A Note on Letters of Yangian Invariants

Motivated by reformulating Yangian invariants in planar ${\cal N}=4$ SYM directly as $d\log$ forms on momentum-twistor space, we propose a purely algebraic problem of determining the arguments of the $d\log$'s, which we call"letters", for any Yangian invariant. These are functions of momentum twistors $Z$'s, given by the positive coordinates $\alpha$'s of parametrizations of the matrix $C(\alpha)$, evaluated on the support of polynomial equations $C(\alpha) \cdot Z=0$. We provide evidence that the letters of Yangian invariants are related to the cluster algebra of Grassmannian $G(4,n)$, which is relevant for the symbol alphabet of $n$-point scattering amplitudes. For $n=6,7$, the collection of letters for all Yangian invariants contains the cluster ${\cal A}$ coordinates of $G(4,n)$. We determine algebraic letters of Yangian invariant associated with any"four-mass"box, which for $n=8$ reproduce the $18$ multiplicative-independent, algebraic symbol letters discovered recently for two-loop amplitudes.


Introduction
Recently, a new computation for two-loop n = 8 NMHV amplitude is performed using the so-calledQ equations [20], which provides a candidate of the symbol alphabet for n = 8; in particular, it provides the space of 18 algebraic letters which involve square roots of the kinematics. After that new proposals concerning the symbol alphabet for n ≥ 8 have appeared [21][22][23], which exploit mathematical structures such as tropical Grassmannian [24] and positive configuration space [25].
In this short note, we make a simple observation that an alternative construction, based on solving polynomial equations associated with plabic graphs for Yangian invariants, may provide another route to symbol alphabet including algebraic letters. We will argue that the alphabet of Yangian invariants, which will be defined shortly, contains not only the (rational) symbol letters of n = 6 and n = 7 amplitudes but also the algebraic letters for two-loop n = 8 NMHV amplitude mentioned above.
We emphasize that generally for any Yangian invariant, the alphabet includes but is not restricted to the collection of poles, which have been explored in the literature so far, e.g. in the context of the so-called cluster adjacency [11,[26][27][28][29][30]. As we will see already for n = 7, by going through all Yangian invariants we find 42 poles in total, which are 7 frozen variables and 35 unfrozen ones, i.e. 7 of the 42 cluster A variables of G(4, 7) are still missing! On the other hand, the alphabet of all possible Yangian invariants consists of 63 letters, which include the 42 cluster variables. The search for alphabets of Yangian invariants is by itself a beautiful open problem related to positive Grassmannian, plabic graphs and cluster algebra [1], and let's first formulate the general problem as suggested by [31].
An N k MHV Yangian invariant is an on-shell function of (super-)momentum twistors {Z|η} 1≤a≤n , which is associated with a certain 4k-dimensional positroid cell of the positive Grassmannian G + (k, n) 1 ; the cell can be parametrized by a k × n matrix C({α} 1≤i≤4k ) where α i 's are positive coordinates, and the on-shell function (which has Grassmann-degree 4k) is given by where the contour of the integral encloses solutions of equations n a=1 C I,a ({α})Z a = 0 (for I = 1, 2, · · · , k). As first proposed in [32] and studied further in [33], it is instructive to write Yangian invariants, or more general on-shell (super-)functions, as differential forms on momentum-twistor space, where we replace the fermionic variables η a by the differential dZ a for a = 1, · · · , n. Remarkably, after the replacement the Yangian invariant defined as a 4k form, Y({Z|dZ}) := Y | ηa→dZa , becomes the pushforward of the canonical form of the cell, i d log α i , to Z space: In other words, we evaluate the form on solutions, α * (Z), of the polynomial equations C(α) · Z = 0, and we have used subscript * as a reminder that there can be multiple solutions in which case Y * becomes a collection of 4k forms, one for each solution (the number of solutions, Γ(C) is given by the number of isolated points in the intersection C ⊥ ∩ Z [1]). The solutions, α * (Z), are GL(1)-invariant functions of Plücker coordinates on G(4, n), a b c d := det(Z a Z b Z c Z d ), and when there are multiple solutions, Γ(C) > 1, they are generally algebraic functions. Since symbol letters of loop amplitudes are arguments of generalized polylogarithms, it is only natural to refer to the arguments of the d log's on the support of C · Z = 0, as the letters of a given Yangian invariant. One can define the alphabet of a Yangian invariant to be the collection of possible letters, i.e. the solutions {α * i (Z)}, which form a 4k × Γ(C) matrix. However, there are ambiguities since under a reparametrization of the cell, C({α}) → C({α }), which leaves the form invariant, the alphabet generally changes. Such reparametrizations are related to cluster transformations acting on variables of (positroid cells of) G + (k, n) [1]. In principle one could attempt to find all possible letters for a given positroid cell, by scanning through all transformations that leave the canonical form invariant, but the resulting collection of letters can easily be infinite for n ≥ 8 ! Therefore, we restrict ourselves to a very small subset of such transformations, namely those generated by equivalence moves acting on plabic graphs associated with a positroid cell. Therefore, we define the alphabet of any given Yangian invariant to be the collection of face or edge variables of all possible plabic graphs. It is obvious that this collection of letters is finite for any Yangian invariant, and we can further take the union of the alphabets for all Yangian invariants of the same n and k, which will be referred to as the alphabet of a given n and k.
There is still a residual redundancy in this definition: any monomial reparametrization of the form α j = i α n i,j i for 1 ≤ j ≤ d with det n i,j = 1 leaves the form invariant. Even for a given plabic graph, we have such redundancy in the definition of face or edge variables, and the invariant content of the alphabet is the space generated by monomials of {α * (Z) i } (or the linear space spanned by {log α * i }), and one can choose some representative letters which give the same space. For example, when α (Z) are rational with Γ(C) = 1, one can always choose a set of irreducible polynomials of Plücker coordinates, which are analogous to A coordinates of cluster algebra of G(4, n). Similar choices can be made for algebraic solutions with Γ(C) > 1, modulo ambiguities from the fact that monomials of these algebraic letters can form rational letters which are irreducible polynomials.
Since the letters of Yangian invariants are generally not DCI, there is no unambiguous way to extract the analog of cluster X coordinates. However, we will see that at least for our main example of algebraic letters, where the Yangian invariants correspond to leading singularities of four-mass boxes, there is a natural way to define such DCI X -like variables, which turns out to give precisely the same 18 algebraic alphabet for two-loop NMHV amplitudes in [34].

Letters of Yangian invariants
In the following, we will illustrate our algorithm by finding (subsets of) alphabets of Yangian invariants for certain n and k. The procedure goes as follows: for any given n and k, we first scan through all possible 4k-dimensional cell of G + (k, n) and list the resulting Yangian invariants. The representation of any Yangian invariant can be obtained using the procedure given in [1], in terms of the matrix C({α}) with canonical coordinates {α} 1≤i≤4k for the cell; equivalently these coordinates can be identified with non-trivial edge variables of a representation plabic graph. Then one can start to apply square moves to the graph, which generate reparametrization of the cell and possibly find more letters. In simple cases, it is straightforward to find all equivalence moves and obtain the complete alphabet of the Yangian invariant; for more involved cases, we will content with ourselves in finding a subset of the alphabet by applying such move once, and the results turn out to be already illuminating.

Letters of NMHV and MHV invariants
First, we present the simplest non-trivial Yangian invariant, which is the only type for NMHV (k = 1). A 4-dimensional cell C ∈ G + (1, n) can be parametrized as where only a, b, c, d and e-th entry are non-zero, and the on-shell function is given by the R invariant [a, b, c, d, e]. The solution of the linear equation C ·Z = 0 is simply and one can easily check that Obviously any reparametrization is trivial in this case, and we can choose representative letters to be the five Plücker coordinates appeared above.
By taking the union of alphabets of all NMHV invariants for n points, we trivially get the collection of all Plücker coordinates. Also note that in this special case the representative letters are exactly the poles of the Yangian invariant, but we will see immediately that this is no longer true beyond NMHV.
Our next example is for MHV (k = n−4), where the unique Yangian invariant for n points is given by the top cell of G + (n−4, n) (or equivalently that of G + (4, n), and the form is the familiar cyclic measure: The poles of the invariant (MHV amplitude) are the frozen Plücker coordinates, as they should be. However, the letters for any plabic graph involve variables other than these frozen ones. Let's start with n = 6 (for n = 5 it is just NMHV invariant [1,2,3,4,5]), and a representative plabic graph is The corresponding C matrix reads and the solution of C(α) · Z = 0 is Note that these are the A coordinates of a cluster in G(4, 6): 6 frozen variables plus 3 unfrozen ones, and they can be arranged as the quiver below. This quiver can be obtained as the dual of the plabic graph [1], and in this case, any square moves on plabic graphs are in 1:1 correspondence with mutations on the cluster. Thus, without explicitly performing square moves, we see the complete alphabet of the Yangian is given by the collection of all A coordinates, which are the 15 Plücker coordinates. For general n, one can always choose a representative plabic graph, which corresponds to an initial cluster of G(4, n) (as a generalization of the quiver above). It is straightforward to see that for such a graph, the letters are given by 4(n − 4) + 1 Plücker coordinates, with n frozen variables and 3(n − 5) unfrozen ones. The square moves still correspond to mutations but only those special ones acting on a node with two incoming and two outgoing arrows (dual to a square face), and such mutations generate new letters that are still Plücker coordinates.

Letters of n = 6 and n = 7 invariants
From the discussions above, one can determine the collection of letters for all possible Yangian invariants for n = 6. In fact, either from the union of alphabets for NMHV invariants or from that of N 2 MHV one, we get 9 + 6 = 15 A coordinates of G(4, 6). Now we move to n = 7, where in addition to NMHV Yangian invariants, we also have two classes of N 2 MHV ones, which are the first cases with letters other than Plücker coordinates. Since N 2 MHV invariants are given by parity-conjugate of the NMHV ones for n = 7, the poles of the former can be obtained as the parity conjugate of poles of the latter (which are Plücker coordinates). It is straightforward to see that parity conjugate of ii+1jj+1 and i−1ii+1j are proportional to Plücker coordinates of the same type, thus for n = 7 the only kind of Plücker coordinates with non-trivial parity conjugate are 1346 and cyclic permutations; under parity, we have 7(12)(34)(56) etc., and we see that 35 unfrozen cluster variables of G(4, 7) appear as poles of Yangian invariants. However, we will see new letters appearing for N 2 MHV invariants, which are not parity conjugate of Plücker coordinates, and they are crucial for getting all cluster variables in this case.
A representative plabic graph of a Yangian invariant in the first class is The corresponding C matrix reads and the solution of C(α) · Z = 0 is By considering cyclic rotations of the labels, we see 7 new letters of the form i (i i+1)(i+2 i+3)(i−2 i−1) for i = 1, · · · , 7, in addition to the 7 + 28 Plücker coordinates. However, if we compare to the 42 (unfrozen) A coordinates of G(4, 7), 7 variables of the form i(i−1 i+1)(i+2 i+3)(i−3 i−2) (for i = 1, · · · , 7) are still missing. Can we find such cluster variables as letters of this Yangian invariant? Very nicely, we will see that by applying square moves, these new letters appear in equivalent plabic graphs for the same Yangian invariant. Recall that it is more convenient to describe the square moves using face variables, which are given by monomials of our coordinates (certain edge variables) [1]. For an internal face with face variable f , after a square move, the only new factor introduced in the new set of face or edge variables is given by 1 + f . We consider square move on either of the two internal faces (the left one is adjacent to α 2 , α 4 , and the right one is adjacent to α 4 , α 6 ); their face variables are and we see that a new letter 6(12)(34)(57) appears. This letter and its cyclic permutations exactly give the missing 7 cluster variables mentioned above! A representative plabic graph of a Yangian invariant in the second class is We obtain the following letters, which also miss the last class of 7 cluster variables: 1267 , 1367 , 2367 , 3456 , 3457 , 3467 , 3567 , 4567 , 3(12)(45)(67) .

Algebraic letters of N 2 MHV invariants
where the two twistors A, B are parametrized as and we have two solutions for these quadratic equations, which we denote as X + and X − for X = A, B. This is the source of letters involving square roots, and the discriminant of the quadratic equations, ∆, reads All the algebraic letters for this invariant involve the square root of ∆. Therefore, we have the following 2 Plücker coordinates and 8 A-like algebraic letters: (2.8) For each algebraic letter, one can plug in two solutions, X + and X − but they are not multiplicative independent since the product is a rational function (which is expected to be given by rational letters from other Yangian invariants), thus we have 8 independent algebraic letters for this plabic graph.
Quite nicely, by a square move on the internal face, we obtain yet another algebraic letter. The face variable of the internal face is 9) and the corresponding square move produces a new factor in the numerator: (2.10) Therefore, from the graph and the one after square move, we find 9 algebraic letters: the 8 in (2.8) and (AB) ∩ (456)128 in (2.10), which are all A-like variables. It is straightforward to check that these 9 algebraic letters generate the same space as the 9 algebraic letters with this ∆ [34], modulo some rational letters.
In fact, we can find all possible algebraic letters for this Yangian invariant by cyclic symmetry. The four-mass box is invariant under cyclic rotation by 2, and one can apply it to the 8 algebraic letters in (2.8), which take the form Aijk and Bi j k for i, j, k ∈ {3, 4, 5, 6} and i , j , k ∈ {7, 8, 1, 2}. Under the rotation, we obtain the following 8 letters: A i + 2 j + 2 k + 2 , B i + 2 j + 2 k + 2 with and α and β are generated by cyclic rotation by 2 of α and β respectively (they share the same square root). Thus, even without performing square moves explicitly, we find 16 distinct algebraic letters (if we rotate by 2 again, no new letters appear); one can check that 9 out of the 16 letters are multiplicative independent.
Alternatively, we can write the algebraic letters similar to cluster X variables, which are dual conformally invariant. In fact, in the above example, we already see two DCI letters which are internal face variable f and the one after square move, 1 + f . Recall the well-known variables z andz defined by the equations: It is easy to see that f = z/(1 − z) and 1 + f = 1/(1 − z).
To find X -like letters more systematically, let's consider DCI ratios of the form f (X + )/f (X − ), where f (X) is any of the 16 A-like letters with X = A, B, A , B . This is a natural way to find 16 X -like letters, which are not multiplicative independent and the rank is 9 as expected. Together with the other half from the box (2,4,6,8), these X -like letters generate precisely the same space as those DCI algebraic letters given in [34]! In fact, the new result gives a nice understanding of the multiplicative relations of [34] and provides a nice basis.
Since this is the only type of n-point N 2 MHV algebraic Yangian invariants, we have exhausted algebraic letters for k = 2: for each four-mass box (a, b, c, d), there are 9 independent algebraic letters which share the same square root.

Discussions
In this note, we have proposed an algebraic problem of finding the collection of letters, or arguments of d log's, of any Yangian invariant using parameterizations given by plabic graphs. For a given plabic graph, the 4k polynomial equations C(α) · Z = 0 provide a map from 4k-dimensional cell of G + (k, n), parametrized by 4k α's, to a collection of functions defined on G(4, n); the alphabet of a Yangian invariant consists of such functions for all plabic graphs related to each other by square moves. In addition to alphabet of NMHV and MHV invariants which are just Plücker coordinates, we show that for n = 6, 7, the union of alphabets for all invariants includes all cluster A variables of G(4, n), and for n = 8 the algebraic letters of N 2 MHV invariants coincide with the 18 multiplicative-independent symbol letters found for two-loop NMHV amplitudes [34].
We have only provided some data for the general problem, and it would be highly desirable to have a general method for constructing letters for any plabic graph, without the need of case-by-case study. This may involve some algorithm directly in Z space, which mimics the construction of cells in G + (k, n) from the map C·Z = 0 [1]. Such a construction would circumvent the bottleneck of our computation, namely the need to scan through all plabic graphs of a Yangian invariant.
Relatedly, another pressing question here is how to systematically understand the transformations of letters under square moves of the plabic graphs. As we have seen in the top-cell (MHV) case, these transformations correspond to certain cluster transformations, and we would like to understand the analog of these for lowerdimensional cells. The appearance of non-cluster variables in the alphabet of certain Yangian invariant is interesting. By studying all possible types of (rational) Yangian invariants with k = 2 [1], we find that only two of them, the one we have seen above for n = 7 and one for n = 8, contain such non-cluster variables. It would be interesting to understand the origin of these non-cluster variables.
Going into fine structures of the alphabet, one can ask "cluster-adjacency" kind of questions not for the poles of a Yangian invariant, but rather the letters which appear in the d log form. That is, what letters can appear as arguments of a 4kdim d log form? When the letters are all cluster variables, we expect them to be in the same cluster, but we will need some new ideas when algebraic letters (and those non-cluster variables) are involved. The answer to this kind of questions may provide new insights for the study of symbol alphabet of multi-loop amplitudes.
Last but not least, we would like to study this problem at higher n and k. Already for n = 8, it would be interesting to find the complete alphabet show that the union of alphabets include all the 180 rational letters of [34]. For n = 9, we find that the 9 × 9 independent algebraic letters from the 9 N 2 MHV four-mass boxes are not enough for just the two-loop NMHV amplitudes [35]. This is not surprising since there are two types of algebraic Yangian invariants for k = 3 (both with Γ(C) = 2), for which we have not been able to determine the alphabet yet. We leave all these fascinating open questions to future investigations.
Notes added: During the preparation of the manuscript, [36] appeared on arXiv which has significant overlap with the result presented in this note.