Symbol Alphabets from Plabic Graphs III: n=9

Symbol alphabets of n-particle amplitudes in N=4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4,n) as well as certain algebraic functions of cluster variables. In this paper we solve the C Z = 0 matrix equations associated to several cells of the totally non-negative Grassmannian, combining methods of arXiv:2012.15812 for rational letters and arXiv:2007.00646 for algebraic letters. We identify sets of parameterizations of the top cell of Gr_+(5,9) for which the solutions produce all of (and only) the cluster variable letters of the 2-loop nine-particle NMHV amplitude, and identify plabic graphs from which all of its algebraic letters originate.

In this paper we continue the program outlined in [11][12][13], which is based on the observation that symbol letters of SYM theory seem to naturally emerge from certain plabic graphs [17] (or equivalently, Yangian invariants). Specifically, if Z is an n × 4 momentum twistor matrix parameterizing the kinematic data for an n-particle scattering process, and if C is a k × n matrix parameterizing a 4k-dimensional cell of the totally non-negative Grassmannian G + (k, n) [18], then solving the matrix equations CZ = 0 [19,20] sets the parameters of C to various rational or algebraic functions of Plücker coordinates on G(4, n) that often turn out to be products of symbol letters of amplitudes.
In [11,12] an example for (k, n) = (2,8) was considered that precisely reproduces all of the 18 algebraic symbol letters known to appear in the 2-loop eight-particle NMHV amplitude [21]. At the same time it was pointed out that if the cell parameterized by C is not the top cell (i.e., the one with dimension k(n−k)), then one generally encounters rational quantities that are not expressible in terms of cluster variables. On the other hand, in [13] it was shown that for any cluster parameterization of the top cell (not necessarily one associated to a plabic graph), this procedure will only give cluster variables.
Here our focus is on the case n = 9, where the most up-to-date symbol alphabet information comes from the computation of the two-loop NMHV amplitude [22]. We show how to obtain all known n = 9 symbol letters from cluster parameterizations of cells of G + (k, 9). First, we provide an explicit list of cluster parameterizations of the top cell of G + (5,9) which collectively provide all 531 of the n = 9 rational letters found in [22] (and no additional letters). Second, we identify a cyclic class of parameterizations of cells of G + (3,9) which collectively provide all 99 of the n = 9 algebraic letters, together with a few additional algebraic quantities.
As already acknowledged in [11,13], we do not as of yet have a "theory" to explain the pattern of which cells are associated to cluster variables (or algebraic functions thereof) that are actually observed to appear in amplitudes. Instead, we view our work as providing some kind of "phenomenological" data in the hope that future work will be able to shed more light on this interesting problem.

n = 8 Extended Rational Alphabet
To date, a total of 180 rational letters, all of which are cluster variables of G (4,8), are known to appear in the eight-particle amplitudes of SYM theory. These letters are tabulated in [21]. By studying a certain fan one can naturally associate to the tropical positive Grassmannian (or, equivalently, its dual polytope), [8][9][10] encountered a larger list of cluster variables that includes these 180, together with 100 more. These additional variables may appear in the symbols of eight-point amplitudes that have not yet been computed. We call this collection of 280 cluster variables the n = 8 extended rational alphabet; it consists of • 68 four-brackets of the form a a+1 b c , Here abcd are Plücker coordinates on G(4, n) and we definē We know from [13] that for any cluster parameterization C of the top cell of G + (4, 8), solving CZ = 0 expresses the parameters of C in terms of products of powers of G(4, 8) cluster variables. Our aim is to identify a set of parameterizations that collectively involve precisely the 280 letters of the extended rational alphabet (and no other letters).
We begin by taking C to be the boundary measurement of the plabic graph shown in Fig. 1 (see [11,13] for more details on our conventions). Then the solution to CZ = 0 is given by By drawing the dual quiver (with arrows clockwise around white vertices and counterclockwise around black vertices) and reading off the adjacency matrix, we can mutate the face variables according to the cluster X -variable mutation rules [23] f where b i,j is the adjacency matrix of the dual quiver. Under mutations, the adjacency matrix transforms as We perform sequences of mutations on internal faces (external faces are considered frozen) and collect all monomial factors that appear in the mutated face variables. We then find a minimal set of mutation sequences for which the mutated face variables collectively contain the entire 280 letter extended rational alphabet (mod cyclic permutations of external labels), and only letters from that alphabet. Note that the cluster algebra associated with the dual quiver of the G + (4, 8) top cell is of infinite type, and we only search far enough to find minimal length mutation sequences that suffice to produce the entire 280-letter alphabet.
We find that considering all mutation sequences of up to length 5 is sufficient, and in particular we find 13 clusters that are sufficient to generate the entire 280-letter n = 8 extended rational alphabet (mod cyclic rotations of external labels). These clusters are obtained from the following 13 mutation sequences: where the sequences should be read as: {a, b, c, . . .} : mutate on the node f a , then mutate on f b , and then mutate on f c , etc. It is important to emphasize that this set of minimal length mutational sequences is not unique. Also, note that at intermediate steps between the initial cluster and the final 13 clusters obtained at the end of these sequences, one can encounter additional cluster variables not contained in the 280-letter alphabet.

Algebraic Letters
In this section we show how to obtain the algebraic letters of the the n = 9 two-loop NMHV symbol alphabet [22] by solving CZ = 0 for plabic parameterizations of nontop cells of G + (4,9). This generalizes the corresponding analysis for n = 8 carried out in [11,12].

n = 9 Two-loop NMHV Algebraic Symbol Letters
In [22] it was found that 99 multiplicatively independent algebraic symbol letters appear in the symbol of the two-loop nine-particle NMHV amplitude. All algebraic letters of two-loop NMHV amplitudes trace their origin to the one-loop four-mass box integral. Here we recall some definitions useful for expressing these letters: (3.1) We will also define 3) , (3.4) where x b−1 abcd , x c−1 abcd differ by exchanging a ↔ a−1 when the superscript is a−1, exchanging b ↔ b−1 when the superscript is b−1, and so on. With this, we can define two classes of algebraic symbol letters where the star corresponds to the six choices a−1, a, b−1, b, c−1, c of the superscript of x abcd . We note that X abcd , X bcda , X cdab and X dabc all depend on the same square root ∆ abcd . With this, we have a total of 4 × 2 × 6 = 48 algebraic letters depending on each ∆ abcd from X abcd andX abcd . In addition to these letters, there are two more letters depending on ∆ abcd bringing us to a grand total of 50 algebraic letters depending on ∆ abcd in the most general case. However, in cases where 0 ≤ m ≤ 4 of the corners of the four-mass box (from which these letters originate) contain only two particles, the number of independent letters containing ∆ abcd , is reduced to 50 − 2m. In addition, there are 33 multiplicative relations between the algebraic symbol letters of (3.5) and (3.6), meaning that the number of independent letters containing ∆ abcd is reduced to 17 − 2m. Thus in the nine-particle case, where we always have four-mass boxes with three corners containing two particles and one containing three, we have m = 3 and thus 17 − 6 = 11 letters for each ∆ abcd . There are nine different square roots at n = 9, so there are in total 11 × 9 = 99 independent algebraic symbol letters at n = 9.
3.2 n = 9 Algebraic Letters from Plabic Graphs  At n = 9, there are two cyclic classes of positroid cells with intersection number 2 and dimension 4k. We recall from [11] that the latter condition is necessary for CZ = 0 to admit solutions for generic Z, and the former condition is necessary for the solution to involve algebraic functions (and specifically, square roots). These two classes of cells are represented by the decorated permutations {2, 6, 5, 8, 7, 10, 9, 13, 12} , and {2, 6, 4, 8, 7, 10, 9, 12, 14} . (3.7) Performing all possible mutations on the internal faces of the plabic graph in Fig. 3, we find an additional 12 unique factors, which can be expressed in terms of (3.8) as , (3.10) Altogether, we therefore encounter a total of 25 algebraic factors associated to this cell of G + (3,9). We find that 20 of these 25 factors are multiplicatively independent; including, of course, the 11 √ ∆ 3579 -containing algebraic letters that appear in the 2loop nine-particle NMHV amplitude. The additional algebraic letters that we find may appear in higher, not yet computed nine-particle amplitudes, or they may be analogs of the "non-cluster variable" rational quantities that generally appear when solving CZ = 0 for non-top cells (see [11,12] for examples).