Notes on cluster algebras and some all-loop Feynman integrals

We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is D2≃A12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_2\simeq {A}_1^2 $$\end{document}, we show that penta-box ladder has an alphabet of D3 ≃ A3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D4 cluster algebra. We relate the symbol letters to the u variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D5 and D6 cluster functions respectively.

On the other hand, N = 4 SYM has proved to be an extremely fruitful laboratory for the study of Feynman integrals.For example, important ideas and powerful tools such as symbol and co-products [27,28], integrals with unit leading singularity and d log forms [29] and even differential equations [30][31][32], have all more or less originated from the study in N = 4 but they have much wider applications.One of the most recent examples of this kind, which was motivated by [33], is the so-called Wilsonloop d log forms for a large class of Feynman integrals, based on the duality between amplitudes and Wilson loops in the theory [34][35][36][37][38][39].Various ladder integrals, and e.g. the generic double-pentagon integrals for two-loop MHV and NMHV (component) amplitudes [40,41], can be computed efficiently in this way, and we believe it to be closely related to the differential-equation method.
Remarkably, the connection to cluster algebras extend to Feynman integrals as well: e.g. the symbol alphabet of six-point double-penta-ladder integral etc. was given by A 3 cluster algebra [42], and the so-called cluster adjacency condition was observed for certain seven-point integrals in E 6 [43].One can bootstrap Feynman integrals [44,45] based on such knowledge (see also [46]).Very recently, the authors of [47] have argued that cluster algebra structures appear for rather general Feynman integrals which go way beyond planar N = 4 SYM.They have provided strong evidence that four-point Feynman integrals with an off-shell leg is controlled by a C 2 cluster algebra, and found cluster-algebra alphabets for various one-loop integrals, as well as the general five-particle alphabet.A very natural question is how the alphabet may change as we go to higher loops for certain Feynman integrals: for six-point double-penta-ladder integrals, the alphabet stays as A 3 as mentioned [42], and the main goal of the paper is to extend this to more general cases.
In this paper, we mainly show that (generic, eight-point) penta-box-ladder and (seven-point) double-penta-ladder have alphabets which correspond to cluster algebra D 3 ≃ A 3 and D 4 respectively; as a toy example, we also present the trivial case of (eight-point) box-ladder which has alphabet D 2 ≃ A 2 1 .For the nontrivial ladders, we make the claim based on explicit calculations up to five loops (including all odd-weight cases in between).As shown in Fig. 1, let us denote these three classes of integrals at L-loop as I (L) b (x 1 , x 3 , x 5 , x 7 ), I (L) pb (x 1 , x 2 , x 4 , x 5 , x 7 ) and I (L) dp (x 1 , x 2 , x 4 , x 5 , x 6 , x 7 ) which depend on 4, 5 and 6 dual points, respectively.Before we give the precise definition of these classes of ladder integrals, let us first review the kinematics.Recall that the dual points are related to n ordered, massless momenta by p µ i = x µ i+1 − x µ i , thus they form an null polygon with n edges, labelled by external legs i = 1, 2, • • • , n.It is convenient to introduce (supersymmetric) momentum twistors [48], Z := Z A i with the SL(4) label A = 1, 2, 3, 4, defined as Each point x i in dual point space (vertex of the null polygon) corresponds to a line (i−1i) determined by two momentum twistors Z i−1 and Z i .Each loop momentum is represented by a point y ℓ in the dual space, which also becomes a line/bi-twistor ℓ := (AB) in twistor space.Squared-distance of two dual points then reads Similarly we also have the definition The box-ladder integral, I b , which has no non-trivial numerator, is defined as (in terms of dual points and momentum twistors): .
( dp , it is more convenient to directly write them using momentum twistors, especially for the "wavy-line" numerators [29,30]: where we have introduced the shorthand notation ℓi := ℓi−1i ℓii+1 .Note that we can alternatively adopt the "dashed-line" numerator ℓij which is proportional to our ℓ ī∩ j (for I dp we need to replace both wavy-lines by dashed-lines to have a pure function).The alphabet of I pb and I dp is not affected by such a parity conjugation.All these integrals can be evaluated relatively straightforwardly: in addition to the well-known box ladder-integrals, in [40] we have proposed a recursive formula for the other two classes of integrals, in terms of the so-called Wilson-loop d log representation; explicitly the chiral-pentagon on the right-end can be written as twofold d log integrals of a (L−1)-loop integral where the right-end is again a pentagon with deformed legs, e.g. for I (L) pb we have the following recursion with X 1 = Z 8 −τ X Z 2 and Y 1 = Z 3 − τ Y Z 5 (more details can be found in [40] and below): (1.4) After we finished the first version of the manuscript, we noticed that our doublepenta-ladder integral I dp has been evaluated up to L = 3 (denoted as heptagon A) in [49].It is interesting to note that the authors of [49] have also evaluated examples of other double-penta-ladder integrals for heptagon and octagon cases using Feynman parametrization.We will discuss such integrals in detail in section 3, and let us briefly summarize the main result here, which shows the remarkable universality of cluster algebra structures for these Feynman integrals.For our purposes, the most generic case is the nine-point double-penta-ladder integrals: and we remark that there are several natural choices of numerators ("wavy lines") which make the integral pure, and we write two of them explicitly in sec.3 (see [49]).We conjecture that the alphabet of these nine-point integrals (independent of the choice of numerator) is given by (a subset of) cluster algebra D 6 .We can take collinear limit 3 → 2 (and relabel i+1 → i for i = 3, • • • , 8) to obtain eight-point integral that has been denoted as octagon A) (and evaluated to L = 4) in [49].
Note that the kinematics of our nine-point case is the same as three-mass-easy hexagon [50], and in the collinear limit, it becomes two-mass-easy case.There is no smooth limit in 9 → 8 or 6 → 5 (for reasons similar to that in [49]), but one can define such eight-point double-penta-ladder integrals where we have two massive corners with legs 2, 3 and 5, 6 (we denote it as octagon C).We can similarly define seven-point integrals which are different than our I dp (heptagon A) above, such as heptagon B in [49] (all these integrals are shown in figure. 2 without specifying numerators).We conjecture that the alphabet of any octagon integrals is given by D 5 cluster algebra and that of any heptagon ones is given by D 4 cluster algebra.It is remarkable that the alphabet for these integrals seem to be independent of any detail: not only the numerators but also different propagator structures (see the comparison of two types of octagons and heptagons).It is tempting to say that we can simply associate the three-mass-easy, two-mass-easy and one-mass hexagon kinematics (for n = 9, 8, 7) with cluster algebras D 6 , D 5 and D 4 , respectively!Such integrals evaluate to (linear combinations of) multiple polylogarithms, and there is a well-known Hopf algebra structure [51], which has led to the notion of symbol [27,28].For any multiple polylogarithm G (w) whose differential reads where w is called the weight of the polylogarithm and {G (w−1) i } are polylogarithms of lower weight w − 1, its symbol of G (w) is defined by and G (0) := 1.For example, Therefore, the symbol of a polylogarithm of weight w is a tensor of length w, whose entries are called letters.The collection of all letters is called the alphabet.
Given our recursion for the ladder integrals above, we can directly read off the symbol by the following rules [19]: Suppose we have an integral where F (t) ⊗ w(t) is a integrable, linear reducible symbol in t, i.e. its entries are products of powers of linear polynomials in t, and w(t) is the last entry.The total differential of this integral is the sum of the following two parts: (1) the contribution from endpoints: Then we can recursively compute the symbol of the lower-weight integrals and obtain the symbol of the iterated integral of d log forms.We emphasize that it is also straightforward to compute the functions rather than the symbol, but to get the alphabet and other symbolic information we still need to take the symbol map.We have computed I (L) pb up to L = 5 and I (L) dp up to L = 4, as linear combinations of multiple polylogs1 .We find that the functions are actually quite nice at least using "good variables" motivated by cluster algebras (see below), e.g. from L = 2 to L = 3, the number of mutliple polylogs in the answer grows from a dozen to a few hundreds at most.As an illustration we present a rather compact expression for I (L) pb at L = 3.

Review of cluster algebra and cluster configuration space
Cluster algebras [53][54][55][56] are commutative algebras with a particular set of generators A i , known as the cluster A-coordinates; they are grouped into clusters which are subsets of rank d.From an initial cluster, one can construct all the A-coordinates by mutations acting on A's (the so-called frozen coordinates or coefficients can also be included, which do not mutate).Alternatively one can define cluster X coordinates which are given by monomials of A's.
There is a natural space of polylogarithm functions associated with a cluster algebra, given a set of cluster-A or (X ) coordinates.A cluster function F (w) [57,58] of transcendental weight w is defined such that its differential has the form where F (w−1) i are cluster functions of transcendental weight w − 1 and A i are cluster-A coordinates.We see that if a multiple polylogarithm is a cluster function, then the alphabet can be identified with the corresponding cluster algebra.
For the purpose of this paper, it suffices to know that all finite-type cluster algebras, i.e. those with finite number of A-cluster coordinates (the dimension of the cluster algebra, denoted as N), has been classified in terms of Dynkin diagrams.There are series To identify an alphabet with certain finite-type cluster algebra, it is convenient to parametrize the coordinates of the latter in a nice way.For example, for type A d and D d , we have the following N = d(d+3)/2 and N = d 2 letters respectively [47]: In other words, once we find an alphabet which can be written as a collection of N polynomials (of d variables), the remaining task would be to look for some birational change of variables such that they become multiplicative combinations of letters in e.g.Φ A d or Φ D d (or a subset of them).
On the other hand, without any smart parametrization, there is a totally gaugeinvariant way for describing any finite-type cluster algebra, known as the cluster configuration spaces [59,60].One can simply represent Φ by N variables called u variables, where β|α are integers known as compatibility degrees [59].It is remarkable that the u equations are consistent and give a d-dimensional solution space we call cluster configuration space.For example, for type A d we have N = d(d+3)/2 u variables (one for each diagonal of (d+3)-gon): which satisfy N equations of the form where on the RHS we have the product of all u k,l with (k, l) crossing (i, j) (or u k,l incompatible with u i,j , with compatibility 1).Similarly for D d we have N = d2 variables which we denote as which satisfy u equations as explicitly given in [59,60].
Note that such a configuration space can be viewed as a "binary geometry" for the corresponding cluster polytope (or generalized associahedra).If we ask all u to be positive, we have {0 < u α < 1|α = 1, • • • , N}, which cuts out a "curvy" cluster polytope.Each of the N boundaries of the space is reached by exactly one u α → 0, and the u equations force all incompatible (i.e.those with β|α > 0) u β → 1, and we obtain the configuration space of the corresponding sub-algebra, which factorizes according to which node of the Dynkin diagram we remove.Note that though the complex configuration space is no longer a polytope, we still have such boundary structures exactly as any u α → 0.
How do we find the u variables given an alphabet of N polynomials?This has been proposed in [61], and the basic idea is to use any positive parametrization such that the N polynomials can be expressed as subtraction-free Laurant polynomials of positive coordinates x 1 , • • • , x d , which we denote as p I ({x}) for I = 1, • • • , N; then we study the so-called stringy canonical forms [61]: where we integrate i d log x i in the positive domain, and we take polynomials p I (to the power α ′ s I as "regulators" for potential divergences.The domain of the convergence for I {p} ({s}) is given by a polytope in the exponent space, which is defined as the the Minkowski sum of Newton polytopes of p I 's 2 .One can generally define u variables (and configuration spaces) for such an integral following the procedure in [61], and quite beautifully all finite-type cluster algebras belong to a special case that the corresponding polytope has exactly N facets and it is cut out by X α ({s}) ≥ 0 for α = 1, • • • , N, known as the ABHY realization of cluster polytopes; for type A, B, C, D, the canonical form of these polytopes, or α ′ → 0 limit of the integrals, have nice interpretation as planar φ 3 amplitudes through one-loop [62,63].We can recombine the exponents into X α 's and the regulator becomes N α=1 u α ′ Xα α , where the polynomials combine into exactly the N u α variables!Thus the u variables can be obtained by a Minkowski-sum calculation given any positive parametrization.
For example, by using any positive parametrization of polynomials in Φ A d , we recognize I {p} as the usual (d+3)-point open-string integral, and the Minkowski sum gives exactly the ABHY associahedron in the kinematic (Mandelstam) space; the u ij variables are then dihedral coordinates of M 0,d+3 [64], given by cross-ratios of the world-sheet coordinates z i 's (with three additional fixed at (0, −1, ∞)).
2 Cluster algebras for three classes of ladder integrals 1 for box-ladder integrals Let us start with the well-known ladder integral, I (L) b (x 1 , x 3 , x 5 , x 7 ) which depends on two cross-ratios and it is convenient to introduce z and z variables defined by x 2 15 x 2 37 . (2.1) The ladder integrals have been evaluated in [65], and one way to do so is by solving differential equations they satisfy, which nicely relate I .Recall that the integral has a natural overall normalization factor ) becomes pure function of weight 2L.The functions f (L) satisfy second-order differential equations: with "tree-level source" There is a closed-form expression for f (L) from solving the differential equations: which is a single-valued, analytic function of z (In Euclidean signature, z and z are complex conjugates to each other).From (2.3) it is obvious that the alphabet of the symbol consists of 4 letters, {z, z, 1 − z, 1 − z}, which we can immediately identify as that of D 2 ≃ A 2 1 .We are mostly interested in positive external kinematics, where momentum twistors Z ∈ G + (4, n), and it is easy to see that for such kinematics we have z < 0 and z < 0. To relate this alphabet to the positive u variables of D 2 , we make the change of variables u := z/(z − 1) and ū := z/(z − 1).The D 2 alphabet can be alternatively written in these variables 3 : where all the letters are positive (between 0 and 1); this is the u space of D 2 which is literally a quadralateral.The variables z, z (or equivalently u, ū) are algebraic functions of momentum twistors, since they are two roots of the above quadratic equation.Without loss of generality for this specific problem, we can "rationalize" the square root explicitly by reducing the kinematics to two dimensions [66].Recall that when external kinematics lie in two-dimensional subspace, the polygon can only have even number of edges (which we denote as 2n) and take a zigzag shape: edges with even and odd labels go along two light-like directions respectively.It is convenient to reduce momentum twistors as ), which reduces the conformal group SL(4) to SL(2) × SL (2).The kinematics are encoded in even and odd SL(2) invariants i j (or [i j]) for odd (or even) i, j, which are in fact onedimensional distances along odd (or even) direction.
Any cross-ratio factorizes into the product of an even and an odd cross-ratio, e.g. for i, j, k, l all even, we have denote familiar cross-ratios of A n−3 ∼ G(2, n)/T in the even sector (and similarly an A n−3 in the odd sector).Now for the box-ladder with 2n = 8, we see that the 2d kinematics naturally require two A 1 's (for even and odd sectors), and we the square root disappear to give [66] u = u 2d 1,3,5,7 , ū = u 2d 2,4,6,8 and 1 − u = u 2d 3,5,7,1 etc.. Thus we see that the u variables are literally the u variables for the two A 1 ∼ G(2, 4)/T .
Moreover, it is trivial to see that we have A 1 sub-algebras of D 2 which can be reached when any of the u variables goes to zero.This is well known since e.g. at one-loop, as u → 0 the box function diverges, but we can look at the "finite part" Li 2 (1 − ū) which has the A 1 alphabet {ū, 1 − ū}.Although this particular integral I (L) b diverges at any "boundary" of the D 2 , more generic D 2 cluster functions can have such A 1 functions in these limits.

D 3 ≃ A 3 for penta-ladder integrals
Next we consider a more non-trivial example, I (L) pb (x 1 , x 2 , x 4 , x 5 , x 7 ), which depends on 3 cross-ratios defined as x 2 14 x 2 25 . (2.5) As shown in [40], we obtain a two-step recursion relation for I pl (u, v, w): We remark that the actual integrals with even weight are symmetric in exchange of u and v (the integral has a symmetry axis); we introduce odd-weight objects which break the symmetry, but we can alternatively write down recursion with u, v swapped, which give different odd-weight functions but will not affect the even-weight integrals.Nicely, the recursion applies to L = 0, where the tree case is defined to be I The recursion makes it manifest that the result will always be pure functions starting L = 1.We emphasize that by using the algorithm of [19], it is straightforward to compute the symbol of pb to any loop order.We are mainly interested in the alphabet of the resulting symbol.As we have seen at L = 1 (weight 2), and in fact also for L = 3 2 (weight 3) as obtained using the first line of (2.6), the alphabet consists of eight letters, u, v, w, 1 − u, 1 − v, 1 − w, 1 − uw, 1 − vw.However, these are just degenerate cases, and starting at L = 2 we find 9 letters where the additional one is nothing but the tree-level factor 1 − u − v + uvw: We have checked up to L = 5 (weight 10), and in sec.3 we will give an all-order proof using the recursion that this 9-letter alphabet is true to all loops with L ≥ 2. Now we identify the alphabet A[I pb ] with that of D 3 ≃ A 3 cluster algebra, and we do so in two ways.First, similar to [47], we find the bi-rational change of variables and, up to multiplicative redefinition, the alphabet (2.7) becomes which we immediately recognize as that of D 3 (second line of (1.5) with d = 3).A trivial change of variables turns it into that of A 3 (first line of (1.5) with d = 3).These changes of variables may seem a bit arbitrary, but as mentioned earlier we can reach at the conclusion in a totally invariant way.Pick any positive parametrization of the 9 letters in (2.7); it does not matter what positive parametrization we choose as long as they give subtraction-free polynomials p I for I = 1, . . ., 9, and we write the stringy canonical form (2.10) Without being smart, we can simply compute the Minkowski sum of Newton polytopes of p I 's which gives the convergence domain of I {p} , and we find that the result is nothing but a 3-dimensional associahedron!It is given by 9 inequalities of the form X a ({s}) ≥ 0, each of which can be written as a linear combination of the s I 's.With these 9 linear combinations, we can identify the 9 u variables of A 3 by writing . With a bit hindsight, we label the u's by diagonals of a hexagon as in (1.6), and they automatically satisfy the 9 u equations in (1.7).This gives a description of the A 3 alphabet that is totally parametrization-independent, A[I (L) pb ] = {u i,j |1 ≤ i < j − 1 < 6, (i, j) = (1, 6)}, where the u variables are multiplicative combinations of the original letters: One can easily check that (2.11) satisfies u equations and with any positive parametrization all u variables are between 0 and 1.As mentioned such a description using u variables is very useful, e.g. each of the 9 boundaries can be obtained by sending exactly one u to zero.There are 6 boundaries by u i,i+2 → 0 (for i = 1, • • • , 6) which are A 2 (pentagon), and 3 boundaries by u i,i+3 → 0 (for i = 1, 2, 3) which are D 2 ≃ A 2 1 (quadrilateral).However, for I (L) pb many boundaries are too degenerate as the symbol vanishes identically, only the following 4 boundaries correspond to non-trivial limits: u 1,3 → 0, u 1,5 → 0, u 3,5 → 0, which are A 2 , and u 2,5 → 0, which is A 1 .
The first A 2 is given by w → 0, and it is the familiar collinear limit which gives seven-point penta-box ladder with alphabet {u, v, 1−u, 1−v, 1−u−v} [14]; the next two A 2 are also collinear limits reached by u → 0 and v → 0 respectively, though the integral diverges in these limits.Finally, the last limit is given by w → 1 which can be nicely reached by reducing the kinematics to two dimensions!In such a limit, we have u → u 2d 2,4,6,8 and v → u 2d 1,3,5,7 , and it turns out that the resulting D 2 function is even simpler than box ladder (2.3).As first noted in [40] from resummation, pentabox ladder in 2d is perhaps the simplest A 2 1 function: it is given by the product of weight-L classical polylogarithm function of u and that of v, e.g. for L = 1 we have the chiral pentagon in 2d: I p = log(u) log(v).

D 4 for double-penta-ladder integrals
Finally, we move to I (L) dp (x 1 , x 2 , x 4 , x 5 , x 6 , x 7 ), which depends on 4 cross-ratios . (2.12) As shown in [40] for L ≥ 1 it satisfies a similar recursion5 : where the one-loop case dp , is a seven-point chiral hexagon evaluating to Note that as also noticed in [40], although we can similarly find L = 1 2 and 0 cases, the latter (tree case) will not be simply weight-0 object but also involves log term.
We have computed up to L = 4 and find the alphabet of penta-box-ladder integrals as (for L ≥ 2) (2.14) where we have defined 6 Only the first 10 letters in (2.3) appear for L = 1, and as can be obtained from (2.13), an additional letter 1 − v 3 − v 2 v 4 appears at L = 3 2 (weight 3); these are all degenerate cases of the alphabet (2.14), which become generic starting L = 2.
To identify the alphabet with that of D 4 , it is crucial to have any change of variables which gets rid of the square root.Clearly this can be done using momentum twistors (c.f.[50]).Here we adopt the following parametrization of the 7 momentum twistors involved: By plugging (2.15) into (2.12)we find ) , and ∆ 7 above becomes a perfect square.The upshot is that the alphabet becomes multiplicative combinations of exactly 16 polynomials: From here it is straightforward to find a positive parametrization: we simply need to find positive variables which guarantees all a i < 0 for i = 1, 2, 3, 4 as well as a 2 > a 3 , e.g. (2.17) where on the last line we have u ↔ ũ as well.Explicit expressions of u variables in terms of v 1 , • • • , v 4 involve square roots but they simplify in terms of a 1 , • • • , a 4 : and we can easily check that they satisfy the 16 equations (2.17Not surprisingly, this amounts to embed the D 4 alphabet naturally in the E 6 ∼ G(4, 7)/T , and we remark that the embedding is not unique.In fact, we can first write the 42 u variables for E 6 cluster algebra, in terms of Plücker coordinates in G(4, 7).Then we express our 16 letters above as monomials of the 42 variables, and we find there are four solutions, each corresponding to a co-dimension 2 boundaries of the E 6 space.What we have found above is just one of the four solutions and the other three can be obtained by cyclic rotation in D 4 .
Let us also identify all 16 boundaries of the D 4 alphabet, each reached by one u → 0. There are 12 A 3 corresponding to the first 12 u variables of (2.17) and 4 A 3 1 for the last 4.For I (L) dp , we find that the symbol vanishes for 4 boundaries, namely those for u 2 , u 4 , u 12 , u 41 , which are too degenerate.We believe that all the remaining 12 non-trivial boundaries have certain physical interpretation.Note I (L) dp diverges at those for u 23 ,u 34 (A 3 1 ) and those for ũ2 , ũ3 , ũ4 , u 13 , u 31 (A 3 ), thus it remains finite at 5 boundaries (all A 3 ): u 1 → 0, ũ1 → 0, u 3 → 0, u 24 → 0 and u 42 → 0.
At least the physical interpretation of the first two boundaries are very clear: the first one corresponds to v 3 → 0, and the integral reduces to I (L) pb with with (u, v, w) = (u 1 , u 2 , u 4 ), while the second one correspond to v 4 → 0 and it reduces to the six-point double-penta ladder [30], which is the famous A 3 for hexagon function [4].As one can see from (2.17), the first D 3 ≃ A 3 sub-algebra has 9 remaining u variables, and in the order of (2.11) they read ũ1 , u 41 , u 31 , u 4 ,u 3 , u 2 ,u 13 , u 12 , u 42 .The 9 remaining variables for the second A 3 are obtained by swapping u and ũ, which are combinations of the familiar 9 letters for hexagon bootstrap (in terms of original variables, they are Last but not least, we note that our variables a 1 , a 2 , a 3 , a 4 are just relabelling of y 2 , y 3 , y 4 , y 5 in [47], thus alternatively one can find the following bi-rational transformation which sends them to the familiar z variables for D 4 : and the alphabet becomes the 16 polynomials on the second line of (1.5) with d = 4.

Comments on "adjacency" constraints on cluster function spaces
Let us describe constraints on adjacent entries that we observe for the symbol of the ladder integrals, which may have their origins in extended Steinmann relations [9,10,13] or cluster adjacency [11,43], and briefly comment on their consequences on cluster function spaces.
To describe these constraints, it is important to use "good" variables: it turns out that for both I (L) pb and I (L) pb , the function/symbol expressed in the z variables is much shorter than that in terms of cross-ratios (or u variables).We also note that the z variables also exactly give 3 and 4 combinations that appear in the last entry of these integrals (this point will be important when considering differential equations or resummation).Therefore, we consider such constraints using z variables, first for pb .As we have checked through L = 5, 12 pairs never appear next to each other in its symbol: there is no a ⊗ a for a = 1 + z 1 , 1 + z 2 , 1 + z 3 or z 1 + z 2 z 3 , and there is no a ⊗ b or b ⊗ a for {a, b} equals Note that these constraints imply that for the original 16 letters, none of {w, w}, − vw} is allowed, but these are much weaker than those in z var.One can construct the corresponding cluster function space at symbol level to relatively high weight, and we focus on the A 3 space with physical first-entry condition, i.e. the collection of all weight-k integrable symbols with only u, v, w in the first entry.The first observation is that the dimension of the space (denoted as d k ) is 3,11,40,146,538,2006, • • • , and we conjecture in general it reads which is a special case of a more general observation on the dimension of such spaces.Now if we impose the adjacency conditions in terms of z variables, we find that the dimension of the space is drastically reduced to 3, 8, 20, 44, 88, 171, • • • (note that for weight 6 the space is reduced by more than 90 percent!).At least up to L = 3, it should be easy to bootstrap penta-ladder integrals by imposing other conditions.Similarly, for I (L) pb through L = 4, we find that there are 37 pairs that cannot appear next to each other in the symbol (some of them may become allowed at higher loops).First z 1 − z 2 − z 1 z 2 + z 1 (z 3 + z 4 ) − z 3 z 4 cannot appear next to itself, z 2 , 1 + z 2 , z 3 , z 4 or z 1 + z 3 z 4 ; z 1 + z 3 z 4 also cannot appear next to 1 + z 1 , z 2 , 1 + z 2 , z 2 − z 3 or z 2 − z 4 ; z 2 + z 3 z 4 cannot appear next to 1 + z 1 , 1 + z 2 , z 1 − z 3 or z 1 − z 4 ; the remaining pairs that are not allowed read We have not attempted to find physical explanation for these adjacency conditions, which may have origin from extended-Steinmann relations; they could be alternatively explained by re-expressing our variables in terms of the E 6 variables suitable for cluster adjacency [11,43].We leave a more systematic study of these conditions and their consequences for cluster function spaces to a future work.
3 Cluster-algebra alphabets from recursive d log forms Having identified alphabets of three classes of ladder integrals with D 2 , D 3 , D 4 , a natural question is why these alphabets stay invariant as L increases?We do not have a complete answer, but here we would like to sketch an argument based on the recursive d log forms, which also allows us to further extend our explorations.
Let us illustrate the idea by the following example.Suppose one of the terms in the symbol integration reads d log(t+b 1 )(t+b 2 ) ⊗ a ⊗ (t+b 3 ) ⊗ (t+b 4 ) + • • • .We can make an estimation for the alphabet of any integral of this type without performing the symbol integration explicitly.If we have an t-deformed alphabet However, as mentioned we have checked that through weight 8 the alphabet of I (L) dp stays invariant, so that the new ones must be spurious at least up to L ≤ 4. Take I (2) dp as an example: the last two letters there are trivially spurious, since the corresponding t + b i and t + b j that produce them never appear in the same term; we also find the first two letters got cancelled in the final result.Similarly, if we start again with 16 letters of I (L+ 1 2 ) dp , the recursion for I (L+1) dp also produces four new letters which we have checked to be spurious at least through weight 8.It remains an important open question if these new letters are spurious to arbitrary L, but we emphasize that it is already interesting that our estimation does not grow with L. dp using our recursion, and it is natural to try and apply it to more examples and to see if the alphabets are related to cluster algebras.We expect it to be a general phenomenon for a large class of Feynman integrals, especially for those referred to as generalized penta-ladder integrals [40].Once we know the one-loop case, we can apply the recursion to obtain symbol at higher-loops, and we conjecture that the alphabet will stay invariant starting a certain order.In the first version we have studied a mathematical experiment by applying the recursion relation to (weight-3) hexagons in 6d, including one-mass, two-mass-easy and three-mass cases.We have found that by applying recursion relation to these three hexagon integrals: I 6d 3me [50], I 6d 2me , and I 6d 1m [67,68] (see figure.We have not found any physical interpretation of these series of functions; given the results of [49], it is natural to wonder if those double-penta-ladder integrals (for n = 7, 8, 9), which share the same kinematics as these 6d hexagons, also have alphabet of D 4 , D 5 , D 6 respectively.Let us first consider the most general kinematics, the three-mass-easy (nine-point) hexagon (which does not involve any square root in 4d), and it depends on the following 6 cross-ratios: .
We can parametrize the 9 momentum twistors as In terms of these variables, the cross-ratios v i for i = 1, • • • , 6 read: , v 2 = − a 4 − a 5 (a 3 − 1) (a 4 − 1) , v 3 = − a 1 − a 6 (a 1 − 1) (a 4 − 1) , v 4 = − a 2 (a 3 − 1) a 2 − a 3 , v 5 = (a 4 − 1) a 5 a 4 − a 5 , v 6 = (a 1 − 1) a 6 a 1 − a 6 As shown in [49], there are four natural choices of the numerators, while in our formalism, we fix the numerator of the right-most pentagon to be the wavy line N 1 = 1 ∩ 4. Then for the left-most pentagon, we have two choices Since our recursion is derived using the right-most pentagon, it turns out to be independent of the choice of numerators N 2 ; we collectively denote such integrals as F (L) 3me (the source at L = 1 differs by the choice of numerators), and the recursion universally reads: with numerator either (N 1 , N 1 2 ) or (N 1 , N 2 2 ) up to L = 4, we find 35 out of 36 letters which form the alphabet of D 6 7 .For example we can use the following change of variables to arrive at d = 6 case of (1.5) but with the letter 1 + z 4 missing: Now we move to various degenerations.As pointed out in [49], F (L+1) 3me does not have smooth limit u 5 → 0 or u 6 → 0. The only degeneration we can consider is u 4 → 0, i.e. collinear limit 3 → 2 or limit a 2 → 0 in as variables, after which we get octagon A. Up to L = 4, its alphabet has 24 letters, which forms D 5 with one letter missing as well.To see D 5 , we can follow the same procedure and find the u variables, or alternatively a degeneration z 1 → z 2 from (3.6).The missing one of D 5 alphabet is then the degeneration image of 1 + z 4 .
Moreover, we can consider octagon C in Fig. 2, which shares the same kinematics as the two-mass-easy hexagon but has different massive corners as octagon A. There are also two natural choices of the numerator N 2 , after fixing N 1 = 1 ∩ 4: Explicit computation shows that up to L = 3, its alphabet has 25 letters which forms the full D 5 , corresponding to the degeneration z 4 → −z 5 z 6 of D 6 alphabet.Finally, when taking limit z 1 → z 2 furthermore, we obtain heptagon B with 16 letters, forming a full D 4 alphabet.

Conclusion and Discussions
In this paper, motivated by [47] we have studied relations between Feynman integrals and cluster algebras using the recursive d log forms [40].Our main examples are the penta-box-ladder integral I (L) pb which has an alphabet of D 3 ≃ A 3 and seven-point double-penta-ladder integral I (L) dp which has an alphabet of D 4 .We have also found that such D n -type cluster algebras seem to be rather universal for (double-penta) ladder integrals: various such integrals with one-mass, two-mass-easy and threemass-easy hexagon kinematics are associated with cluster algebras D 4 , D 5 and D 6 respectively, which is independent of details of the integrals.We have identified the u variables of cluster configuration space for a gauge-invariant description.
The most pressing open question is to understand why alphabets of such ladder integrals can be identified with cluster algebras.For n = 6, 7 cases, it is not surpris-

Figure 2 :
Figure 2: Degenerations of nine-point double-penta-ladder t + c) (F (t) ⊗ w(t)), − u − v + uvw.By applying the first equation of (2.6) to this tree result, we obtain a weight-1 function I log(uw), and by applying the second equation, we arrive at the well-known one-loop chiral-pentagon I (L=1) pb := I p (u, v, w):

4
and the above 16 polynomials become subtraction-free.Then we can simply follow the procedure by computing the Minkowski sum of the Newton polytopes of such 16 polynomials, and remarkably we find a D 4 polytope!It is then straightforward to work out the 16 u variables as an invariant description of A[I (L) dp ], (1.8) for n = 4, provided that they satisfy the following 4 + 4 + 8 = 16 u equations:

(3. 1 )
Here a, b 1 , b 2 , b 3 , b 4 are different constants, t is integrated over the region R + .According to the algorithm of symbol integration, generically the result depend on letters of the form:{a; b 1 , b 2 , b 3 , b 4 ; b 1 −b 2 , b 1 −b 3 , b 1 −b 4 , b 2 −b 3 , b 2 −b 4 , b 3 −b 4 }.Note that constant a, which shows up in the alphabet, does not mix with any b i 's, and in addition to b i from each t+b i , each pair of factors t+b i and t+b j contributes a b i −b j to the final alphabet.This follows directly from our algorithm reviewed earlier.Each t + b i contributes b i to the alphabet of the result when evaluated at the end point, following the first part of the algorithm.A constant a is produced both from end-point value and from the situation of (F (t) ⊗ a)d log(t + b i ).Finally, whenever the last entry reads t + b i with d log(t + b 1 ) as differential form, it contributes b i − b 1 as the new last entry, with the differential form changed to d log t−b i t−b 1 .Since t + b i becomes a new entry of d log form, recursively all the mixing letters b i − b j should appear in the final alphabet.

3. 1
More ladder integrals, D 4 , D 5 , D 6 cluster algebras and universality So far we have studied I