All-order differential equations for one-loop closed-string integrals and modular graph forms

We investigate generating functions for the integrals over world-sheet tori appearing in closed-string one-loop amplitudes of bosonic, heterotic and type-II theories. These closed-string integrals are shown to obey homogeneous and linear differential equations in the modular parameter of the torus. We spell out the first-order Cauchy-Riemann and second-order Laplace equations for the generating functions for any number of external states. The low-energy expansion of such torus integrals introduces infinite families of non-holomorphic modular forms known as modular graph forms. Our results generate homogeneous firstand second-order differential equations for arbitrary such modular graph forms and can be viewed as a step towards all-order low-energy expansions of closed-string integrals.


Introduction
The low-energy expansion of string amplitudes has become a rewarding subject that carries valuable input on string dualities and that has opened up fruitful connections with number theory and particle phenomenology. The key challenge of low-energy expansions resides in the integrals over punctured world-sheets that are characteristic for string amplitudes and typically performed order-by-order in the inverse string tension α . The α -expansion of such string integrals then generates large classes of special numbers and functions -the periods of the moduli space M g,n of n-punctured genus-g surfaces.
In particular, one-loop closed-string amplitudes are governed by world-sheets with the topology of a torus and the associated modular group SL 2 (Z). The α -expansion of such torus integrals introduces a fascinating wealth of non-holomorphic modular forms known as (genusone) "modular graph forms" which have been studied from a variety of perspectives . 1 These modular graph forms satisfy an intricate web of differential equations with respect to the modular parameter τ of the torus, namely: first-order Cauchy-Riemann equations relating modular graph forms to holomorphic Eisenstein series [9,13,17,21] and (in-)homogeneous Laplace eigenvalue equations [4,10,14,27].
As the main result of this work, we derive homogeneous Cauchy-Riemann and Laplace equations for generating series of n-point one-loop closed-string integrals and the associated modular graph forms. In contrast to earlier approaches in the literature, our results are valid to all orders in α and do not pose any restrictions on the topology of the defining graphs. As part of our construction, we also propose a basis of closed-string integrals in one-loop amplitudes of the bosonic, heterotic and type-II theories.
More specifically, we study torus integrals over doubly-periodic Kronecker-Eisenstein series and Koba-Nielsen factors which are shown to close under the action of the Maaß and Laplace operator in τ . These Kronecker-Eisenstein-type integrals are shown to generate modular graph forms at each order of their α -expansion, and they additionally depend on formal variables η 2 , η 3 , . . . , η n . The expansion of the generating series in the η k -variables retrieves the specific torus integrals that enter the massless n-point one-loop amplitudes of bosonic, heterotic and type-II strings. At each order in α , the differential equations of individual modular graph forms can be extracted from elementary operations -matrix multiplication, differentiation in η j and extracting the coefficients of suitable powers in s ij and η k .
We stress that the terminology "closed-string integrals" or "open-string integrals" in this work only refers to the integration over the world-sheet punctures. The resulting modular graph forms in the closed-string case are still functions of the modular parameter τ of the torus world-sheet and need to be integrated over τ in the final expressions for one-loop amplitudes. A variety of modular graph forms have been integrated using the techniques of [2,4,16,25,26,35], and the differential equations in this work are hoped to be instrumental for integrating arbitrary modular graph forms over τ .
The motivation of this work is two-fold and connects with the analogous expansions of one-loop open-string integrals: • The α -expansion of one-loop open-string integrals can be expressed via functions that depend on the modular parameter τ of the cylinder or Möbius-strip world-sheet. These functions need to be integrated over τ to obtain the full one-loop string amplitude and were identified [36,37] as Enriquez' elliptic multiple zeta values (eMZVs) [38]. A systematic all-order method to generate the eMZVs in open-string α -expansions [39,40] is based on generating functions of Kronecker-Eisenstein type, similar to the ones we shall introduce in a closed-string setting.
More specifically, the α -expansions in [39,40] are driven by the open-string integrals' homogeneous linear differential equations in τ and their solutions in terms of iterated integrals over holomorphic Eisenstein series [38,41,42]. Similarly, the new differential equations obtained in the present work are a first step 2 towards generating the analogous closed-string α -expansions to all orders in terms of modular forms built from iterated Eisenstein integrals and their complex conjugates.
The first-order differential equations for closed-string integrals in this work turn out to closely resemble their open-string counterparts [39,40]. This adds a crucial facet to the tentative relation between closed strings and single-valued open strings at genus one. The resulting connections between modular graph forms and eMZVs and a link with the non-holomorphic modular forms of Brown [52,53] will be discussed in the future [54].
In summary, the long-term goal is to obtain a handle on the connection between modular graph forms and iterated integrals in the context of the α -expansion of closed-string oneloop amplitudes. The new results reported in this paper arise from the strategy to study 2 Closed-string integrals pose additional challenges beyond the open string in solving their differential equations order-by-order in α . These challenges stem from the expansion of modular graph forms around the cusp and the interplay of holomorphic and anti-holomorphic eMZVs, which will be addressed in follow-up work. See Section 6 for an initial discussion of this point. the modular differential equations satisfied by suitable generating functions of modular graph forms. Together with appropriate boundary conditions they will imply representations of amplitudes in terms of iterated integrals.

Summary of main results
This section aims to give a more detailed preview of the main results and key equations in this work. The driving force in our study of new differential equations for modular graph forms is the matrix of generating series of one-loop closed-string integrals. The integration domain for the punctures z 2 , z 3 , . . . , z n is a torus with modular parameter τ , and translation invariance has been used to fix z 1 = 0. The integrand involves doubly-periodic functions ϕ τ η ( z) of the punctures that are built from Kronecker-Eisenstein series and depend meromorphically on n − 1 bookkeeping variables η 2 , η 3 , . . . , η n . The rows and columns of W τ η (σ|ρ) are indexed by permutations ρ, σ ∈ S n−1 that act on the labels 2, 3, . . . , n of both the z j and η j . Note in particular that the permutations ρ and σ acting on ϕ τ η ( z) and the complex conjugate ϕ τ η ( z) may be chosen independently, so (1.1) defines an (n−1)! × (n−1)! matrix of generating integrals. Finally, G(z, τ ) denotes the standard closed-string Green function on the torus to be reviewed below, and the Mandelstam invariants are taken to be dimensionless throughout this work: As will be detailed in Section 2.3, the Kronecker-Eisenstein-type integrands ϕ τ η ( z) will be viewed as Laurent series in the η j variables. The accompanying coefficient functions from the η j -expansion of the Kronecker-Eisenstein series are building blocks for correlation functions of massless vertex operators on a torus [21,36,55]. By independently expanding (1.1) in the η j andη j variables, one can flexibly extract the torus integrals in one-loop closed-string amplitudes with different contributions from the left-and right movers -including those of the heterotic string, see Appendix C for more details.
The component integrals at specific (η j ,η j )-orders of (1.1) in turn generate modular graph forms upon expansion in α , i.e. in the dimensionless Mandelstam invariants (1.2). The modular graph forms in such α -expansions have been actively studied in recent years, and their differential equations in τ were found to play a crucial role to understand the systematics of their relations [9,13]. The generating functions (1.1) will be used to streamline the differential equations for infinite families of arbitrary modular graph forms, without any limitations on the graph topology. As will be demonstrated in Sections 3 to 5, the W τ ηintegrals close under the action of first-order Maaß operators and the Laplacian. Like this, one can extract differential equations among the n-point component integrals and therefore for modular graph forms at all orders in α .

Open-string integrals and differential equations
The definition and Cauchy-Riemann equations of the closed-string integrals W τ η are strongly reminiscent of recent open-string analogues [39,40]: The generating functions of cylinder integrals in one-loop open-string amplitudes involve the same doubly-periodic ϕ τ η ( z) as seen in their closed-string counterpart (1.1). However, instead of the complex conjugate ϕ τ η ( z) in the integrand of W τ η , the open-string integrals (1.3) are characterized by an integration cycle C(σ), σ ∈ S n−1 which imposes a cyclic ordering of the punctures on the world-sheet boundaries. In case of a planar cylinder amplitude, the punctures can be taken to be on the A-cycle of an auxiliary torus 3 , that is why we will refer to the open-string quantities (1.3) as "A-cycle integrals". Accordingly, the open-string Green function G A (z i −z j , τ ) is essentially obtained from the restriction of the closed-string Green function G(z, τ ) in (1.1) to the A-cycle z ∈ (0, 1).

Outline
This work is organized as follows: Section 2 combines a review of background material with the introduction of the generating functions W τ η of closed-string integrals. In Section 3, we set the stage for the differential equations of W τ η -integrals by introducing the relevant differential operators and illustrating the strategy by means of two-point examples. Then, Sections 4 and 5 are dedicated to the n-point version of the Cauchy-Riemann and Laplace equations, respectively. In Section 6, we comment on the problems and perspectives in uplifting the differential equations of this work into all-order α -expansions of the W τ η -integrals. The concluding section 7 contains a short summary and outlook. Several appendices provide additional background material, examples or technical aspects of some of the derivations.

Basics of generating functions for one-loop string integrals
In this section, we review the basic properties of one-loop string integrals. The starting point is provided by the doubly-periodic version of a well-known Kronecker-Eisenstein series whose salient features we exhibit. Based on this we then define the generating integrals whose differential equations will be at the heart of our subsequent analysis.

Kronecker-Eisenstein series
The standard Kronecker-Eisenstein series is defined in terms of the (odd) Jacobi theta function as [57,58] where τ ∈ C lives on the upper half-plane (Im τ > 0) and labels the world-sheet torus Σ τ = C/(Z[τ ] + Z) and z = uτ + v with u, v ∈ [0, 1) is a point on that torus. The parameter η can be used for a formal Laurent-series expansions with Ω(z + mτ + n, η, τ ) = Ω(z, η, τ ) for all m, n ∈ Z. Expanding this function in η defines doubly-periodic but non-holomorphic functions f (a) via The first two instances are given explicitly by and f (1) is the only function among the f (a) with a pole: f (1) (z, τ ) = 1 z +O(z,z) and f (2) (z, τ ) is ill-defined at the origin z = 0 as its expansion contains a termz z .
For several insertion points z i we introduce z ij = z i − z j and use as well the shorthand f (a) ij := f (a) (z ij , τ ). The function f (a) (z, τ ) is even/odd in z for even/odd a, such that f ji .

Derivatives of Kronecker-Eisenstein series
Since F (z, η, τ ) is meromorphic in z, the derivative ∂z of Ω is easy to evaluate and given by The first contribution stems from the additional phase in (2.3) and the δ (2) contribution is due to the simple pole of F (z, η, τ ) at z = 0. 4 Expanding (2.6) in η leads to When taking a derivative with respect to τ , the meromorphic Kronecker-Eisenstein series satisfies the mixed heat equation [58] There are two different forms of the corresponding equation for the doubly-periodic Ω, keeping either z or u and v fixed, Our convention for the delta function on the torus Στ is d 2 z Im τ δ (2) (z,z) = 1 Im τ and we also have that δ(u)δ(v) = Im τ δ (2) (z,z) so that du dv δ(u)δ(v) = 1.

Fay identities
The Kronecker-Eisenstein series obeys the Fay identity [58,59] 12) and the same identity also holds for the doubly-periodic version: Expanding (2.13) in η 1 and η 2 generates relations between products of f (a) functions. For explicit expressions, see Appendix A. Since the Koba-Nielsen integrals we are considering contain products of f (a) functions as integrands, Fay identities will be crucial in the derivation of Cauchy-Riemann equations. By taking the limit z 1 → z 2 in the meromorphic (2.12) and then passing to the doublyperiodic version, one obtains (2.14) As detailed in Appendix A, this can be used to derive the following identity [40]: which is central in the derivation of Cauchy-Riemann equations. Here, ℘(η, τ ) is the Weierstraß function, given by the following lattice sum over (2.16) and the functions G k for k ≥ 4 are the usual holomorphic Eisenstein series on the upper half-plane that are only non-vanishing for even k. For k = 2, (2.17) is only conditionally convergent and we define G 2 to be the expression obtained from using the Eisenstein summation convention: Since this function is not modular, we define for later reference also G 2 as the non-holomorphic but modular function

Lattice sums and modular transformations
Many of the properties of the Kronecker-Eisenstein series mentioned in the last sections can be checked conveniently using the formal lattice-sum representation Note that the sum over the lattice is unconstrained. The term with (m, n) = (0, 0) corresponds to the singular term η −1 f (0) = η −1 in (2.4). From the lattice form (2.20) and the expansion (2.4) we see also that formally where we have introduced the short-hand notation for the torus momentum p and the realvalued pairing between the torus insertion z and momentum p For a > 2, the sum (2.21) converges absolutely at the origin z = 0 and is given by We shall also encounter the complex conjugate functions e 2πi p,z p a (2.24) and the following combined functions which are in fact single-valued elliptic polylogarithms [8,60,61] for (a, b) = (0, 0) and include the special cases We also note that, using an auxiliary intermediate point z 0 , we can write where we have used the same short-hand for C (a,b) ij ij := f (a) (z ij , τ ) as for f (a) . The integral is over the torus with a measure normalized to yield d 2 z Im τ = 1. Note that the function C (a,b) (z, τ ) is even/odd in z for even/odd a+b, i.e. C For later reference, we also define Hence, f (a) transforms like a Jacobi form of vanishing index and weight a.

Koba-Nielsen factor
A central ingredient in the study of closed-string amplitudes is the n-point Koba-Nielsen factor [62] KN τ n := n 1≤i<j exp (s ij G(z ij , τ )) , is the real scalar Green function on the world-sheet torus and the dimensionless Mandelstam variables s ij for massless particles were defined in (1.2). 5 The Green function can also be written as the (conditionally convergent) lattice sum [1] G(z, τ ) = Im τ π p =0 e 2πi p,z |p| 2 , (2.33) using the notation (2.22). Hence, the Green function is a special case of the more general lattice sums in (2.25) and related to the functions f (1) and f (1) by Moreover, its τ -derivative satisfies The derivatives of the Koba-Nielsen factor following from (2.35) and (2.36) are For the z-derivative we have used the anti-symmetry of f (1) (z, τ ) in its argument z. We note that (2.37a) implies for any 1 ≤ k ≤ n that (2.38) 5 We note that we are not using momentum conservation at this stage and the variables sij are symmetric in i and j but otherwise unconstrained.

Component integrals and string amplitudes
The W -integrals in (2.39) are engineered to generate the integrals over torus punctures in closed-string one-loop amplitudes upon expansion in the η j andη j variables. The expansion (2.4) of the doubly-periodic Kronecker-Eisenstein integrands introduces component integrals for the integral over the n−1 punctures. Note that the z ij arguments of the f (a k ) ij with weights from the first index set A = a 2 , a 3 , . . . , a n are permuted with the permutation ρ in the second slot of the argument of W τ (A|B) and vice versa. This is to ensure consistency with the notation Here and in the rest of this work, we use the abbreviating notation Component integrals of the type in (2.43) arise from the conformal-field-theory correlators underlying one-loop amplitudes of closed bosonic strings, heterotic strings and type-II superstrings [21]. More specifically, the f (a) ij were found to appear naturally from the spin sums of the RNS formalism [36] and the current algebra of heterotic strings [55]. 6 For these theories, the (n−1)! × (n−1)! matrix in (2.43) is in fact claimed to contain a basis of the integrals that arise in string theory 7 for any massless one-loop amplitude. Moreover, massive-state amplitudes are likely to fall into the same basis.
The main motivation of this work is to study the α -expansion of component integrals kl . At each order in α , these integrals fall into the framework of modular graph forms to be reviewed below. As will be demonstrated in later sections, the W -integrals (2.39) allow for streamlined derivations of differential equations for infinite families of modular graph forms.

Relations between component integrals
It is important to stress that the component integrals W τ (A|B) (σ|ρ) are not all linearly independent. There are two simple mechanisms that lead to relations between certain special cases of component integrals. Still, component integrals W (A|B) with generic weights A, B are not affected by the subsequent relations, that is why they do not propagate to relations between the (n − 1)! × (n − 1)! generating series in (2.39).
Firstly, there can be relations between different W τ (A|B) (σ|ρ) stemming from the fact that the functions f kl . Since these parity properties interchange points they intertwine with the permutations ρ and σ. For instance, if the last two entries of A and B are A = (a 2 , . . . , a n−2 , 0, a n ) and B = (b 2 , . . . , b n−2 , 0, b n ), respectively, then the only places where the points z n−1 and z n 6 Also see e.g. [65][66][67] for earlier work on RNS spin sums, [68][69][70][71][72] for g in one-loop amplitudes in the pure-spinor formalism and [73,74] for applications to RNS one-loop amplitudes with reduced supersymmetry. 7 String theory integrals can also involve integrals over that are not in the form of (2.43) but can be reduced to the conjectural basis by means of Fay identities and integration by parts w.r.t. the punctures. Similar reductions should be possible for products of ∂z i f i k i 1 by adapting the recursive techniques of [75][76][77] to a genus-one setup.
appear are f (an) n−1,n , f (bn) n−1,n and in the permutation invariant Koba-Nielsen factor. Applying the parity transformation to the these factors of f (an) and f (bn) therefore can be absorbed by composing the permutations ρ and σ with the transposition n − 1 ↔ n and an overall sign (−1) an+bn . This yields a simple instance of an algebraic relation between the component integrals and we shall see an explicit instance of this for three points in Section 4.2 below.
The second mechanism is integration by parts -integrals of total z-derivatives (orzderivatives) vanish due to the presence of the Koba-Nielsen factor. Such derivatives produce sums over s ij f (1) ij from the Koba-Nielsen factor (see (2.37a)) and may also involve ∂ z i f  ij from the Koba-Nielsen derivatives. We note that these integration-by-parts relations can mix component integrals of different modular weight as they can also contain explicit instances of Im τ . A two-point instance of such an integrationby-parts identity among component integrals can be found in (2.67) below.

Modular graph forms
Modular graph forms are a compact way of denoting certain classes of torus world-sheet integrals that will be shown in Section 2.5 to arise also in the low-energy expansion of the component W -integrals (2.43). The starting point of a modular graph form is a decorated graph Γ on n vertices, corresponding to insertion points z i (1 ≤ i ≤ n), and with directed edges of loop momentum p e , where e runs over the set of edges E Γ . The decoration corresponds to a pair of integers (a e , b e ) for each edge e.
The modular graph form associated with the decorated graph Γ is then the following function of τ [9] where our normalization conventions differ from similar definitions in the literature 8 . From the definition one sees that the decorations (a e , b e ) on the edges label the powers of the holomorphic and anti-holomorphic momenta p e = m e τ + n e andp e = m eτ + n e (with m e , n e ∈ Z) propagating through the edge. At each vertex there is a momentum conserving delta function as indicated by summing over all momenta touching the vertex. The two terms with opposite signs in the momentum conserving delta function distinguish the incoming and 8 More specifically, the right-hand side of our definition (2.53) does not include the factor of e∈E Γ ( Im τ π ) 1 2 (ae+be) from [9,13,20,22] and the factor of e∈E Γ ( Im τ π ) be from [21].
outgoing momenta at a vertex. This definition implies in particular that if an edge connects a vertex to itself, C Γ factorizes. If for any two edges e and e the sum of weights a e + b e + a e + b e > 2, (2.49) is absolutely convergent. Furthermore, due to the symmetry under p e → −p e (for all e ∈ E Γ simultaneously), it follows immediately from the definition (2.49) that C Γ (τ ) vanishes if the sum of all exponents e∈E Γ (a e + b e ) is odd.
Under an SL 2 (Z) transformation a modular graph form transforms as where the integers |A| and |B| defined by 9 are commonly referred to as holomorphic and anti-holomorphic weights, respectively, and often written as a pair (|A|, |B|). Similarly, Im τ transforms as Im τ → (γτ + δ) −1 (γτ + δ) −1 Im τ and hence carries modular weight (−1, −1). We note that the modular graph form (2.49) vanishes when there are vertices with a single edge ending on them and more generally when the graph Γ is one-particle reducible. As a consequence, one of the delta functions in (2.49) is redundant due to overall momentum conservation, i.e. there is always one vertex whose in-and outgoing momenta are already fixed by the assignments of the momenta at all other vertices.
The modular graph form (2.49) can also be written in terms of the lattice sums (2.25) as where we have denoted by z e the difference between the starting and final points of an edge e and use (2.28) for C (0,0) . The delta function in (2.49) originates from the integral over the punctures z i on the torus and the phase factors e 2πi pe,z i for all edges touching the vertex z i . By translation invariance on the torus we could also set z 1 = 0 and integrate only over n−1 points as we have done for the W -integrals (2.39). This translation invariance corresponds to overall momentum conservation. Modular graph forms with symmetric decorations a e = b e for all edges are known as modular graph functions [8]. They arise if the integrand in (2.43) is solely made of Green functions and can be rendered modular invariant upon multiplication by a suitable power of Im τ . More generally, modular invariant completions of this type can be attained under the weaker condition |A| = |B| on (2.51) with a e = b e for some edges. Figure 1: Decorated dihedral graph with associated notation for modular graph form.

Dihedral examples
As an example, we consider dihedral graphs shown in Figure 1 that consist of two vertices connected by R lines. For the associated modular graph forms we use the notation where our normalization conventions differ from [9,13,[20][21][22], see Footnote 8. In the above expression we have suppressed the redundant delta function from overall momentum conservation. Following (2.52) the integral representation of this is There are two special cases of dihedral modular graph forms that will play an important role in our examples. These correspond to cases with only two edges: where the zeros in the second column stem from C a 1 a 2 b 1 b 2 = (−1) a 2 +b 2 C a 1 +a 2 0 b 1 +b 2 0 and E k denotes the non-holomorphic Eisenstein in the normalization convention that converges absolutely for Re(k) > 1. The non-holomorphic Eisenstein series is invariant under modular transformations by virtue of (Im τ ) k in the prefactor. Note that in the above formulas the column with two zeroes is essential in order to have a non-vanishing modular graph form. From the convergence conditions in (2.55) we see that there are two special cases that can be considered separately. We formally define where the G 2 denotes the non-holomorphic but modular function in (2.19) while E 1 is strictly divergent but can be regularized using the (first) Kronecker limit formula [81]. For later reference, we furthermore define the combinations which capture the independent two-loop modular graph functions at weight four and five [4] and were tailored to satisfy particularly simple differential equations [9,17]. They correspond to so-called depth-two iterated integrals over holomorphic Eisenstein series [17].

Differential equations of modular graph forms
We will frequently express modular graph forms at the leading s ij -orders of the W -integrals through E k , E 2,2 , E 2,3 and their derivatives w.r.t. the operator 10 seen in earlier literature [9,13,17,21]. Non-holomorphic Eisenstein series give rise to the closed formula for two-column modular graph forms whereas the derivatives of their simplest generalizations in (2.58) are determined by Higher iterations of ∇ DG give rise to holomorphic Eisenstein series [9], e.g.
Note, however, that the Cauchy-Riemann and Laplace equations of the W -integrals take a more convenient form when trading ∇ DG for the differential operator that will be introduced in Sections 3.1 and 3.2. Figure 2: The graph associated to the trihedral modular graph form C A 12

More general graph topologies
While the dihedral modular graph forms (2.54) accommodate any order in the α -expansion of two-point component integrals (2.43), higher multiplicities n ≥ 3 introduce more general graph topologies. Their three-point instances additionally involve the trihedral topology depicted in Figure 2 and that we shall encounter in examples in this paper. For trihedral modular graph forms, the integral representation (2.52) in terms of the building blocks (2.25) takes the form where z 1 = 0, and the collective labels A ij , B ij have length R ij , e.g. A ij = (a 1 ij , a 2 ij , . . . , a R ij ij ). In special cases, this simplifies to dihedral modular graph forms, as detailed in Appendix B.1.
More generally, α -expansions of n-point component integrals generate modular graph forms associated with n-vertex graphs due to n i<j ( ).

Low-energy expansion of component integrals
The component integrals W τ (A|B) (σ|ρ) introduced in (2.43) depend on the Mandelstam variables s ij through the Koba-Nielsen factor (2.31). In this section, we will study the expansion of the W τ (A|B) (σ|ρ) in the Mandelstam variables. Since the Mandelstams as defined in (1.2) carry a factor of α , this expansion is also an expansion in α and therefore corresponds to the low-energy expansion of the corresponding amplitude in the string-theory context.

Expanding the integrands of the component integrals
where the contributions to the (α ) w -order are characterized by n ij and G(z ij , τ ) in the integrand are identified with the doubly-periodic functions C Conversely, any convergent modular graph form can be realized through the α -expansion (2.64) of suitably chosen component integrals. The topology of the defining graph determines the minimal multiplicity n that admits such a realization. For instance, the spanning set ij . By the local behavior KN τ n ∼ |z ij | −2s ij of the Koba-Nielsen factor as z i → z j , the integration region over |z ij | 1 yields kinematic poles ∼ s −1 ij . Still, component integrals W τ (A|B) (σ|ρ) with integrands ∼ |f (1) ij | 2 can be Laurent-expanded via suitable subtraction schemes as reviewed in Appendix D.1. Since the residues of the kinematic poles 12 are expressible in terms of Koba-Nielsen integrals at lower multiplicity, any contribution to such subtraction schemes can be integrated in the framework of modular graph forms.
Note that the differential-equation approach of the next sections does not require any tracking of kinematic poles, and our results do not rely on any subtraction scheme. 11 In particular, the odd two-loop modular graph forms [22] arise at the α → 0 limit of the component integrals Im . The simplest odd modular graph form A1,2;5 which is not expressible in terms of ∇ n DG E k in (2.60) and their complex conjugates is generated by the component integral Im W τ (4,1|3,2) (3,2|2,3) over Im(f 32 ). 12 In the same way as |f ..ip in multiparticle channels. Pole structures of this type can still be accounted for via analogous subtraction schemes, and the residues are again expressible in terms of lower-multiplicity integrals.

Two-point examples
At two points, the generating function and the component integrals do not depend on permutations and are given by where we have denoted η = η 2 for simplicity. By identifying the Green function as Im τ π C (1,1) (z, τ ) as in (2.64), one can easily arrive at closed formulae for the α -expansion of W τ (a|b) in terms of dihedral modular graph forms (2.53) where 1 k denotes the row vector with k entries of 1. The expansion of W τ (0|b) can be obtained by complex conjugating the expansion of W τ (a|0) . The α -expansion of W τ (1|1) requires extra care since the singularity f 12 ∼ 1 |z 2 | 2 of the integrand leads to a kinematic pole in s 12 as mentioned above. We can bypass the need for subtraction schemes through the integration-by-parts-identities Note that similar integrations by parts should suffice to rewrite higher-multiplicity component integrals W τ (A|B) (σ|ρ) with kinematic poles in terms of regular representatives with a Taylor expansion in s ij . The kinematic poles will then appear as the expansion coefficients such as the factor of s −1 12 in the first step of (2.68), see Appendix D.2 for a three-point example. Although the α -expansion in terms of lattice sums could be quickly generated from (2.64), the representation in (2.66) is not optimal since many non-trivial identities between modular graph forms exist [2,4,8,9,13]. The types of identities in the references are reviewed in Appendix B and can be used to reduce the above expansions into a putative basis of lattice sums.
At the lowest orders, these conjectured bases are given by the non-holomorphic Eisenstein series E k in (2.56), their higher-depth analogue E 2,2 in (2.58) and derivatives (2.60) or (2.61) and we have for example where the simplified α -expansion of W τ (1|1) follows from inserting the first line of (2.69) into (2.68). Note that none of the modular graph forms on the right-hand sides of (2.66) is amenable to holomorphic subgraph reduction reviewed in Appendix B.4. In fact, since the z ij arguments of the chiral or anti-chiral Kronecker-Eisenstein integrands of the W -integrals do not form any cycles, none of the modular graph forms in the n-point α -expansions (2.64) will allow for holomorphic subgraph reduction.

Three-point examples
From three points onward, the generating functions and component integrals start depending on a chiral and an anti-chiral permutation. Following (2.43), we introduce three-point component integrals by where ρ, σ ∈ S 2 act on the subscripts i, j ∈ {2, 3} of the η andη in (2.70) and of f (n) and f (n) in (2.71) but not on those of a i and b j . As in the two-point case, kinematic poles arise if the integrand develops a 1 |z| 2 singularity in (some combination of) the punctures. The details of how to treat these poles using integration-by-parts-identities are spelled out in Appendix D.2.
In contrast to the two-point case, the three-point α -expansions also contain trihedral modular graph forms as defined in (2.63). Nevertheless, using the identities from Appendix B, the leading orders displayed below can also be brought into the basis spanned by the E k , their higher-depth generalizations and derivatives: Note in particular that although W τ (2,0|2,0) (2, 3|2, 3) and W τ (2,0|2,0) (2, 3|3, 2) differ just in their chiral permutations, their α -expansions are very different.

From modular graph forms to component integrals
The closed formulae (2.66) for two-point component integrals allow to identify infinite families of modular graph forms within their α -expansion. Similarly, we shall now list possible realizations of more general modular graph forms in (n ≥ 3)-point component integrals. Like this, the differential equations of the modular graph forms in Table 1 can be extracted from the differential equations of W -integrals in later sections. It is straightforward to extend the list Modular graph form to arbitrary graph topologies, where the multiplicity of the associated component integrals will grow with the complexity of the graph. Note that the k powers of Green functions G(z ij ) instruct to extract the coefficients of s k ij from the component integrals.

Modular differential operators, Cauchy-Riemann-and Laplace equations
In this section, we introduce the Cauchy-Riemann and Laplace operators that we will use to derive the differential equations for generating functions of world-sheet integrals.

Maaß raising and lowering operators
We define the following standard Maaß differential operators [82]: These have the property that they act on functions transforming with modular weights (a, b) as 13 when using that the derivatives transform as ∂ τ → (γτ + δ) 2 ∂ τ and ∂τ → (γτ + δ) 2 ∂τ under modular transformations. The Maaß raising and lowering operators satisfy the product rule 13 Note that the differential operator ∇DG, defined in (2.59) to compactly write modular graph forms, is related to the Maaß operators by ∇DG = Im τ ∇ (0) . Therefore its image has only nice modular properties if ∇DG acts on a modular invariant. For this reason, we will not use ∇DG to derive Cauchy-Riemann or Laplace equations in the following sections. We note also that, acting on modular invariant functions f , one has ∇ n DG f = (Im τ ) n ∇ (n−1) · · · ∇ (1) ∇ (0) f , a version of Bol's identity [83].
for any a and a and similarly for ∇ (b) . Here, f and g can be any real analytic functions and do not need to have definite modular transformation properties. The differential operators (3.1) should be thought of as raising and lowering operators of the action of SL 2 (R) on the space spanned by functions transforming with modular weights (a, b) for some a, b. The diagonal element of this action obtained by commutation is given by the scalar acting on any function of modular weight (a, b). 14 We define a Laplace operator on functions of modular weight (a, b) by 15 where we have chosen the overall normalization to be such that ∆ (a,b) reduces to the ordinary hyperbolic Laplacian on modular invariant functions with (a, b) = (0, 0). Let us consider the action of the Maaß operators on a modular graph form (2.49) under the assumption of absolute convergence so that differentiation and summation can be interchanged freely. Recall from (2.50) that the total modular weight of a general modular graph function C Γ (τ ) is given by (|A|, |B|). Hence, on C Γ (τ ), the action of the raising and lowering operators (3.1) is given by where Γ+s e is the same graph as Γ but with the decoration on edge e shifted to (a e +1, b e −1).
Similarly Γ+s e is the same graph Γ but with the decoration on edge e shifted to (a e −1, b e +1). In deriving (3.7) we have used the product rule (3.3) and that The connection to SL2(R) representation can also be seen by noting that E k and G k are trivially eigenfunctions of the corresponding h (a,b) , namely h (0,0) for E k and h (k,0) for G k . The function G k satisfies moreover ∇ (k) G k = 0 and is thus a lowest-weight vector, namely that of a discrete series unitary representation. E k is the so-called spherical vector in a principal series representation of SL2(R); see [81] for more information on this connection. 15 This differs by an overall sign from the one in [52,53] and from the standard second-order SL2(R)-invariant . Both operators reduce to the standard scalar Laplacian (3.6) and satisfy ∆ (a,b) f = ∆ (b,a) f since their difference is symmetric under the interchange a ↔ b.

Differential operators on generating series
Equipped with the differential operators introduced in the previous section, we will derive and study differential equations satisfied by the generating integrals W τ η defined in (2.39) in the remainder of this work.
These differential equations describe the dependence of W τ η on τ at all orders in α and in the series parameters η = (η 2 , η 3 , . . . , η n ). As W τ η is defined as an integral over the worldsheet torus with complex structure parameter τ we first have to clarify how the τ -derivative acts on such integrals. Our convention will always be to treat such torus integrals as such that they are taken to not depend on τ when the torus coordinate z = uτ + v is written in terms of two real variables along the unit square. The τ -derivative is then always taken at constant u and v and will only act on the integrand. While the definitions (3.1) and (3.5) of the Maaß operators and the Laplace operator are tailored to functions of definite modular weights (a, b), the W -integrands in (2.39) mix different modular weights in their expansion w.r.t. η j andη j . More precisely, the component integrals W τ (A|B) (σ|ρ) in (2.43) have modular weights (|A|, |B|), using the notation (2.45). Hence, it remains to find a representation of the Maaß operator (3.1) such that its action on the expansion (2.46) of n-point W -integrals is compatible with the modular weights of the component integrals. The modular weights of the W τ (A|B) (σ|ρ) correlate with the homogeneity degrees in the η j andη j that is measured by the differential operators n j=2 η j ∂ η j and n j=2η j ∂η j , respectively. We therefore define the following operators on functions depending on τ and η ∇ (k) Due to the shift in the expansion of the component integrals (2.46), there is an offset between the eigenvalues of ( n j=2 η j ∂ η j , n j=2η j ∂η j ) and the weights (|A|, |B|) according to that relates the raising operator on the generating series W τ η correctly to the raising operator on the component integrals W τ (A|B) of definite modular weight (|A|, |B|). In (3.11), we have also used the S n−1 permutation invariance of the raising and lowering operators (3.10) that descends to the specific sums (2.40) of the η j -variables in the expansion of the W -integrals via n j=2 η j ∂ η j = n j=2 η j,j+1...n ∂ η j,j+1...n = n j=2 ρ η j,j+1...n ∂ η j,j+1...n . (3.13) The Laplace operator can be defined in a similar fashion to (3.10) as such that it acts on component integrals W τ η for any σ, ρ ∈ S n−1 , and valid at any order in the (η j ,η j )-and α -expansions of W τ η at n points. In the remainder of this section we work out the first-order Cauchy-Riemann equation and the second-order Laplace equation satisfied by W τ η for two points in order to illustrate the basic manipulations. We will dedicate Sections 4 and 5 to the Cauchy-Riemann equations and Laplace equations of n-point W -integrals.

Two-point warm-up for differential equations
For the simplest case of n = 2, there are no permutations to consider, and the (n−1)!×(n−1)! matrix in (2.39) reduces to the real scalar where we have denoted η = η 2 for simplicity.

Cauchy-Riemann equation
Under the two-point instance ∇ (1) η = (τ −τ )∂ τ + 1 + η∂ η of the operator (3.10a), the two-point W -integral (3.16) satisfies 12 Ω(z 12 , η, τ ) Ω(z 12 , η, τ ) KN τ where the first line on the right-hand side stems from (2.11) (for fixed coordinates u and v in the torus integral) and the second one from the Koba-Nielsen derivative (2.37b). In passing to the third line, we have integrated ∂ z 2 by parts in the first term 16 and used (2.6), (2.37a) and the fact that ∂ η only acts on the Ω factor in the product. For the next equality, one simplifies (f 12 )Ω(z 12 , η, τ ) via (2.15) that produces a Weierstraß function ℘(η, τ ). Since the differential operator in η does not depend on z 2 , we have moved it out of the integral and defined svD τ η := s 12 in passing to the last line of (3.17). Note that this operator is meromorphic in η and τ . The notation svD τ η in (3.18) is motivated by the analogous differential equations for Acycle integrals in (2.41). Their two-point instance Z τ η = 1 0 dz 2 Ω(z 12 , η, τ )e s 12 G A (z 12 ,τ ) was shown in [39,40] to obey the differential equation The open-string differential operator D τ η only differs from its closed-string counterpart (3.18) through the additional term −2ζ 2 s 12 . As a formal prescription to drop the ζ 2 -contribution to D τ η , we refer to the single-valued map for (motivic) MZVs sv : ζ 2 → 0 [50,51] in the notation for svD τ η in (3.19).

Laplace equation
According to (3.14), the representation of the Laplacian on the two-point W -integral (3.16) is η −(1 +η∂ η )η∂η. In order to evaluate the action of the Maaß operators, 16 There are no boundary terms arising in this process since they are suppressed by the Koba-Nielsen factor.

Two-point warm-up for component integrals
We shall now translate the Cauchy-Riemann-and Laplace equations (3.17) and (3.24) of the generating integral W τ η to the equations satisfied by its component integrals W τ (a|b) defined in (2.65).

Cauchy-Riemann equation
At the level of component integrals (2.65), the Cauchy-Riemann equations (3.17) are equivalent to with the understanding that W τ (a|−1) = W τ (−1|b) = 0 for all a, b ≥ 0. The simplest examples for low weights a, b include These equations are obtained by direct evaluation of (3.17) and are valid at all orders in α . In Section 3.4.3, we shall analyze their α -expansion and relate them to equations for modular graph forms.

Laplace equation
The Laplace equation (3.24) of the generating integrals W τ η implies the following component relations for the W τ (a|b) of modular weight (a, b) in (2.65): The simplest examples include These are again valid to all orders in α and we shall discuss their α -expansion below. We note that the evaluation of the differential operators in (3.24) a priori leads to double poles in η orη on the right-hand side. These do not occur on the left-hand side and therefore have to cancel. While this is not necessarily manifest, their appearance can be traced back to the integration by parts that was used in the derivation (3.17) of the differential equation. For this reason, the residues of the putative double poles vanish by the integration-by-parts identities (2.67), that is why the η-expansions on both sides of (3.24) are identical.

Lessons for modular graph forms
In Section 2.5.1, we calculated the leading orders in the α -expansions of two-point component integrals in terms of modular graph forms. Using these expansions, (3.25) and (3.27) imply Cauchy-Riemann and Laplace equations for modular graph forms.
As an example, using (2.66) to expand the right-hand side of the Cauchy-Riemann equation (3.26) for W τ (2|0) leads to Similarly, the right-hand side of the Laplace equation (3.28) expands to 17 . 17 We have not yet inserted the simplified form (2.69) of the α -expansions which are obtained after using identities between modular graph forms, since we want to illustrate that (3.25) and (3.27) can be used to generate these kinds of identities.
In particular, the differential equations (3.25) and (3.27) of the component integrals bypass the need to perform holomorphic subgraph reduction (see Appendix B.4) to all orders in α . This is exemplified by the identities for modular graph forms C A c d B 0 0 in (3.33) and becomes particularly convenient at n ≥ 3 points, where the state-of-the-art methods for holomorphic subgraph reduction [20] are recursive and may generate huge numbers of terms in intermediate steps.

Cauchy-Riemann differential equations
In this section, we derive the general first-order differential equation in τ on the generating series W τ η (ρ|σ) for n points. The steps will generalize the two-point derivation in Section 3.3.1 with some additional steps due to the permutations ρ, σ ∈ S n−1 . After deriving the general n-point formula we exemplify it by studying in detail the cases n = 3 and n = 4.

Cauchy-Riemann differential equation at n points
In order to act with ∇ (n−1) η on the generating series W τ η (σ|ρ) defined in (2.39), we observe that the Maaß raising and lowering operator distributes correctly according to (3.3) and acts only on the product of chiral Ω-series and on the Koba-Nielsen factor. Moreover, the differential operator and the Koba-Nielsen factor are invariant under ρ ∈ S n−1 as can be seen from the definition (2.31) and the property (3.13). Using the mixed heat equation (2.9b) for the τ -derivative of Ω as well as (2.37b) for the τ -derivative of the Koba-Nielsen factor this leads to where we have introduced the following short-hand: 18 In each of the terms in the i-sum in (4.1) one can replace ∂ z i → ∂ z i +∂ z i+1 +. . .+∂ zn = n j=i ∂ z j as the function it acts on does not depend on the other z-variables. This has the advantage that one can integrate by parts all z-derivatives without producing any contribution from the other chiral Kronecker-Eisenstein series since they all depend on differences such that the corresponding terms cancel. This leads to two contributions: In the first the z-derivatives act on the Koba-Nielsen factor, and the second contribution comes from the action on the anti-chiral Kronecker-Eisenstein series. These two contributions are of different kinds and we first focus on the one when the z-derivative acts on the anti-chiral Ω.
A partial z-derivative acting on a single anti-chiral Ω was given in (2.6) and generates the correspondingη. It can be checked that the combination of all terms does not depend on the permutations ρ and σ that one started with and in total produces the operator 2πi n i=2η i ∂ η i acting on the whole expression. We emphasize that this is the only part that does not involve only holomorphic or anti-holomorphic η-parameters but mixes them. Carrying out the full integration by parts, we can therefore rewrite (4.1) as  shuffle shuffle (a 1 , . . . , a k−2 , a k−1 , a k , a k+1 , . . . , a n ) Figure 3: The shuffles appearing in the s ij -form on the right-hand side of (4.6) for a fixed i < k ≤ j. The values i and j are not included in the indicated ranges. The middle two intervals are reversed to descending order before the shuffles. The origin of this reversal is (E.14). All sequences obtained in this way constitute the set S n (i, j, k) defined in (4.5).
We next focus on analyzing the terms inside the ρ-permutation by using (2.38) for evaluating the z-derivative acting on the Koba-Nielsen factor:  As shown in Appendix E, the cyclic product of Kronecker-Eisenstein series can be brought into a form such that the differential operator in square brackets has a simple action, see (E.10), generalizing (2.15). The result can be written in terms of certain shuffles explained in detail in the Appendix and illustrated here in Figure 3. At n points for 1 ≤ i < j ≤ n and i < k ≤ j one has to consider all sequences (a 1 , . . . , a n ) in the set   Here, we have set ∂ η 1 = 0 in the case i = 1 to avoid a separate bookkeeping of the terms ∼ n j=2 s 1j ∂ 2 η j . We see that there is a 'diagonal term' containing the differential operators that goes back to the standard ordering of points. The terms including the Weierstraß functions mix the standard ordering with other orderings described by the set S n (i, j, k). Since neither the differential operators nor the Weierstraß functions depend on z they can be pulled out of the world-sheet integral.
We note that these operations are also related to the so-called S-map [84,85] which enters the expressions of [40] for the τ -derivatives of A-cycle integrals (2.41). In fact, the (n ≥ 6)-point instances of the open-string differential operator D τ η in (1.4) were conjectural in the reference, and (4.6) together with Appendix E furnish the missing proof. Equation (4.6) is expressed in an over-complete basis since a sequence (a 1 , . . . , a n ) ∈ S n (i, j, k) can have the index 1 at any place. Assume that the index 1 appears at position m > 1, i.e. a m = 1 and write (a 1 , . . . , a n ) = (A, 1, B) with A = (a 1 , . . . , a m−1 ) and B = (a m+1 , . . . , a n ). The index 1 can be moved to the front using the fact that, as a consequence of the Fay identity (2.13), products of Kronecker-Eisenstein series obey the shuffle identity [86]  where A t = (a m−1 , a m−2 , . . . , a 1 ) denotes the reversed sequence and we have set c 1 = 1 always. Applying this identity replaces one sequence (a 1 , . . . , a n ) ∈ S n (i, j, k) by a sum of sequences but the resulting integrals are then all of W -type but with different orderings of the n − 1 unfixed points. This replaces the second term in (4.6) by a sum over all possible permutations α ∈ S n−1 multiplying W τ η (σ|α) with coefficients T τ η (ρ|α) constructed out of Mandelstam invariants and Weierstraß functions. We write the total contribution of (4.6) to the Cauchy-Riemann derivative, up to an overall (τ −τ ), as the operator (4.8) Explicit expressions for svD τ η (ρ|α), detailing in particular the coefficients T τ η (ρ|α) will be given in Section 4.2 and 4.3 below for 3 and 4 points. At two points, one can identify T τ η = −s 12 ℘(η, τ ) from the expression (3.18) for svD τ η . The 'single-valued' notation here again instructs to drop the diagonal term ∼ −2ζ 2 s 12...n δ ρ,α in the analogous open-string differential operator D τ η (ρ|α) in (1.4) [39,40], i.e. svD τ η (α|ρ) = D τ η (α|ρ) = D τ η (α|ρ) + 2ζ 2 s 12...n δ ρ,α . (4.9) We have further separated svD τ η (α|ρ) into a part that contains the holomorphic derivatives with respect to η and terms T τ η (α|ρ) that are completely meromorphic in η and τ and contain no derivatives.
Putting everything together we conclude that (2.39) obeys the Cauchy-Riemann equation defining a short-hand for the action of the Maaß operator and with svD τ η given in (4.8). By expanding this equation in the η-parameters one can obtain systems of Cauchy-Riemann equations for the component integrals which in turn yield Cauchy-Riemann equations for modular graph forms.
One can then work out the three-point Cauchy-Riemann equations (4.10) that read for three points 16) by substituting in the matrix elements of svD τ η 2 ,η 3 (ρ|α) given in (4.14) and (4.15). We carry out the derivation of the Cauchy-Riemann equations for component integrals (2.71) in detail in Appendix F where we explain a subtlety in translating (4.16) to the component level: Both sides of (4.16) have to be expanded in the same η variables (e.g. η 23 = η 2 +η 3 and η 3 ) but other permutations naturally come with different η variables that have to be rearranged using the binomial theorem.
A general formula for the Cauchy-Riemann equation of the components (2.71) can be found in (F.6). It can be specialized to yield and further examples listed in (F.7). The very simplest instance of this is for where we have used the corollary 32 . This is an example of the first type of linear dependence between component integrals mentioned in Section 2.3.2.

Lessons for modular graph forms
We now consider one instance of such a Cauchy-Riemann equation to probe its contents in the α -expansion. The example we shall look at involves the component integral where we have also written out the leading α -order of the right-hand side. Alternatively, we could have applied directly the differential operator to the expansion (4.20) using (3.7) which yields Equating this to (4.21) again leads to a non-trivial identity between modular graph forms, now mixing trihedral and dihedral type. This identity can be checked by using various identities for modular graph forms to bring both (4.21) and (4.22) into the form Similarly to the two-point results outlined in Section 3.4.3, also the three-point Cauchy-Riemann equations imply infinitely many relations between modular graph forms, now also including trihedral topologies. In particular, these identities allow to circumvent the convoluted trihedral holomorphic subgraph reduction [20]. In the example above, when (4.22) is simplified by means of the factorization and momentum conservation identities spelled out in Appendices B.2 and B.3, one obtains In this expression, the first three trihedral modular graph forms have to be simplified using holomorphic subgraph reduction. In (4.21), by contrast, no holomorphic subgraph reduction is necessary which exemplifies a general feature of the Cauchy-Riemann equations generated by (4.16): They avoid a large number of iterated momentum conservations and all instances of holomorphic subgraph reductions.

Four-point examples
At four points we restrict ourselves to providing the expression for the operators in (4.8). The following expressions for svD τ η (σ|ρ) can be obtained by applying the general method with the same steps as for three points: They agree with the corresponding open-string expressions D τ η (σ|ρ) in [39,40] after dropping the term −2ζ 2 s 1234 in the diagonal entries which is annihilated by the single-valued map.

Laplace equations
In this section, we extend the first-order Cauchy-Riemann equation (4.10) to a second-order Laplace equation. This is first done in general for n points and then examples are worked out for a low number of points. The derivation follows the ideas of Section 3.3.2 on two-point Laplace equations.

Laplace equation at n points
In order to extend the Cauchy-Riemann equation (4.10) to the Laplacian we need to act with ∇ (n−2) η from (3.10b) on (4.10) and subtract an appropriate combination of weight terms according to (3.14). The action of ∇ (n−2) η = ∇ (n−1) η −1 on (4.10) is simple since the differential operator passes through most terms in Q τ η (ρ|α) except for the explicitη i in the diagonal term and the explicitτ in front of svD η (ρ|α), leading to the simple commutation relation generalizing (3.21) Taking the complex conjugate of (4.10) leads to 19 which implies (see (3.22) for the analogous two-point calculation) This expression can be expanded further by moving all η-differential operators in Q τ η to the right to act directly on W τ η since most terms commute. The only extra contributions come where the last line is part of 2πi(τ −τ ) n k=2 η k ∂η k svD τ η (ρ|α), and therefore 19 Note that one has W τ η (σ|ρ) = W τ η (ρ|σ), leading to the summation over the first permutation labeling W τ η in the complex conjugate equation.
According to (3.14), the Laplacian differs from this by terms proportional to the weights that are also given by differential operators in η. The final result for the general Laplace equation is then The term in the second line is due to the fact that the contractions of the summation variables i, j = 2, 3, . . . , n of the second-derivative terms are different: One is (η∂η)(η∂ η ) while the other is (η∂ η )(η∂η), so that only the diagonal terms cancel and one is left with a rotation-type term that contributes for n > 2, as does the first line. These terms were not visible in the twopoint example (3.24). Above we still set ∂ η 1 = ∂η 1 = 0. We note that the equation (5.6) has the correct reality property under complex conjugation associated with a real Laplacian at n points. Similar to the discussion at the end of Section 3.4.2 the consistency of the η-expansions of the left-hand side and right-hand side of (5.6) follows from integration-by-parts-identities for the component integrals. The η-expansion around different variables is also analyzed in Appendix F in a three-point example.
The general formula (5.6) can be evaluated for any number of points n, any permutations ρ, σ ∈ S n−1 and for any component integral W τ (A|B) (σ|ρ). The complexity of doing so grows very rapidly, therefore we restrict ourselves her to giving only a few low-weight examples.
In principle, this kind of direct computation involving the Koba-Nielsen derivative (5.16) can also be used beyond the simplest cases, e.g.

The open-string analogues
We reiterate that the open-string integral is over the boundary of the cylinder with a certain ordering σ of the punctures and we restrict to the planar case of all punctures on the same boundary for simplicity. As shown in [39,40], these integrals satisfy the differential equation (1.4) with the differential operator D τ η that is linear in the Mandelstam variables and whose single-valued version appears in (4.8). This homogeneous first-order differential equation can be solved formally by Picard iteration (with q = e 2πiτ ) The summation variable in the last line tracks the orders of α carried by the D τ j η matrices. The important point here is that the initial values Z i∞ η (σ|α) are by themselves series in α that have been identified with disk integrals of Parke-Taylor type at n + 2 points [39,40]. Their α -expansion is expressible in terms of MZVs [43,[87][88][89][90], and the dependence on s ij can for instance be imported from the all-multiplicity methods of [91,92]. Hence, any given α -order of the A-cycle integrals is accessible from finitely many terms in the sum over , i.e. after finitely many steps of Picard iteration.

An improved form of closed-string differential equations
We shall now outline a starting point for the corresponding procedure to expand closed-string generating series W τ η (σ|ρ). One can first rewrite the Cauchy-Riemann equation (4.10) as The terms in the first line are independent of the Mandelstam variables and also mix the holomorphic and anti-holomorphic orders in the variables of the generating series. This obstructs a direct link between Picard iteration and the α -expansion in analogy with the open-string construction. In the following, we will present a redefinition of the W τ η integrals such that one can still obtain each order in the α -expansion of the component integrals through a finite number of elementary operations.
The contributions 1 − n − n i=2 η i ∂ η i to the α -independent square-bracket in (6.6) can be traced back to the connection term in the Maaß operator (3.10a) which simply adjusts the modular weights. In general, one can suppress the connection term in (3.1) by enforcing vanishing holomorphic modular weight on the functions it acts on, which is always possible by multiplication with suitable powers of (τ −τ ). Hence, we will consider a modified version of the W -integrals, where each component integral in (2.46) of modular weight (|A|, |B|) is multiplied by (τ −τ ) |A| such as to attain the shifted modular weights (0, |B| − |A|).
Since the component integrals W τ (A|B) defined in (2.43) have modular weights (|A|, |B|), the desired modification of (2.46) is given by Given that the entire Y -integral has holomorphic modular weight zero, the action of the Maaß operator (3.10a) reduces to (τ −τ )∂ τ , and the Cauchy-Riemann equation (6.6) simplifies to where we have expanded the closed-string differential operator in terms of Eisenstein series in analogy with (6.2). The operator R η ( 0 ) contains also the s ij -independent term ∼η j ∂ η j in its diagonal components whereas R η ( 0 ) ρ α = r η ( 0 ) ρ α for ρ = α. For k ≥ 4, we have agreement with the open-string expression R η ( k ) = r η ( k ) and these matrices should again form a matrix representation of Tsunogai's derivations [93]. We have checked that they preserve the commutation relations of the k [41,95,96] to the same orders as done for their open-string analogues, see Section 4.5 of [40]. The appearance of the term ∼η j ∂ η j in R η ( 0 ) does not obstruct the α -expansion of component integrals from finitely many steps 20 in a formal solution with the structure of (6.4). Note that the asymmetric role of the η j andη j variables in the definition (6.7) of the Y -integrals will modify the Laplace equation (5.6) and obscure its reality properties. For this reason we have worked with the generating series W τ η (σ|ρ) in earlier sections of this work.

A formal all-order solution to closed-string α -expansion
In analogy with the open-string α -expansion (6.4), one can construct a formal solution to (6.8) from the infinite sum over words in the alphabet k j ∈ {0, 4, 6, 8, . . .}: This ansatz obeys (6.8) with the correct initial conditions at τ → i∞ if the quantities β sv (k 1 , k 2 , . . . , k |τ ) satisfy the following initial-value problem However, an iterative solution of (6.11) does not completely determine them since one can still add anti-holomorphic functions of τ that vanish at the cusp at each step. We expect these antiholomorphic 'integration constants' to be fixed by the reality condition W τ η (σ|ρ) = W τ η (ρ|σ) and the modularity properties of the component integrals. In particular, the anti-holomorphic differential equation of the generating series (6.7) might be a convenient starting point. Still, the ansatz (6.10) may not yet be the optimal formulation of closed-string α -expansions since the initial value (6.11b) is in general incompatible with the usual shuffle relations for the β sv (k 1 , k 2 , . . . , k |τ ). Another important point in this context is the determination of the initial values Y i∞ η from its α -expanded Laurent polynomial and the connection with sphere integrals. The recent all-order results for the Laurent-polynomial part of the two-point component integral W τ (0|0) [24,28] and the results on the τ → i∞ degeneration of the A-cycle integrals [39,40] should harbor valuable guidance. The correct prescription to iteratively determine the β sv (k 1 , . . . |τ ) 20 This follows from the fact that none of the R η ( k ) has a contribution that lowers the powers ofηj. Hence, any component integral at a given order ofηj in (6.7) can only be affected by finitely many instances of the termηj∂η j in R η ( 0) which is the only contribution to the R η ( k ) without any factors of sij.
is expected to involve a notion of 'single-valued' integration along the lines of [47,49] and may be related to the non-holomorphic modular forms of [52,53]. We plan to report on this in the near future.

Summary and outlook
In this paper, we have derived first-and second-order differential equations for generating functions W τ η (σ|ρ) of general torus integrals over any number n of world-sheet punctures. The torus integrands involve products of doubly-periodic Kronecker-Eisenstein series and Koba-Nielsen factors that are tailored to n-point correlation functions of massless one-loop closedstring amplitudes in bosonic, heterotic and type-II theories. These differential equations, given in (4.10) and (5.6), are exact in α . The differential operators appearing in the differential equations coincide with the single-valued versions of similar operators appearing in the open string [39,40].
The flavor of open-closed string relations motivating the present work is different from the Kawai-Lewellen-Tye (KLT) relations [97] that express closed-string tree-level amplitudes via squares of open-string trees. Tree-level KLT relations do not manifest the disappearance of MZVs outside the single-valued subspace, and loop-level KLT relations in string theory are unknown at the time of writing 21 . The long-term goal of this work is to directly relate closedand single-valued open-string amplitudes at one loop, generalizing the tree-level connection of [43][44][45][46][47][48][49]. Hence, it remains to be seen to which extent the present results and follow-up work [54] relate to a tentative one-loop KLT construction.
The generating series W τ η (σ|ρ) can be expanded in powers of the bookkeeping variables η 2 , η 3 , . . . , η n to yield equations for the component integrals that are modular forms of fixed weights. These in turn can be expanded in powers of Mandelstam variables, corresponding to a low-energy expansion, to generate systematically differential equations for modular graph forms that arise in explicit string scattering calculations. Our generating function approach yields Cauchy-Riemann and Laplace equations for modular graph forms associated with an arbitrary number of vertices and edges. The equations generated in this way can be used to deduce non-trivial identities between modular graph forms. Our method in particular never generates modular graph forms with negative entries and does not necessitate holomorphic subgraph reduction to expose holomorphic Eisenstein series.
It would be interesting to generalize the methods of this work such as to generate differential equations for higher-genus modular graph forms [30][31][32][33][34]. Moreover, in view of the recent 21 See Section 5.3 of [98] for a recent account on the key challenges in setting up loop-level KLT relations among string amplitudes from the viewpoint of intersection theory [99]. A field-theory version of one-loop KLT relations among loop integrands in gauge theories and supergravity has been derived from ambitwistor strings [100,101]. advances in integrating modular graph forms over τ [2,4,16,25,26], our results are hoped to yield useful input for the flat-space limit of string amplitudes in AdS 5 × S 5 [102][103][104][105][106].
In the following, we want to extend (A.2) to the case where the arguments of the two f (a) coincide. To this end, we take the limit z 1 → z 2 in the meromorphic version (2.12) of the Fay identity and obtain (2.14), which we quote here for reference: 22 Expanding this in η 1 and η 2 and using leads to a version of (A.2) with both f (a) evaluated at the same point: where a 1 , a 2 ≥ 0 and we set f (a) = 0 for a < 0. Note that the manifest exchange symmetry a 1 ↔ a 2 is broken just by the different summation ranges. However, the terms with k = a 1 +1, . . . , a 1 +a 2 −1 in the first sum all vanish due to the binomial coefficient and the only remaining term is due to k = a 1 +a 2 and given by (−1) a 2 G a 1 +a 2 , which is invariant under a 1 ↔ a 2 . We did not make this manifest since G a 1 +a 2 does not appear if a 1 +a 2 < 4. Specializing (A.5) to a 1 = 1, 2 yields Here, a ≥ 0 and again we set f (a) = 0 for a < 0. Using (A.6a) and (A.6b) we can derive 22 One might be tempted to take the coincident limit directly in (A.2), however, this is not possible since the limit lim →0 f (a) ( , τ ) is ill-defined for a = 2.
(2.15) in the main text by expanding in η: We note that the terms involving z-derivatives of the f (a) cancel in this particular combination and we are left with a purely algebraic expression in these functions. In passing to the last line of (A.7), we used the expansion (2.16) of the Weierstraß function. Note that the Weierstraß function also satisfies the identity Finally, the identity (A.6a) can also be derived by expanding the following relations for derivatives of Kronecker-Eisenstein series:

B Identities between modular graph forms
There are many non-trivial identities relating modular graph forms and these are crucial in the simplification of α -expansions of Koba-Nielsen integrals [2,4,9,13,14,21,23]. In this appendix, we will review the most important techniques by which these identities can be obtained. The most straightforward approach to simplifying modular graph forms is probably to manipulate their lattice-sum representation. This was done e.g. in the case of C[ 1 1 1 1 1 1 ] by Zagier (see [4]), who proved that However, this kind of direct manipulation is hard and only possible in a limited number of cases. In the following subsections we exhibit some more systematic ways of obtaining identities between modular graph forms.

B.1 Topological simplifications
The simplest way to derive identities arises when the graph associated to the modular graph form has certain special topologies [9]. For example, if the graph contains a two-valent vertex, momentum conservation at that vertex together with the form of the propagators implies that the vertex can be removed and the weights of the adjacent edges added: A special case of this arises when the graph is dihedral. In this case, the simplification (B.2) implies where the [ 0 0 ] column is necessary to be consistent with the general notation for dihedral graphs introduced in (2.53). For a trihedral graph with a two-valent vertex, (B.2) implies the following simplification to the dihedral topology: Furthermore, if removing one vertex is sufficient to make the graph disconnected, the associated modular graph form factorizes: For a trihedral graph with a vertex pair that is not connected by any edges, (B.5) implies the following identity: where the empty first column reflects that vertices 1 and 2 are not connected.

B.2 Factorization
The simplifications in the previous section dealt with identities arising from special topologies, i.e. absent edges. If an edge is not absent but carries weight (0, 0), one can perform the sum over the momentum flowing through this edge explicitly [9]. The result has two contributions according to the two-term expression (2.28) for C (0,0) : One factorizes, the other one is of lower loop order. Explicitly, for the dihedral case, the resulting identity is The corresponding identity for trihedral graphs reads (B.8)

B.3 Momentum conservation
Momentum conservation implies that the sum of all momenta flowing into a vertex (with sign) is zero. Writing this vanishing momentum sum into the numerator of the modular graph form and expanding the result leads to non-trivial identities [9]. For dihedral graphs, they are Similarly, in the trihedral case we have where S i is the row vector whose j th component is δ ij .

B.4 Holomorphic subgraph reduction
Finally, if a modular graph form contains a closed subgraph of only holomorphic or antiholomorphic edges, the sum over the momentum flowing in this subgraph can be performed explicitly by means of partial-fraction decomposition [9]. For dihedral graphs with two holomorphic edges, this leads to The derivation of (B.11) involves the evaluation of the following conditionally convergent sums using the Eisenstein summation prescription [9]: Higher-point generalizations of these sums were worked out in [20] and hence higher-point generalizations of (B.11) can be obtained. A lengthy, but closed, formula for trihedral graphs is available in [20].

B.5 Verifying two-point Cauchy-Riemann equations
As an example of the power of the identities outlined the previous sections, we will prove the two-point Cauchy-Riemann equations (3.25) for component integrals in this section. The proof relies on applying momentum conservation (B.10) and holomorphic subgraph reduction (B.11) order by order in α . The α -expansion of two-point component integrals (2.65) can be expressed in closed form (cf. (2.66)) where 1 k denotes the row vector with k entries of 1. In the case a = 0, b ≥ 0, it is easy to check that (3.25) is satisfied without using any identities for modular graph forms. The case a > 0, b = 0 follows along the lines of the derivation below by dropping the [ 0 b ] columns everywhere and adjusting the overall sign. The final case a = b = 1 can be reduced to the a = b = 0 case by using the integration-by-parts identity (2.68).
Focusing on the coefficient of s k 12 in (B.13), we compute the action of the Cauchy-Riemann operator ∇ (a) by means of (3.7), leading to where we have used (B.9) to remove the -1 entry in the first term and (B.11) to simplify the second term. Using holomorphic subgraph reduction again in C[ a+1 0 1 1 k−1 0 b 0 1 k−1 ] and reorganizing the terms leads to (recall that a, b > 0 and (a, b) = (1, 1)) Comparing this to the coefficient of s k 12 in (3.25) shows that they agree.

C Component integrals versus n-point string amplitudes
In this appendix, we specify the component integrals W τ (A|B) (σ|ρ) (2.43) that enter n-point one-loop closed-string amplitudes of the bosonic, heterotic and type-II theories in more detail.
Even though the zero modes of the world-sheet bosons couple the chiral halves of the closed string at genus g > 0, it is instructive to first review the analogous open-string correlators: • The n-point correlators of massless vertex operators of the open superstring are strongly constrained by its sixteen supercharges -by the sum over spin structures in the RNS formalism [107,108] or by the fermionic zero modes in the pure-spinor formalism [109,110]. As a result, these correlators comprise products k f (a k ) i k j k of overall weight k a k = n − 4 as well as admixtures of holomorphic Eisenstein series G w k f (a k ) i k j k with w + k a k = n − 4 and w ≥ 4 [36,65]. When entering heterotic-string or type-II amplitudes as a chiral half, open-superstring correlator introduce component integrals W τ (A|B) (σ|ρ) with holomorphic modular weights |A| ≤ n − 4.
• In orbifold compactifications of the open superstring that preserve four or eight supercharges, the RNS spin sums are modified by the partition function, see e.g. [111] for a review. The spin-summed n-point correlators of massless vertex operators may therefore depend on the punctures via k f i k j k of weight k a k = n−2 or w+ k a k = n−2 [73,74]. The resulting component integrals W τ (A|B) (σ|ρ) in closedstring amplitudes with such chiral halves have holomorphic modular weights |A| ≤ n−2.
• Open bosonic strings in turn allow for combinations k f i k j k of weight k a k = n and w + k a k = n. The same is true for the npoint torus correlators of Kac-Moody currents entering the gauge sector of the heterotic string [21,55]. Accordingly, closed-string amplitudes of the heterotic and bosonic theory comprise component integrals W τ (A|B) (σ|ρ) with modular weights |A| ≤ n or |B| ≤ n in one or two chiral halves.
The pattern of f (a k ) i k j k obtained from a direct evaluation of the correlators may not immediately line up with the integrands of W τ (A|B) (σ|ρ). First, contractions among the world-sheet bosons introduce spurious derivatives ∂ z f (1) (z, τ ). Second, one may encounter arrangements of the labels i j in the first argument in "cycles" f (a 1 ) i n−1 in characteristic to (2.43). In both cases, combinations of integration by parts and Fay identities of Appendix A are expected to reduce any term in the above correlators to the integrands of W τ (A|B) (σ|ρ). For instance, the methods in Appendix D.1 of [21] reduce the cycle f with a 2 + a 3 + a 4 = 4 and permutations. The above properties of chiral halves set upper bounds on the modular weights |A|, |B| seen in the component integrals W τ (A|B) (σ|ρ) of the respective closed-string amplitudes. However, additional contributions with (non-negative) weights (|A| − k, |B| − k), k ∈ N arise from interactions between left and right-movers: Both the direct contractions between left-and right-moving world-sheet bosons and the contribution of (2.7) to integrations by parts convert one unit of holomorphic and non-holomorphic modular weight into a factor of (Im τ ) −1 , see e.g. [3,74,79].

D Kinematic poles
In this appendix, we spell out an example of the subtraction schemes that were mentioned in Section 2.5 to capture kinematic poles of component integrals in terms of modular graph forms.

D.1 Subtraction scheme for a two-particle channel
For simplicity, we begin with the example where the integrand Φ(z j ,z j ) is tailored to admit no kinematic pole different from s −1 12 . This can be reconciled with the general form (2.43) of component integrals if exhibits no singularities of the form |z kl | −2 which for instance imposes i 3 = j 3 .
The key idea of the subtraction scheme is to split the Koba-Nielsen factor of (D.1) into The combinations on the right-hand side ij (s ik f ik + s jk f Note that we use here Since the notation is a bit involved, we also give a more intuitive description of (E.1). For a fixed k in the range i < k ≤ j we have a permutation (a 1 , . . . , a n ) of the range (1, . . . , n) with the following boundary conditions: (iii) The subsequence (a k+1 , . . . , a n ) to the right of j (at position k) is a rearrangement of {k, . . . , n} \ {j} such that it is obtained as a shuffle in {j − 1, . . . , k}¡{j + 1, . . . , n}.
The situation is also illustrated in Figure 3. To each such sequence (a 1 , . . . , a n ) there is a product in (E.1) such that the arrangement of η-arguments in (E.1) is correct to represent a different sequence of points with corresponding generating arguments. We note that the change of variables at n points ξ p = n =p η p ⇐⇒ η p = ξ p − ξ p+1 for 1 < p < n ξ n for p = n (E. 8) implies that for the differential operator one has for all 1 ≤ i < j ≤ n when setting ∂ η 1 = 0. Thus the differential operator j k=i+1 ∂ ξ k acts only as ∂ ξ k on Ω(z ij , ξ k ) and vanishes on all other factors in (E.1). This is true since ξ k only contains η j but not η i . The term ξ k − η a k = ξ k − η j and all other terms to the right of Ω(z ij , ξ k ) are free of η j while ξ k + η a k−1 = ξ k + η i and all other terms to the left of Ω(z ij , ξ k ) always contains η i + η j and are therefore also annihilated by (E.9). Thus the s ij -form (E.1) is the correct form for evaluating for 1 ≤ i < j ≤ n that using (2.15). Here, we set again ∂ η 1 = 0 and have introduced the shorthand S n (i, j, k) defined in (4.5) for the sequences obtained by all possibles shuffles occurring in (E.1) and illustrated in Figure 3.
The proof of the s ij -form (E.1) of the product of Kronecker-Eisenstein series proceeds by several lemmata.

E.2 Extending left and right
We now want to prove (E.1) by induction on n which means extending the product on the left and on the right, beginning with the right. We first consider keeping i and j fixed and extend (E.1) by multiplying by Ω(z n,n+1 , ξ n+1 ) on the right. If k = n there is nothing to do since the factors just multiplies correctly at the end of the sequence: If the original sequence is (a 1 , . . . , a n ) with a n = n the new sequence is (a 1 , . . . , a n , a n+1 ) with a n+1 = n + 1 and the ξ-factors are correct for the action of the differential operator.
Note that the index n+1 always ends up to the right of n in this product and so the order is preserved for them. This means that (E.3) is the correct shuffle prescription for extending from j = n to any j < n by shuffling in the additional indices into the reversed indices to the right of Ω(z ij , ξ k ) in (E.14).
By similar methods one can also show that multiplying by Ω(z 01 , ξ 1 ) to extend on the left shuffles in the index 0 in all possible places to the left of the index 1. After renaming the indices, we conclude from that (E.2) is the correct shuffle prescription for extending from i = 1 to any i > 1 by shuffling in the indices {1, . . . , i − 1} into the reversed indices to the left of Ω(z ij , ξ k ) in (E.14).
The sequences constructed by this inductive method constitute the set S n (i, j, k) defined in (4.5) and illustrated in Figure 3. This concludes the proof of (E.1).

F Derivation of component equations at three points
In this appendix, we present additional details on the Cauchy-Riemann and Laplace equation for the component integrals of the generating series W τ η (σ|ρ) at three points, cf. Sections 4.2 and 5.2.
This means that we need to expand W τ η 2 ,η 3 (σ|3, 2) = In the main text we discuss one particular instance of this involving trihedral modular graph forms at lowest order.

F.3 Further examples for Laplace equations at three points
In this section, we provide two more examples of Laplace equations for three-point component integrals similar to (5.10) for W τ (0,0|0,0) (2,3|2,3). By expanding both sides of (5.7) in the η andη variables as discussed in Section 5.2, we obtain