KK-like relations of $\alpha^{\prime}$ corrections to disk amplitudes

Inspired by the definition of color-dressed amplitudes in string theory, we define analogous {\it color-dressed permutations} replacing the color-ordered string amplitudes by their corresponding permutations. Decomposing the color traces into symmetrized traces and structure constants, the color-dressed permutations define {\it BRST-invariant permutations}, which we show are elements of the inverse Solomon descent algebra and we find a closed formula for them. We then present evidence that these permutations encode KK-like relations among the different $\alpha'$ corrections to disk amplitudes refined by their motivic MZV content. In particular, the number of linearly independent amplitudes at a given ${\alpha^{\prime}}$ order and motivic MZV content is given by (sums of) Stirling cycle numbers. In addition, we show how the superfield expansion of BRST invariants of the pure spinor formalism corresponding to ${\alpha^{\prime}}^2 f_2$ corrections is encoded in the descent algebra.


Introduction
It is well-known that superstring n-point color-ordered disk amplitudes satisfy monodromy relations which imply that the number of linearly independent amplitudes is (n−3)!, for all α ′ corrections [1,2]. These relations involve coefficients that depend on Mandelstam variables [3] and are famously related to the Bern-Carrasco-Johansson (BCJ) color-kinematics amplitude relations in the field-theory limit. In this paper, we investigate a weaker set of relations, called KK-like relations [4], of higher α ′ corrections to disk amplitudes [5] refined by their motivic MZV content written in the f -alphabet of [6]. More precisely, writing the motivic superstring color-ordered disk amplitude as 1 In this paper, this result and its generalization will be obtained as n − 1 1 + n − 1 3 + · · · + n − 1 2m + 1 , m ≥ 2 .
We will see that the general KK-like relations are closely related to the mathematical framework of the Solomon descent algebra [8,9,10,11,12,13,14]. To see this we will define the color-dressed permutation Tr(T σ ) σ , T σ := T σ(1) T σ(2) · · · T σ(n) (1.5) which is inspired by the expression of the color-dressed disk amplitudes, where T i := T a i denotes a Chan-Paton factor. When the closed formula from [15] for the color-trace decomposition [16] is plugged into (1.5), the permutations appearing as coefficients with respect to a basis of color factors define what we call BRST invariant permutations γ 1|P 1 ,...,P k with 1 ≤ k ≤ n−1. We conjecture the following closed formula γ 1|P 1 ,...,P k = 1.E(P 1 )¡E(P 2 )¡ . . . ¡E(P k ) (1.6) where E(P ) satisfying E(R¡S) = 0 for R, S = ∅ is the Berends-Giele idempotent, defined in section 2.2 from mapping the permutations of the Solomon idempotent [17] into their inverses. Then in section 3.1 we will find evidence that these BRST-invariant permutations encode the general KK-like relations as
In the appendices we review the descent algebra and collect various proofs and explicit expansions omitted from the main text.

Conventions
Words from the alphabet N = {1, 2, . . .} will be denoted either by capital Latin letters or, especially when viewed as elements of the permutation group, by lower case Greek letters.

Color-dressed permutations
In this section we will investigate the combinatorics of the color-dressed permutations P n P n = σ∈S n ,σ(1)=1 Tr(T σ ) W σ , T σ := T σ(1) T σ(2) · · · T σ(n) , (2.1) arising from decomposing [16] the traces of color factors into symmetrized traces d 12...k and structure constants f abc of the gauge group [15] Tr(T 0 T 1 · · · T n−1 ) = where σ = σ 1 · σ 2 · . . . · σ k denotes the decreasing Lyndon factorization of σ to be defined below, F σ a for a word σ and a letter a is defined recursively by F P j a = F P b f bja with base case F i a = δ i a . The coefficients κ σ are defined in (A.20). In addition, the symmetrized trace and the structure constant are given by The decreasing Lyndon factorization (dLf) of σ is defined as [20,9] where σ 1 · · · σ k is the decreasing Lyndon factorization of σ, in the color-dressed permutation (2.1), i.e.
The reason for this terminology will become clear in (3.23) when γ 1|σ 1 ,σ 2 ,σ 3 will be associated to BRST invariants superfields in the pure spinor formalism. For example, plugging Repeating the same exercise for n = 4 using (2.2) we obtain where the BRST-invariant permutations are given by For n = 5 we obtain where the various γ 1|A 1 ,A 2 ,...,A k are listed in the appendix C.

Relating the BRST-invariant permutations with the descent algebra
The permutations in each γ 1|A 1 ,A 2 ,...,A k turn out to be related to the descent algebra reviewed in the appendix A. To see this consider γ 1|23,4 from (2.8), relabel i → i − 1 and strip off the leading "0" (denoted by ×) to obtain The above permutations are not in the descent algebra D 3 since permutations in the same descent class have different coefficients (see also proposition 2.1 from [21]). However, the inverse permutations in θ(γ ×|12,3 ) do belong to the same descent classes: (2.11) as can be verified using (A.3). This suggests that the BRST-invariant permutations belong to the inverse descent algebra D ′ n := θ(D n ). To find an algorithm that generates these permutations, we will consider the inverse of the Eulerian idempotent (A.20).

The Berends-Giele idempotent
Define the Berends-Giele idempotent 2 E n as the inverse of the Eulerian idempotent (A.20): The reason for this terminology is the correspondence with the standard Berends-Giele current of Yang-Mills theory [22], see section 3.2.
A few examples of (2.12) are while the expansion of E(1234) can be found in the appendix C.1.
Proposition (Shuffle Symmetry). The Berends-Giele idempotent (2.12) satisfies (2.14) Proof. 3 Since the sum in (2.12) is over all permutations we rename P • σ = τ and sum over τ . Notice that where the last equality follows from (A.22) and the crucial observation in (1.5) of [9] that where σ −1 (R) denotes the word obtained by replacing each letter in R by its image under σ −1 .
2 The inverse of an idempotent is also idempotent. 3 We know from [23] that any Lie polynomial can be expanded as σ M σ σ with M R¡S = 0 for nonempty R, S, so it follows that if Γ is a Lie polynomial then the word function F (P ) := P • θ(Γ) satisfies the shuffle symmetry F (R¡S) = 0 for R, S = ∅.

Inverse idempotent basis and BRST-invariant permutations
Following the realization in section 2.1.1 that the BRST-invariant permutations are related to the inverse of the descent algebra, we define the inverse of the idempotent basis I p as where the map θ is defined in (A.14). For example I 21 (12, 3) = 1 2 W 123 + W 132 − W 213 − W 231 + W 312 − W 321 . See (B.4) for the explicit permutations in I 22 (12,34).
An alternative representation is proven in the appendix B where #even(p) denotes the number of even parts in the composition p. To see this note that if a function satisfies the shuffle symmetry or, in other words, belongs to the dual space of Lie polynomials [24], thenF (P ) = (−1) |P|−1 F (P ) and (2.19) follows from (2.17).

BRST-invariant permutations and orthogonal idempotents
Since the BRST-invariant permutations have been related to the idempotent basis of the (inverse) descent algebra in (2.20) we may construct orthogonal idempotents as in sec- From the discussion of section A.3 it follows that (2.22) are orthogonal idempotents in the inverse descent algebra D ′ n satisfying (δ ij is the Kronecker delta) For example, from the BRST-invariant permutations in (2.6) we get It is straightforward but tedious [26] to check that the above satisfy (2.23). At multiplicity five, the orthogonal idempotents are given by whose expansions can be found in the appendix C and can be checked to obey (2.23).

KK-like relations of α ′ corrections to disk amplitudes
The color-dressed string motivic disk amplitude is a sum over disk orderings of the (φ map of) open string color-ordered amplitude weighted by traces of Chan-Paton factors. The explicit form of the disk amplitudes is a linear combination of field-theory amplitudes A SYM of ten-dimensional super-Yang-Mills [27] given by [28,29] A string (P ) = where S(P |Q) 1 is the field-theory KLT kernel [30,31,32] conveniently computed recursively [33,34,24]. In addition, Z(P |Q) are the non-abelian Z-theory amplitudes of [35,29]. The color-ordered motivic amplitudes [5] are decomposed as where M runs over all words composed of odd positive integers ≥ 3 and f M = [6] for words of length p. From now on we will use the shorthand motivic MZV amplitudes for the A f m 2 f M components of (3.3), and we will see below that components with the same α ′ but different motivic MZV content satisfy different relations.
For an example KK-like relation, one can verify that the f 2 amplitudes satisfy [18], Based on explicit computations, we find that motivic MZV amplitudes with different f m 2 content 5 satisfy the following KK-like relations: which constitute the descent algebra decomposition of KK-like relations. To count the basis dimensions implied by (3.9) we recall that # γ 1|P 1 ,...,P k = n−1 k and n−1 k=1 The string monodromy relations [1,2] give rise to deformations of the field-theory BCJ relations by powers of α ′ 2m f m 2 [33,37]. Since the BCJ-satisfying f M components lead to the minimum (n − 3)! dimension, they are not expected to modify the dimensions within a given f m 2 f M class.
(n − 1)!. Thus, subtracting the dimensions of the BRST-invariant permutations from the number of cyclically symmetric n-point amplitudes 6 leads to The dimension (3.10) corresponds to the number of independent amplitudes under the for the BCJ-satisfying f M corrections. The dimension (3.11) was obtained in [18] following a similar discussion of the all-plus one-loop amplitudes of [4] (see also [38]).
The basis dimension formula (3.12) and the corresponding amplitude relations in (3.9) are new. Interestingly, they imply that the basis dimension of the f m 2 f M components with m ≥ 2 is given by 1 2 (n − 1)! only when n ≤ 2m + 3. More explicitly, For a deviation from the 1 2 (n − 1)! dimension, we have, for example with m = 2 which was confirmed by a long brute-force search using FORM [26]. The additional relation on top of (3.7) is seen to be A f 2 2 (γ 1|2,3,4,5,6,7, with the analysis of [33,39]. At nine points and m = 2, the formula (3.12) predicts a mismatch due to 8 7 = 28 additional relations etc. The evidence for the descent algebra decomposition of KK-like relations in (3.9) was collected from explicit calculations of The results are summarized in Table 1 ..,P k ) at n points would appear to grow rapidly, but 6 Note that 1 2 (n − 1)! is equal to the sum over n−1 k with even k. Since k even is included in (3.9), the Stirling cycle numbers are subtracted from (n − 1)!. 7 This data was collected using the α ′ corrections to disk amplitudes obtained in [28,40,41,29] (see also [42,43,44,45] and references therein for earlier work and [46] for a discussion on MZVs). Table 1. Overview of the descent algebra symmetries of higher α ′ corrections to string disk amplitudes of up to n = 8 points displayed by their motivic MZV content of weight w ≤ 7.
The entries depend only on the number of parts k of the composition of n−1. However, a partition with k parts cannot be probed by disk amplitudes with fewer than k+1 points. to the α ′ -corrected abelian Z-theory amplitudes A string (γ 1|2,3,...,n ) ∼ A NLSM (1, 2, . . . , n) of [33,39]. As a consistency check, the proof (D.1) implying that A f m 2 f M (γ 1|2,...,k ) vanishes when k is even (so n is odd) agrees with the vanishing of NLSM odd-point amplitudes.

The field-theory and α ′ 2 corrections
The SYM amplitudes are computed in pure spinor superspace from the expression M 1 E P , where E P is a superfield satisfying the same shuffle symmetry E R¡S = 0 for R, S = ∅ of the standard Berends-Giele current J m P [47,48]. The A f 2 amplitudes [49] can be computed in pure spinor superspace [18] using BRST-closed combinations of superfields C 1|X,Y,Z symmetric under exchanges of any pairs X ↔ Y, Z and satisfying C 1|R¡S,Y,Z = 0 for R, S = ∅. For convenience, define the BRST-closed combination C 1|P := M 1 E P so that 8

15)
where 2...n is a shorthand for the sum over the deconcatenations of 2 . . . n = XY Z. In terms of these BRST invariants, the color-dressed amplitudes at four and five points can 8 The angular brackets . . . denotes the pure spinor zero-mode integration of [50], but it plays no role in the subsequent discussions.

BRST-invariant permutations and BRST-invariant superfields
The linearity condition M n = φ A string (P n ) implies that BRST-invariant permutations in P n are mapped to kinematics in M n leading to a correspondence between the BRSTinvariant permutations (2.4) and a series of motivic MZV corrections from the motivic string disk amplitude in the f -alphabet γ 1|P 1 ,P 2 ,...,P k ↔ φ A string (γ 1|P 1 ,P 2 ,...,P k ) .
(3.21) whose precise content follows from Table 1  with similar expansions at higher points. The data presented in Table 1 and the discussion in section 3.2 suggest that the BRST-invariant permutations can be associated to a series of higher-mass BRST-invariant superfields by defining The motivic MZV amplitudes then follow from where 23...n is a shorthand notation for the deconcatenations of P 1 P 2 . . . P k = 23 . . . n.
For example,  (1, 2, 3, 4) were used. The superfield representation of the above BRST invariants is known only for the two simplest cases, C 1|P = M 1 E P [51] and C f 2 1|P 1 ,P 2 ,P 3 [19,52]. Note that C 1|P 1 = A SYM (1, P 1 ) while the S-map algorithm of [19] gives rise to a purely combinatorial translation between C f 2 1|P 1 ,P 2 ,P 3 and sums of s 2 ij A SYM . It remains to be seen whether there exists a general algorithm to rewrite C f m 2 f M in terms of SYM amplitudes.

BRST invariants from A f 2
Another consequence of the duality (3.18) is that the representation of A f 2 in terms of C 1|P 1 ,P 2 ,P 3 given in (3.16) is invertible, as argued indirectly in [18]. This follows from Theorem 4.2 of [9], E µ • I p = I p if λ(p) = µ where λ(p) is the shape of the composition p and E µ is defined in (A.30). This implies (1 · I p ) • (1 · θ(E µ )) = (1 · I p ) for λ(p) = µ or, using the function interpretation of the right action σ • F := F (σ) with a partition with three parts k(µ) = 3 where we used the parity relation (3.4) in the RHS.

The superfield expansion of C 1|P,Q,R from BRST-invariant permutations
The so-called BRST invariants C 1|P,Q,R of the pure spinor formalism [50] play an important role in the mapping between BRST-invariant permutations and kinematics, see section 3.3.
They were firstly derived at low multiplicities in [18] and were subsequently studied in different contexts and given general recursive algorithms, see [19,52,53]. Their superfield expansions in terms of Berends-Giele currents follow from These terms can be extracted from the permutations of the BRST-invariant permutations γ 1|P,Q,R of the inverse descent algebra as follows: 1. Sum over the cyclic permutations of all permutations in γ 1|P,Q,R : 2. Decompose W σ into all possible four-word deconcatenations: 3. Move label 1 to the front by repeatedly commuting W C .W A1B = W A1B .W C if necessary and write the result in terms of Berends-Giele superfields: The resulting expressions have been explicitly checked 11 for all topologies of BRST invariants up to eight points. In addition, using the descent duality (3.18) one may also derive the change of basis identities for C i =1|... = C 1|... from [52,53] by choosing a different label to be singled-out in the color-dressed permutation (2.1) [54].

Conclusion
In this paper we investigated the combinatorial properties of the permutations appearing in the color-dressed permutations (2.1) using the tools from the descent algebra. In particular, we defined BRST-invariant permutations, found a closed formula, and related them to orthogonal idempotents which sum to the identity permutation [10,55,9].
We then considered the color-dressed motivic string disk amplitudes of [5] within this framework. This led to the discovery of the relations (3.9) obeyed by the motivic MZV amplitudes, dubbed the descent algebra decomposition of KK-like relations. The basis dimensions of linearly independent motivic MZV amplitudes are given by sums of Stirling cycle numbers. These claims have been explicitly checked using various data points in string theory up to n = 8 and α ′ 7 . 11 The shuffle symmetry AiB = (−1) |A| iÃ¡B [21] is needed to rewrite words in a Lyndon basis.
Inspired by [18], we proposed a correspondence between the permutations from the (inverse) descent algebra and kinematics from the motivic string disk amplitudes in terms of higher-mass BRST invariants. In the particular case of α ′ 2 , we exploited a theorem from the mathematics literature on descent algebra to prove certain claims in [18] and to systematically express BRST invariants as linear combinations of A f 2 corrections to motivic disk amplitudes in (3.26).
And finally, we found an algorithm to extract the superfield content of the BRST invariants in the pure spinor formalism from the BRST-invariant permutations in the inverse descent algebra. It will be interesting to obtain superfield realizations of the higher-mass BRST invariants defined in section 3.3. They can probably be extracted from a perturbiner series of amplitudes at the appropriate mass level. For instance, the superfields in C SYM 1|P = M 1 E P are related to the series Tr(VVV) [56,27]. The superfields in C f 2 1|P 1 ,P 2 ,P 3 are related to the series Tr(V 1 (λγ m W)(λγ n W)F mn ), while the superfields in C f 3 1|P should follow from Tr (λγ mnpqr λ)(λγ s W)F mn F pq F rs . It would also be interesting to find combinatorial algorithms to directly translate the higher-mass BRST invariants C f m 2 f M 1|P 1 ,...,P k into linear combinations of super-Yang-Mills amplitudes and powers of Mandelstam invariants, generalizing the S-map algorithm of [19] for C f 2 1|P 1 ,P 2 ,P 3 . This would give rise to a combinatorial description of the P n and M n matrices of [5].
Acknowledgements: I thank Ruggero Bandiera for discussions during an early stage of this project. I also thank Oliver Schlotterer for sharing notes containing intriguing observations about the permutations in the color-dressed amplitude that served as the motivation for this paper, for discussions, for collaboration on related topics, and for comments on the draft (including suggesting a v3 to be uploaded). CRM is supported by a University Research Fellowship from the Royal Society.

Appendix A. The Solomon descent algebra
We review the salient features of the Solomon descent algebra [8,9,10,11,12,13,14]. In particular, we discuss different bases and highlight the orthogonal idempotents discovered by Reutenauer, as they will be related to α ′ corrections to string amplitudes.

A.1. Descent classes and the Solomon descent algebra
The descent set D(σ) and the and the descent number d σ of a permutation σ = σ 1 σ 2 . . . σ n in S n are defined by where the coefficients c S,T,U are non-negative integers [8]. The descent classes therefore form a 2 n−1 dimensional algebra, the so-called Solomon's descent algebra D n [8,9,11,12,13,14].
As an example of (A.4), consider the permutations in S 4 . Its 24 elements are organized into 8 descent classes as follows It is straightforward to multiply the permutations among these descent classes using the right-action of the symmetric group (1.10). For example, where the last line follows from the remarkable property (A.4) which ensures that the permutations in (A.6) are themselves a sum of descent classes.

A.2. Bases of the descent algebra
Apart from the descent classes D S indexed by descent sets S, there are other convenient bases of the descent algebra [9].
where the inverse map θ is given by For example, if p = (1, 1, 2) then X 1 = 1, X 2 = 2 and X 3 = 34 and we get There is a closed formula for the multiplication of B p •B q [12,9,57]. where we used the conversions (A.11).

A.2.3. The Eulerian idempotent
The Eulerian (or Solomon) idempotent is defined by [17,10,55,58] (see also [59]) where d σ denotes the descent number (A.1) of the permutation σ. For example, Apart from being an idempotent satisfying E n • E n = E n , the definition (A.20) is also a Lie polynomial [10]. Therefore its coefficients κ σ must satisfy the shuffle symmetry [23] κ R¡S = 0, R, S = ∅ . (A. 22) As usual, the definition (A.20) in terms of the fixed alphabet N in S n can be turned into a function of an arbitrary word P by the right action (1.10) of the symmetric group [14,60],

A.2.4. The idempotent basis I p
The idempotent basis I p of the descent algebra D n satisfying I p • I p = I p was introduced in [9] and it is indexed by the compositions of n where the sum is constrained by the length of X i being equal to the corresponding p i in the composition p and E X i denote the Eulerian idempotent function (A.23 The idempotent basis elements I p for p = p 1 p 2 . . . p k can be expanded in terms of compositions B q using an algorithm discussed in [9]. First one defines moments e m as a polynomial in non-commuting variables t i for i = 1, 2, . . . from the generating series where x is a commuting parameter. For example, from (A.26) it follows that Then to convert the I p basis elements to the composition basis B q one uses [9] For example,

A.3. Reutenauer orthogonal idempotents
A partition λ of n, denoted λ ⊢ n, is a k-tuple of positive integers with sum n satisfying λ 1 ≥ λ 2 ≥ . . . ≥ λ k . If p |= n is a composition of n, the shape λ(p) of p is the partition of n obtained by rearranging the parts of p in decreasing order. Also, k(p) is the number of parts of the composition p. For example, p = (2, 3, 1, 2) implies λ(p) = 3221 and k(p) = 4.
Given a partition λ = (λ 1 , λ 2 , . . . , λ k ) into k parts, theorem 3.1 of [9] shows that Note that when the partition λ of n has only one part, E λ = I n coincides with the Eulerian idempotent E n (A.20), so this notation is not ambiguous. For example, E 1 = I 1 and It was shown in [9,10] that (A.32) are orthogonal idempotents which sum to the identity An alternative definition of the Reutenauer idempotents in terms of a generating function can be found in [14].

Appendix B. The inverse of the idempotent basis
In this appendix we will prove (2.17), that is: Proposition. The inverse of the idempotent basis (2.16) satisfies I p 1 p 2 ...p k (P 1 , P 2 , . . . , P k ) = E(P 1 )¡E(P 2 )¡ . . . ¡E(P k ) , where P = P 1 . . . P k is the factorization of P with P i of length p i .
Proof. The proof will be based on the following observations collected from [14], which should be consulted for more details as the equation numbers below refer to it. First, the adjoint of an arbitrary function F (P ) = P • F of a word P is given by θ(F )(P ) = P • θ(F ), see (3.3.5). Second, the adjoint of and ⋆ ′ are the convolution operators defined in (1.5.7) and (1.5.8) and θ(F j ) is the adjoint of F j when viewed as a function by the right-action (1.10), see proof of Lemma 3.13. Third, for permutations F p i of length p i one can show (by adapting the proof of Lemma 3.13) where the functions are defined via a right action as F p i (P i ) := P i • F p i . The proof of (B.1) then follows from the observation by (1.5.4) and (1.5.7) that the idempotent basis I p (A.24) can be rewritten as a convolution I p 1 ...p k (P ) = E p 1 ⋆ . . . ⋆ E p k P where E p is the Eulerian idempotent (A.20). Therefore its adjoint θ(I p 1 ...p k )(P ) is given by where we used (B.2) and E(P i ) = θ(E p i )(P i ).
Proof. The parity of A string at n points can be written as A string (1, σ) = (−1) n A string (1,σ) by cyclicity. This means, by (2.20), that A string (γ 1|P 1 ,...,P k ) will vanish whenever the parity of A string at n points is opposite to the parity of I p for p |= n−1.  (23,4).
The proposition can now be proven by considering the two cases when n is even or odd.
For n even the parity of the n-point disk amplitude is + so A string (γ 1|P 1 ,...,P k ) vanishes if the parity of I p is − for a composition p of n−1. By (2.19) this means that there must be an odd number of even parts in the composition p (which sum to even). But since n−1 is odd, there must be an odd number of odd parts in p (which sum to odd). Therefore the number of parts k(p) is even (= odd + odd). Similarly, when n is odd the number of parts k(p) in the composition of p is also even (from even + even). This finishes the proof.