Towards the n-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics

This is the second installment of a series of three papers in which we describe a method to determine higher-point correlation functions in one-loop open-superstring amplitudes from first principles. In this second part, we study worldsheet functions defined on a genus-one surface built from the coefficient functions of the Kronecker--Einsenstein series. We construct two classes of worldsheet functions whose properties lead to several simplifying features within our description of one-loop correlators with the pure-spinor formalism. The first class is described by functions with prescribed monodromies, whose characteristic shuffle-symmetry property leads to a Lie-polynomial structure when multiplied by the local superfields from part I of this series. The second class is given by so-called generalized elliptic integrands (GEIs) that are constructed using the same combinatorial patterns of the BRST pseudo-invariant superfields from part I. Both of them lead to compact and combinatorially rich expressions for the correlators in part III. The identities obeyed by the two classes of worldsheet functions exhibit striking parallels with those of the superfield kinematics. We will refer to this phenomenon as a duality between worldsheet functions and kinematics.


Introduction
This is the second part of a series of papers [1] (henceforth referred to as part I, II and III) in the quest of deriving the one-loop correlators of massless open-and closed-superstring states using the pure-spinor formalism [2,3]. As detailed in the introduction of part I, the goal of these papers is to determine the correlators from first principles including gauge invariance, supersymmetry, locality and single-valuedness. The present work is dedicated to the implication of single-valuedness on how the correlators may depend meromorphically on the punctures on a genus-one worldsheet. The key results are the following i) We present a bootstrap program to construct worldsheet functions for the correlators that share the differential structure and relations of their superspace kinematics.
These parallels will be referred to as a duality between kinematics and worldsheet functions, and they endow one-loop amplitudes of the open superstring with a double-copy structure [4].
ii) We establish the notion of generalized elliptic integrands (GEIs) which mirror the combinatorics of BRST invariant kinematic factors in the spirit of the duality between kinematics and worldsheet functions.
These results will come to fruition in the assembly of one-loop correlators in part III, also see appendix C for their representation that manifests their double-copy structure.
Since we will often refer to section and equation numbers from the papers I and III, these numbers will be prefixed by the roman numerals I and III accordingly.

Worldsheet functions at one loop
This section introduces the elementary worldsheet functions used in part III as building blocks of multiparticle genus-one amplitudes. These functions are meromorphic and defined as the coefficients of a recent expansion [5] of the classical Kronecker-Eisenstein series [6,7].
They are quasi-periodic under z → z + τ and therefore live on the universal cover of an elliptic curve.
However, our goal is to study string scattering amplitudes that require functions on an elliptic curve. For this purpose, we will later on consider meromorphic functions defined on an enlarged space parameterized by the standard vertex-insertion coordinates z i and the loop momentum ℓ m (with vector indices m, n, p, . . . = 0, 1, . . . , 9 of the ten-dimensional Lorentz group). Following the chiral-splitting formalism [8,9,10], ℓ m represents certain zero modes associated with the worldsheet field x m (z, z), cf. (I. 2.24). The interplay between z j and ℓ m will then lead to the definition of generalized elliptic integrands (GEIs) [4], which become doubly-periodic under z → z + 1 and z → z + τ upon integration of loop momenta.
The properties and explicit construction of GEIs will be the subject of the subsequent discussions.
As reviewed in more detail in section I.2.2, chiral splitting allows to derive open-and closed-string amplitudes from the same function K n (ℓ) of the kinematic data. Open string n-point amplitudes at one loop descend from worldsheets of cylinder-and Moebius-strip topologies with punctures z j on the boundary, see [11] for the integration domains D top and the associated color factors C top . Closedstring one-loop amplitudes in turn are given by where F denotes the fundamental domain for the modular parameters τ of the torus worldsheet. As a universal part of the underlying correlation functions, both (2.1) and (2.2) involve the Koba-Nielsen factor (with s ij ≡ k i · k j and conventions where 2α ′ = 1 for open and α ′ = 2 for closed strings) I n (ℓ) ≡ exp n i<j s ij log θ 1 (z ij , τ ) + n j=1 z j (ℓ · k j ) + τ 4πi ℓ 2 . (2. 3) The leftover factors of K n (ℓ) in the loop integrands carry the dependence on the superspace polarizations and are referred to as correlators, see part III for their construction. The brackets . . . in the above integrands denote the zero-mode integration of the spinor variables λ α and θ α of the pure-spinor formalism [2], and the odd Jacobi theta function in (1 − q n ) 1 − q n e 2πiz 1 − q n e −2πiz . (2.4) Note that the open-string worldsheets relevant to (2.1) can be obtained from a torus via suitable involutions [12,12], that is why the subsequent periodicity requirements will be tailored to the torus topology.

Weight counting
The integrand of n-point one-loop open-string amplitudes (2.1) can be written in terms of loop momenta, holomorphic Eisenstein series (2.7) excluding G 2 and the above g (m) ij (possibly including their z-derivatives) [13,14]. As a necessary condition for modular invariance of the closed-string amplitude (2.2), the overall powers of ℓ, g (m) ij and G k have to obey the following selection rule: Once we assign the following weights to these constituents, each term in the n-point open-string correlator K n (ℓ) must have weight n−4. The notion of weight in the table is conserved in each term of the monodromies (2.12), and the same will hold in the subsequent Fay relations and total derivatives.
Note that the label 2 (corresponding to z 2 ) appears twice in the monomials of the left-hand side in the above identities while appearing at most once in the monomials of the righthand side. This property can be exploited to rewrite arbitrary products of g (n) ij -functions in a canonical way. Since any repeated label can be eliminated this way, for convenience in a product g (n) ij g (m) jk one can use the Fay identities if the repeated label j is the smallest among i, j and k (which can be obtained from a relabeling of (2.18)). In addition, Fay identities involving z-derivatives of g (n) (z, τ ) are easy to obtain from (2.17) and (2.18), and can be similarly written in a canonical way.

Total derivatives
Correlators K n (ℓ) are always accompanied by the Koba-Nielsen factor I n (ℓ) given by (2.3), when they enter open-and closed-string amplitudes, see (2.1) and (2.2). One can show that its derivatives with respect to worldsheet positions z i and modulus τ are given by ∂ ∂z i I n (ℓ) = ℓ · k i + n j =i s ij g (1) ij I n (ℓ) , (2.20) ∂ ∂τ I n (ℓ) = 1 2πi where (2.8) and n i<j s ij = 0 have been used in (2.21). Given the integrations over z j and τ in the amplitudes (2.1) and (2.2), one can therefore set the following total derivatives to zero within one-loop correlators, where f (z, τ, . . .) is an arbitrary function on the worldsheet.
The absence of boundary terms w.r.t. z j follows from the short-distance behavior 3 |I n (ℓ)| → |z ij | s ij of the Koba-Nielsen factor (2.3) as z i → z j . It is well known from discussions of the anomaly cancellation in the open superstring that the boundaries of moduli space can give non-vanishing contributions from individual worldsheet topologies [16,17]. Hence, blindly discarding total derivatives w.r.t. the modulus τ would generically lead to inconsistencies. However, when summing over the different worldsheet topologies these inconsistencies are canceled for the gauge group SO (32); since this will always be the case for the open superstring we may freely discard total derivatives in τ .

Generalized elliptic integrands
When using the chiral-splitting method [8,9,10] to handle the joint zero mode ℓ m of 4 ∂x m (z) and ∂x m (z), superstring scattering integrands of (2.1) and (2.2) involve a loopmomentum dependent Koba-Nielsen factor (2.3). As explained in [10], the integrands of superstring amplitudes containing the loop momentum ℓ m do not need to be single-valued as functions of z i . Instead, it is sufficient to attain single-valuedness after the loop momentum is integrated out. Here "single-valued" is used in its conventional sense; it refers to functions f (z i ) left invariant as the coordinates z i are transported around the A and B homology cycles of fig. 1. In this work, the chiral-splitting method will be used but the concept of single-valuedness will be extended to invariant functions of (z i , ℓ m ) under a simultaneous variation of both z i and ℓ m along the cycles. Let us now present the reasoning that motivated this idea.

Motivating and defining generalized elliptic integrands
As we will see in section III.3.2, the evaluation of the five-point one-loop amplitude of the open superstring using the standard rules of the pure-spinor formalism (and some mild assumptions) gives rise to the following integrand: 12 +(2 ↔ 3, 4, 5) + V 1 T 23,4,5 g 23 +(2, 3|2, 3, 4, 5) . (3.1) 3 The cancellation of |z ij | s ij as z i → z j is obvious in the kinematic region where Re(s ij ) > 0 and otherwise follows from analytic continuation. 4 In the pure-spinor formalism, the worldsheet fields ∂x m (z) and ∂x m (z) enter the vertex operators in their spacetime-supersymmetric combinations Π m (z) and Π m (z) [2].
The kinematic factors V 1 , V 12 , T m 2,3,4,5 , T 3,4,5 in pure-spinor superspace [18] are reviewed in section I.4. Throughout this work, the notation +(a 1 , . . . , a p |a 1 , . . . , a p+q ) instructs to sum over all ordered combinations of p the labels a i taken from the set {a 1 , a 2 , . . . , a p+q }, leading for instance to a total of six permutations of V 1 T 23,4,5 g (1) 23 in (3.1). Having obtained (3.1), it was natural to ask about its B-cycle monodromies using the relations (2.13). Ignoring the term with the loop momentum for a moment, it is easy to see that the correlator (3.1) changes by −2πi V 12 T 3,4,5 + (2 ↔ 3, 4, 5)] as z 1 goes around the B-cycle. Recalling the vanishing of k m 1 V 1 T m 2,3,4,5 + V 12 T 3,4,5 + (2 ↔ 3, 4, 5) in the BRST cohomology, see (I.4.23), suggests the following speculation: if the loop momentum changed as ℓ m → ℓ m − 2πik m 1 at the same time as z 1 goes around the B-cycle, then the integrand (3.1) would be single valued as a function of both z 1 and ℓ m .
As it stands the above speculation is not compelling enough as we did not consider how the Koba-Nielsen factor (2.3) behaves under these changes. Luckily, the quasi-periodicity θ 1 (z+τ, τ ) = −e −iπτ −2πiz θ 1 (z, τ ) of the odd Jacobi theta function (2.4) implies that the absolute value of the Koba-Nielsen factor is invariant under the simultaneous transformation of z 1 → z 1 +τ and ℓ m → ℓ m − 2πik m 1 , Hence, the loop-integrated open-and closed-string expressions d D ℓ |I n (ℓ)| K 5 (ℓ) and d D ℓ |I n (ℓ)| 2 K 5 (ℓ) K 5 (−ℓ) will still lead to single-valued functions of the punctures in the conventional sense of [10]. But the above reasoning suggests that one can even talk about single-valued chirally-split superstring integrands by also letting the loop momentum change along the B-cycle. Furthermore, the same analysis can be performed for shifts along the A-cycle (without any modification of the loop momentum as z 1 → z 1 + 1), motivating the following definition:

Definition 1 (GEI). A generalized elliptic integrand (GEI) is a single-valued function
f (z i , ℓ, τ, k j ) of the lattice coordinates z j , j = 1, . . . , n, the loop momentum ℓ m , the modular parameter τ and the external momenta k m j such that as z j and ℓ m go around the A and B cycles By their dependence on ℓ m and k m j , GEIs may have free vector indices f m 1 m 2 ... (z j , ℓ, τ, k j ).
As the absolute value of the Koba-Nielsen factor is by itself a GEI, the five-point example (3.1) suggests that superstring correlators are given by GEIs in the above sense, We will see that this observation harbors valuable constructive input to the derivation of correlators from first principles. Furthermore, the argument above suggests a deeper connection between BRST invariance of pure-spinor superspace expressions and GEIs. As we will see in the following sections, this synergy is quite powerful and leads to many interesting results.
Integrands depending on ℓ, k, z and τ satisfying the key property (3.3) were used for the first time in [4], where the acronym GEI was coined. As detailed in section 7, integrating the GEIs in n-point closed-string integrands over d D ℓ |I n (ℓ)| 2 yields modular forms of weight (n−4, n−4) and leads to modular invariant closed-string amplitudes (2.2).

The linearized-monodromy operator
Given a monomial in g (n) ij , the monodromies as z j → z j + τ are polynomials in 2πi by (2.12). We will be interested in combinations of g (n) ij and the loop momentum such that the monodromies are compensated by shifts ℓ → ℓ − 2πik j and the defining property (3.5) of GEIs is attained. In order to efficiently identify GEIs, we formally truncate the combined transformations of g (n) ij and ℓ to the linear order in 2πi and study the operator where δ j g (0) jm = 0 and δ j g (n) im = 0 for all i, m = j. This operator probes the linearized monodromy w.r.t. a given puncture δ j : z j → z j + τ with the accompanying shift ℓ → ℓ − 2πik j . Accordingly, it is understood to obey a Leibniz property for arbitrary functions f i of the loop momentum and the punctures. It is convenient to assemble the linearized monodromies w.r.t. all of z 1 , z 2 , . . . , z n into a single operator as where we have introduced formal variables Ω j to track the contribution of the j th puncture.
Then, (3.6) and the shorthand notation Ω ij ≡ Ω i − Ω j give rise to where momentum conservation k m 1 = −k m 2 − · · · − k m n has been used in the last relation. For example, 12 , D g 12 ℓ m = Ω 12 ℓ m + g Note that D will be later on argued to play a role similar to the BRST operator Q of the pure-spinor formalism. One can enforce that D shares the nilpotency Q 2 = 0 by defining the formal variables Ω j to be fermionic 5 . However, the choice of statistics for the Ω j won't affect any calculation done in this work, so we defer this decision to follow-up research.
Since the linearized monodromy operator D only picks the terms linear in 2πi that arise from the transformation z j → z j + τ and ℓ → ℓ − 2πik j , invariance DE = 0 is only a necessary condition for E to be a GEI. It remains to check if the higher orders in 2πi also drop out from the image of E under the above shift of z j and ℓ. For all solutions to DE = 0 studied in this work, we have checked that they constitute a GEI on a case-by-case basis, and it would be interesting to find a general argument. In many cases, single-valuedness can be seen from the generating-function techniques in later sections.

Bootstrapping shuffle-symmetric worldsheet functions
In this section we will construct a system of worldsheet functions Z for superstring correlators on a genus-one Riemann surface by analogies with kinematic factors. When the latter are organized in terms of Berends-Giele superfields as detailed in part I, their variation under the pure-spinor BRST operator will be used as a prototype to prescribe monodromy variations for the Z-functions. As a consequence, the combinatorics of BRST-invariant 5 The conditional nilpotency of D for fermionic formal variables Ω j follows from the fact that linearized monodromies (3.6) w.r.t. different punctures commute, δ i δ j = δ j δ i . This commutativity property follows from (3.6) and (3.7).
kinematic factors can be borrowed to anticipate D-invariant combinations of Z-functions, i.e. GEIs.
The correspondence between the pure-spinor BRST charge Q acting on superfields and the monodromy operator D acting on functions is the first facet of a duality between kinematics and worldsheet functions. Further aspects of the duality will be presented in section 5 that lead to a variety of applications. In particular, the duality between kinematics and worldsheet functions implies a double-copy structure of open-superstring one-loop amplitudes discussed in [4] and expanded in part III.

Shuffle-symmetric worldsheet functions
In the computation of tree-level correlators for n-point open-string amplitudes [19], the nested OPE singularities were captured by worldsheet functions of the following form 6 It follows from partial-fraction relations such as (z 12 z 23 ) −1 + cyc(1, 2, 3) = 0 that the tree-level functions satisfy shuffle symmetries 7 (e.g. Z tree 1¡23 = Z tree 123 +Z tree 213 +Z tree 231 = 0) [22] Z tree Since the appearance of shuffle-symmetric worldsheet functions (4.1) at tree level can be traced back to the short-distance behavior of vertex operators, the same structure must persist at higher genus. Therefore we assume that the short-distance singularities at one loop arise from analogous chains built from functions g (1) 12 g 23 . . . g (1) p−1,p .  At multiplicity p = 2, antisymmetry of g 21 suffices to make it shuffle-symmetric. However, for the tentative one-loop counterpart g (1) 12 g (1) 23 of Z tree 123 it is easy to see that the Fay identity (2.17) prevents the shuffle relation Z tree 1¡23 = 0 from generalizing. Luckily, the same Fay identity also suggests how to restore the shuffle symmetry without altering the pole at z i → z j by adding non-singular g (2) ij -functions. One can check via (2.17) that both of 12 g 23 + g 12 + g 23 − g share the desired shuffle symmetry of Z tree 123 . Also at higher multiplicity, the non-singular functions g (n) ij with n ≥ 2 admit various shuffle symmetric completions of g 23 . . . g (1) p−1,p which reproduce the singularity structure of (4.1) and qualify as one-loop counterparts of Z tree 123...p . From the availability of two shuffle-symmetric multiplicity-three candidates in (4.5), one can anticipate that many more options arise at higher multiplicities. In the next subsection we will identify a guiding principle to prefer Z

Duality between monodromy and BRST variations
We will now prescribe the monodromy variation DZ 12...p of shuffle-symmetric worldsheet functions by analogies with Berends-Giele superfields M 12...p that share the shuffle symmetry and are reviewed in section I.5. The idea is to impose the combinatorics of the BRST variation QM 12...p to carry over to the worldsheet functions, Z 12...p ↔ M 12...p . This relationship is at the heart of an emerging proposal for a duality between worldsheet functions kinematics -monodromy variations are taken to be dual to BRST variations. where the length of the word A = a 1 a 2 . . . a |A| is denoted by |A|, and the bookkeeping variables Ω j of (3.8) always follow the special label of E j|... , e.g.  The E i|... on the right-hand sides will be defined in analogy 8 with C i|... , and this analogy will be reflected by the notation: The duality between superfields and Z A,B,C,D as well as the resulting correspondence between Q and D imply that BRST invariants C i|A,B,C should be dualized to GEIs. By the vertical-bar notation, the symmetries C i|A,B,... = C i|B,A,... where the second variation will later be shown to be equal to Ω 13 E 1|23,4,5 . Since it is easy to see that DZ

Scalar monodromy variations
123 is not single-valued, only the second option has the required structure (4.7) on the right-hand side. In order to reconcile the expression for Z , from now on we use the notation Z 123,4,5,6 = Z (ii) 123 , also see section 4.4.3. 8 We note the mismatch between the slots of E i|A,B,C defined in analogy with the BRST invariants and the slots of E i|A,B,C,D appearing in the right-hand side of (4.7). This difference is inconsequential for functions up to multiplicity nine and can be bypassed by defining the extension of scalar GEIs by E i|A,B,C,D ≡ E i|A,B,C and by adding extra permutations to the tensorial GEIs.

Tensorial monodromy variations
The same ideas can be reused at higher tensor ranks r to infer tensorial worldsheet functions and we will use the following results for the left-hand side [23],  Here and in the following, Lorentz indices are (anti)symmetrized such that each inequivalent term has unit coefficient, e.g. k . . k m r r + perm(m 1 , . . . , m r ), for a total of r! terms. In case of symmetric tensors, imposing unit coefficients leads to fewer terms such as δ (mn k p) ≡ δ mn k p +δ mp k n +δ np k m , and expanding the symmetrization   The defining property of pseudo-invariants is that their BRST variation is entirely expressible in terms of anomaly superfields [23].
where the anomalous superfield in the first line of (4.11) does not have any worldsheet

Refined bootstrap equations
The duality between superspace kinematics and worldsheet functions suggests to introduce a notion of refined Z-functions defined via monodromies where the anomalous superfields in the first line of (4. 16 (4.1). This requirement fixes the most singular term to be (g (1) a 2 a 3 . . . g (1) a |A|−1 a |A| ) and should prevent the addition of non-constant functions with vanishing monodromies, as they would necessarily modify this singularity structure. At the level of unrefined scalar GEIs, this follows from the fact that non-constant elliptic functions always involve singularities as z i → z j , and we expect this property to carry over to tensorial and refined GEIs.
However, this requirement cannot determine the presence (or absence) of terms proportional to a holomorphic Eisenstein series G n , for they are monodromy invariant (DG n = 0) as well as constant functions on the worldsheet ( ∂G n ∂z j = 0). The construction of Z A,B,... and GEIs from g ij automatically qualifies holomorphic Eisenstein series G n = −g (n) ii as possible constituents. Moreover, G n are known to arise in (n ≥ 8)-point one-loop correlators from the spin sums in the RNS formalism [13,14].
By the weight counting of section 2.1.2, the first instance where the above ambiguity may affect the expressions for shuffle-symmetric functions happens at eight points. And indeed, we will see in section III.3.5 that the eight-point correlator is plagued by unwanted appearances of G 4 whose kinematic coefficient remains undetermined in this work.

Lie-symmetric worldsheet functions
From the discussion in section I.5.1, Berends-Giele superfields M A,B,C subject to shuffle symmetries can be translated to local building blocks T A,B,C that satisfy Lie symmetries (cf. section I.3.4). The dictionary in (I.5.8) boils down to the KLT-matrix S(·|·) i [24] (also known as the momentum kernel [25]) that cancels the kinematic poles of the Berends-Giele currents and is recursively defined by for instance In analogous fashion, one can also define worldsheet functions that satisfy Lie symmetries.
To this effect we define, in analogy with (I.5.8),

Worldsheet dual expansions of BRST pseudo-invariants
In this section we will see the first non-trivial consequence of the conjectural duality between worldsheet functions and kinematics: the systematic construction of GEIs. This is done by exploiting the analogy between monodromy variations of Z-functions and the BRST variations of Berends-Giele currents put forward in section 4.2. The tentative idea is to assemble GEIs or "worldsheet invariants" following the same combinatorics used in building kinematic BRST invariants C 1|A,B,C and C m 1|A,B,C,D in (I.5.20) and (I.5.21) from Berends-Giele currents. It turns out that the worldsheet invariants constructed in this way give rise to GEIs as defined in section 3, i.e., their monodromy variations vanish.
10 In order to see that (4.27) defines a GEI as well, one can either employ the explicit representation assembled in (4.37) or insert the integration-by-parts identity (5.1) among GEIs into the D variation obtained from (4.18).

The bootstrap
At first glance, the discussion in sections 4.2 and 4.3 seems to suffer from a chicken-andegg dilemma; in section 4.2, to obtain the monodromy variations of the shuffle symmetric functions one needs the associated GEIs from section 4.3, while the expressions of the GEIs require the shuffle-symmetric functions from section 4.2.
The way out of this conundrum is to note that this self-recursive structure can be exploited to bootstrap the shuffle-symmetric Z-functions order by order in multiplicity, starting with the four-point solution which is taken to be a constant. We will see how this works in practice in the following subsections.
Note that the functions obtained below will be used inside one-loop correlators of the open superstring, and as such, are considered to be multiplied by the overall Koba-Nielsen factor (2.3). Therefore functions that differ by derivatives of the Koba-Nielsen factor given in (2.22) and (2.23) are considered equivalent as will be indicated by the symbol ∼ =.

Four-point worldsheet functions
From the computation of the four-point correlator in [26,3], it follows that the four-point shuffle-symmetric worldsheet function is a constant. Similarly, the expansion (4.25) implies that also its corresponding GEI is a constant. Both are normalized to one, To proceed to the next level we define the slot extension of (4.28) as E 1|2,3,4,5 ≡ 1.

Assembling five-point GEIs
We can now assemble associated GEIs from the expansions in (4.25), 12 + (2 ↔ 3, 4, 5) . It is easy to check that (4.31) and (4.32) are indeed invariant under monodromy variations (using momentum conservation in the latter case). Before proceeding to the next multiplicity, we define the slot extension of (4.31) and (4.32), including an extra permutation 2 ↔ 6 in the vector GEI. These extensions are natural from the generating functions for GEIs to be given in a later work and they will be used on the right-hand sides of the monodromy variations of six-point Z-functions below.
In accordance with the discussion in section 4.1, their behavior as the vertex insertions collide corresponds to their tree-level counterparts. For instance, the short-distance behavior Z 123,4,5,6 → (z 12 z 23 ) −1 is the same as that of Z tree 123 .

Seven-point worldsheet functions
At seven points, the monodromy variations for the scalar shuffle-symmetric functions following from (4.13) and (4.38 The solutions for the tensorial functions will be presented in Appendix A.2, see in particular In addition to the above unrefined solutions, the monodromy variations of the three seven-point topologies of refined worldsheet functions following from (4.18) read 13 − g 14 − g 35 + g 12 are considered non-singular as they don't generate kinematic poles when integrated along with the Koba-Nielsen factor.
Therefore all functions in (4.42) are in fact non-singular upon integration over z j .
Having the shuffle-symmetric worldsheet functions we can now assemble seven-point GEIs as discussed in the previous section. The results are displayed in Appendix A, see in particular (A.31) to (A.34). Also, the building blocks of section 6 turn out to admit the compact representations (6.22) or (6.23).

Eight-point shuffle-symmetric worldsheet functions
The system of monodromy variations can be solved explicitly at eight points following the bootstrap approach. This will be done in the appendix A.3.

Duality between worldsheet functions and kinematics
In this section, we will illustrate various further facets of the duality between worldsheet functions and kinematics. It will be exemplified that GEIs E ... Like this, we support the double-copy structure of one-loop open-string amplitudes [4] up to and including seven points. At eight points we will sometimes encounter terms proportional to the holomorphic Eisenstein series G 4 that do not have a corresponding kinematic companion. Accommodating these terms with the duality between worldsheet functions and kinematics is left for a future work.

The GEI dual to BRST-cohomology identities
The appearance of the correlators K n (ℓ) in open-and closed-string amplitudes is insensitive to BRST-exact terms. This has been exploited in [23] to derive so-called Jacobi identities in the BRST cohomology that relate momentum contractions k m 1 C m... to (pseudo-)invariants of lower tensor rank, see section I.5.4. We will now exemplify that GEIs E m... 1|A,B,... obey the same Jacobi identities between different tensor rank and degree of refinement, where BRST-exact terms translate into total derivatives.
As in (2.22), the ∼ = notation is a reminder that z j -derivatives have been discarded in passing to the right-hand side. Note that momentum conservation reduces the identities for contraction with k 1 to combinations of the remaining ones involving k m A E m... 1|A,... .

Higher multiplicity
More generally, the elliptic identities that are dual to the BRST-cohomology identities in section 9 of [23] can be written as all non-empty X = a 1 a 2 . . . a j and Y = a j+1 . . . a |A| with j = 1, 2, . . . , |A|−1. We have verified all of (5.3) up to and including eight points, and their higher-point generalizations are plausible by the dual kinematic identities given in (I. 5.43) and [23]. Note the absence of elliptic-function duals to the BRST-exact anomaly terms ∆ m 1 ... 1|A 1 ,... without refined slots. Following the worldsheet duals of the higher-rank identities in section 9 of [23], one arrives at for some a priori undetermined GEIs G 1|... in the trace component. The latter can be thought of as a tentative GEI dual of the refined anomaly superfields ∆ m 1 ...
At higher degree of refinement, appropriate choices of GEIs should obey the dual of the most general Jacobi identity (I. 5.45) on the kinematic side that are checked up to and including eight points. These proposals will serve as a key input for the all-multiplicity construction of GEIs from generating series. Note that the first term does not have any refined slots at d = 1 and should vanish by the duality with the BRST-exact unrefined anomaly superfields. In fact, we even observe stronger identities among seven-and eight-point GEIs such as with a single momentum contraction, which implies (5.7) upon symmetrization in 2 ↔ 3.

The GEI dual to BRST change-of-basis identities
In section 11 of [23] several identities among (pseudo-)invariants were derived using BRSTcohomology manipulations that implement a change of basis 11  where the right-hand side is written in terms of the canonical basis of C 1|A,B,C and P 1|A|B,C,D,E with leg 1 in the first position of the subscript. The BRST-exact terms in the ellipses are spelled out in [23]. Naturally, these identities have an elliptic dual under C → E as well as its "refined" version 12 P → E. 11 They were referred to as "BRST-canonicalization" identities in [23]. 12 The pseudo-invariant P i|A|B,... should really be denoted C i|A|B,... , as it would unify this and countless other formulas.

Five points
It is straightforward to show that the five-point GEIs in (6.16) obey change-of-basis identities dual to (5.9),  More general cases such as expanding E 2|13,45,67,89 in terms of E 1|... have no matching analogous identities in terms of C 1|A,B,C , so the required change-of-basis identities are not readily available from [23]. These identities can, however, be generated using the general algorithm described in the appendix I.A.3. Note that τ -derivatives acting on both the Koba-Nielsen factor and ℓ m or g (1) 12 have been discarded in (5.16), using the mixed heat equation (2.9) for the latter. 13 Note that the building blocks with d = 0 are denoted by M rather than J .

The worldsheet analogue of kinematic anomaly invariants
The vanishing of the six-point function Z 1|2,3,4,5,6 can be understood as a correspondence between refined worldsheet functions at multiplicity n and unrefined Y superfields at multi- In the following we will use the notation Z ∆ 1|A,B,C,D,E to denote the worldsheet counterpart of ∆ 1|A,B,C,D,E that follows the same combinatorics (with obvious generalizations to tensors and refined cases). We will see that the six-and seven-point Z ∆ vanish up to total derivatives (confirming the suggested duality) whereas subtle contributions ∼ G 4 may arise at eight points.

Seven points
The natural next step is to check whether the seven-point refined functions following from the combinatorics of the six-point BRST-exact superfields [23],  As will become clear in the discussion of the eight-point correlator in section III.3.5, the subtleties associated to the presence or absence of G 4 terms are responsible for the difficulties in obtaining a BRST-closed eight-point correlator.

Seven points
Similarly, we have checked that the seven-point tensor traces of GEIs obey relations analogous to the dual (pseudo-)invariants, In fact, the first three relations of (5.33) are independent on the choice of Z orẐ, while the other two change (but do not vanish in either case).

Higher multiplicities
At higher multiplicity, suitable choices of the GEIs are expected to admit the dual of the kinematic relation (I.5.29),

Simplified representations of GEIs
In this section, we review and extend the construction of elliptic functions from the Kronecker-Eisenstein series [27,28] and identify ubiquitous building blocks for GEIs. These building blocks turn out to yield compact expressions for the GEIs in section 4.4 and will be used to present explicit all-multiplicity formulae for unrefined GEI of tensor rank r ≤ 2.

Elliptic functions and their extensions
One can show via (2.11) that the cyclic product F (z 12 , α)F (z 23 , α) . . . F (z n−1,n , α)F (z n,1 , α) of Kronecker-Eisenstein series (2.5) is an elliptic function of the punctures z 1 , z 2 , . . . , z n [27], where the dependence on τ is kept implicit for ease of notation. Since this property is independent on α, each term on the right-hand side of (6.1) is an elliptic function V w in n punctures z 1 , z 2 , . . . , z n by itself. At the level of linearized monodromies (3.8), we have DF (z ij , α) = αΩ ij F (z ij , α) and therefore Given that shuffle symmetry is shared by Berends-Giele currents and (pseudo-)invariant kinematic factors, the V w (1, 2, . . . , n) with w = n−2 will play a key role for the duality between worldsheet functions and kinematics.

Derivative extension of elliptic functions
Compact representations of vectorial and tensorial GEIs will require extensions of the set of V w -functions (6.1) that are covariant rather than invariant under linearized monodromies.
Functions with these properties can be constructed by inserting a derivative with respect to the bookkeeping variable α into their generating series: (1, 2, . . . , n) . (6.7) The notation ∂V w for the functions on the right-hand side reminds of the α-derivative on the left-hand side and should not be confused with ∂ ∂z j . Based on D∂ α F (z ij , α) = Ω ij α∂ α F (z ij , α) + F (z ij , α) , the monodromy variations of the ∂V w -functions in (6.7) can be written as D∂V w (1, 2, . . . , n) = Ω n1 V w (1, 2, . . . , n) . (6.8) Given that their D-variation is expressible in terms of the elliptic V w -functions of (6.1), the ∂V w -functions are said to be monodromy-covariant.
The desired expressions for the Z-functions and GEIs turn out to only involve ∂V w (1, 2, . . . , n) with w = n−2.
With the covariant monodromy variation (6.8) of ∂V 1 at hand, it is easy to verify that DE m 1|23,4,5,... = 0. One may identify the above g (1) 1j and g (2) 1j as −∂V 0 (1, j) and 1 2 ∂ 2 V 0 (1, j), respectively, to arrive at a uniform presentation for the coefficients of k m j . In view of ∂ 1 g (1) 12 = V 2 (1, 2) − G 2 , the refined GEI (4.37) is also expressible in terms of elliptic functions where the last line again follows from integration by parts.
Similarly, the ∂ 2 V w -functions in (6.12) allow for compact representations of the two-and three-tensors in (A.33) 15 , 1j ℓ (m k n) j V 1 (1, 2, 3) (6.21) 15 Note that our conventions lead to ℓ (m ℓ n k p) j = ℓ m ℓ n k p j + ℓ m ℓ p k n j + ℓ n ℓ p k m j .
Using the monodromy variation (6.13) of ∂ M V w and the shuffle symmetries (6.6) and (6.10) of V n−2 and ∂V n−2 , all the above E ... 1|... can be verified to be GEIs with pen-andpaper effort. Similarly, one can show that the refined GEIs (A.34) whose combinatorics mimic the Berends-Giele expansion of the refined superfields P from [23] can be rewritten more compactly as  Alternatively, using integration-by-parts identities leads to 12 g (1) 23 (g (2) 12 + g 12 .

Closed formulae for vectors and two-tensors
The above examples of vector and two-tensor GEIs can be lined up with the closed formulae  (a j−1 . . . a 2 a 1 ¡a j+1 . . . a |A| ), 1) 1 (1, D) . . . + (A, B|A, B, C, D . . .) (6.29) l and higher-rank terms. This shortcoming motivates the development of more powerful tools for all-multiplicity and all-rank constructions of GEIs, which we leave for a future work.

Integrating the loop momentum and modular invariance
The purpose of this section is to set the stage for integrating the one-loop correlators of part III over the loop momentum. We will see below that loop-integrated GEIs yield manifestly single-valued worldsheet functions that largely conspire to modular weight (n−4, 0). The loop integrals of individual GEIs at (n≥6) points also feature terms of different modular weights that (as will be shown in part III) cancel from the amplitude by kinematic identities among their coefficients. Such modular anomalies will be illustrated to follow the patterns of BRST anomalies of pseudo-invariants. Like this, we extend the duality between worldsheet functions and kinematics to anomalies.
The doubly-periodic but non-holomorphic functions f (n) in (7.1) are related to the holomorphic g (n) with B-cycle monodromies (2.12) via [14] where the simplest examples are f (0) = 1 and where a, b, c, d form an SL 2 (Z) matrix. Similarly, each holomorphic derivative in z adds holomorphic weight (1, 0) to the f (n) . However, meromorphicity of the g (n) is replaced by the condition ∂ ∂τ + Im z Im τ ∂ ∂z f (n) (z, τ ) = 0 (7.6) following from It will be convenient to extend the shorthand notation (2.14) for g (n) ij to their doublyperiodic counterparts,

Integrating out the loop momentum
In this section, we set the stage for loop integrals over both the Koba-Nielsen factor and ℓ-dependent open-and closed-string correlators in the amplitudes (2.1) and (2.2). For closed-string correlators independent on ℓ, the result of the Gaussian loop integral has already been spelled out in (I.2.26). Zero-mode integration at n ≥ 5 points, however, requires generalizations of (7.10) to additional polynomials p(ℓ) in the loop momentum besides |I n (ℓ)| 2 . We will use the square-bracket notation where momentum conservation has been used to eliminate k m 1 = −k m 2 − · · · − k m n from the definition of L m 0 . As a result of straightforward Gaussian integration, we have (recall the convention (I.2.3) where all terms generated by (anti)symmetrization of indices have unit coefficient, e.g., δ (mn k p) ≡ δ mn k p + δ mp k n + δ np k m ) which are sufficient to integrate open-string correlators at n ≤ 8 points and closed-string correlators at n ≤ 6 points. In general, following standard Gaussian integration rules, one has to sum over all possibilities to perform pairwise contractions ℓ m ℓ n → − π Im τ δ mn on a subset of the loop momenta in the integrand while setting the others to ℓ m → L m 0 . The open-string analogue of (7.11) reads In summary, (7.11) and (7.14) are tailored to express the open-and closed-string amplitudes (2.1) and (2.2) in the following form

Integrating unrefined GEIs
In section 3, GEIs E ... 1|... have been introduced as meromorphic functions that are doublyperiodic up to shifts of the loop momentum. Hence, upon integration over ℓ, GEIs are guaranteed to become doubly-periodic, and the functions f (n) in (7.1) turn out to be the natural framework to represent the dependence of [[E ... 1|... ]] on z j . Unrefined scalar GEIs E 1|A,B,C were found to be elliptic functions in the conventional sense and expressible in terms of the V w -functions of (6.1), see e.g. (6.16) and (6.17).

Integration by parts
The integration-by-parts relations of meromorphic correlators K n (ℓ) were governed by the derivatives of the ℓ-dependent Koba-Nielsen factor I n (ℓ), see section 2.3. Accordingly, the loop-integrated Koba-Nielsen factorÎ n in (7.10) gives rise to a modified set of integrationby-parts relations. The z j -derivatives (2.22) straightforwardly generalize to while the τ -derivative (2.23) requires more adjustments after integration over ℓ. After momentum conservation, the Koba-Nielsen exponent in (7.10) has the following τ -derivative where the admixtures of f (1) ij cancel from the action of the differential operator Im z j1 Im τ ∂ ∂z j (7.23) depending on n punctures z j . The operator ∇ τ obeys the usual Leibniz property and appears naturally in the following generalization of the mixed heat equation (2.9), Then, after taking the prefactor ofÎ n ∼ (Im τ ) −D/2 in (7.10) into account, a convenient analogue of the τ -derivative (2.23) after loop integration reads where we will set the number of spacetime dimensions to D = 10 henceforth. The operator (7.23) can be aligned into the following boundary term with h(z, τ ) denoting an arbitrary function on the worldsheet. Since both of (7.21) and see (2.22) and (2.23) for their chirally-split analogues. The simplest example of (7.28) with h(z, τ ) = 1 has been used in [29] to identify the BRST variation of the (n = 6)-point closed-string amplitude as a boundary term.
Note that the holomorphic derivative ∂ ∂z i in (7.27) acts non-trivially on the contributions f (w) ij from the opposite chiral half in closed-string amplitudes. This follows from the complex conjugate of (7.7) and gives rise to examples such as [30,31] The differential operator (7.23) in turn annihilates undifferentiated f (w) ij and only acts on z-derivatives of the f (w) ij from the opposite chiral half in closed-string amplitudes

Integrating refined GEIs
After loop integration, the integration-by-parts equivalent representations of the simplest refined GEI E 1|2|3,4,5,6 in (4.37) translate into 12 (s 23 f 23 + s 24 f 24 + s 25 f 12 − 3f where we reiterate that the elliptic V w -functions are unchanged under the global replace- ij . One can perform integrations by parts (7.27) similar to (7.32) to avoid the appearance of ∂f (n) ij on the right-hand side. Similar to (7.19) and (7.20), the factors of π Im τ on the right-hand sides of (7.32) and (7.33) signal a modular anomaly: They depart from the purely holomorphic modular weights (n, 0) and (n+1, 0) of the f  is a consequence of (7.26) at n = 6 and h(z, τ ) = 1, the seven-point analogues require a specialization of (7.28) to 17 c k d k with m+ k w k = n−6. In these cases, the second line of the following equivalence relation (7.28) vanishes (we are suppressing the

Modular anomalies versus BRST anomalies
The above instances of modular anomalies furnish another incarnation of the duality be-

Conclusions
In this paper we continued setting up the ingredients that will be needed to build up one-loop correlators for massless open-and closed-string amplitudes in the pure-spinor formalism. We have introduced two classes of worldsheet functions that will manifest different aspects of the correlators to be assembled in part III. Both of them are constructed from loop momenta and combinations of Jacobi theta functions g (n) ij = g (n) (z i −z j , τ ) that are the coefficients in the Laurent expansion of the Kronecker-Eisenstein series [5].
The first class of worldsheet functions, denoted by Z, is designed to capture the worldsheet singularities arising when the vertex operators approach each other on a genusone surface. These singularities are straightforward to handle via an OPE analysis, and their behavior when the vertices are close together is the same as products of 1/z ij = 1/(z i −z j ) functions well-known from the tree-level correlators.
However, the OPE analysis is not enough to completely determine the one-loop Zfunctions as there can be non-singular pieces that do not vanish on a genus-one surface 18 .
Instead, our starting point to constrain the non-singular pieces is the following observation on tree-level correlators: The products of singular functions 1/z ij at genus zero end up assembling chains 1/(z 12 z 23 . . . z p−1,p ) [19] that obey shuffle symmetries among their labels 1, 2, . . . , p. By imposing the same shuffle symmetries among the labels of their one-loop counterparts Z and using Fay identities one proves the existence of non-singular pieces in the one-loop worldsheet functions.
The algorithmic determination of these non-singular pieces follows from another surprising feature of these functions; their properties mimic those of superfield building blocks 18 These non-singular parts are absent at tree level where the knowledge of the singular behavior is enough to fix the whole function. A multitude of identities among Z-functions and GEIs has been discussed in this paper that support their duality connection with superfield building blocks. However, we observed that holomorphic Eisenstein series lead to departures from a strict duality between functions and kinematics starting at eight points. The solution to this puzzling behavior, for instance through systematic redefinitions of Z-functions and GEIs via Eisenstein series, will be left for the future. Furthermore, a preliminary analysis indicates that the functions considered in this paper admit compact generating-series representations whose detailed presentation we also leave for future work.
The relevance of both the Z-functions as well as GEIs for the assembly of one-loop correlators will become apparent in the sequel part III of this series of papers.

Appendix A. Bootstrapping the shuffle-symmetric worldsheet functions
This appendix complements the results of the bootstrap techniques for Z-functions outlined in section 4.4 with derivations based on the system of monodromy variations. The key steps will be presented in detail for six points and for some selected seven-and eight-point functions; the results from the omitted derivations can be obtained with reasonable effort [32] and do not require any new methods.
In the derivations below we will use the representations of GEIs obtained in section 6 as they lead to considerably shorter results; in some cases, they even suggest pattern-driven general closed formulae.

A.1. Six points
The starting point at six points is given by the extended GEIs To solve these equations using the generating-series techniques of section 6 it will be convenient to rewrite the above GEIs in a basis where leg 1 is in the special slot. This can be done by exploiting the duality with the BRST invariants and using the identities of The scalar equations are easily solved using cyclic symmetry of V w (1, 2, . . . , n) and the monodromy variations D∂V w (1, 2, . . . , n) = −Ω 1n V w (1, 2, . . . , n). We get, 13 + Ω 13 g 12 − Ω 23 V 1 (1, 2, 3)) + (3 ↔ 4, 5, 6) .
This completes the bootstrapping of the vectorial shuffle-symmetric functions for seven points.
Note that the coefficient of k m 1 k n 2 k p 3 in the last line is totally symmetric in 1, 2, 3. Again, their singularity structure within a given word is the same as in their tree-level counterparts, see section 4.1.
The above GEIs, in turn, will be used as input in the monodromy-variation equations to bootstrap the eight-point shuffle-symmetric functions.
Repeating the same steps as in the previous analyses yields,

A.3.2. Tensorial and refined functions
Given their sizes, the tensorial and refined shuffle-symmetric functions will be omitted.
For convenience, in this appendix we quote from part III of this series of papers a representation of the one-loop correlators utilizing GEIs for n = 4, 5, 6, 7, as they are frequently referred to in this work.