Symbol Alphabets from Plabic Graphs

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 suggest an algorithm for computing these symbol alphabets from plabic graphs by solving matrix equations of the form C Z = 0 to associate functions on Gr(m,n) to parameterizations of certain cells of Gr(k,n) indexed by plabic graphs. For m=4 and n=8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-point amplitude from four plabic graphs.


Introduction
A central problem in studying the scattering amplitudes of planar N = 4 super-Yang-Mills (SYM) theory is to understand their analytic structure. Certain amplitudes are known or expected to be expressible in terms of generalized polylogarithm functions. The branch points of any such amplitude are encoded in its symbol alphabet-a finite collection of multiplicatively independent functions on kinematic space called symbol letters [1]. In [2] it was observed that for n = 6, 7, the symbol alphabet of all (then-known) n-point amplitudes is the set of cluster variables [3,4] of the Gr(4, n) Grassmannian cluster algebra [5]. The hypothesis that this remains true to arbitrary loop order provides the bedrock underlying a bootstrap program that has enabled the computation of these amplitudes to impressively high loop order and remains supported by all available evidence (see [6] for a recent review).
For n > 7 the Gr(4, n) cluster algebra has infinitely many cluster variables [4,5]. While it has long been known that the symbol alphabets of some n > 7 amplitudes (such as the two-loop MHV amplitudes [7]) are given by finite subsets of cluster variables, there was no candidate guess for a "theory" to explain why amplitudes would select the subsets that they do. At the same time, it was expected [8,9] that the symbol alphabets of even MHV amplitudes for n > 7 would generically require letters that are not cluster variables-specifically, that are algebraic functions of the Plücker coordinates on Gr (4, n), of the type that appear in the one-loop four-mass box function [10,11] (see Appendix B). (Throughout this paper we use the adjective "algebraic" to specifically denote something that is algebraic but not rational.) As often the case for amplitudes, guesses and expectations are valuable but explicit computations are king. Recently the two-loop eight-point NMHV amplitude in SYM theory was computed [12], and it was found to have a 198-letter symbol alphabet that can be taken to consist of 180 cluster variables on Gr (4,8) and an additional 18 algebraic letters that involve square roots of four-mass box type. (Evidence for the former was presented in [9] based on an analysis of the Landau equations; the latter are consistent with the Landau analysis but less constrained by it.) The result of [12] provided the first concrete new data on symbol alphabets in SYM theory in over eight years. We will refer to this as the "eight-point alphabet" in this paper since (turning again to hopeful speculation) it may turn out to be the complete symbol alphabet for all eight-point amplitudes in SYM theory.
A few recent papers have sought to explain or postdict the eight-point symbol alphabet and to clarify its connection to the Gr (4,8) cluster algebra. In [13] polytopal realizations of certain compactifications of (the positive part of) the configuration space Conf 8 (P 3 ) of eight particles in SYM theory were constructed. These naturally select certain finite subsets of cluster variables, including those in the eight-point alphabet, and the square roots of four-mass box type make a natural appearance as well. At the same time, an equivalent but dual description, involving certain fans associated to the tropical totally positive Grassmannian [14], appeared simultaneously in [15,16]. Moreover [15] proposed a construction that precisely computes the 18 algebraic letters of the eight-point symbol alphabet by (roughly speaking) analyzing how the simplest candidate fan is embedded within the (infinite) Gr (4,8) cluster fan.
In this paper we show that the algebraic letters of the eight-point symbol alphabet are precisely reproduced by an alternate construction that only requires solving a set of simple polynomial equations associated to a certain plabic graph. This raises the possibility that symbol alphabets of SYM theory could be encoded more generally in certain plabic graphs. In Sec. 2 we introduce our construction with a simple example and then complete the analysis for all graphs relevant to Gr(4, 6) in Sec. 3. In Sec. 4 we consider an example where the construction yields non-cluster variables of Gr (3,6) and in Sec. 5 we apply it to a graph that precisely reproduces the algebraic functions on Gr (4,8) that appear in the symbol of [12].

A Motivational Example
Motivated by [17], in this paper we consider solutions to sets of equations of the form which are familiar from the study of several closely related or essentially equivalent amplitude-related objects (leading singularities, Yangian invariants, on-shell forms; see for example [18][19][20][21][22]). For the application to SYM theory that will be the focus of this paper Z is the n×4 matrix of momentum twistors describing the kinematics of an n-particle scattering event, but it is often instructive to allow Z to be n × m for general m.
The k × n matrix C(f 0 , . . . , f d ) in (2.1) parameterizes a d-dimensional cell of the totally non-negative Grassmannian Gr(k, n) ≥0 . Specifically, we always take it to be the boundary measurement of a (reduced, perfectly oriented) plabic graph expressed in terms of the face weights f α of the graph (see [22,23]). One could equally well use edge weights, but using face weights allows us to further restrict our attention to bipartite graphs and eliminates some redundancy; the only residual redundancy of face weights is that they satisfy a f α = 1 for each graph.
For an illustrative example consider which affords us the opportunity to review the construction of the associated C-matrix from [23]. The graph is perfectly oriented because each black (white) vertex has all but one incident arrows pointing in (out). The graph has two sources (1, 2) and four sinks (3, 4, 5, 6) so we begin by forming a 2 × (2 + 4) matrix with the 2 × 2 identity matrix occupying the source columns: The remaining entries are given by where s is the number of sources strictly between i and j, the sum runs over all allowed paths p from i to j (allowed paths must traverse each edge only in the direction of its arrow) and the product runs over all faces α to the right of p, denoted by p. In this manner we find where ijkl = det(Z i Z j Z k Z l ) are Plücker coordinates on Gr(4, 6). We pause here to point out two features evident from (2.6). First, we see that on the solution of (2.1) each face weight evaluates (up to sign) to a product of powers of Gr(4, 6) cluster variables, i.e. to a symbol letter of the six-point alphabet of amplitudes in SYM theory [1]. Moreover, the set of cluster variables that appear ( 2346 , 2356 , 2456 , and the six frozen variables) constitute a single cluster of the Gr(4, 6) algebra.
The fact that cluster variables of Gr(m, n) seem to arise, at least in this example, raises the possibility that the symbol alphabets of amplitudes in SYM theory might be given more generally by the face weights of certain plabic graphs evaluated on solutions of C · Z = 0. A necessary condition for this to have a chance of working is that the number of independent face weights should equal the number of equations (both 8 in the above example); otherwise the equations would have no solutions or continuous families of solutions. For this reason we focus exclusively on graphs for which (2.1) admits isolated solutions for the face weights as functions of generic n × m Z-matrices; in particular this requires that d = km. In such cases the number of isolated solutions to (2.1) is called the intersection number of the graph.
The possible connection between plabic graphs and symbol alphabets is especially tantalizing because it manifestly has a chance to account for both issues raised in the introduction: (1) while the number of cluster variables of Gr(4, n) is infinite for n > 7, the number of (reduced) plabic graphs is certainly finite for any fixed n, and (2) graphs with intersection number greater than 1 naturally provide candidate algebraic symbol letters. Our showcase example of (2) is presented in Sec. 5.

Six Particle Cluster Variables
The problem formulated in the previous section can be considered for any k, m and n. In this section we thoroughly investigate the first case of direct relevance to the amplitudes of SYM theory: m = 4 and n = 6. Although this case is special for several reasons, it allows us to illustrate some concepts and terminology that will be used in later sections.
Modulo dihedral transformations on the six external points, there are a total of four different types of plabic graph to consider. We begin with the three graphs shown in Fig. 1 (a)-(c), which have k = 2. These all correspond to the top cell of Gr(2, 6) ≥0 and are related to each other by square moves. Specifically, performing a square move on f 2 of graph (a) yields graph (b), while performing a square move on f 4 of graph (a) yields graph (c). This contrasts with more general cases, for example those considered in the next two sections, where we are in general interested in lower-dimensional cells.
The solution for the face weights of graph (a) (the same as (2.2)) were already given in (2.6), and those of graphs (b) and (c) are derived in (A.2) and (A.4) of Appendix A. The latter two can alternatively be derived from the former via the square move rule (see [22,23]). In particular, for graph (b) we have with f 0 , f 1 , f 7 and f 8 unchanged. 16     To every plabic graph one can naturally associate a quiver with nodes labeled by Plücker coordinates of Gr(k, n). In Fig. 1 (d)-(f) we display these quivers for the graphs under consideration, following the source-labeling convention of [24,25] (see also [26]). Because in this case each graph corresponds to the top cell of Gr(2, 6) ≥0 , each labeled quiver is a seed of the Gr(2, 6) cluster algebra. More generally we will have graphs corresponding to lower-dimensional cells, whose labeled quivers are seeds of subalgebras of Gr(k, n).
Henceforth we refer to a labeled quiver associated to a plabic graph in this manner as an input cluster, taking the point of view that solving the equations C · Z = 0 associates a collection of functions on Gr(m, n) to every such input. At the same time, there is a natural way to graphically organize the structure of each of (2.6), (A.2) and (A.4) in terms of an output cluster, as we now explain.
First of all we note from (A.2) and (A.4) that, like what happened for graph (a) considered in the previous section, each face weight evaluates (up to sign) to a product of powers of Gr(4, 6) cluster variables. Second, again we see that for each graph the collection of variables that appear precisely constitute a single cluster of Gr(4, 6): suppressing in each case the six frozen variables, we find 2346 , 2356 and 2456 for graph (a); 1235 , 2356 and 2456 for graph (b); and 1456 , 2346 , and 2456 for graph (c). Finally, in each case there is a unique way to label the nodes of the quiver not with cluster variables of the "input" cluster algebra Gr(2, 6), as we have done in Fig. 1 (d)-(f), but with cluster variables of the "output" cluster algebra Gr (4,6). We show these output clusters in Fig. 1 (g)-(i), using the convention that the face weight (also known as the cluster X -variable) attached to node i is j a b ji j where b ji is the (signed) number of arrows from j to i.
For the sake of completeness, we note that there is also (modulo Z 6 cyclic transformations) a single relevant graph with k = 1: for which the boundary measurement is and the solution to C · Z = 0 is given by (3.4) Again the face weights evaluate (up to signs) to simple ratios of Gr(4, 6) cluster variables, though in this case both the input and output quivers are trivial. This graph is an example of the general feature that one can always uplift an n-point plabic graph relevant to our analysis to any value of n > n by inserting any number of black lollipops. (Graphs with white lollipops never admit solutions to C · Z = 0 for generic Z.) In the language of symbology, this is in accord with the expectation that the symbol alphabet of an n -point amplitude always contains the Z n cyclic closure of the symbol alphabet of the corresponding n-point amplitude.
In this section we have seen that solving C · Z = 0 induces a map from clusters of Gr(2, 6) (or subalgebras thereof) to clusters of Gr(4, 6) (or subalgebras thereof). Of course, these two algebras are in any case naturally isomorphic. Although we leave a more detailed exposition for future work, we have also checked for m = 2 and n ≤ 10 that every appropriate plabic graph of Gr(k, n) maps to a cluster of Gr(2, n) (or a subalgebra thereof), and moreover that this map is onto (every cluster of Gr(2, n) is obtainable from some plabic graph of Gr(k, n)). However for m > 2 the situation is more complicated, as we see in the next section.

Towards Non-Cluster Variables
Here we discuss some features of graphs for which the solution to C · Z = 0 involves quantities that are not cluster variables of Gr(m, n). A simple example for k = 2, m = 3, n = 6 is the graph whose boundary measurement has the form (2.3) with The solution to C · Z = 0 is given by leaving eight variables which (together with 126 and 345 , which are missing) comprise a cluster of Gr (3,6). However, it is not possible to associate a labeled "output" quiver to (4.5) in the usual way, as we did for the examples in the previous section.
To turn this story around: had we started not with (4.1) but with its squaremoved partner, we would have encountered (4.5) and then noted that performing a square move back to (4.1) would necessarily introduce the multiplicative factor into four of the face weights. The example considered in this section provides an opportunity to comment on the connection of our work to the study of cluster adjacency for Yangian invariants. In [27,28] it was noted in several examples, and conjectured to be true in general, that the set of factors appearing in the denominator of any Yangian invariant with intersection number 1 are cluster variables of Gr(4, n) that appear together in a cluster. This was proven to be true for all Yangian invariants in the m = 2 toy model of SYM theory in [29] and for all m = 4 N 2 MHV Yangian invariants in [30]. We recall from [22,31] that the Yangian invariant associated to a plabic graph (or, to use essentially equivalent language, the form associated to an on-shell diagram) is given by d log f 1 ∧ · · · ∧ d log f d . One of our motivations for studying the conjecture that the face weights associated to any plabic graph always evaluate on the solution of C · Z = 0 to products of powers of cluster variables was that it would immediately imply cluster adjacency for Yangian invariants. Although the graph (4.1) violates this stronger conjecture, it does not violate cluster adjacency because on-shell forms are invariant under square moves [22]. Therefore, even though 1 × 4, 2 × 3, 5 × 6 appears in individual face weights of (4.3), it must drop out of the associated on-shell form because it is absent from (4.5).

Algebraic Eight-Point Symbol Letters
One reason it is obvious that the solutions of C · Z = 0 cannot always be written in terms of cluster variables of Gr(m, n) is that for graphs with intersection number greater than 1 the solutions will necessarily involve algebraic functions of Plücker coordinates, whereas cluster variables are always rational.
The simplest example of this phenomenon occurs for k = 2, m = 4 and n = 8, for which there are two cyclic classes of relevant plabic graphs. One of these is with 3) The solution to C · Z = 0 for generic Z ∈ Gr (4,8) can be written as  (12)(34)(78) , where a(bc)(de)(f g) ≡ abde acf g − abf g acde (5.8) and the nine a i provide a (multiplicative) basis for the algebraic letters of the eightpoint symbol alphabet that contain the four-mass box square root √ ∆ 1357 , as reviewed in Appendix B.
The nine face weights shown in (5.7) satisfy f α = 1 so only eight are multiplicatively independent. It is easy to check that they remain multiplicatively independent if one sets all of the Plücker coordinates and brackets of the form (5.8) to one. Therefore, the f α (multiplicatively) only span an eight-dimensional subspace of the full nine-dimensional space. We could try building an eight-point alphabet by taking any subset of eight of the face weights as basis elements (i.e., letters), but we would always be one letter short.
Fortunately there is a second plabic graph relevant to √ ∆ 1357 : the one obtained by performing a square move on f 3 of (5.1). As is by now familiar, performing the square move introduces one new multiplicative factor into the face weights: which precisely supplies the ninth, missing letter! To summarize: the union of the nine face weights associated to the graph (5.1), and the nine associated to its square-move partner, multiplicatively span the nine-dimensional space of √ ∆ 1357 -containing symbol letters in the eight-point alphabet of [12].
The same story applies to the graphs obtained by cycling the external indices on (5.1) by one-their face weights provide all nine algebraic letters involving √ ∆ 2468 . Of course it would be very interesting to thoroughly study the numerous plabic graphs relevant to m = 4, n = 8 that have intersection number 1. In particular it would be interesting to see if they encode all 180 of the rational (i.e., Gr (4,8) cluster variable) symbol letters of [12], and whether they generate additional cluster variables such as those obtained from the constructions of [13,15,16].
Before concluding this section let us comment briefly on "k" since one may be confused why the plabic graph (5.1), which has k = 2 and is therefore associated to an N 2 MHV leading singularity, could be relevant for symbol alphabets of NMHV amplitudes. The symbol letters of an N k MHV amplitude reveal all of its singularities, including multiple discontinuities that can be accessed only after a suitable analytic continuation. Physically these are computed by cuts involving lower-loop amplitudes that can have k > k. Indeed the expectation that symbol letters of lower-loop higher-k amplitudes influence those of higher-loop lower-k amplitudes is manifest in the Q-bar equation technology [7,32,33] underlying the computation of [12]. Moreover there is indirect evidence [34] that the symbol alphabet of the L-loop n-particle N k MHV amplitude in SYM theory is independent of both k and L (beyond certain accidental shortenings that may occur for small k or L). This suggests that for the purpose of applying our construction to "the n-point symbol alphabet" one should consider all relevant n-point plabic graphs regardless of k.

Discussion
The problem of "explaining" the symbol alphabets of n-particle amplitudes in SYM theory apparently requires, for n > 7, a mechanism for identifying finite sets of functions on Gr(4, n) that include some subset of the cluster variables of the associated cluster algebra, together with certain non-cluster variables that are algebraic functions of the Plücker coordinates.
In this paper we have initiated the study of one candidate mechanism that manifestly satisfies both criteria and may be of independent mathematical interest. Specifically, to every (reduced, perfectly oriented) plabic graph of Gr(k, n) ≥0 that parameterizes a cell of dimension mk, one can naturally associate a collection of mk functions of Plücker coordinates on Gr(m, n).
We have seen that for some graphs the output of this procedure is naturally associated to a seed of the Gr(m, n) cluster algebra; for some graphs the output is a cluster's worth of cluster variables that do not correspond to a seed but rather behave "badly" under mutations (this means they transform into things which are not cluster variables under square moves on the input plabic graph); and finally for some graphs the output involves non-cluster variables including, when the intersection number is greater than 1, algebraic functions.
We leave a more thorough investigation of this problem for future work. The "smoking gun" that this procedure may be relevant to symbol alphabets in SYM theory is provided by the example discussed in Sec. 5, which successfully postdicts precisely the 18 multiplicatively independent algebraic letters that were recently found to appear in the two-loop eight-point NMHV amplitude [12]. Our construction provides an alternative to the similar postdiction made recently in [15].
It is interesting to note that since for m = 4, n = 8 there are no other relevant plabic graphs having intersection number > 1, beyond those already considered Sec 5, our construction has no room for any additional algebraic letters for eight-point amplitudes. Therefore, if it is true that the face weights of plabic graphs, evaluated on the locus C · Z = 0, provide symbol alphabets for general amplitudes, then it necessarily follows that no eight-point amplitude, at any loop order, can have any algebraic symbol letters beyond the 18 discovered in [12].
At first glance this rigidity seems to stand in contrast to the constructions of [13,15,16] which each involve some amount of choice-having to do with how coarse or fine one chooses one's tropical fan, or equivalently how many factors to include in the Minkowski sum when building the dual polytope. But in fact our construction has a choice with a similar smell. When we say that we start with the C-matrix associated to a plabic graph, that automatically restricts us to very special clusters of Gr(k, n)those that contain only Plücker coordinates. Clusters containing more complicated, non-Plücker cluster variables are not associated to plabic graphs. One certainly could contemplate solving the C · Z = 0 equations for C given by a "non-plabic" cluster parameterization of some cell of Gr(k, n) ≥0 and it would be interesting to map out the landscape of possibilities.
It has been a long-standing problem to understand the precise connection between the Gr(k, n) cluster structure exhibited [22] at the level of integrands in SYM theory and the Gr(4, n) cluster structure exhibited [2] by integrated amplitudes. It was pointed out in [17] that the C · Z = 0 equations provide a concrete link between the two, and our results shed some initial light on this intriguing but still very mysterious problem. In some sense we can think of the "input" and "output" clusters defined in Sec. 3 as "integrand" and "integrated" clusters. (See the last paragraph of Sec. 5 for some comments on why k "disappears" upon integration.) Although we have seen that the latter are not in general clusters at all, the example of Sec. 5 suggests that they may be even better: exactly what is needed for the symbol alphabets of SYM theory.
where the x's are defined in (13) of [12]. While the overall sign of a symbol letter is irrelevant, we have taken the liberty of putting a minus sign in front of a 3 , a 4 and a 5 to ensure that each of the nine a i , indeed each individual factor appearing in in (5.7), is positive-valued for Z ∈ Gr(4, 8) >0 .