Towards the n-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure

In this final part of a series of three papers, we will assemble supersymmetric expressions for one-loop correlators in pure-spinor superspace that are BRST invariant, local, and single valued. A key driving force in this construction is the generalization of a so far unnoticed property at tree-level; the correlators have the symmetry structure akin to {\it Lie polynomials}. One-loop correlators up to seven points are presented in a variety of representations manifesting different subsets of their defining properties. These expressions are related via identities obeyed by the kinematic superfields and worldsheet functions spelled out in the first two parts of this series and reflecting a duality between the two kinds of ingredients. Interestingly, the expression for the eight-point correlator following from our method seems to capture correctly all the dependence on the worldsheet punctures but leaves undetermined the coefficient of the holomorphic Eisenstein series ${\rm G}_4$. By virtue of chiral splitting, closed-string correlators follow from the double copy of the open-string results.


Introduction
This is the third part of a series of papers [1] towards the derivation of one-loop correla- tors of massless open-and closed-superstring states using techniques from the pure-spinor formalism [2,3]. We often refer to section and equation numbers from part I & part II and then prefix these numbers by the corresponding roman numerals I and II. The main result of this paper is the assembly of local one-loop correlators in pure-spinor superspace [4] up to eight points. This will be done by combining two main ingredients: 1. local kinematic building blocks introduced in part I that capture the essentials of the pure-spinor zero-mode saturation rules and transform covariantly under the BRST charge 2. worldsheet functions introduced in part II capturing the singularities generated by OPE contractions among vertex operators. In particular, their monodromies as the vertex positions are moved around the genus-one cycles also follow a notion of "covariance". More precisely, the monodromies are described by a system of equations that share the same properties of the so-called BRST invariants and naturally lead to a duality between kinematics and worldsheet functions.
The fundamental guiding principle that will act as the recipe to combine these two ingredients will correspond to the one-loop generalization of a symmetry property obeyed by the analogous tree-level correlators derived in [5] and reviewed in section 2.1 below.
More precisely, the tree-level correlators are composed from products of Lie-symmetric kinematic building blocks V 123...p and shuffle-symmetric worldsheet functions Z 123...p = (z 12 z 23 . . . z p−1,p ) −1 . Given the similar structure between these symmetries and the composing elements in a theorem of Ree concerning Lie polynomials [6], we dubbed the correlators obtained in this way as having a Lie-polynomial form. We will see that this line of reasoning leads to a key assumption of this paper, that the local n-point one-loop correla- tors of the open superstring can be written as Definitions of the kinematic building blocks T m 1 ...m r A 2 ,...,A r+4 and the worldsheet functions Z m 1 ...m r A 1 ,...,A r+4 can be found in part I and part II, respectively 1 . The notation for the permutations in terms of partitions of words addresses the kind of permutations resulting from the interplay between shuffles and Lie symmetries, and is explained in the appendix A. As discussed in part II, a beneficial side effect of requiring shuffle symmetry for the worldsheet functions is that the resulting functions automatically contain non-singular pieces that are invisible to an OPE analysis. Lie symmetries in turn refer to the generalized Jacobi identities satisfied by the kinematic building blocks, in lines with the Bern-Carrasco-Johansson duality between color and kinematics [9].
The notions of locality, BRST invariance and single-valuedness will then lead to a discussion for why the "+ corrections" are needed starting at n ≥ 7 points. In section 3, a multitude of representations for the correlators with n = 4, 5, 6, 7 (including the "+ corrections" at n = 7) will be given that expose different subsets of their properties. While the n = 8 correlator following from the proposal (1.1) satisfies many non-trivial constraints, it fails to be BRST invariant by terms proportional to the holomorphic Eisenstein series G 4 . In the future, we expect to address this challenging leftover problem in order to extend our results to arbitrary numbers of points. Section 4 is dedicated to manifesting the modular properties of the open-and closedstring correlators by integrating out the loop momentum. We will relate a double-copy structure of the open-string correlators [10] to the low-energy limit of the closed-string amplitudes. This incarnation of the duality between kinematics and worldsheet functions is checked in detail up to multiplicity seven, and we describe the problems and perspectives in the quest for an n-point generalization at the end of section 4.

One-loop correlators of the open superstring: general structure
In this section, we set the stage for assembling one-loop correlators K n (ℓ) from the system of kinematic building blocks and worldsheet functions introduced in part I and II. By their definition in section I.2.2, correlators K n (ℓ) carry the kinematic dependence of one-loop open-string amplitudes among n massless states with . . . denoting the zero-mode integration prescription of the pure-spinor formalism [2]. The integration domains D top for the modular parameter τ and vertex positions z j are tailored to the topologies of a cylinder or a Möbius strip with associated color factors C top , see [11] for details. The integration over loop momenta ℓ is an integral part of the chiral-splitting method [12,13,14], which allows to derive massless closed-string one-loop amplitudes from an integrand of double-copy form with F denoting the fundamental domain for inequivalent tori w.r.t. the modular group.
Both of (2.1) and (2.2) involve the universal one-loop Koba-Nielsen factor with lightlike external momenta k j , where we use the shorthands and conventions where 2α ′ = 1 for open strings and α ′ 2 = 1 for closed ones. In trying to calculate multiparticle one-loop amplitudes using the pure-spinor prescription (I.2.4), one soon realizes that most efforts tend to be hampered by the complicated nature of the b-ghost (I.2.5). This difficulty, however, motivates a less direct approach which illuminates the structure of the answer in a somewhat unexpected way; the organizing principle will be drawn from the tree-level correlators of [5].

Lessons from tree-level correlators
Recall that n-point open-string tree amplitudes in the pure-spinor formulation require the evaluation of the following n-point correlation function [2], One of the crucial steps in the calculation of [5] was showing that the multiparticle vertex operators V P [15] could be used as the fundamental building blocks of the correlator; for example, in terms of the function Z tree P defined by Z tree 123...p ≡ 1 z 12 z 23 . . . z p−1,p , (2.6) we have (2.7) where the symbol ∼ = is a reminder that the above relations are valid up to total derivatives and BRST-exact quantities. As reviewed in section II.4.1, the accompanying functions exhibit shuffle symmetry such as Z tree 12 + Z tree 21 = 0, Z tree 123 − Z tree 321 = 0 and Z tree 123 + Z tree 213 + Z tree 231 = 0 dual to the Lie symmetries V 12 = −V 21 , V 123 = −V 213 and V 123 + V 231 + V 312 = 0, cf. (I. 3.25). When combined with (2.7), these symmetries lead to the following generalization: which eventually gives rise to the solution found in [5]. As detailed in section I.3.1, the summation range |A| = n in (2.8) refers to the n! words A formed by permutations of a 1 a 2 . . . a |A| with |A| = n.
At this point one may realize that the right-hand side of (2.8) has the structure of a Lie polynomial [16,6] and that the expressions for the n-point correlators at tree level obtained in [5] can be written in terms of their products. More precisely, K tree n is given by two copies of (2.8) with n−2 deconcatenations AB = 23 . . . n−2 and an overall permutation over (n−3)! letters for a total of (n−2)! terms: For example (Z tree i ≡ 1), to each zero-mode saturation pattern, see e.g. (I.3.23) and (I. 3.24). The corresponding Lie polynomials will therefore differ with respect to these features but will preserve their mathematical characterization as sums over products of shuffle-and Lie-symmetric objects.
We will be concerned with the particulars of the expressions (2.11) in section 3; for the moment we note that their growing number of terms calls for a more convenient notation. In the subsequent discussion we will distill the combinatorial properties of these permutation sums and propose an intuitive notation for them.

Stirling cycle permutation sums
In order to grasp the combinatorics of (2.11), note that the symmetries of the Lie polynomial A V A Z A imply that only (p−1)! permutations are independent for words of length p.

This is true for each word
For an n-point correlator these words A i must encompass all particle labels, that is |A 1 |+|A 2 |+|A 3 |+|A 4 | = n.
Therefore the sums of in the correlators of (2.11) can be interpreted as being all the permutations of n labels that are composed of 4 cycles, or p cycles in the general case of tensorial ..,A p . This is the characterization of the Stirling cycle numbers 3 n p . Using the above interpretation, the scalar building blocks in (2.11) are generated by the following combinatorial notation Similarly, the vector contribution to K 5 (ℓ) and K 6 (ℓ) in (2.11) follows the same combinatorial pattern as the scalars and its contribution is captured by extending the Stirling cycle permutations to five slots in a similar manner, 14) The generalization of the above sums to more slots is straightforward.

Unrefined Lie polynomials
The Stirling cycle permutations allow for a straightforward generalization of the correlators in (2.11) to multiplicities n ≥ 4, where the summand with r = 0 and r = 1 reproduces (2.13) and (2.14), respectively. The reason for the superscript in K (0) n (ℓ) will become clear below, and this is related to the corrections in (1.1). Expanding the sum yields, ..,A 6 + 12 . . . n|A 1 , . . . , A 6 . . .
We will see in section 3 that (2.15) gives the correct form of the one-loop correlators up to and including six points, i.e., K n (ℓ) = K (0) n (ℓ) for n ≤ 6. By "correct" we mean that the resulting correlators satisfy a number of requirements detailed in section 2.4, the most stringent ones being BRST invariance and single-valuedness.
So the question to consider is whether the expression (2.16) provides the complete answer for correlators with seven or more external states. Unfortunately this is not the case; the explicit construction of the seven-point correlator indicates that the proposal (2.16) needs to be amended by terms involving superfields with higher degrees of refinement defined in section I.4.4. This will be done below and leads to an expression for K 7 that passes all consistency checks. At eight points and beyond, however, the appearance of Eisenstein series in the correlators cannot be determined by the methods in this work.
Hence, we will only propose an expression for K 8 up to an unknown kinematic factor multiplying G 4 while completely fixing its dependence on the z j .

Including refined building blocks
The reason why K (0) n (ℓ) in (2.16) cannot be the full expression for the one-loop correlator for n ≥ 7 is related to BRST invariance; it is not difficult to show that the seven-point instance is not BRST invariant using the worldsheet functions discussed in part II. However, the desired invariance can still be achieved by adding corrections containing refined superfields J A|B,C,D,E and their tensorial generalizations, cf. section I.4.4. The patterns encountered at multiplicities seven and eight suggest the following organization; the n-point correlator contains contributions with varying degree d of refinement according to, These expressions generalize to n ≥ 7 points at generic degree d of refinement as ..,A 6 + (A 2 ↔A 3 , . . . , A 6 ) + 1 . . . n|A 1 , . . . , A 6 (2.22) The collection of K We will see that up to and including eight points, the BRST variation of (2.17) is purely anomalous (it is written in terms of the anomalous superfields Y , see section I.4.3.1) and it is natural to conjecture that this behavior is valid for arbitrary n.

BRST variation of the Lie-polynomial correlator
By BRST covariance of their kinematic building blocks in section I.4, the Q variations of the above K Lie n (ℓ) boil down to ghost-number four superfields .. . As will be detailed in the next section, the coefficients of these ghost-number four combinations read as follows in the simplest non-vanishing variations, where K Lie n=5,6 (ℓ) already furnish the complete correlators K n (ℓ). Note that we have disregarded the vanishing of Z 2|1,3,4,5,6 for later convenience, and one can compactly absorb the Mandelstam invariants in (2.23) into the S[A, B] map defined in (I.5.13), e.g. s 31 Z 312,4,5,6 − s 32 Z 321,4,5,6 = Z S [3,12],4,5,6 . Based on these examples and analogous observations on QK Lie n (ℓ) for higher values of n, it is possible infer a general pattern and propose closed formulae. We organize the general conjecture on the BRST variation of the correlator (2.17) into the following Stirling permutation sums, where the suppressed terms T (d,r) and Y (d,r) in . . . refer to higher degree of refinement d ≥ 2 and start to contribute at n = 9. The case r = 0 is understood as containing no vector indices in the superfields, and a upper negative integer in the sum must be discarded;   4 Recall that the J A|... building block is naturally identified as a d = 1 refined version of T .
suggest their analogues at d ≥ 2. Finally, the shorthands Θ (d) and Ξ (d) stand for the following linear combinations of worldsheet functions with degree d of refinement that capture the right-hand sides of (2.23), (recall that S[A, B] denotes the S-map defined in (I.5.13)), and Hence, after modding out by Lie symmetries of the superfields, combining (2.25) and Two comments are in order here. First, notice that the presentation of the BRST variation as a Stirling permutation sum (with the conventions of the appendix A) is essential to fix the ambiguity of B|A,... . For example, the conventions of the appendix A fix the relative ordering between the cycles (1)(234) in the permutation sum such that we get V 1 V 234 T 5,6,7 Θ (0) 234|1,5,6,7 rather than V 234 V 1 T 5,6,7 Θ (0) 1|234,5,6,7 . And second, although a bit surprising, the BRST variation leads to crossing-symmetric definitions such as Θ and for some K Y n (ℓ) to be determined. Such an anomaly sector K Y n (ℓ) is plausible by the kinematic identities of section I.5.4, as they mix anomalous and non-anomalous terms. Up to and including six points, we have K Y n (ℓ) = 0 , for n ≤ 6 . From multiplicity seven on, we need to find an expression for the anomaly sector K Y n (ℓ) such that the full correlator satisfies the criteria summarized below. Even though we will find the proper K Y n (ℓ) in the seven-point example of section 3, this is done case-by-case, so it would be desirable to understand the general pattern behind them.

Final assembly of one-loop correlators
The general form of the one-loop correlators (2.33) was suggested by analogy with the Lie-polynomial structure observed at tree level [5]. The one-loop correlators K n (ℓ) are expressions in the cohomology [18] of pure-spinor superspace that depend on the loop momentum ℓ m and the zero modes of the pure spinor λ α and of the superspace coordinate θ α . Moreover, they are also expanded in terms of worldsheet functions that have to be integrated over the vertex operator insertions points as well as over the moduli space that parametrize the different genus-one surfaces. Given this setting, the final assembly of one-loop correlators K n (ℓ) as defined in (2.1) must satisfy the following fundamental requirements: 1. The correlator must be in the cohomology of the BRST operator; 2. The correlator must be a single-valued function with respect to both z i and ℓ m ; 3. The correlator must admit a local representation; 4. The correlator must be manifestly 5 symmetric in the labels (2, 3, . . . , n).
These conditions arise from general CFT considerations applied to the pure-spinor amplitude prescription (I.2.4), and they are compatible with the tree-level arguments that led to the Lie-polynomial proposal (2.33). The notion of single-valuedness in 2. is defined in (II.3.3), and 3. refers to the absence of kinematic poles s −1 P in a local representation of K n . The combination of 1. and 3. turns out to be particularly constraining: Any BRST-invariant linear combination of the building blocks of section I.4 has been checked to vanish in the cohomology at 5 ≤ n ≤ 8 points (see appendix I.B for further details). Therefore there is no freedom of adding BRST-invariant local terms multiplying single-valued functions at these multiplicities.
In the next section we write down explicit examples of one-loop correlators fulfilling the above criteria up to seven points. Moreover, we propose an expression at eight points with mild violations of 1. and 3.: Its BRST variation vanishes only up to local terms proportional to the Eisenstein series of modular weight four, G 4 , and certain terms in the anomaly sector K Y 8 violate locality. We expect that the eight-point proposal to be given in section 3.5 differs from the correct correlator K 8 by G 4 multiplying an unknown kinematic factor, i.e. it correctly captures all dependence on the z i .

One-loop correlators of the open superstring: examples
We will now apply all the techniques developed in the previous sections to obtain explicit expressions for the one-loop correlators of the open superstring in a manifestly supersymmetric fashion. The correlators at four, five, six and seven points meet all the requirements described in section 2.4, and we will elaborate on the aforementioned issues with the eightpoint correlator below.

Four points
The four-point correlator is uniquely determined by the zero-mode integration over the pure-spinor variables and it was firstly computed by Berkovits in [3]. Using the definition (I.4.1) its correlator can be written as the manifestly local pure-spinor superspace expression K 4 (ℓ) = V 1 T 2,3,4 . (3.1) Note that there are no worldsheet singularities among the vertex positions nor an explicit dependence on the loop momentum ℓ m . This is in accordance with the general discussion in section I.2.1.3 that a n-point correlator K n (ℓ) is a polynomial in loop momenta of degree n−4 and that the maximum number of OPE contractions is also n−4. It has been shown in [19] using BRST cohomology identities in pure-spinor superspace that the one-loop correlator (3.1) is proportional to its tree-level counterpart (2.10), Therefore it reproduces the well-known [11,20] supersymmetric completion of t 8 F 4 and the one-loop amplitudes of Brink, Green and Schwarz with bosonic external states [21].

Five points
The reasoning behind the derivation of the five-point correlator will be presented in detail as it constitutes the prototype for similar derivations at higher points. Not surprisingly, the outcome of the following analysis is in accordance with the general features of one-loop correlators summarized in section 2.3.
As discussed in section I.2.1.3, the pure-spinor prescription [3] implies that the fivepoint correlator K 5 (ℓ) is a polynomial of degree one in ℓ with at most one OPE singularity.
Therefore the correlator is composed of two classes of terms containing: (i) one OPE contraction, (ii) one loop momentum. Let us consider them in turn.
It will be rewarding to rewrite the correlator (3.6) in a slightly more abstract manner, since the higher-point generalization will become more natural in this way. The correlator lines up with the Lie-polynomial structure of (2.15), where the notation for the permutations is explained after (2.13) and in the appendix A.
Expanding the above permutations leads to the following 5 5 + 5 4 = 1 + 10 = 11 terms, In summary, the five-point one-loop correlator (3.8) is a manifestly local expression of superfields that was obtained using general arguments based on the amplitude prescription of the pure-spinor formalism. If we want to argue that it is also the correct correlator, it must be BRST invariant and single-valued as well. Indeed, a short calculation yields

BRST invariance
where in the second line we used the shorthand defined in (2.28). At first sight (3.10) appears to be different than zero, but luckily this particular arrangement of integrands turns out to be a total worldsheet derivative, where we used the expansions (3.9) and the identity (II.2.22). Therefore the five-point correlator (3.8) is BRST invariant.

Single-valuedness
From the discussion in section II.2.1.1, it follows that the monodromies around the A-cycle vanish for any combination of ℓ m and g (n) ij , so the correlator (3.8) will be single valued if its monodromies around the B-cycle also vanish. In this case, the variations (II.3.9) yield Since the anomalous superfield ∆ 1|2,3,4,5 was shown to be BRST-exact in [22], the monodromy variation (3.13) vanishes in the cohomology of the pure-spinor superspace (indicated by ∼ = 0), and the correlator (3.8) is therefore single-valued.

Duality between worldsheet functions and BRST invariants
The vanishing of (3.11) is a clear indication of the duality between worldsheet functions and BRST invariants discussed in section II.4 and pointed out in [10]; it corresponds to the BRST-exact linear combination of superfields in (3.12) under the replacement (II.4.10), Next, the integration-by-parts identity (3.11) can be used to eliminate all functions of the 13 , g 14 , g 15 ). Doing this leads to

The T · E representation: manifesting single-valuedness
Since the five-point correlator (3.8) is single valued, it is worthwhile to spell out a representation that manifests this property. To do this, we rewrite the terms containing a factor of V 1A with non-empty A using the BRST cohomology identity which follows from the (I.5.41) and the BRST-exactness of ∆ 1|2,3,4,5 . Doing this replacement in the correlator (3.8) and collecting terms leads to The combinations of Z-functions the round brackets can be identified with the GEIs E m 1|2,3,4,5 and E 1|23,4,5 from (II.4.31) and (II.4.32), respectively. Using these functions, the correlator (3.19) takes the manifestly single-valued form: which reproduces the double-copy expression for the five-point correlator proposed in [10] and manifests both BRST invariance and single-valuedness.

Summary of representations
As shown above, there are multiple Lie-polynomial representations of the five-point correlator according to which features are chosen to be manifested: In addition to the above, the single-valued representation of the five-point correlator obtained by explicit integration over the loop momentum will be presented in section 4.
We remark that the one-loop five-point amplitude in the open superstring has been computed with the RNS and GS formalisms for states in the Neveu-Schwarz sector [23,24,25] and in the Ramond sector [26,27]. Manifestly supersymmetric expressions were obtained in [28] using the non-minimal pure-spinor formalism [29] and later in [30] using the minimal pure-spinor formalism.
Note that a possible contribution of a d = 1 refined sector according to (2.21) is suppressed since the monodromy variations (II.4.34) are compatible with Z 1|2,3,4,5,6 = 0. The explicit expansion of the Stirling cycle permutations in (3.23) generates a total of 6 6 + 6 5 + 6 4 = 1 + 15 + 85 = 101 terms, We will now prove that the six-point correlator (3.23) is both single-valued and BRST invariant, in accordance with the expectations outlined in section 2.4.

BRST invariance
As anticipated in section 2.3.4, the BRST algebra of the building blocks leads to the following Q-variation of the correlator (3.23), where the shorthands Θ and conspire to total derivatives in z j . Therefore the BRST variation is proportional to the trace Z mm 1,2,3,4,5,6 , where the total τ derivative of the Koba-Nielsen factor has been identified in (II.5.15).
Thus, the BRST variation is a boundary term in moduli space [31], and the usual mechanism of anomaly cancellation [32] implies that the amplitudes computed from the correlator (3.23) are BRST invariant.

The C · Z representation: manifesting BRST invariance
Now that BRST invariance of the six-point correlator is proven, let us rewrite it using the BRST invariants from section I.5.2, in a similar spirit as done with the five-point correlator in the previous section. There are different ways to achieve this, one uses the trading identity (2.12) to rewrite the Lie polynomial (3.23) as, The idea now is to exploit the fact that terms of the form A,B,... , which feature the single-particle Berends-Giele current M 1 , are the leading terms in the expansion of the BRST (pseudo-)invariants from section I.5.2.1. Therefore they can be rewritten as To arrive at (3.31) the following three topologies of terms (and their permutations) were discarded as they are total derivatives: Expanding the Stirling cycle permutations in (3.31) yields the following 5 5 + 5 4 + 5 3 = 1 + 10 + 35 = 46 terms, Note one important difference between the expansion above and an earlier representation; unlike the local Lie polynomial (3.23) in which six labels are distributed among the available slots, in the non-local representation (3.31) only five labels participate in the Stirling permutations. Like this, the initially 101 terms in (3.25) conspire to the considerably smaller number of 46 terms in (3.33).
As a consistency check, we note that the scalar C 1|A,B,C and vectorial C m 1|A,B,C,D are manifestly BRST closed while the BRST variation of the two-tensor C mn 1|A,B,C,D,E is proportional to δ mn [22]. Hence, we arrive at the same conclusion as in (3.28)

The C · E representation: manifesting BRST invariance & single-valuedness
As another application of the duality between kinematics and worldsheet functions, we shall now derive a manifestly BRST-invariant and single-valued representation of the sixpoint correlator. The idea is to start from the C · Z representation (3.33) and to exploit the dual of (3.30): Each Z-function with leg one in a single-particle slot is taken as a leading term of a GEI, see (II.4.26), and the additional terms in the ellipsis of (3.42) are of the form In this way, a long sequence of BRST cohomology identities given in section I.5.4 leads to the following manifestly BRST-invariant and single-valued Liepolynomial form of (3.33), The GEIs have been expressed in terms of the Lie symmetric E ij in (II.4.37). By the vanishing of Z 2|1,3,4,5,6 , the C ·Z-representation (3.33) of the six-point correlator does not feature any analogue of the terms P 1|2|3,4,5,6 E 1|2|3, 4,5,6 in the first line of (3.44).

The T · E representation: manifesting locality & single-valuedness
The C · E representation (3.43) is not manifestly local, but it is written in terms of GEIs manifesting monodromy invariance. However, by construction, we know that (3.43) is equivalent to the local representation (3.23), so all the non-localities within the pseudoinvariants C and P must be spurious. In the following discussions we exploit this reasoning to find a new representation that is both manifestly local and monodromy invariant.
We can do this starting from (3.44), plugging in the Berends-Giele expansion of the pseudo-invariants and separating terms according to their kinematic poles. The non-local terms turn out to vanish (as will be exemplified below) while the local terms conspire to produce the full correlator K 6 (ℓ). After going through the algebra we obtain the following manifestly local and monodromy-invariant form of the six-point correlator K 6 (ℓ), The non-local terms from the kinematic side turn out to vanish due to identities obeyed by their accompanying worldsheet functions. For instance, one such class of terms (featuring an uncancelled s 12 pole) is given by, Note that the T · E representation (3.47) is related to the C · Z representation (3.33) through the duality between kinematics and worldsheet functions: In order to see this, one needs to adjoin the vanishing terms − Z 2|1,3,4,5,6 P 1|2|3,4,5,6 + (2 ↔ 3, 4, 5, 6) to the latter.
Note that the representation of Z 12|3,4,5,6,7 in the second line of (3.56) manifests that K 7 (ℓ) can be written without any derivatives ∂g ij with coinciding arguments. This observation should play an important role for the transcendentality properties upon integration over z j .

BRST invariance
In order to show that the full correlator (3.54) is BRST invariant, let us first consider its non-anomalous part, QK Lie 7 (ℓ). This computation can be organized according to the ghostnumber four products of superfields it generates; this general structure was anticipated in section 2.3.4 but it is instructive to see it again in this particular case: It is evident from the above permutations that the general compact expression (2.25) leads to involved combinatorics resulting in many terms present in the seven-point BRST variation (3.58), even when written using the shorthands Θ (d) and Ξ (d) defined in (2.28) and (2.29). Fortunately, the analysis of the outcome is also greatly simplified by this very same organization, as it suffices to check only a handful of different topologies of Θ (d) and Ξ (d) rather than all their permutations. In fact, it is straightforward to check that all T (d,r) terms above vanish due to and therefore vanish as well.
Using the results above the BRST variation of (3.54) is purely anomalous Therefore the full correlator (3.54) is BRST invariant up to total derivatives, Before showing that (3.54) is also monodromy invariant, it will be convenient to rewrite it using the pseudo-invariants of section I.5.2, as that will simplify the proof considerably.  1|A,B,... + · · · and collect the terms containing a factor of M 1P with P = ∅. A long but straightforward analysis using integration-by-parts relations (2.30) for the Z-functions shows that all terms proportional to M 1P vanish and we arrive at ..,A 6 + 234567|A 1 , . . . , A 6 (3.68) Note that only six legs participate in the Stirling permutations, and Z (s) ... are defined in (II. 4.22). To compute the BRST variation of (3.68) it will be convenient to recall that [22]

Single-valuedness
We will take the manifestly BRST-invariant representation ( 7 + · · · + Ω 7 δK   7 by relabeling of 1 ↔ j in both the kinematics and GEIs of (3.72). To verify this last statement one uses the change-ofbasis identities for pseudo-invariants derived in [22]. This is because the relabeling of δK It is crucial to note that only unrefined building blocks ∆ 1|... arise, whose BRST exactness is discussed in section I.5.3. Since the other δK (j) 7 are relabellings of (3.74), it follows that the complete monodromy variation DK 7 (ℓ) in (3.71) is BRST-exact and therefore vanishes in the cohomology; DK 7 (ℓ) ∼ = 0.

The C · E representation: manifesting BRST invariance & single-valuedness
Having derived the C · Z representation and shown that it is single valued, we can reexpress it to manifest both BRST and monodromy invariance. We proceed similarly as in the six-point case by inserting Z m.. .  1,A,B,... = E m... 1|A,B,... + · · · into the C · Z representation (3.68) and using a long sequence of BRST cohomology identities described in [22]. Doing this leads to a manifestly BRST-invariant and single-valued expression neatly summarized by the following Stirling permutation sums for a total number of 326 terms with pseudo-invariants C and 81 terms with P . This is the double-copy expression for the seven-point correlator implicitly proposed in [10]. Similar to (3.68), only six legs participate in the Stirling permutations, but there is no analogue of the terms ∆ 1|2|3,...,7 Z 12|3,...,7 in the last line of the C · Z representation.

The T · E representation: manifesting locality & single-valuedness
From the C · E representation (3.76) one can derive a manifestly local and single-valued representation following the same ideas as explained for the six-point case in section 3.3.2.2.
Similar to the six-point case (3.47), this T · E representation is related to the C · Z representation through the duality between kinematics and worldsheet functions, up to the fact that (3.77) does not exhibit any dual of the terms ∆ 1|2|3,...,7 Z 12|3,...,7 in (3.68). Moreover, the combinatorial structure of (3.77) is identical to that of the C · E representation (3.76).
In addition, proving BRST invariance of the representation (3.77) requires the same elliptic worldsheet identities used to generate (3.77) from (3.76).

Eight points
Following the general structure of one-loop correlators presented in (2.17) and (2.33), the manifestly local Lie-series part of the eight-point correlator is proposed to be which will later receive a purely anomalous correction K Y 8 (ℓ). The unrefined part with d = 0 follows the general pattern indicated in (2.16),      In terms of the undeformed functions, the BRST variation is given by

Purely anomalous sector
The strategy to cancel the terms (3.86) in a bid to achieve BRST invariance is similar to the seven-point case; we propose to add a purely anomalous contribution to the eight-point correlator (3.78), By analogy with the expression (3.57) for K Y 7 (ℓ), we start from an ansatz comprising anomalous ∆ superfields of (I.C.1) and some unknown worldsheet functions U , appendix I.C), so (3.90) amounts to a mild violating of manifest locality.
In order to determine the U -functions in (3.90) we start by noting that Q 2 K Lie 8 (ℓ) = 0 implies that QK Lie 8 (ℓ) is BRST closed. Therefore all the ghost-number-four superfields V A Y m 1 m 2 ... B 1 ,B 2 ,... in (3.87) must combine to ghost-number four BRST invariants given by Γ defined in [22] (also see the alternative algorithm in appendix I.A.2 for the unrefined cases). This can be seen by rewriting the local superfields in (3.86   To solve these equations it will be convenient to exploit the vanishing of Z ∆ according to Given that the expressions (3.100) for the U -functions in K Y 8 (ℓ) solve all of (3.94) to (3.98), the BRST variation of the overall correlator (3.89) reduces to The R-functions from (3.88) are all proportional to the holomorphic Eisenstein series G 4 , i.e. any dependence of the BRST variation (3.101) on ℓ or the z j has cancelled.
Unfortunately, we explicitly checked [8] that there is no manifestly local deformation of the correlator that can be used to cancel the remaining terms in (3.101). Therefore, even though the BRST variation of K Lie 8 (ℓ) + K Y 8 (ℓ) turns out to be a local expression, its component expansion is non-local, see appendix I.C for the kinematic poles of the ∆ 1|... superfields in (3.90). This suggests that there may be another non-local sector whose BRST variation cancels (3.101), although we have not been able to pinpoint it yet. We leave the quest for finding such a completion to future investigations.

Modular forms: Integrating out the loop momentum
This section is dedicated to the integration over the loop momentum which will lead to manifestly single-valued one-loop correlators. In this way, the correlators acquire welldefined weights under modular transformations, namely holomorphic weight n−4 for the loop integral of K n (ℓ).
At the same time, closed-string correlators are no longer chirally split after integration over the loop momenta [12,13,14]. We will describe the systematics of the interactions between left-and right movers that arises from loop integration of the holomorphic squares After factorizing these universal quantities in the worldsheet integrand of open-and closedstring amplitudes (2.1) and (2.2),

Five-point open-string correlators
Starting from this section, we apply the techniques of integrating the loop momentum to the correlators K n (ℓ) of section 3. We will complement the direct integration of GEIs with a study of the T · Z and C · Z representations where the origin of the kinematic factors from the OPEs is more transparent.

The T · F representation: manifesting locality & single-valuedness
As discussed in [14], integration over the loop momentum leads to manifestly single-valued representations of chirally-split correlators. We therefore integrate out the loop momentum from the representation (3.8) using (II.7.13) to obtain

The C · F representation: manifesting BRST invariance & single-valuedness
It is also possible to obtain a representation without the loop momentum which manifests both BRST invariance and single-valuedness. This can be achieved in at least two ways: integrating out the loop momentum from the C · Z representation (3.17) or using integration-by-parts identities to eliminate all f This form reproduces the five-point one-loop correlator proposed in [28] and rederived in [30,10]. Alternatively, one can arrive at the representation (4.6) using integration-byparts identities (II.7.27) in the local and single-valued representation (4.4). In fact, this is how (4.6) was originally derived in [30]. where we used the shorthand F which will be tensorial at higher multiplicity. We see that integrating out the loop momentum from the functions Z in (3.8) has the same effect as sending ℓ → 0 and g (1) ij → f (1) ij . However, these replacement rules are tied to the present open-string context and no longer apply to the closed-string five-point correlators of section (4.20).

Lessons from the T · E and C · E representations
As an alternative to the earlier computations, one can start from the representations (3.20) or (

The T · F representation: manifesting locality & single-valuedness
We already know that the six-point correlator (3.23) is single-valued, and in this section this will be manifested by integrating out the loop momentum and checking that the generated variables ν ij combine into single-valued functions f (n) ij according to (II.7.3) ij .
with manifestly single-valued worldsheet functions given by To see this, we use the integration-by-parts identity (II.7.27) obtained from ∂ 2 (ν 2Î6 ) = 0, ij . Furthermore, we see from (II.7.5) that the non-holomorphic six-point correlator (4.12) is a modular form of weight two.

The C · F representation: manifesting BRST invariance & single-valuedness
There are several alternatives to deriving a manifestly BRST-invariant form of the correlator without the loop momentum. The most straightforward way is to use integration-byparts identities (II.7.27) in the representation (4.12). A long calculation very similar to the derivation of (3.49)  Also, note that the C · F representation (4.16) results from straightforward regroupings of terms in the integrated C · E representations (3.44): There is no need to perform integration by parts on the f (n) ij , and the coefficients of f (1) 1j , j = 2, 3, 4, 5, 6 are easily seen to vanish after using Fay relations and cohomology identities of section I.5.4.

Closed-string correlators
One of the major motivations for chiral splitting is that closed-string correlators are literally the square of open-string correlators before integration over ℓ, cf. (2.2). Performing the loop integral reveals modular invariance of the closed-string amplitude representation (4.2), at the expense of introducing interactions between left-and right-movers. We will now illustrate these interactions based on examples up to six points.
Most obviously, the expressions of section II.7.2 for integrated GEIs are augmented by additional terms involving π Im τ when the opposite-chirality sector contributes additional loop momenta, e.g.
13 ) 12 + (2 ↔ 3, 4, 5, 6) Once these additional loop momenta are regrouped into complex conjugate GEIs, the net effect of the additional L m 0 is to recombine the g (n) functions to The first term exemplifies that factors of π Im τ are not necessarily associated with modular anomalies in a closed-string setup: Both π Im τ and the remaining terms f ij and g (1) ij − ν ij = f  This representation has been firstly given in [33], based on a long sequence of integrationby-parts identities in (4.20) and carefully tracking all ∂ i f

Six points
A manifestly BRST invariant closed-string six-point correlator has been proposed in [33] [[K 6 (ℓ)K 6 (−  .22) has not yet been derived from first principles but was inferred by indirect arguments including properties of the low-energy limit [33]. In appendix D, we will demonstrate the terms |P 1|2|3,4,5,6 | 2 in (4.22) to follow from a careful analysis of integration-by-parts identities.
Our derivation of (4.22) starts from the C · E representation (3.44) of K 6 (ℓ) and a convenient organization of the loop integrals in the closed-string case according to the number of contractions ℓ m ℓ n → − π Im τ between left-and right-movers BRST anomaly of (4.22) was shown in [33] to yield a boundary term in τ , based on a special case of (II.7.28).

Higher multiplicity
The organization of the closed-string loop integration in the six-point example (4.24) readily generalizes to higher multiplicity. The left-right contractions in the seven-point correlator can be captured via ∂ℓ m δ mn ∂K 7 (−ℓ) ∂(−ℓ n ) (4.28) where the last line evaluates to 1 3! ( π Im τ ) 3 C mnp Note that the all-multiplicity generalization of (4.28) reads .

Closed-string low-energy limits versus open-string correlators
The one-loop low-energy effective action 9 of type-IIB and type-IIA superstrings features a supersymmetrized higher-curvature operator 10 R 4 at its leading order in α ′ [21]. Hence, ij , their low-energy limit can be conveniently extracted through the techniques of [25,35]. The idea is to perform the α ′ -expansion of the integrals in (4.2) over the punctures while keeping τ finite 11 in this process. Then, the leftover τ -integration at the leading order in α ′ straightforwardly yields the volume π 3 of moduli space. 9 See [36] for the exact coefficient of the R 4 operator in the type-IIB effective action, including all perturbative and non-perturbative contributions. 10 While the R 4 operators in the tree-level effective action of the type-IIA and type-IIB theories are identical, at one loop they differ by a contribution proportional to ǫ 10 ǫ 10 R 4 [37,38]. As detailed in [35], the type-IIB matrix elements of this section are proportional to the α ′3 ζ 3 -order of the respective tree amplitudes, where the proportionality constant depends on the R-symmetry charge of the components (say gravitons or dilatons). 11 This approach yields a power series in α ′ that is tailored to infer the one-loop low-energy effective action. The branch cuts of the overall amplitude due to the τ → i∞ limit of the modulispace integral are disentangled when integrating over the z j at fixed values τ . Still, in analyzing effective interactions beyond the low-energy limit, a subtle interplay between the branch cuts and the power-series part has to be taken into account [39,40].
In this setup, the representations (4.21) and (4.22) of the closed-string correlators are tailored to extract the following low-energy limits [33] M R 4 4 = C 1|2,3,4C1|2,3,4 (4.30) These representations of the matrix elements are tailored to connect with the Feynman diagrams of the effective action proportional to R + R 4 : All the propagators stem from the right-moving pseudo-invariants whose expansion in terms of Berends-Giele currents is reviewed in section I.5.2. These Berends-Giele constituents manifest that each term in (4.32) has at most n−4 propagators, reflecting at least one vertex of valence ≥ 4 in each diagram.
The equivalence of (4.30) and (4.32) can be checked without any further calculation by exploiting the duality between kinematics and worldsheet functions: Given that K n (ℓ) and M R 4 4 are related by exchange of pseudo-invariants and GEIs, the manipulations that connect the T · E and C · E representations of the correlators apply in identical form to the matrix elements of R 4 . This follows from the observations of section II.5.1 that all the integration-by-parts identities among GEIs at n ≤ 6 points have a counterpart in the BRST cohomology, relating pseudo-invariants of different tensor rank.
In summary, the low-energy limit of closed-string one-loop amplitudes results in supersymmetrized matrix elements of R 4 that share the structure of open-string correlators, cf. (4.31). Like this, the duality between kinematics and worldsheet functions connects the representations (4.30) and (4.32) and implies that the matrix elements are both local and BRST invariant.

Seven points
As explained in section 4.3.3, the low-energy limit of the closed-string seven-point amplitude may not be readily available from the expression (4.28) for the loop-integrated correlator. Still, it is tempting to invoke the connection between matrix elements of R 4 and open-string correlators to propose a candidate expression on the basis of the C · E representation (3.76) of K 7 (ℓ):

Higher multiplicity
In order to obtain a more general perspective on its BRST invariance, we note that ( such that all of the accompanying V 1A with A = ∅ are set to zero, we recover (4.36) from Note, however, that the vanishing kinematic factors on the right-hand side of (4.39) are the result of translating Z-functions to kinematic factors via (4.37). At the level of open-string correlators, i.e. before applying the (non-invertible) map (4.37), the Ξ (d) are generically non-zero, cf. (3.84).
Up to these open questions on the pseudo-invariants, it appears likely to arrive at BRST-invariant and local expressions for n-point matrix elements of R 4 by applying the map (4.37) to K Lie n (ℓ). Then, the leftover task to generate BRST-invariant and local correlators in a T · E representation would be to identify a suitable system of GEIs: Such T · E representations of K n (ℓ) would follow from M R 4 n through the duality between pseudoinvariants and GEIs if the latter can be made to • satisfy all the trace relations dual to those of the pseudo-invariants • obey the analogue of the condition Θ (d) = 0 (possibly up to analogues of the BRST non-exact anomaly superfields ∆ 1|... , cf. the objects G 1|... in (II.5.6) and (II.5.7)), In case one succeeds in generating local and BRST invariant T · E representations for ij . As a drawback of the GEIs in T · E representations, their slot structure does not expose the singularities of the g (1) ij . In spite of the large list of open questions, we are optimistic that the above ideas will on the long run guide a path towards an explicit n-point open-string correlator.

Conclusions
It is appropriate to summarize the achievements and future directions arising from this series of three papers [1]. We have presented a method to determine manifestly local one-loop correlators of the pure-spinor superstring. Their dependence on the external polarizations is organized in terms of BRST-covariant building blocks discussed in part I.
A bootstrap procedure is introduced to assemble the accompanying worldsheet functions from loop momenta and coefficients of the Kronecker-Eisenstein series. As a key input of the bootstrap, the monodromies of the worldsheet function around the B-cycle are taken to mirror the BRST variations of the associated kinematic factors. This is a first example for a multifaceted duality between kinematics and worldsheet functions described in part II.
The bootstrap approach results in shuffle-symmetric worldsheet functions that conspire with the Lie symmetries of the kinematic factors: The two kinds of ingredients combine into a Lie-polynomial structure which leads to a natural ansatz for manifestly local n-point correlators. Up to six points, the Lie polynomials are BRST-invariant by themselves and reproduce the non-local correlators known from earlier work [33]. At seven points, the Lie-polynomial ansatz exhibits a simple BRST variation which can be cancelled by adding a local collection of certain anomalous superfields to the full correlator. Starting from eight points, however, an anomalous BRST variation along with the holomorphic Eisenstein series G 4 remains uncancelled. Like this, we can only give an incomplete proposal for the eight-point correlator, leaving a single kinematic factor along with G 4 undetermined. We leave it as an open problem for the future to understand the systematics of Eisenstein series G k in (n≥8)-point correlators.
Further aspects of the duality between kinematics and worldsheet functions concern the BRST-(pseudo-)invariants obtained from certain non-local combinations of kinematic building blocks [22]. By exporting their underlying combinatorial pattern to the shufflesymmetric worldsheet functions, one is led to the notion of generalized elliptic integrands (GEIs) whose B-cycle monodromies cancel upon integration over the loop momentum.
GEIs are observed to share the relations of the dual kinematic factors up to seven points, but the preliminary definition of eight-point GEIs are found to violate certain trace relations by a factor of G 4 . Hence, it remains to incorporate G k into (n≥8)-point GEIs in order to realize the duality between kinematics and worldsheet functions at all multiplicities.
We rewrite the (n≤7)-point correlators in terms of (pseudo-)invariants and/or GEIs such as to manifest the respective kinds of invariances. When both of BRST-invariance and monodromy invariance are manifested, the (pseudo-)invariants and GEIs are found to enter on completely symmetric footing. This kind of exchange symmetry between kinematics and worldsheet functions is reminiscent of the disk amplitudes of [5,41], where gauge-theory trees and Parke-Taylor integrands are freely interchangeable. Hence, the observed duality between kinematics and worldsheet functions up to and including seven points induces a double-copy structure in one-loop open-superstring amplitudes [10]. In the same way as disk amplitudes are dual to supergravity trees when replacing worldsheet integrals by kinematics, the duality maps one-loop open-superstring amplitudes to matrix elements of the supersymmetrized higher-curvature operator R 4 .
The results of this work result suggest a variety of follow-up directions.
Higher genus: Most obviously, the systems of BRST-covariant kinematic building blocks and shuffle-symmetric worldsheet functions call for an extension to higher genus.
First instances of BRST-covariant vectorial superfields have been studied in the low-energy regime of two-loop five-point [42,43] and three-loop four-point amplitudes [44,45]. The principle of BRST-covariance should guide their systematic generalizations to higher tensor ranks as well as analogues of the refined and anomalous building blocks of this work.
As to the worldsheet functions, one would need to identify a higher-genus generalization of the Kronecker-Eisenstein series and its expansion coefficients, where the elliptic functions of [46] may play a role. It would be interesting to extend the duality between worldsheet functions and kinematics -in particular between monodromy and BRST variations -to the multiloop level.
Gravitational operators versus open-string correlators: There is an intuitive reason to find the matrix elements of R 4 and no other gravitational operator as the kinematic duals to one-loop open-string correlators: The supersymmetrized higher-curvature operator R 4 governs the low-energy limit of the corresponding closed-string amplitudes.
Accordingly, the supersymmetrized matrix elements of 12 D 4 R 4 and D 6 R 4 are likely to imprint their double-copy structure on two-loop and three-loop open-string correlators.
In one-loop string amplitudes with reduced supersymmetry in turn, the closed-string low-energy limit results in matrix elements of R 2 [47]. Hence, the open-string one-loop correlators with half-and quarter-maximal supersymmetry of [48] should share the doublecopy structure of R 2 involving GEIs similar to the ones in this work.
Field-theory limits and ambitwistors: The framework of chiral splitting is a natural starting point to determine loop integrands of super-Yang-Mills and supergravity in momentum space from the field-theory limit. We leave it to follow-up work to investigate the τ → i∞ degeneration of GEIs relevant to field-theory amplitudes and the emergence of new color-kinematics dual representations.
Moreover, the superstring correlators of this work can be exported to the one-loop amplitudes of the ambitwistor string [49,50]. It will then be interesting to explore the 12 The shorthands D 4 R 4 and D 6 R 4 are understood to comprise the companion terms D 2 R 5 +R 6 and D 4 R 5 + D 2 R 6 + R 7 of the same mass dimension determined by non-linear supersymmetry.
interplay of GEIs with the color-kinematics dual field-theory amplitudes obtained from the methods of [51,52]. The same questions will arise at higher genus [53,54].
GEIs and scalar amplitudes: The double-copy structure of open-string tree amplitudes [5,41] motivated the interpretation of Parke-Taylor-type disk integrals as scattering amplitudes in effective field theories of scalars. Indeed, the low-energy limit of disk integrals reproduces the tree amplitudes of bi-adjoint scalars with a φ 3 interaction [55] and Goldstone bosons [56,57]. Similarly, higher orders in their α ′ -expansion suggest higher-massdimension deformations of the respective Lagrangians collectively referred to as Z-theory [56,58,57].
In one-loop string amplitudes, GEIs are found to play a role similar to the Parke-Taylor factors at tree level. Hence, it is tempting to compare the moduli-space integrals of GEIs with loop integrands in scalar field theories -for worldsheets of both toroidal and cylinder topology. Also, it will be interesting to compare such integrated GEIs with the forward limits of Z-theory amplitudes.
Connections with combinatorics: After observing that several patterns and identities obeyed by the BRST pseudo-invariants are also satisfied by the GEIs, one is left wondering if these kinematic and worldsheet-function invariants could be a manifestation of a more fundamental mathematical property of objects constructed from building blocks subject to the shuffle symmetries. After all, the combinatorics of these "invariants" can be generated by linear maps acting on words that also feature prominently in the free-Liealgebra literature. We suspect that many combinatorial algorithms on words have direct relevance to the study of scattering amplitudes and in particular string-theory correlators, and that many relations among amplitudes can be understood in terms of free-Lie-algebra structures.

Appendix A. Stirling cycle permutation sums
In order to explain the Stirling cycle permutation sums used throughout this work it is convenient to start by briefly recalling the definition, using the notation and terminology proposed in [17], of the Stirling cycle numbers n p and Stirling set numbers n p . The Stirling set number n p represents the number of ways to partition a set of n elements into p non-empty sets [59]. For example, 4 2 = 7 because there are seven ways to split the set {1, 2, 3, 4} into two non-empty subsets: The Stirling cycle number n p is closely related and represents the number of ways to split n objects into p cycles 13 . It is easy to write down the different arrangements of cycles once the Stirling set partitions have been worked out: simply convert a given k-element subset into its (k−1)! distinct cycles as {1, 2, . . . , k} → (123 . . . k) + perm(2, 3, . . . , k). For example, using the above subset decomposition of 4 2 we obtain 4 2 = 11: Since there is no unique way of representing a product of disjoint cycles we fix this ambiguity by ordering the cycles as follows: i. each cycle is written with its smallest element first, (A. 3) ii. the cycles are written in increasing order of its smallest element. and it represents the sum over all n p ways to partition the set {1, 2, . . . , n} into p cycles, ordered according to (A. 3), and that are distributed to S A 1 ,...,A p as follows, (a 1 . . . a n a )(b 1 . . . b n b ) · · · (p 1 . . . p n p ) → S a 1 ...a n a ,b 1 ...b n b ,...,p 1 ...p n p .  For some typical numbers appearing in this work, we note that the total number of terms in the local representation (2.15) of K (0) n (ℓ) is given by T n ≡ n 4 + n 5 + · · · + n n , while in the corresponding manifestly BRST-invariant representation they become C n ≡ n−1  It is straightforward but tedious to see that there are 932 terms above, reproducing the number T 7 = 932 discussed above.

A.1. Lie polynomials
There are several characterizations of a Lie polynomial in the mathematics literature, see for example [16]. For our purposes, Lie polynomials are composed by linear combinations of nested commutators in a given set of non-commutative indeterminates. For example, if t a 1 , t a 2 , t a 3 are non-commutative, P = [t a 1 , [t a 2 , t a 3 ]] + 3[[t a 1 , t a 2 ], t a 3 ] is a Lie polynomial while N = t a 1 t a 2 t a 3 is not.
The identification of a Lie-polynomial structure within the correlators of this work stems from a theorem proved by Ree [6]. Using the notation of section I.3.1, the theorem states that if M A satisfies shuffle symmetries (i.e., M R¡S = 0, for any R, S = ∅) and t a i are non-commutative indeterminates with t A ≡ t a 1 t a 2 . . . t a p , a sum over all words A of length p of the form gives rise to a Lie polynomial of degree p. For example, at degree two the shuffle symmetry on M a 1 a 2 implies that M a 2 a 1 = −M a 1 a 2 and the sum (A.7) becomes P = M a 1 a 2 t a 1 t a 2 + M a 2 a 1 t a 2 t a 1 = M a 1 a 2 [t a 1 , t a 2 ]. Hence P is a Lie polynomial.
In general, the sum in (A.7) can be rewritten as a sum proportional to where ℓ(A) is the Dynkin map defined in (I.3.8). Thus, the Lie polynomial arising from (A.7) has the form of a sum over products of objects satisfying shuffle symmetries and objects satisfying generalized Jacobi symmetries. Schematically we have P = (shuffle)(Lie). This is precisely the structure within each word (slot) in the local form of the one-loop correlators found in this work, see for example (2.15).

Appendix B. Monodromy invariance of the six-point correlator
In this appendix we demonstrate the monodromy invariance of the six-point correlator in its local representation (3.23) and thereby provide an alternative to the proof in section 3.3.2 with manifest BRST invariance. It will be convenient to define the following shorthands: 6 + · · · + Ω 6 δK   which vanishes in the cohomology for the same reason as above. Therefore, the six-point correlator (3.23) is confirmed to be single valued.
The above proof can be extended to higher-point correlators, but since it is simpler to prove monodromy invariance using a non-local representation with manifest BRST invariance (see section 3.3.2), we will omit further discussions.

Appendix C. Vanishing linear combinations of worldsheet functions
In this appendix we write down a few explicit expansions of the vanishing linear combinations of worldsheet functions given by Θ (d) from (2.28).
At six points, the three topologies of worldsheet functions were expanded in (3.27) and are easily checked to be zero.

C.1. Seven points
At seven points, the inequivalent topologies of Θ (0) are given by Θ  and can also be verified to be zero up to total derivatives.

C.2. Eight points
At eight points, the following topologies can be shown to vanish up to total derivatives:  The purpose of this appendix is to deliver intermediate steps in deriving the manifestly BRST-invariant representation (4.22) of the six-point closed-string correlator that has been proposed in [33].