Recall that the SE map is given by d log R k,n = 0, and under the limit that approaches a co-dimension 1 boundary, d log R k,n becomes proportional to d log ǫ, and the residue at ǫ → 0 is nothing but the linear function for the corresponding facet 1 where we have denoted the boundary configuration as γ, and the corresponding facet S[γ]. This is a direct consequence of our SE map, and has also been shown in the context of higher-k CHY formula in [14]. Therefore, once we have obtained a boundary configuration we can directly read off the facet by computing the residue of d log R k,n . On the other hand, already in the simplest examples of n = 5, 6 with k = 3, we have seen that in order to describe these facets in an invariant fashion, i.e. independent of parametrization, it is natural to introduce the notation S A , S (B) and S A(B) . As we will see shortly, such a notation exactly reflects the underlying boundary configurations for general n, and the latter all have very natural geometric interpretations. Restricting to k = 3 case for now, note that d log R 3,n = a<b<c s a b c d log(a, b, c), and one way to describe a boundary configuration γ is to list at which order every minor (a, b, c) vanishes in the limit ǫ → 0. We will show that each facet for P(3, n) corresponds to a boundary configuration of G + (3, n)/T , which can be encoded geometrically by how the n points on a plane (columns of the 3 × n matrix with first components fixed to be 1) become collinear or colliding with each other.
The simplest example is the configuration for the facet S A which turns out to corre-spond to a set of points becoming collinear, i.e.
where a 1 , a 2 , a 3 are any 3 different labels in A, and we will use a line to denote this configuration. For example, for A = {1, 2, 3, 4}, we have 1 2 3 4 , and under the SE Similarly, we can have a set of points in B collide or coincide with each other, which means that the minors involving 3 or 2 of these labels vanish as O(ǫ 2 ) or O(ǫ), respectively We denote the configuration by a black dot B , and the corresponding facet reads By momentum conservation it is easy to show that in fact S (B) is equal to SB whereB denotes the complementary set of B: Thus we say that the configuration where points in B collide is equivalent to the one where points inB become collinear B ≃B . We emphasize that such an equivalence between boundary configurations of G + (k, n)/T can also be verified by direct computation, but here we see how it directly follows from momentum conservation. For example The last class of configurations which have appeared for G + (3, 6)/T are those for the facets S A(B) . As the notation suggests, such a configuration corresponds to the case where points in B collide and at the same time they are also collinear with points in A. For example, for S 12 (34) which is exactly the definition of S A(B) given in (2.11).
Note that we also have equivalence relation for this class of configurations, which can be diagrammatically shown by . (3.11) Here we comment that we have assumed there are at least two points in each of the set A, B, C in (2.11) and (3.10). However when any of them are empty, such as B = ∅, the identity (2.16) as well as equivalence relation (3.10) still holds and will reduce to the case where S A = S (Ā) . Besides, when there is only one point in a set, such as B = {b}, then S A(B) is nothing but SÃ withÃ = A ∪ {b} as implied by the geometry of the boundary configuration (3.10).
In the rest of the section, we will present all the facets of P(3, n) and corresponding boundary configurations of G + (3, n)/T up to n = 9, where more types of configurations appear as n increases. It turns out that all boundary configurations up to n = 8 can be nicely described using lines and black dots, which we will refer to as normal configurations, and via (3.2) they give all facets of the polytopes. Starting at n = 9 we will see even more complicated configurations, which requires "blowing up" the black dot etc.. We will refer to those configurations which are not normal ones as exceptional configurations. We also got all of the 314 × 10 facets of P(3, 10) by SE map, which are put in the auxiliary file G310facets.txt. As we have emphasized repeatedly, once the facets are known, the polytope can be cut out directly. We will also comment on other aspects of the result, such as various types of vertices, the computation of the dual volume and connections to cluster polytopes up to n = 8.

P(3, 7)
Using Minkowski sum or SE map, we obtain all 6 × 7 = 42 facets for the (3,7) case. All of them can be obtained by cyclic permutations of point labels of the following 6 seeds S 123 , S 1234 , S 12345 , S 12 (34) , S (12)34 , S (12)3(45) . (3.12) The first 5 types of facets have been explained before, so we will focus on the last type.
As the notation suggests, the configuration can be drawn as 1, 2 3 In fact, on the support of momentum conservation, one can check S (3.14) Here the configuration means (456) where the last line explicitly reads S 45(67) +S (67)12 −S (67) . By momentum conservation, it is easy to show that (3.13) equals to (3.15). Again, one can check that the two configurations are equivalent: 1, 2 3 4, 5 Let us spell out this type of facets and corresponding configurations more generally. We start with the configuration where points in A and C collide respectively, and additionally points in A and B are collinear, as well as those in B and C. Using the map we obtain the facet where we have used the notation S B|C defined in (2.11). It is an interesting exercise to show that on the support of momentum conservation, it equals to the facet of a different configuration: where we have defined the new facet as Diagrammatically, the equivalence of the two configurations reads (3.20) Here we comment that we have assumed there are at least two points in each of the sets A, C, D in (3.18)   Each simple vertex would correspond to a simplex as a facet on dual polytope P • (3, 7). Likewise in P • (3,6), their contribution to the whole amplitudes is the volume of the convex hull of this facet and the reference point (1, 0, 0, 0, 0, 0, 0), and just equals to the inverse of the production of the facets involved.
As for those remaining non-simple vertices, their corresponding facets on the dual polytope P • (3, 7) are non-simplicial. Similarly to those two non-simplicial facets in P • (3, 6), we can triangulate these non-simplicial facets in P • (3, 7) into simplices to calculate their contribution to the whole amplitudes. Here we give one example where a vertex of P(3, 7) is intersected by such 7 facets:  7) and we obtain the generalized amplitude A 3,7 . The result is given in the auxiliary file G37.nb. In [1], it has been shown that P(3, 7) is a degeneration of the generalized associahedron of type E 6 , which we denote as P(E 6 ). The latter is a simple polytope, with in total 833 simple vertices. A set of extra identities among the boundaries of P(3, 7) such as (3.22) lead to the degeneration from P(E 6 ) to P(3, 7) by making various simple vertices merge into non-simple ones.

P(3, 8)
The same construction is also valid to build the polytope P (3,8). From Minkowski sum or SE map, we found that there are in all 15 × 8 = 120 facets. All of them can be obtained by cyclic permutation of point labels of the following 12 seeds, where the facets in the first two lines has been explained before, so we focus on the three new types in the third line. For convenience we denote them as S 12 (34) Let's spell out these types of facets and corresponding configurations more generally. We start with the configurations where points in B and D collide respectively, and additionally points in A and B are collinear, as well as those in B and C, and those in C and D. Using the map (3.2) we obtain the facet, (3.26) Likewise, with the reflection of labels, we have, We can also have the configurations where points in B and D collide respectively, and additionally points in A and B are collinear, as well as those in B and C, those in C and D, and those in D and E. Using the map (3.2) we obtain the facet, (3.28) Here A and E could share one point. See  As in the case of (3, 6) and (3,7), each simple vertex corresponds to a simplex as a facet on dual polytope P • (3,8), and their contribution to the generalized amplitude is the volume of the convex hull of this facet and the reference point (1, 0, 0, 0, 0, 0, 0, 0, 0), which equals to the inverse of the production of the facets involved. Summing over all the 13612 facets of the dual polytope P • (3, 8) as above, we get its entire volume, which gives the generalized amplitude A 3,8 . The result is given in the auxiliary file G38.nb. As shown in [1], P(3, 8) is a degeneration of the generalized associahedron of type E 8 , which we denote as P(E 8 ). The latter is a simple polytope with 128 facets and 25080 vertices. Note that there are only 120 facets for P (3,8), so the degeneration from P(E 8 ) to P(3, 8) has made 8 facets disappear. We find that the intersection of two facets S (12)345 , S 456(78) of P (3,8), which is a codim-2 boundary, can actually be obtained by shrinking a facet of P(E 8 ). The remaining seven disappearing facets are completely analogous. See Appendix A for more details.

P(3, 9)
Now we move to the polytope P (3,9), and by the construction of Minkowski sum or SE map, we have obtained 471 facets in total, which are put in the auxiliary file G39facets.txt. Out of these facets, there are 38 × 9 + 3 × 3 = 351 facets which correspond to normal configurations. All of them can be obtained by cyclic permutation of 41 seeds. Let's first list the 38 cyclic classes of length 9, where the seeds are,  There are 3 more cyclic classes of length 3, where the seeds are S 123(456) , S (123)456 , S (12)3(45)6(78)9 (12) . (3.33) All the facets above have be explained before except for S (12)3(45)6(78) and S (12)3(45)6(78)9 (12) . As the notation suggest, their configurations can be drawn as Apart from these 351 facets, there are 120 remaining facets which come from configurations that require "blowup". Before, when we have two colliding points a, b and another point i that doesn't collide to them, their minor (a, b, i) vanishes at order O(ǫ). However their minor can also vanish at order O(ǫ 2 ) when b collides to a in the direction determined by a and i. New method is needed to study this feature. However some "blowup" of the normal configurations for S 12(345)6(78) , S 12(345)6(78)91 and S (12)3(45)6(78)9 (12) indeed produce the seeds of new boundary configurations for the remaining 120 facets of (3,9). First we have six kinds of configurations containing the new structure (3.38),

Polytopes, amplitudes and boundary configurations for k = 4
In this section, we move to k = 4 and present our results for P(4, n) up to n = 9 using Minkowski sum or SE map. Again since the subspace is given, it suffices to provide linear functions for all the facets (or poles of the amplitudes), which cut out the polytope P(4, n). Similar to the case in k = 3, each facet of P(4, n) corresponds to a boundary configuration of G + (4, n)/T where we take limit of one or more web variables controlled by a parameter ǫ → 0. Each facet is given by the residue at ǫ → 0 via the SE map, (3.2).
We will again provide a gauge-invariant way to describe these facets and the corresponding boundary configurations, which are now configurations of n points in 3-dimensional space. As a warm-up, we will first take a look at these configurations in G + (4, 7)/T . The "parity" duality between G + (4, 7)/T and G + (3, 7)/T allows us to directly obtain all the facets of P(4, 7) from those of P(3, 7) by the isomorphism s i j k → s a b c d where {i, j, k, a, b, c, d} = {1, 2, ..., 7}.
This isomorphism, when applied to the facets of P (3, 7), provides the dictionary between their facets in table 3. We will explain the notation for facets of P(4, 7) which is encoding the corresponding boundary configurations of G + (4, 7)/T . This will serve as our starting point for presenting facets and configurations for higher points with k = 4.  Table 3: Dictionary between facets in P(3, 7) and those in P (4, 7).
The simplest facets of P(3, 7) are like S 123 = s 123 , which map to s 4567 for P (4, 7). We denote the facet as S 4567 , and in general S A for a set A is defined as: (4.1) The boundary configuration of G + (4, n)/T corresponding to S A is given by a configuration where all points in A are coplanar. This means where a 1 , a 2 , a 3 , a 4 are 4 labels in A. We draw this configuration by putting all points in A on a plane. For example, for A = {1, 2, 3, 4, 5}, we have The next case concerns facets like S 1234 in P (3,7). Under the isomorphism, it becomes a facet of P(4, 7), which reads s 123 + s 124 + s 134 + s 234 → s 4567 + s 3567 + s 2567 + s 1567 . (4.4) We denote the facet as S [567] , where in general S [B] for a set B is defined as: The boundary configuration corresponding to the facet S [B] is one where the points in B are collinear, i.e. for any b 1 , · · · , b 4 ∈ B and i ∈B, We draw points in B on a line (which is of course on a plane) to denote the configuration. (4.7) Next we move to facets like S (12) , which maps to a facet of P(4, 7) which reads We denote such a facet as S (12) , and in general S (C) for a set C is defined as: The configuration corresponding to S C is that all the points in C collide, i.e. for any c 1 , · · · , c 4 ∈ C and i, j ∈C, We draw all points in C on a black dot to denote this configuration. For example. for C = {1, 2, 3, 4}, we have  Similar to the identity S (B) = SB for k = 3, here for k = 4 we have the following identities on the support of momentum conservation a,b,c =i s a b c i = 0 (for i = 1, 2, · · · , n): (4.12) The corresponding configurations are equivalent for k = 4: points in A are coplanar is equivalent to the points inĀ colliding, and points in A collinear is equivalent to points in A collinear. Before proceeding, note that a natural basis where the polytope lives can be written using these facets already. We have X = {S 1234 , S 12345 , S [123] , S [1234] , S (12) , S (123) } for (4,7), and for general n the basis X = {X 1 , · · · , X 3(n−5) } can be written as  which corresponds to a configuration where points 1, 2 collide and they become coplanar with points 5, 6, 7 at the same rate, and we denote the facet as S 567 (12) . Recall that for P (3, 7), we have the identity S (4.15) As the notation suggests, the first configuration denotes that points 3, 4 collide and they become collinear with 1 and 2; the second configuration denotes that points 5, 6, 7 are collinear, and become coplanar with 3 and 4. Thus we have an identity for (4, 7) as well: where points in C collide, and they are collinear with points in B, and finally points in B, C altogether become coplanar with points in another set A. Via the map (3.2), the facet corresponding for such a configuration becomes is also a facet given by We denote such a facet as S (45)67 (12) , and the corresponding configuration is that points 1, 2 and 4, 5 collide respectively at the same rate, and both of them become coplanar with points 6 and 7. Diagrammatically the configuration is denoted as 1, 2 4, 5 7 6 .
Recall that for (3, 7) we have S (12)3(45) = S 45(67)12 , and the facet of (4, 7) corresponds to the latter reads   Besides, there is one identity as an analog of (3.18) on the support of the momentum conservation,    respectively. Under cylclic permutations, they give 4 classes of length 8, and by reflection 4 more classes, thus in total we have 64 more facets.
Apart from these 184 + 20 + 64 facets, there are 92 remaining facets which come from exceptional configurations that require "blowup".
First, when two points a, b collide and at the same time they are collinear with two more points i, j at different positions, we can require a collide to b in a particular direction paralleling to the plane i j l determined by i, j and another point l that is not collinear with them, which makes the minors (a, b, i, l), (a, b, j, l) vanish at order O(ǫ 2 ) instead of O(ǫ).
Similarly to the case (3,9), we introduce a ball to show the new structure, where the radius of the ball is understood to be at order O(ǫ). We use a parallel plane to indicate that a is colliding to b along a particular plane paralleling to i j l. By computing the residue of d log R 4,n (ǫ) for the above new configuration graph (4.39), one can easily find that it is directly related to that of normal configuration by adding several generalized Mandelstam variables, where we "blow up" the point c to a ball of radius ǫ, and we see that the distance of c to the plane abi is at the order O(ǫ 2 ), thus as ǫ → 0, c is still on the plane abi. Similarly to (4.40), we have where in the second configuration, 5 and 6 collide and at the same time they are collinear with 3 and 4, but 5 is collinear with them along a partuclar plane 134 determined by 1,3 and 4 (obviously 2 is also on the plane), while 6 is collinear with them along a generic direction. Compared to the last configuration in (4.41), we see the second configuration here involves a different way to "blow up" the same black dot of the second configuration in (4.37). Via the map the two configurations in (4.45) give the following two seeds of facets: Under the cyclic permutations, they give 2 classes of length 8, and by reflection 2 more classes, thus in total we have 32 more facets.

P(4, 9)
The same construction is also valid to build the polytope P (4,9). From SE map, we found that there are in all 2155 × 9 = 19395 facets, with 2155 cyclic classes of length 9. The seeds are listed in the auxiliary file G49seeds.txt. Given all these facets F a of P (4,9), the polytope can be cut out by these inequalities F a ≥ 0 for a = 1, 2, · · · , 19395. where the first one is an analog of the first configuration given in (4.32) while the remaining two do not have analogs in (4,8). Via the map the first configuration in above equation gives the facet S [12 (3)4(567)] for (4,9) and the remaining two give respectively The remaining 2017 cyclic classes of facets come from exceptional configurations which can be obtained by "blowing up" some black dots of normal configurations. For example, for the first boundary configuration of (4.49), we can require that point 6 and 7 collide to point 5 along a particular plane, such as the one parallel to the plane 348 determined by 3,4,8, which can be drawn as Another way to "blow up" the same black dot is to require additionally that 6 collides to 5 in a particular direction parallel to the line 34, while 7 collides to 5 along a generic direction, which can be drawn as  There are many more exceptional boundary configurations than normal ones in (4,9), since there are many new ways to "blow up" black dots here. Starting from any normal configuration, one could get the first series of exceptional configurations by "blowing up" a black dot, and then "blow up" another black dot to get more exceptional ones. Repeat this process and in the end one finds all 19395 configurations for (4,9).

Outlook
In this note we study the polytopes P(k, n) which control the convergence and leading order as α ′ → 0 of natural generalizations of open-string integrals, namely the Grassmannian stringy integrals (1.1). The computation via Minkowski sum or SE map for such polytopes is straightforward (albeit tedious for higher k and n), which are cut out by inequalities F a ≥ 0 for facets F a . However, as we have emphasized, the main novelty is that we recast the linear functions F a in a manifestly gauge-invariant and cyclic manner, which are the analog of the usual planar variables for k = 2 case. Given our results up to n = 9, one can easily compute the canonical functions by e.g. triangulating the dual polytopes, which produce the generalized bi-adjoint amplitudes A k,n . This is also a straightforward computation, as no extra information is needed beyond our construction for the polytope.
More importantly, as our notation for the facets already suggests, it is very natural to find all the boundary configurations of G + (k, n)/T which are in bijection with the facets of P(k, n) via the SE map. Such a boundary configuration is controlled by a single parameter ǫ → 0, and can be specified by given at which order in ǫ certain minors vanish. It is intriguing that these configurations have nice geometric interpretations in terms of n points in (k−1)-dimensional space. As we have seen, for k = 3, each configuration concerns which of the n points are collinear or colliding, and for k = 4, which of the n points are coplanar, collinear or colliding; new configurations which require "blowup" are also needed starting at n = 9 for k = 3 and n = 8 for k = 4.
Our preliminary investigations have left many open questions unanswered. The most pressing question is of course how to find all facets and boundary configurations for higher k and n. In particular, it is conceivable that all n results can be obtained for k = 3 at least, and it would be highly desirable to compare these results with the configurations of G + (k, n)/T from e.g. hypersimplex decomposition [32]. On the other hand, there are other methods for studying these facets, configurations and the related fan N (P(k, n)), which can be derived from the tropical positive Grassmannian. The generalized amplitudes A k,n can be directly obtained from tropical positive Grassmannian, as shown in [14][15][16][17], where cluster variables are used to find a natural triangulation of the fan N (P(k, n)). Each ray of the fan, which corresponds to a vertex of the dual polytope P • (k, n), is associated with a pole of A k,n . The rays of the fan are closely related to cluster variables [33,34] and it would be highly desirable to connect the cluster variables to boundary configurations of G + (k, n)/T , which may help us to find the general pattern.
Very recently, inspired by the connections between tropical Grassmannian for k = 3 and metric arrangements of trees explained in [35], Borges and Cachazo proposed [36] that planar collections of trivalent Feynman diagrams can be used to compute the generalized bi-adjoint amplitudes A 3,n . This idea has been generalized to planar arrays of trivalent Feynman diagrams for any k [37]; such a planar array of Feynman diagrams is believed to correspond to a cone of the fan N (P(k, n)), thus a facet of the dual polytope P • (k, n), or a vertex of our P(k, n). As a strong check for both our computation and this idea, we have seen that the number of vertices of P(k, n) is equal to that of planar arrays, for (3,6), (3,7), (3,8), (3,9) and (4,8). Moreover, the amplitudes up to eight points also agree, and it would be interesting to compare for k = 4, n = 9 and even beyond.
Based on the idea of planar arrays of Feynman diagrams, we can look at the polytope P(k, n) in a different way. Recall that for k = 2, each vertex of ABHY associahedron, P(2, n), corresponds to a trivalent Feynman diagrams, while higher-dimension boundaries correspond to diagrams where some cubic vertices are merged to higher-valency vertices. For higher k, each vertex of P(k, n) corresponds to a set of planar trivalent Feynman di-agrams and one can expect that the a higher-dimension boundary of P(k, n) correspond to degenerate planar arrays where cubic vertices of Feynman diagrams get merged. In other words, the polytope P(k, n) provides a geometry description of the relations between different (degenerated) planar arrays, and the combinatorial study of (degenerated) Feynman diagrams in (degenerated) planar arrays and their relations [38] can also help us have a better understanding of the polytope P(k, n). Very recently more progress has been made [39,40] in the study of tropical positive Grassmannian using the method of matroid subdivision [35]. It would be interesting to see how all these methods can help us to push further the understanding of P(k, n) and tropical positive Grassmannian.
There are also various interesting avenues for future investigations on P k,n , which generalize aspects of the kinematic associahedron and string amplitudes for k = 2. For example, it is natural to study generalized amplitudes m (k) n (α|β) with α and β the orderings of the form and integration domain, respectively, which is the analog of m n (α|β) [2]. For k = 2, the Parke-Taylor factor is the simplest leading singularity and all other leading singularities can be linearly expanded onto Parke-Taylor factors [41]. For k ≥ 3, there are new kind of leading singularities that can not be expanded into Parke-Taylor factors., and it'd be interesting to find their corresponding polytopes. Relatedly, it may be worth to find the analog of polytope in [6] and those in [42]. Besides, one can also study α ′ expansion of Grassmannian string integrals or even the structure at finite α ′ , such as the recurrence relation [1] and integration-by-parts reduction as in [43,44] for k = 2.
Finally, as already mentioned in [12], the ultimate question is how can one ask questions that are related to G + (4, n)/T and P(4, n), to which the answer are the all-loop scattering amplitudes in planar N = 4 SYM? where S ′ (12)3(45)6(78) is the facet of P(E 8 ) corresponding to S (12)3(45)6(78) and the dark shadow shows the region inside the polytope. We can then see the facet S (12)3(45)6(78) does not contribute anything to the polytope P (3,8).
Now that those eight facets of P(E 8 ) are true poles in the amplitudes A(E 8 ). However, as they are no longer facets of P (3,8), they should turn to be "spurious poles" of the amplitudes A 3,8 . Let us see how the change happens. We take S (12)3(45)6(78) as the example. By carefully checking all of the 25080 simplicial facets of the dual polytope P • (E 8 ), we have found that: (1) if a facet is surrounded by the vertex corresponding to S ′ Likewise, for the other 7 disappeared facets in the cyclic class of S (12)3(45)6(78) , also totally "lose their role" in P (3,8) and P • (3,8). They are also no longer poles in A 3,8 .