Two dialects for KZB equations: generating one-loop open-string integrals

Two different constructions generating the low-energy expansion of genus-one configuration-space integrals appearing in one-loop open-string amplitudes have been put forward in \rcites{Mafra:2019xms, *Mafra:2019ddf, Broedel:2019gba}. We are going to show that both approaches can be traced back to an elliptic system of Knizhnik--Zamolodchikov--Bernard(KZB) type on the twice-punctured torus. We derive an explicit all-multiplicity representation of the elliptic KZB system for a vector of iterated integrals with an extra marked point and explore compatibility conditions for the two sets of algebra generators appearing in the two differential equations.


Introduction and summary
During the last years we have been experiencing a significant growth in understanding the mathematical concepts leading to recursion relations for scattering amplitudes in quantum field and string theory. A multitude of languages and approaches is available for various quantum field theories, see for instance [3][4][5][6][7][8][9][10] and references therein. Recent progress on string amplitudes in turn was driven by disentangling their polarization degrees of freedom from moduli-space integrals over punctured worldsheets and finding separate recursions for both types of building blocks. The low-energy expansion of string amplitudes exposed by such recursions at tree and loop level contains a wealth of information on relations between gauge theories and gravity, string dualities and counterterms including their non-renormalization theorems. For the moduli-space integrals in open-string tree-level amplitudes, a recursion based on the Knizhnik-Zamolodchikov equation was already identified in ref. [11] based on refs. [12,13] and later complemented by other methods put forward in refs. [14,15].
The problem of finding a one-loop (or genus-one) analogue of the open-string tree-level recursions was long-standing. A first mathematical challenge was to thoroughly understand iterated integrals on the elliptic curve and their associated special values, elliptic multiple zeta values [16][17][18][19]. Then, the cooperation of mathematicians and physicists was instrumental to investigate and understand the relation of those iterated integrals to one-loop open-string amplitudes and their differential equations [20][21][22]. The closed-string counterparts of these genus-one integrals lead to an intriguing system of non-holomorphic modular forms [23,24] that inspired mathematical research lines including refs. [25][26][27][28][29].
These structural considerations paved the way for two recent methods [1,2] to systematically evaluate the integrals over punctures on the boundary of a genus-one surface order by order in the inverse string tension α . These integrals to be referred to as genus-one configurationspace integrals 1 form the backbone of one-loop open-string amplitudes. Both algorithms rely on differential equations of Knizhnik-Zamolodchikov-Bernard(KZB) type on a genus-one surface with boundaries.
• In ref. [1], a KZB-type differential equation with respect to the modular parameter τ , which encodes the geometry of genus-one surfaces, was established. Acting on a vector of generating functions for one-loop configuration-space integrals, the τ -derivative can be expressed as a linear operator that mixes the components in different vector entries. In particular, this exposes finite-dimensional conjectural matrix representations of a special derivation algebra with corresponding generators k . Using Picard iteration, the equation can be solved starting from a particular value which is conveniently chosen as the limit τ → i∞ where the genus-one configuration-space integrals degenerate to their genus-zero counterparts with two additional legs.
• In ref. [2], a KZB-type differential equation with respect to the position of an auxiliary point z 0 was identified. Facilitating a vector of configuration-space integrals with the auxiliary point (genus-one Selberg integrals), a solution can be obtained using the KZB associator: it relates two regularized boundary values, which emerge when sending the auxiliary point to the poles of the differential equation in two distinct ways. At one boundary value one obtains the one-loop configuration-space integrals, while at the other boundary one recovers again genus-zero configuration-space integrals with two additional legs. The main players in the construction are infinite-dimensional matrix representations of an algebra with generators x k , which can be cut off to finite size when calculating up to a certain order in the α -expansion of the string amplitudes.
The two algorithms relate open-string tree-level and one-loop amplitudes in the same way: both are capable of determining the n-point configuration-space integrals at genus one from (n+2)point configuration-space integrals at genus zero. On the contrary, the representations of the KZB equations and underlying algebra generators are quite distinct. The relation between the two approaches can be best understood and investigated by considering a formalism combining the advantages of each of the previous methods: the central object to be considered in this article is a length-n! vector of generating functions for planar n-point one-loop configurationspace integrals to be denoted by Z τ 0,n with an auxiliary point z 0 : In particular a) we will find an all-multiplicity expression for the τ -derivative of Z τ 0,n in order to connect with the approach in ref. [1]. This will be an equation of the form 2πi∂ τ Z τ 0,n = D τ 0,n ({ k }) + B τ 0,n ({x j }) Z τ 0,n , k = 0, 4, 6, 8, . . . , j = 1, 2, 3, . . . , (1.1) where the operators D τ 0,n and B τ 0,n are n!×n! matrices with entries proportional to α .
b) we will rewrite the formalism of ref. [2] in terms of the vector of generating series Z τ 0,n , leading to finite-size matrix representations and an all-multiplicity expression for the z 0 -derivative of Z τ 0,n of the form ∂ z 0 Z τ 0,n = X τ 0,n ({x k }) Z τ 0,n , k = 0, 1, 2, . . . , (1.2) where X τ 0,n is a n!×n! matrix proportional to α as well. The constituents x k are related to the braid matrices that govern the genus-zero counterparts of Z τ 0,n [30].
Hence, the generating functions Z τ 0,n of genus-one configuration-space integrals to be introduced in this work furnish integral representations for solutions to the elliptic KZB system. Having two differential equations (1.1) and (1.2) at our disposal, we can demand commutativity of the two derivatives. This implies consistency conditions for the two classes of algebra generators involved. We have checked on a case-by-case basis that our realizations of the generators satisfy these relations.
In section 2 we are going to provide the mathematical and physical setting: we will discuss genus-zero and genus-one configuration-space integrals contributing to tree-level and one-loop open-string scattering amplitudes, respectively. This will set our conventions and incorporate a review of general properties of configuration-space integrals, iterated integrals and (elliptic) multiple zeta values. Section 3 is devoted to the discussion of several types of differential equations allowing for recursive solutions: in subsection 3.1 we review the genus-zero recursion from ref. [11] and bring it into the context of the later genus-one results. In subsections 3.2 and 3.3 we discuss the τ -based and z 0 -based genus-one recursions from ref. [1] and ref. [2], respectively. The central object to be discussed in section 4 is the vector of configuration-space integrals Z τ 0,n with an auxiliary point. After introducing the vector, we will perform the two steps a) and b) lined out above, resulting in an all-multiplicity representation of the elliptic KZB system on the twice-punctured torus. By considering the regularized boundary values for the elliptic KZB system, we will relate the different approaches in section 5, before we conclude in section 6.

Open-string scattering amplitudes and configuration-space integrals
In this review section we will introduce several mathematical objects and concepts necessary for the description of open-string scattering amplitudes at genus zero and genus one. Rather than providing yet another thorough and detailed introduction, we will just mention and collect the key concepts here and provide numerous links to elaborate discussions. The structure of scattering amplitudes in open-string theories can be most easily captured and understood when disentangling the results from evaluation of a conformal worldsheet cor-relator: the latter depends on the external polarizations through a kinematical part which we will separate from the moduli-space integrals that encode string corrections to field-theory amplitudes through their series expansion in α . Moduli-space integrals are dimensionless as they depend on dimensionless Mandelstam variables where n denotes the number of external particles. Their integrands are calculated as conformal correlators of vertex positions z i on Riemann surfaces, whose genus refers to the loop order in question. In the next two subsections, we are going to collect the basic formalism for the integration over open-string punctures at genus zero and one: tree level and one loop, respectively. Since we do not perform the integral over the modular parameter τ of genus-one surfaces in this work, the integrals over the z i will be referred to as configuration-space integrals in contradistinction to the full moduli-space integrals entering one-loop string amplitudes. The Mandelstam variables defined above in eq. (2.1) are going to take a role as (complex) parameters in the configuration-space integrals to be considered in this article. Naturally, the convergence behavior of those integrals depends on the values of the Mandelstam variables. Convergent integrals are obtained, when the Mandelstam variables are taken to satisfy the conditions listed below, though one can analytically continue to different regions. The conditions are formulated in terms of Mandelstam variables whose indices are related to consecutive insertion points on the disk or cylinder boundary.
• For the augmented genus-one configuration-space integrals Z τ 0,n , we relax momentum conservation and consider all the s ij with i < j as independent. Still, the above condition (2.2) for convergence carries over to the genus-one configuration-space integrals, where the notion of consecutive insertion points is adapted to an auxiliary puncture z 0 between z n and z 1 . The associated auxiliary Mandelstam invariant s 01 is furthermore taken to obey Re(s i 1 i 2 ...ir ) < Re(s 01 ) < 0 for all consecutive labels (i 1 , i 2 , . . . , i r ) = (0, 1) , (2.3) which is no restriction in the applications to one-loop open-string amplitudes since s 01 will drop out from the final results. In the context of one-loop open-string amplitudes, integrals of the type in Z τ 0,n are analytically continued from their region of convergence to physically sensible situations. The resulting singularities in the form of poles and branch cuts have been for example explored in [33].

Tree-level: genus zero
Calculating open-string amplitudes at tree level amounts to the evaluation of configuration-space integrals on a genus-zero surface with boundary. The corresponding genus-zero Green's function is a plain logarithm G tree ij = log |z ij | = G(0; |z ij |) (2.4) of the distance z ij = z i − z j (2.5) of two insertion points. The notation G refers to the iterated integrals defined in eq. (2. 10) below. In the configuration-space integrals, the Green's function appears in terms of the genuszero Koba-Nielsen factor All configuration-space integrals for the calculation of open-string scattering amplitudes at tree level can be expressed as linear combinations of the integrals [34][35][36] Z tree n (a 1 , a 2 , . . . , a n |1, 2, . . . , n) = −∞<za 1 <...<za n <∞ dz 1 · · · dz n vol SL 2 (R) KN tree 12...n z 12 z 23 · · · z n−1 n z n1 , (2.7) where the labels a 1 , . . . , a n fix a certain succession of the insertion points on the disk boundary. An independent cyclic ordering selects the permutation of the inverse z 12 z 23 · · · z n−1 n z n1 , the socalled Parke-Taylor factor. For a given multiplicity and particular choice of the labels a i in the first slot, the collection of all integrals obtained for all permutations of the ordering 1, 2, . . . , n in the second slot is not independent: integrals over different Parke-Taylor factors are related by partial fraction and integration by parts [34,35]. A convenient basis choice, which we are going to use throughout this article, consists of fixing the position of three of the labels in the Parke-Taylor factor, for example Z tree n = Z tree n (a 1 , a 2 , . . . , a n |1, σ, n, n−1) for σ ∈ P(2, 3, . . . , n−2) . (2.8) For the choice of fixing the SL 2 (R) redundancy via (z 1 , z n−1 , z n ) = (0, 1, ∞) and the ordering to (a 1 , a 2 , . . . , a n ) = (1, 2, . . . , n), the integrals are explicitly given by Z tree n (1, 2, . . . , n|1, σ, n, n−1) = − 0<z 2 <...<z n−2 <1 dz 2 · · · dz n−2 1≤i<j≤n−1 |z ij | −s ij z 1σ(2) z σ(2)σ(3) · · · z σ(n−3),σ(n −2) . (2.9) The dimension, that is, the length of the basis vector Z tree n for a fixed integration domain, is (n−3)!, which is precisely the number predicted by twisted cohomology and BCJ relations [37]. The basis dimension follows from results in twisted de Rham theory [38], which have been interpreted in a string-theory context recently [39,30].
After taking its kinematic poles into account [40,35,32], a Z-integral as defined in eq. (2.7) above is calculated by expanding the Koba-Nielsen-factor in α (cf. eq. (2.1)) and then evaluating each iterated integral separately. In particular, each of the Z-integrals can be expressed in terms of iterated integrals (multiple polylogarithms) 2 G(a 1 , a 2 , . . . , a r ; z) = z 0 dz 1 z 1 − a 1 G(a 2 , . . . , a r ; z 1 ) (2.10) with a i ∈ {0, 1} as well as G(; z) = 1 and z ∈ C \ {0, 1}. For tree-level open-string integrals, the outermost integration variable, e.g. one of the insertion points, can always be chosen to equal one by fixing the volume of SL 2 (R) in eq. (2.7). Thus we will have to evaluate integrals of type (2.10) at z = 1. Fortunately, all integrals appearing can be related to well-known representations of multiple zeta values (MZVs) using the identity: (2.11) The integrals defined in eq. (2.10) exhibit endpoint divergences if a r = 0 or a 1 = z. Therefore, they will have to be regularized, which implies corresponding regularizations for MZVs and may have an echo in the kinematic poles of the Z-integrals defined in eq. (2.7). Throughout this article, we will always assume to work with regularized iterated integrals. For instance, the multiple polylogarithms G(1, . . . ; 1) and G(. . . , 0; 1) in (2.10) will be shuffle-regularized based on G(0; 1) = G(1; 1) = 0 which assigns regularized values to divergent MZVs (2.11) with n r = 1 such as ζ 1 = 0 [43].
As an example, let us state the first couple of orders of the series expansion of a typical integral Z tree n : Z The analogous expressions for arbitrary orders in the α -expansion of n-point disk integrals can for instance be generated from the Drinfeld associator [11,44] or Berends-Giele recursions [14]. 3 The Berends-Giele method in ref. [14] applies to Z-integrals with arbitrary pairs of permutations Z tree n (a 1 , . . . , a n |b 1 , . . . , b n ) whose decomposition in the (n−3)! bases expanded in [11,44] can be generated from the techniques in ref. [51].

One-loop level: genus one
The calculation of one-loop open-string amplitudes requires consideration of configuration-space integrals on a genus-one surface with boundary. The latter can be constructed by starting from a genus-one Riemann surface (an elliptic curve or torus) whose geometry is usually parametrized by a modular parameter τ ∈ C with Im τ > 0. The two homology cycles of the torus can be mapped to the boundaries of the fundamental domain of a lattice Z + τ Z, where τ is the ratio of the length of B-and A-cycle respectively (see figure 1).
Frequently, the modular parameter is used in an exponentiated version, which appears in the Fourier expansions of the τ → τ + 1 periodic functions to be used below.
One-loop open-string amplitudes receive contributions from worldsheets of cylinder and Moebius-strip topology which can be obtained from a torus through involutions described for instance in ref. [52]. The cylinder worldsheet with two boundaries at Im z = 0 and Im z = 1 2 Im τ then arises from cutting the torus in two parts. When all insertion points z i are located at one boundary only, the resulting situation is called planar, while insertion points on two boundaries lead to non-planar integrals [53] (see figure 2).  [53,52], the cylinder is obtained from a torus with purely imaginary value for τ .
The frameworks of elliptic multiple zeta values (eMZVs) [18] and elliptic polylogarithms [16,17] allow to systematically perform the integrals over open-string punctures order by order in α [20]. For this purpose, the genus-one Green's function for planar open-string integrals is written as 14) see ref. [21] for their non-planar counterpart 4 . The corresponding Koba-Nielsen factor is given by with Γ(; z|τ ) = 1 and z ∈ R. In the same way as MZVs can be obtained as special values of iterated integrals on a genus-zero Riemann surface, see (2.11), one can relate Enriquez' A-cycle eMZVs 6 as special values of the elliptic iterated integrals defined in eq. (2.16): The integration kernels f (k) (z|τ ) in (2.16) are generated by a doubly-periodic version of a Kronecker-Eisenstein series [58,17], where θ is the odd Jacobi theta function and θ (0|τ ) its derivative in the first argument. The double-periodicities of the series and the integration kernels are Given the simple pole of f (1) (z|τ ) = ∂ z log θ(z|τ ) + 2πi Im z Im τ at z ∈ Z + τ Z, the integrals in eq. (2.16) and thus eMZVs exhibit endpoint divergences analogous to those in the tree-level scenario. Throughout this work, we will employ shuffle-regularization based on the prescription in section 2.2.1 of [20] which assigns the following q-expansion to the constituents of the Green's function (2.14), From this q-expansion, the asymptotic behaviour for 0 < z < 1 can be read off: the limit z → 0 yields the logarithmic divergence Apart from the constant f (0) (z|τ ) = 1 and f (1) (z|τ ) with a simple pole, the Kronecker-Eisenstein series (2.18) generates an infinity of kernels f (k≥2) (z|τ ) that do not have any poles in z. Hence, 5 Because of the non-holomorphic terms ∼ Im z Im τ appearing in f (k) (z|τ ), the iterated integrals (2.16) by themselves are not homotopy-invariant but can be lifted to homotopy-invariant iterated integrals by the methods of [17] (also see section 3.1 of ref. [20]). 6 Changing the integration path in eq. (2.17) to (0, τ ) in the place of (0, 1) gives rise to B-cycle eMZVs [18] whose properties have for instance been discussed in refs. [55][56][57].

Z τ -integrals at genus one
In analogy to the genus-zero integrals Z tree n defined in eq. (2.7), let us now define a suitable class of integrals at genus one [1], where we will always fix translation invariance by setting z 1 = 0. When writing a Kronecker-Eisenstein series where the first argument is of the form z ij , we use the shorthand notation as well as both of which will prove very handy below. Similar to the genus-zero case, the labels 1, a 2 , . . . , a n in the first slot refer to an integration domain. We have adapted (2.26) to planar genus-one integrals (cf. eq. (2)), where (1, a 2 , . . . , a n ) specifies a cyclic ordering of insertion points on a single cylinder boundary 7 . As a genus-one analogue of the so-called Parke-Taylor factor (z 12 · · · z n−1,n z n1 ) −1 in eq. (2.7), the labels in the second slot of eq. (2.26) indicate products of the form The absence of a factor f (kn) n,1 to close the cycle is reminiscent of Parke-Taylor factors in an SL 2frame with z n → ∞, where they reduce to open chains like (z 12 z 23 · · · z n−2,n−1 ) −1 as in eq. (2.9). Instead of individual products (2.29), the integrands in (2.26) involve their generating series (2.18) where the combinations n j=i η j of expansion variables are chosen for later convenience. As a major advantage of the generating-series approach, the relations between different permutations of (2.26) take a simple form: by analogy with the genus-zero case, a basis of integrands can be found by taking the genus-one analogue of partial fraction 8 into account, the 7 The integration domain in the non-planar situation is encoded by one cyclic ordering for both cylinder boundaries which can for instance be addressed by two-line labels Z τ n ( b 1 ,b 2 ,...,br c 1 ,c 2 ,...,cs |1, 2, . . . , n) as in [1]. 8 The genus-one analogue of integration-by-parts relations among Parke-Taylor factors in (2.7) does not relate permutations of the products (2.29) for generic choices of ki. Instead, integration by parts at genus one will play an important role in later sections to find differential equations for various Koba-Nielsen integrals.
While the Fay identities among the products in (2.29) shift the overall weight k 1 +k 2 between the two factors, their series in (2.26) are simply related via eq. (2.24). After performing a simultaneous expansion of (2.26) in α and η j , specific string integrals corresponding to particular integrands in eq. (2.29) can be retrieved by isolating suitable coefficients.

Graphical notation
All configuration-space integrals for string amplitudes appearing in this article exhibit the following features: they have a Koba-Nielsen-factor and a collection of integration kernels, which are labeled by (at least) the difference of two vertex positions: z ij . Furthermore, there are vertex positions z i , which are integrated over, and others, which remain unintegrated. For the discussion to follow, it is useful to define a graphical representation for the corresponding integrands, extending the graphical notation of ref. [44] to genus one: we are going to represent each occurring label as a vertex and each integration kernel as a directed edge respectively. In this graphical notation, both SL 2 -fixed Parke-Taylor factors (cf. eq. (2.7)) and the integrals Z τ n with fixed cyclic symmetry at genus one (cf. eq. (2.26)) exhibit a chain structure. As will be elaborated on below, partial-fraction relations and their one-loop analogue, the Fay relation (2.24), allow to reduce tree-structures to chain-structures. The Fay identity (2.24), for example, takes the following graphical form k i j ηa which is -not surprisingly -equivalent to the graphical representation of partial fraction (here: The graphical representation of Kronecker-Eisenstein integrands described above will play a major role in the calculations of section 4.

eMZVs versus iterated Eisenstein integrals
For the A-cycle eMZVs (2.17), another representation as iterated integrals of holomorphic Eisenstein series is available which exposes their relations over Q[MZV, (2πi) −1 ]. While eMZVs have been defined in eq. (2.17) in terms of special values of iterated integrals, which featured repeated integration in insertion points z i , it is possible to write them in terms of τ -iterated integrals. This is possible, because τ -derivatives and z-derivatives of their integration kernels are related by the mixed-heat equation (2.25). Further details in converting the integrals into each other including integration constants at τ → i∞ can be found in ref. [19]. Here we would like to limit our attention to writing down the basic definitions and properties of two types of iterated Eisenstein integrals, which will be made use of below [19], and for k 1 = 0 where γ(|q) = γ 0 (|q) = 1 and the number r of integrations will be referred to as the length of either γ and γ 0 . The integration kernels are holomorphic Eisenstein series 9 or their modifications G 0 with the constant term 2ζ 2k removed for k = 0, respectively. We will interchangeably refer to the arguments of G k , G 0 k and related objects as τ or q.
(2.38) 9 The case of G2 requires the Eisenstein summation prescription Further identities implied by Fay relations as well as the precise relation between the spaces spanned by the respective iterated integrals have been investigated and spelt out in refs. [19,61,62].

Two-point example
In order to wrap up this section, let us provide an example of a genus-one Z-integral (2.26) and express the leading orders of its expansion in α and η = η 2 in two of the languages above: it does contain MZVs as well as eMZVs, which are still a function of the modular parameter τ . This will be crucial for the constructions to be reviewed and discussed below.

Differential equations for one-loop open-string integrals
In the last section, Z-integrals for tree-level and one-loop open-string amplitudes have been introduced. Most importantly, these integrals can be expressed in terms of iterated integrals G and Γ over punctures z i (cf. eqs. (2.10) and (2.16)), which -if evaluated at special points -lead to MZVs and eMZVs, respectively (cf. eqs. (2.11), (2.17) and (2.35)). For iterated integrals with a particular class of differential forms, it is straightforward to infer differential equations -for example does eq. (2.34) immediately imply while eq. (2.16) leads to Starting from those simple equations, one can consider differential equations for complete Zintegrals. In particular, we will study augmented variants of Z-integrals where an additional unintegrated puncture z 0 serves as a differentiation variable. This will require the evaluation of the action of derivatives on the integrands and in particular on the Koba-Nielsen factor. Suitable manipulations, partial fraction and integration by parts for Z tree n integrals as well as Fay identities and integration by parts for the one-loop integrals Z τ n , allow to frame differential equations as matrix equations, acting on a vector Z basis whose elements form a (sometimes conjectural) basis of Z-type integrals and their augmented versions to be defined below: Here ν i are suitable differential forms in the alphabet for the iterated integrals that occur in the α -expansion of the respective Z-integral, whereas r(D i ) denotes a particular square matrix representation of the coefficients of ν i , tailored to the basis choice. The most crucial point of the game is the following: for all Z-type integrals we are going to consider, the representations r(D i ) turn out to be linear in the parameters s ij , and thus in α , entering the Koba-Nielsen factors in eqs. (2.6) and (2.15). This will allow to solve the differential equation of the above form order by order in α , leading to the α -expansion of the Z-integrals. Note that the linear appearance of α in the above r(D i ) is analogous to the -form of differential equations for Feynman integrals, see e.g. [63,64], with α taking the role of the dimensional-regularization parameter .
Considering the integrals Z tree n defined in eq. (2.7), the final result, i.e. the α -expansion, will contain numbers exclusively. In turn, a differential equation with respect to a variable which disappears during the evaluation of the iterated integral, is not very useful. The solution to this problem has been spelt out in both mathematics [12,13] and physics [11,44] literature: one can introduce an additional auxiliary insertion point z 0 and establish a differential equation with respect to z 0 for a basis vector of augmented integrals Z tree 0,n . For the integrals Z τ n at genus one, a similar augmentation can be introduced leading to augmented one-loop integrals Z τ 0,n whose constituents will be reviewed in subsection 3.3 and whose differential equations in section 4 are a central result of this work. However, since the result in eq. (2.39) does still depend on the modular parameter τ , one can readily use τ as a variable for differentiation when considering a vector Z τ n of one-loop integrals eq. (2.26) without z 0 . By the choice of differential forms ν i on the right-hand side of eq. (3.3), the resulting system of differential equations is of Fuchsian type. Even more, on closer inspection one will find the equations to be of Knizhnik-Zamolodchikov(KZ) or Knizhnik-Zamolodchikov-Bernard type for Z tree 0,n and Z τ 0,n , respectively, whose solution theory is well known [65,12,66,13,[67][68][69]. By solving these differential equations along with suitable boundary conditions, one can then evaluate Zintegrals Z tree n and Z τ n at tree-level and one-loop order by order in α . Moreover, the matrix representations r(D i ) we will encounter are linear in the Mandelstam variables s ij each of which comes with a parameter α (cf. eq. (2.1)). Hence, one can obtain solutions to all Z-type integrals in eq. (3.3) in terms of regularized iterated integrals, where the number of integrations is correlated with the power of α . 10 A major advantage of this concept is that the series expansion in α follows from simple matrix algebra for the r(D i ). Once the initial value for some limit of the differentiation variable is known, no integral has to be solved and the recursive nature of the solution algorithms allows to infer all higher-multiplicity Z-integrals at tree-level and one-loop from the knowledge of a single trivial tree-level three-point Z-integral.
In the following subsections, we are going to review the main structural points of three languages and corresponding algorithms: the z 0 -language at genus zero in subsection 3.1 and τ -and z 0 -languages at genus one in subsections 3.2 and 3.3. For each one, there is a basis of (augmented) Z-type integrals, a differential equation of type (3.3) with suitable matrix representations and boundary values, which together allow to solve the differential equation recursively.

z 0 -language at genus zero
The simplest instance of the algorithm described above is the recursive formalism for the evaluation of tree-level configuration-space integrals Z tree n . It has been put forward in refs. [11,44] and is based on refs. [12,13]. At n = 4, 5 points, the augmented versions of tree-level integrals eq. (2.7) with an extra marked point z 0 are given by see the references for higher-multiplicity generalizations. One can write down a differential equation with two types of poles: The above differential equation is of KZ-type and can be solved by considering regularized boundary values C tree 1,n = lim (1−z 0 ) −r tree 0,n (e 1 ) Z tree 0,n , (z 0 ) −r tree 0,n (e 0 ) Z tree 0,n (3.6) which are connected by the Drinfeld associator [65,66] C tree 1,n = r tree 0,n (Φ(e 0 , e 1 ))C tree 0,n . where 11 r tree 0,n (e i e j ) = r tree 0,n (e i )r tree 0,n (e j ) and r tree 0,n (e i + e j ) = r tree 0,n (e i ) + r tree 0,n (e j ). As discussed in ref. [11], the vector C tree 0,n can be shown to contain the integrals Z tree n−1 defined in eq. (2.7) for the (n−1)-point amplitude, while C tree 1,n contains integrals Z tree n for the n-point amplitude. In this formalism, the size of the matrix representations r tree 0,n (e i ) is (n−2)!. In order to calculate for example the four-point disk integral, one would use the differential equation In the kinematic limit s 0i → 0 employed for the recursions of ref. [11], one can read off r tree 0,4 (e 0 ) = − s 12 s 23 0 0 , r tree 0,4 (e 1 ) = − 0 0 s 12 s 23 (3.10) by matching eq. (3.9) with eq. (3.5), and the regularized boundary values turn out to be The first entry of C tree 1,4 may be recognized as the SL 2 -fixed disk integral −Z tree 4 (1, 2, 3, 4|1, 2, 4, 3) via eq. (2.9), and the more subtle computation of the second entry * will not be needed here. The first entry of C tree 0,4 combines the kinematic poles s −1 12 from the integration region z 2 → 0 in eq. (3.4) with the three-point integral Z tree 3 (1, 2, 3|1, 2, 3) = 1 at its residue. With the leading orders of the Drinfeld associator in eq. (3.8) and the matrix representations in eq. (3.10), one is indeed led to the known four-point α -expansion at tree level in the first component of the vector C tree 1,4 :

τ -language at genus one
In this subsection, we are going to review the differential equation of a vector Z τ n of genus-one Z-integrals eq. (2.26) without augmentation through an extra puncture. The basis of these Kronecker-Eisenstein-type integrals w.r.t. Fay relations and integration by parts is (n−1)! dimensional and spanned by [1] Z τ n = Z τ n I n |1, ρ(2, 3, . . . , n) .
The permutations ρ ∈ S n−1 of {2, 3, . . . , n} act on both the punctures z j and the expansion variables η j in the factors of Ω i−1,i (η i...n |τ ) in eq. (2.26). The ordering I n = 1, 2, . . . , n in the first slot refers to a fixed planar integration domain 0 = z 1 < z 2 < . . . < z n < 1 on the A-cycle of the torus (see figure 1). We will usually sort the permutations in lexicographic order, e.g. , (3.14) The (n−1)!-component vector eq. (3.13) was conjectured to generate an integral basis for arbitrary massless one-loop open-string amplitudes [1] as supported by its closure under τ -derivatives to be reviewed below.

Differential equation
Based on the mixed heat equation (2.25) and the differential equation of the Koba-Nielsen factor Finally, the combination of f (2) ij and f (1) ij due to the Koba-Nielsen derivatives can be rewritten via such that the integrand of the τ -derivative only depends on z i via Kronecker-Eisenstein series. Moreover, repeated use of the Fay identity (2.26) allows to rearrange their first arguments such that the τ -derivatives are expressed in terms of the (n−1)! components in eq. (3.13). As a result, the vector of genus-one Z-integrals in eq. (3.13) obeys the linear and homogeneous differential equation [1] 2πi∂ τ Z τ n = D τ n Z τ n (3.18) with an (n−1)!×(n−1)! matrix D τ n . The entries of the latter are linear in α by the Mandelstam invariants in the Koba-Nielsen derivatives in eqs. (3.15), (3.16) and may comprise second derivatives in η j from an expansion of eq. (3.17) around ξ = 0. Most importantly, the matrix D τ n solely depends on τ via holomorphic Eisenstein series eq. (2.36) and can therefore be uniquely decomposed into where G 0 = −1 has been introduced for the τ -independent piece. The appearance of the G k can be traced back to the expansion of the Weierstraß ℘-function in eq. (3.17), The (n−1)!×(n−1)! matrices r n ( k ) are still linear in α , comprise second derivatives in η j at k = 0 and are conjectured to furnish matrix representations of Tsunogai's derivation algebra [76]. By the appearance of these derivations in the τ -derivative of the KZB associator [67][68][69], their commutator relations encode the combinations of iterated Eisenstein integrals eq. (2.34) that occur among eMZVs [19]. The relations in the derivation algebra have been studied in [77,78,19] and were checked to be preserved by the r n ( k ) in eq. (3.19) for a wide range of n and k. An all-multiplicity proposal for r n ( k ) (with a detailed derivation in [79]) can be found in section 4 of [1], and its explicit form is encoded in the later eq. (4.25). Note that the matrix D τ n does not depend on the choice of planar or non-planar integration cycle, i.e. it takes a universal form for any I n → a 1 , a 2 , . . . , a n in eq. (3.13).

Solution via Picard iteration
Given that the matrix D τ n in eq. (3.18) is linear in α , the α -expansion of the entire Z τ n can be conveniently organized by iterative use of Picard iteration of eq. (3.21) leads to a perturbative solution to eq. (3.18) with matrix products of order (α ) in the 'th term of The choice of summation range for the k j already incorporates the vanishing of r n ( k ) at odd k and k = 2 for any n ≥ 2. Generating functions of torus integrals in closed-string oneloop amplitudes obey differential equations similar to eq. (3.19) [79] which can be solved via combinations of derivations similar to eq. (3.23) [71].

Initial value at the cusp
In order to extract the complete α -expansion of Z τ n from eq. (3.23), it remains to analyze the initial values Z i∞ n at the cusp. Given that the torus worldsheet degenerates to a nodal Riemann sphere as τ → i∞, the initial data at n points is expressible in terms of genus-zero integrals with two extra points, i.e. combinations of (n+2)-point disk integrals eq. (2.7) in the open-string case. At n = 2 points, for instance, the genus-one Z-integral (2.39) degenerates to where the Γ-functions stem from a kinematic limit s 23 → − s 12  The η-dependent factor π cot(πη) in eq. (3.24) stems from the τ → i∞ limit of Ω 12 (η). The detailed relation between eqs. (3.24) and (3.25) involving a trigonometric factor sin( π 2 s 12 ) from contour deformations can be found in section 3.4 of [1], also see section 5 of the reference for an all-multiplicity discussion.

Two-point example
In the two-point instance of the above setup, the target vector has a single component Z τ 2 (1, 2|1, 2) and the operator D τ 2 in the differential equation (3.18) is the following scalar instead of a matrix The decomposition (3.19) into holomorphic Eisenstein series allows to read off conjectural scalar representations of the derivation algebra which obey multiplicity-specific relations such as [r 2 ( 4 ), r 2 ( 6 )] = 0 that no longer hold at n ≥ 3 points. With the initial value (3.24) involving a four-point disk integral, the two-point instance of the Picard iteration (3.23) is given by Based on the expansions (3.26) and π cot(πη) = 1 η − 2 ∞ k=1 ζ 2k η 2k−1 , one arrives at the leading orders in α and η spelt out in eq. (2.39).

z 0 -language at genus one
In this subsection, we are going to sketch the formalism introduced in ref. [2]. The formalism is the genus-one generalization of the tree-level recursion discussed in subsection 3.1: it utilizes an auxiliary point z 0 such that In the reference, the differential equation and thus the recursion was formulated for an infinitely long vector of genus-one n-point Selberg integrals, which are defined in terms of a Selberg seed with z 1 = 0 which agrees with an (n+1)-point Koba-Nielsen factor (2.15) upon multiplication by e s 012...n ω(1,0|τ ) . The Selberg seed (3.31) serves as starting point for the recursive definition 12 . . , n as well as z n+1 = z 0 and i ∈ {0, 1, +1, +2, . . . , n} , (3.33) where the shorthand notation for the integration kernels f (k) ij was defined in eq. (2.29).

Differential equation and boundary values
The differential equation of the (infinitely long) vector of Selberg integrals can formally be brought into the KZB form where the representations r E 0,n of x k are block-(off-)diagonal and proportional to α . Similar to the tree-level scenario described in subsection 3.1, one now considers the two regularized boundary values which turn out to contain n-point one-loop integrals Z τ n in C E 1,n and (n+2)-point tree-level integrals Z tree n+2 in C E 0,n . As shown in the next subsection, these two boundary values can be related to each other, such that knowing C E 0,n from the tree-level recursion described above, allows to infer the α -expansion of all one-loop Selberg integrals and thus -as will be elaborated on below -all integrals Z τ n .

The elliptic KZB associator
As argued in sec. 3.3 of [2] for a general solution of the elliptic KZB equation of the form (3.35), the regularized boundary values C E 0,n and C E 1,n are related according to by the elliptic KZB associator [80] Since the matrices r E 0,n (x k ) are proportional to α , the associator eq. (3.37) yields the αexpansion of C E 1,n . Furthermore, by the lower triangular block structure of the matrices r E 0,n (x k ) with k ≥ 2, only finitely many terms of r E 0,n (Φ τ ) contribute to each order of α .
Instead of reviewing the formalism in detail here, we will rephrase and discuss it in the next section: therein we are going to replace the infinite-dimensional vector S E (z 0 ) by a finitedimensional vector Z τ 0,n of integrals Z τ 0,n that serve as a generating series of Selberg integrals (see appendix D for details). These vectors Z τ 0,n augment the vectors Z τ n of section 3.2 by an auxiliary point z 0 . The detailed discussion of the differential equation and boundary values will be performed for vectors Z τ 0,n . While being already closer to the formalism to be spelt out in subsection 4.2 below, rewriting in terms of the generating series Z τ 0,n renders the representation of the matrices x k finite-dimensional.

Two-point example
The two-point example elaborated in [2] can be summarized as follows: the genus-one Selberg integrals (3.32) for n = 2 are where k 2 ≥ 0 and i 2 ∈ {0, 1}. Note that not all of these integrals are independent: due to the triviality of f (0) = 1 for k 2 = 0 and integration by parts for k 2 = 1, there are the relations The corresponding vector of independent integrals satisfies the differential equation (3.35) with the block-off-diagonal matrices , . . . (3.42) and the block-diagonal one On the one hand, the only non-vanishing entry of the boundary value C E 0,2 is proportional to the tree-level Veneziano amplitude (3.25) with the two independent four-point, tree-level Mandelstam variables s 12 and s 02 associated to the punctures 0, z 2 , 1 and the well-known gamma function Γ. On the other hand, the first entry of C E 1,2 is proportional to the simplest two-point, one-loop configuration-space integral with Mandelstam variables 12 = s 12 + s 02 that occurs at the η −1 -order of eq. (2.39). Therefore, due to the block-off-diagonality of r E 0,2 (x k ) for k ≥ 2, the three 4 × 4 submatrices shown in eqs. (3.42) and (3.43) of r E 0,2 (x 0 ), r E 0,2 (x 1 ) and r E 0,2 (x 2 ) are sufficient to calculate the first entry 1 0 dz 2 exp(−s 12 Γ 21 ) of C E 1,2 up to the second order in α using the associator equation (3.37), which results in the expansion Upon multiplication by e (s 12 +s 02 )ω(1,0|τ ) , this agrees with the leading orders of eq. (2.39) since [19]. The two-point associator only contributes through its entry r E 0,2 (Φ τ ) 1,2 because the only non-zero entry of C E 0,2 occurs in the second line of eq. (3.44) and the desired one-loop integral occurs in the first line of C E 1,2 in eq. (3.45).

0,n
In order to link the two formalisms described in subsections 3.2 and 3.3 above, we will now introduce genus-one integrals Z τ 0,n augmented by an auxiliary insertion position z 0 . In particular, we will evaluate their derivatives with respect to both z 0 and the modular parameter τ in closed form. While the integrals Z τ 0,n contain all genus-one Selberg integrals 13 from subsection 3.3 in their expansion with respect to the variables η j , they differ from the integrals Z τ n used in subsection 3.2 only by the inclusion of an auxiliary point z 0 : they are ideal to bridge the gap between the different languages, in particular between the KZB-type differential equations in z 0 and τ .

Integrals with auxiliary point
In order to augment the space spanned by the integrals Z τ n defined in eq. (2.26), we introduce an auxiliary point z 0 , which is, however, not integrated over. Thus, we will consider integrals associated to the configuration space of the twice-punctured torus, with punctures z 1 = 0 and z 0 . In order to compactly write down a conjectural 14 basis Z τ 0,n of such integrals, let us define a chain of Kronecker-Eisenstein series by and ϕ τ (a 1 , a 2 , . . . , a p ) is obtained from simultaneous permutations of z i → z a i and η i → η a i . The individual factors of Kronecker-Eisenstein series in such a chain ϕ τ (a 1 , a 2 , . . . , a p ) accumulate their η-variables from right to left and, thus, is said to begin at a p and end at a 1 . Chains of Kronecker-Eisenstein series will turn out to be a very versatile tool in the description of integrals Z τ 0,n , and already the Z τ -integrals (2.26) without augmentation feature ϕ τ (1, 2, . . . , n) in the integrand.
Using the notation in eq. (4.1), the vector of the n! basis integrals is given by where we keep on setting z 1 = 0, and the permutations ρ ∈ P(2, 3, . . . , n) acting on the integrand are again lexicographically ordered. The original Koba-Nielsen-factor eq. (2.15) is extended by additional variables s 0j with j = 1, 2, . . . , n as in eq. (3.31), and we use the following shorthand notation for the integration domain, The basis integrals in the components can be denoted by where A = (a 1 , a 2 , . . . , a p ) and for an empty sequence B and similarly Z τ 0,n ((1), (0, B)) = Z τ 0,n (0, B) for A = ∅. This notation directly exhibits the chain structure of the products of Kronecker-Eisenstein series: the dictionary of section 2.2.2 assigns two rooted chains -trees without branching points and root vertices 0 and 14 Also in presence of the augmentation by z0, Koba-Nielsen-type integrals over cycles of f are expected to be expressible in terms of chain integrands as in eq. (2.29) and the ηj-expansion of eq. (4.3). Since we do not present a proof of this claim here, the Z τ 0,n below are a conjectural basis of augmented Koba-Nielsen integrals (with supporting evidence from their closure under z0-and τ -derivatives). We will not repeat the word "conjectural" when referring to the Z τ 0,n as a basis later on.

Further integrals from Fay identities
The main ingredients to the integrals in the basis (4.6) are chains, which can conveniently be associated with a chain graph using the dictionary from section 2.2.2, for example 16 of the form (4.7) Multiplying chains corresponds to combining chain graphs. Whenever the same label appears in two different chains, the Fay identity (2.24) can be used to link the chains and produce tree graphs. In fact, repeated use of Fay identities implies various identities shown and proven in appendix A satisfied by products of chains ϕ τ . Such identities among chains are not only useful for the translation to basis integrals, but also for expressing differential equations satisfied by these basis integrals in closed form in subsections 4.2 and 4.3 below. For each disconnected pair of tree graphs, the associated product of Kronecker-Eisenstein series can be represented as linear combination of the integrals in eq. (4.6), resulting in integrals of the form in the integrand are related to the basis (4.3) via Fay identities with graphical form eq. (2.32), the associated edges between vertices i k and k form tree graphs. The counting of vertices and edges only admits one or two connected components, and the vertices 0, 1 cannot be in the same connected component. Moreover, each vertex k = 0, 1 has only one outgoing edge, while the vertices k = 0, 1 have none. 15 The augmented integrals Z τ 0,2 at two points are for instance obtained by starting from the integrands Ω12(η20)Ω20(η0) and Ω10(η20)Ω02(η2) of Z τ 3 | η 3 →η 0 z 3 →z 0 and rewriting the former as Ω10(η0)Ω12(η2) − Ω10(η20)Ω02(η2). After peeling off the factors of Ω10(η0) and Ω10(η20), one is left with the integrands Ω12(η2) and Ω2(η2) in the first and second component of eq. (4.3) at n = 2, respectively. 16 For notational simplicity, in graphs we use the convention ηa 2 a 3 ...ap = ηa 23...p and similarly for other sums of η-variables associated to sequences of the form (a2, a3, . . . ap).
The precise form of the variables ξ k in eq. (4.8) can be deduced from the graph: for the edge pointing away from the vertex k, the associated ξ k is a combination of η 2 , η 3 , . . . , η n obtained by accumulating (adding) all η j -labels from edges higher in the tree pointing towards the vertex k. In other words, ξ k for a given edge is the sum of all the η j of those vertices j which become disconnected from 0 or 1 through deletion of the edge under consideration.
As detailed in appendix D the integrals eq. (4.8) with suitable restrictions on the i k generate the Selberg integrals eq. (3.32) upon expansion in the ξ k . From this observation as well as the aforementioned formal relation between the n! bases Z τ 0,n and Z τ n+1 , one can already anticipate the potential of the augmented Z τ -integrals eq. (4.3) to relate the two approaches of refs. [1,2] to genus-one α -expansions.

Five-point example
A typical example for (4.8) at n = 5 is the integral which is represented by the following diagram: where the η-variables with multiple indices have been defined in eq. (2.28) . In parallel to the Fay identities for genus-one Z-integrals, one can apply Fay identities to rewrite eq. (4.9) as γ 123450 which is based on the following application of the graphical Fay identity (2.32): Here, we will rewrite the differential equation of subsection 3.3 in the language of integrals Z τ 0,n , which will be the main players in section 5. The genus-one Selberg integrals (3.32) with an auxiliary point used in ref. [2] can be obtained by the methods described in appendix D.1.
Starting from the basis choice for the n-point integrals Z τ 0,n in eq. (4.3), we will now demonstrate that the z 0 -derivatives ∂ 0 Z τ 0,n ((1, A), (0, B)), where we denote ∂ 0 = ∂ z 0 , are expressible in terms of the basis integrals Z τ 0,n . In the n!-component vector notation of eq. (4.3), we will derive a differential equation of the form where the entries of the n!×n! matrix X τ 0,n are linear in s ij , comprise first derivatives in the η j and will be explicitly determined at any n. Moreover, the sole z 0 -and τ -dependence of X τ i.e. one can uniquely identify z 0 -and τ -independent n!×n! matrices r 0,n (x k ) that cast (4.13) into the form Hence, the main result of this section to be derived below is that the n!-component vector Z τ 0,n of augmented Z τ -integrals satisfies an elliptic KZB equation in the auxiliary puncture z 0 .

Deriving the n-point formula
The first step in calculating ∂ 0 Z τ 0,n is to use integration by parts such that all the partial derivatives only act on the Koba-Nielsen factor, followed by an application of eq. (3.16) with an extra point z 0 The second step of the calculation consists of rewriting the term in parenthesis using the Fay identity (2.24) and the chain identities (A.6) to (A.14). The rewriting process is cumbersome, but can be cast into an elegant form using a couple of additional notations and tools. The complete derivation can be found in appendix B and the result is the following: for a sequence C = (c 1 , c 2 , . . . , c m ), a sum of η-variables is denoted by then, the η-variables are assigned to the unintegrated punctures z 0 and z 1 = 1. Thus, defining the decomposition of a sequence C into subsequences C ij where a tilde denotes the reversal of a sequence, the following closed formula can be derived as shown in appendix B.2: Moreover, s (1,A),(0,B) , s a k ,(0,B) and s (1,A),b j denote sums of Mandelstam invariants according to the following general definition for sequences P = (p 1 , p 2 , . . . , p l ) and Q = (q 1 , q 2 , . . . , q m ) Upon writing the partial differential equation (4.21) for the vector of integrals Z τ 0,n in matrix form, we arrive at the central result (4.13) previewed above. The entries of the matrix X τ 0,n are determined by the linear combinations in eq. (4.21), and expanding the Kronecker-Eisenstein series therein in terms of the functions f (k) 01 , we arrive at the elliptic KZB equation (4.14) satisfied by the n!-component vector Z τ 0,n of augmented Z τ -integrals. This generalizes the KZB-type eq. (3.35) for the genus-one Selberg integrals to their generating series Z τ 0,n . The matrices r 0,n (x k ) in eq. (4.14) are independent of z 0 and τ , and one can see from eq. (4.21) that they are linear in the Mandelstam variables s ij and of homogeneity degree k−1 in the variables η j . The matrix r 0,n (x 0 ) additionally involves first derivatives in η j that are counted as homogeneity degree −1.

Alternative form in terms of the S-map
The z 0 -derivative in eq. (4.21) can be compactly rewritten using the so-called S-map defined by (4.23) The S-map (4.23) has been firstly studied in ref. [81] to rewrite BCJ relations [37] and can therefore be used to bring integration-by-parts relations among disk integrals (2.7) into the form Z tree n (a 1 , . . . , a n |S[P, Q], n) = 0 (4.24) with arbitrary disjoint sequences P, Q such that P ∪Q = {1, 2, . . . , n−1}. In a genus-one context, the S-map featured in the n-point proposal for the τ -derivatives of Z τ -integrals (2.26) in ref. [1], which was rigorously derived in ref. [79].
As will be derived in appendix B.3, an alternative form of eq. (4.21) is given by the following formula, The individual terms in the shuffle ϕ τ (B 0j . . .) do not necessarily have the label 0 in the first entry of the chain and thereby involve integrands outside the basis of Z τ 0,n ((1, A), (0, B)) in eq. (4.6). Hence, it remains to apply combinations of Fay identities in the Kleiss-Kuijf form [82,83] in order to manifest the n! entries of Z τ 0,n on the right-hand side of eq. (4.26), where |P | denotes the number of labels in P . The combination of eqs. (4.28) and (4.26) encodes all-multiplicity expressions for the matrix X τ 0,n in eq. (4.13).

Two-point example
The simplest example can be found at two points. The basis vector eq. (4.3) of augmented Z τ -integrals at n = 2 points is given by . (4.29) The partial differential equation (4.13) follows from the closed formula (4.21) or its reformulation in section 4.2.2. Both approaches yield and expose the matrix X τ 0,2 in the notation of eq. (4.13). The expansion 01 r 0,2 (x k ) leads to the elliptic KZB equation (4.14), where the matrices r 0,2 (x k ) are given by

Three-point example
The three-point basis vector (4.3) obeys the following differential equation according to eqs. (4.21) and (4.26), see eq. (4.22) for the s P,Q notation in the first line. By matching with the general form (4.14) of the elliptic KZB equation, one can read off the following 6 × 6 matrices r 0,3 (x k ):  36) where k ≥ 2.

0,n
As we will show in this section, the n! basis Z τ 0,n in eq. (4.3) also closes under τ -derivatives. Similar to the homogeneous first-order equation (4.13) in z 0 , the τ -derivative of Z τ 0,n will be cast into the form where the explicit form of the n!×n! matrices D τ 0,n , B τ 0,n will be determined below. We have grouped the matrices according to the τ -and z 0 -dependence which is solely carried by f (k) 01 and Eisenstein series G k , where the entries of D τ 0,n , B τ 0,n and therefore all the r 0,n (b k ), r 0,n ( k ) are again linear in s ij . By construction, the n!×n! matrices r 0,n (b k ), r 0,n ( k ) no longer depend on z 0 and τ . The appearance of f (k) 01 in the differential equation in τ is shared by the operator eq. (4.14) in the z 0 derivative, and we will see from two perspectives that the accompanying n!×n! matrices are related by Note that the additional zero in the subscript distinguishes the n!×n! matrices r 0,n ( k ) in eq. (4.38) from the (n−1)!×(n−1)! matrices r n ( k ) in the differential equation (3.19) of the Z τ -integrals without augmentation.

Deriving the n-point formula
The evaluation of the τ -derivative 2πi∂ τ Z τ 0,n ((1, A), (0, B)) with A = (a 1 , a 2 , . . . , a p ) and B = (b 1 , b 2 , . . . , b q ) follows the same steps as the z 0 -derivative in section 4.2. While the details are shown in appendix C, we give an overview in the following paragraphs.
First, the mixed heat equation (2.25) and integration by parts can be used to find an expression where all the derivatives only act on the Koba-Nielsen factor. Then, for the z j -derivatives eq. (4.15) and for the τ -derivative the equation where the step function θ j≥1 is taken to be 1 for j ≥ 1 and zero for j = 0. From this equation similar identities as for the z 0 -derivative, mainly based on the Fay identity of Kronecker-Eisenstein series, lead to the following expression involving the S-map in eqs. (4.27) and (4.28) 01 Z τ 0,n ((1, A), (0, B)) where Ω ± 01 (±ξ) = ±∂ ξ Ω 01 (±ξ) .
The closed formula (4.44) or (4.42) along with eq. (4.28) lead to the matrix equation The r 0,n (b k ) and r 0,n ( k ) are all linear in s ij and independent of z 0 and τ . Moreover, their instances at k ≥ 2 are both homogeneous polynomials of degree k−2 in the η j , whereas r 0,n ( 0 ) is a combination of 2ζ 2 s 012...n , η −2 j and ∂ 2 η j . As previewed in eq. (4.39), one can confirm from eq. (4.44) or eq. (4.42) that all the r 0,n (b k ) with k ≥ 2 agree with the operators r 0,n (x k−1 ) in the z 0 -derivative eq. (4.14). An alternative derivation of eq. (4.39) will be given in section 4.4.

Elliptic KZB system on the twice-punctured torus
In the previous subsections, the two partial differential equations (4.14) and (4.45) satisfied by the vector Z τ 0,n defined in eq. (4.3) have been identified. Together, they form the system of differential equations which is an elliptic KZB system on the moduli space of the twice-punctured torus, with (fixed) puncture z 1 = 0 and (variable) puncture z 0 [67]. While we have considered the two differential equations separately so far, in this section properties of the corresponding matrices r 0,n (x k ), r 0,n (b k ) and r 0,n ( k ) are determined employing the interplay of both differential equations. In order to investigate these commutation relations, an unspecified system is considered, which has the same structure where unspecified representations of braid matrices x k , derivations k and further generators b k act on an abstract solution I τ 0 . The commutativity of the mixed second derivatives (Schwarz integrability condition) imposes various constraints on the x k , k , b k which will serve as cross-checks for the n!×n! representations r 0,n (·) encoded in eqs. (4.26) and (4.42). By the components 2πi∂ τ f of the mixed heat equation (2.25) for real z 01 , the commutator has to vanish. In particular, the first line of eq. (4.55) has to vanish separately, can be rewritten as [79] 17 and generate terms ∼ G 2 = G 2 − π Im τ that do not occur in the second line of eq. (4.55). By repeating these arguments for the KZB system (4.52) obeyed by the n!-component vector Z τ 0,n in eq. (4.3), we arrive at eq. (4.39) independent of the explicit form of r 0,n (b k ) and r 0,n (x k ) obtained in sections 4.2 and 4.3.
Hence, the leftover integrability constraints after imposing eq. (4.56) are (4.58) 17 The simplest examples of eq. (4.57) are In order to infer relations among the commutators, the products f . have to vanish separately, and one can read off identities like and more generally (with ≥ 1 and k ≥ 4) By iterating the first relation, the adjoint action ad 0 (·) = [ 0 , ·] turns out to be nilpotent when acting on x , ad 0 (x ) = 0 , ≥ 1 , As a consistency check of the results for ∂ z 0 Z τ 0,n and ∂ τ Z τ 0,n , we have confirmed validity of the above relations [r 0,n (x ), r 0,n ( k )] for numerous configurations (n, k).
The nilpotency property (4.62) is known to also hold for Tsunogai's derivations [76], The Z τ -integrals (3.13) without augmentation introduce conjectural (n−1)! × (n−1)! matrix representations r n ( k ) through their differential equations (3.19) which have been tested to preserve eq. (4.64) for a wide range of k and n [1]. For the analogous n!×n! matrices r 0,n ( k ) seen in the KZB system (4.52) of the augmented Z τ 0,n , we conjecture that they furnish another representation of the derivation algebra. This conjecture not only applies to eq. (4.64), conjecture : do not seem to carry over to the r 0,n ( k ), e.g. eq. (4.67) with k → r 0,n ( k ) is already violated at n = 2. Instead, eq. (4.67) and some of its higher-weight analogues [78,19] have been checked to hold upon assigning The deformation in the second line resonates with the additional appearance of b k = x k−1 in the augmented differential eq. (4.52) with respect to τ compared to the non-augmented one in eq. (3.18). As we will see in eq. (5.69), the combination r 0,n ( k ) + r 0,n (x k−1 ) in eq. (4.68) appears naturally when relating the representations r 0,n ( k ) and r n ( k ) of the derivations. Note that depth-two relations such as eq. (4.66) were tested to hold for both assignments eq. (4.68) and k → r 0,n ( k ).

Total differential of Z τ 0,n integrals
In summary of the differential equations in sections 4.2, 4.3 and as a way of manifesting the Schwarz integrability conditions, we will now spell out the total differential of the Z τ 0,n -integrals (4.6), to bring the differential of Z τ 0,n into the form (3.3). After eliminating the r 0,n (b k ), the total differential d = dz 0 ∂ z 0 + dτ ∂ τ following from the KZB system (4.52) takes the form where the characteristic combinations  In the third and sixth line from below, we have used that (4.72) The fact that the f (k =0) 01 on the right-hand side of eq. (4.71) combine to ψ (k) 01 manifests the equality of the operators r 0,n (b k ) and r 0,n (x k−1 ) in the KZB system (4.52). Based on the total differential eq. (4.69), the formalism of [84] can be used to obtain the coaction of the augmented Z τ -integrals, also see section 7.2 of [1] for the coaction of plain Z τ -integrals (2.26).

Identification and translation
While z 0 -and τ -derivatives of the integrals Z τ 0,n have been discussed in full generality in the previous section, let us now compare the two resulting approaches by investigating their boundary conditions, limits and solutions, respectively. As a guiding principle, we will explore how the representations r n and r E 0,n of the algebra generators 18 k and x k in eqs. (3.19) and (3.35), respectively, are related to each other and to the representations r 0,n ( k ) in eqs. (4.14) and (4.37) above.

Overview
Boundary values C τ 0,n and C τ 1,n of Z τ 0,n for the limits z 0 → 0 and z 0 → 1, respectively, are particularly important since they allow for an explicit expansion of the integrals Z τ n in α using the elliptic KZB associator. Simultaneously, it is those boundary values, which finally allow to find the link between the matrices r n ( k ), r E 0,n (x k ), r 0,n ( k ) and r 0,n (x k ). Due to the poles at z 0 = 0 and z 0 = 1 of f (1) 01 in the differential equation (4.14), these limits are singular. The regularization leading to the corresponding non-vanishing finite values will be derived and related to the genus-zero and genus-one Z-integrals Z tree n and Z τ n , respectively, in this section. The boundary values C τ 0,n and C τ 1,n to be considered here are the finite-length cousins of the infinitely long boundary vectors C E 0,n and C E 1,n in eq. (3.36) for the genus-one Selberg integrals. The main results to be derived below are the expressions in terms of bases of genus-zero integrals Z tree n+2 in eq. (2.9) and genus-one integrals Z τ n in eq. (2.26). The entries of the n! × (n−1)! matrices U BCJ n and (n−1)! × n! matrices P n are rational functions of the s ij with 0≤i<j≤n and will be defined in the discussions around eqs. (5.15) and (5.42), respectively.
Based on the associator relation adapted to the n!×n! matrices r 0,n (x k ) constructed in section 4.2, eqs. (5.1) and (5.2) connect the genus-one integrals Z τ n with their genus-zero counterparts Z tree n+2 , Using the expansion of the elliptic KZB associator [80], each term in the expansion of the genus-one integrals in α and η j can be obtained via elementary operations. Eq. (5.4) is the generating-function reformulation of the method in [2], where the matrices r 0,n (x k ) are now finite dimensional. The application of the formalism to integrals Z τ n with kinematic poles (i.e. factors of f (1) ij in the integrand) have not been investigated in ref. [2]. However, when doing so using the method 18 Sometimes these generators are referred to as letters and their respective entirety as alphabets. of ref. [2], the matrices P n allow to project on the desired configuration-space integrals in the same way as they do in the language using generating functions in the current paper.

Lower boundary value C τ 0,n in the z 0 -language
In this subsection, we will derive the expression (5.1) for the lower boundary value C τ 0,n which is defined as the regularized limit The vector Z τ 0,n has been introduced in eq. (4.3), and r 0,n (x 1 ) is the representation of the letter x 1 appearing in the KZB equation (4.14) along with the singular f (1) 01 . Here we are going to derive the main mechanism necessary for evaluating eq. (5.6).
The key observation to be demonstrated below is that the matrix r 0,n (x 1 ) in the exponent of eq. (5.6) can be set to its eigenvalue −s 012...n when acting on the z 0 → 0 asymptotics of Z τ 0,n . With the asymptotic result (5.16), the task is therefore to show that which will turn out to be independent on the BCJ basis vector Z tree n+2 that U BCJ n acts on. Our proof of eq. (5.17) is based on the continuity of (2πiz 0 ) s 012...n 2πi∂ τ Z τ 0,n at z 0 = 0 due to the absence of singular terms in eq. (4.45). We can therefore equate the two orders of performing the limit z 0 → 0 and the τ -derivative.
• On the one hand, eq. (5.16) can be differentiated with respect to τ after taking the z 0 → 0 limit, which only acts via 2πi∂ τ ω(1, 0|τ ) = G 2 −2ζ 2 and yields • On the other hand, exchanging the limit and the partial derivative in eq. (5.18) leadsaccording to eq. (4.45) -to the identity where we have used that r 0,n (b k ) = r 0,n (x k−1 ) (cf. eq. (4.56)) and that for k ≥ 2 (and z 0 ∈ R in case of k = 2) Since the derivative (2πiz 0 ) s 012...n 2πi∂ τ Z τ 0,n is continuous at z 0 = 0, eqs. These equations imply that the columns of U BCJ n are eigenvectors of r 0,n (x 1 ) and r 0,n ( 0 ) for the eigenvalues −s 012...n and 2ζ 2 s 012...n , respectively. This proves the lemma (5.17) and ultimately the main claim (5.1) of this subsection. Moreover, we see that the representations r 0,n ( 0 ) and r 0,n (x 1 ) as well as r 0,n ( k ) and r 0,n (x k−1 ) for k = 4, 6, . . . acting on U BCJ n and thus, on the lower boundary value, are equivalent up to constant factors.

Two-point example
Let us approve the above findings on the two-point example Z τ 0,2 from eq. (4.29). The finite part at z 0 = 0 according to eq. (5.13) and s 23 = s 02 is given by Note that the other two eigenvalue equations of (5.21) are also straightforwardly checked via eqs. (4.31) and (4.47).

Three-point example
At three points, the explicit form of Z τ 0,3 can be found in eq. (4.32), and its finite part at z 0 = 0 is determined as follows by eq. (5.13): . (5.28)

Upper boundary value C τ 1,n in the z 0 -language
In order to derive the claim (5.2) for the upper boundary value C τ 1,n , one will have to evaluate Similar to the procedure in the last subsection, we will evaluate the limit z 0 → 1 separately for the Koba-Nielsen part and the remainder of the integrand, before commenting on the action of the matrix representation r 0,n in the exponent of the regulating factor.
The right-hand side involves the genus-one integrals (2.26) with the shifted Mandelstam variables s ij from eq. (5.32) in the Koba-Nielsen factor. The special cases of eq. (5.34) with B = ∅ yield a particularly simple form for the first (n−1)! components of, where Z τ n defined in eq. (3.13) comprises (n−1)! basis integrals Z τ n (I n |1, A). Also the remaining components of eq. (5.34) fall into this basis: evaluating the shuffles on the right-hand side of eq. (5.34) defines a n! × (n−1)! matrix U n with entries in {0, 1} Note that the shuffle decomposition (5.36) of genus-one integrals in the context of the boundary value C τ 1,n is the analogue of the BCJ decomposition (5.15) relevant for the genus-zero integrals in C τ 0,n . In particular, both n!×(n−1)! matrices U BCJ n and U n feature a unit matrix within their upper (n−1)! × (n−1)! block.

The minimal r 0,n (x 1 ) eigenvalue
What remains to be discussed here is the regulating factor: similar to the previous subsection, it turns out that the finite value in eq. (5.37) emerges from the regularized boundary value C τ 1,n . However, recovering the genus-one Z τ -integrals is more subtle than the tree-level integrals from C τ 0,n , since the validity of eq. (5.34), i.e. the absence of rest terms, relies on the exponent s 01 of the regulating factor (−2πi(1−z 0 )) s 01 and would generally fail if there were further contributions s ij . Therefore, one has to make sure that the dominating eigenvalue of r 0,n (x 1 ) in C τ 1,n , c.f. eq. (5.29), is −s 01 , i.e. the eigenvalue with the minimal real part as opposed to the eigenvalue −s 012...n with maximal real part in the calculation (5.17). This can be achieved by employing a projection to the eigenspace of −s 01 as follows: interchanging limit and τ -derivative of (−2πi(1−z 0 )) s 01 Z τ 0,n using the continuity of the latter leads again on the one hand according to eqs. (1−k) G k r n ( k )|s ij Z τ n |s ij .
On the other hand, we find similar to the calculation in eq. (5.19) such that comparing the coefficients of G k leads to the matrix equations While the second and third equations show the action of the representations r 0,n ( k ), r 0,n (x k ) and r n ( k ) on the vector space spanned by the columns of U n , which will be discussed in more detail in subsection 5.4.3, we shall next elaborate on the first one.

Three-point example
In the three-point case n = 3, the eigenvalue decomposition (5.41) leads to the matrix . . .

1,n
The system of differential equations (4.52) for Z τ 0,n contains the fundamental equation (3.18) for Z τ n as well as the elliptic KZB equation (3.35) for S E n (z 0 |τ ). As a consequence, both corresponding solution strategies can be applied to calculate the integrals Z τ n appearing in the regularized boundary value C τ 1,n given in eq. (5.45). The corresponding calculations presented in the following two subsections are applications of the methods developed in refs. [1,2] and reviewed in subsection 3.2 and subsection 3.3 which builds upon the analysis of C τ 0,n , C τ 1,n in the previous subsections.

The elliptic KZB associator
Having the elliptic KZB equation (4.14) at hand, analogously to the discussion in subsection 3. The associator equation (5.56) is the backbone in calculating the α -expansions of Z τ n from differential equations in z 0 . It relates the n-point, genus-one integrals Z τ n containing the planar, one-loop configuration-space integrals with (arbitrary) Mandelstam variabless ij for 1 ≤ i < j ≤ n to the (n+2)-point, tree-level Z-integrals Z tree n+2 : the elliptic KZB associator can be represented by the generating series of eMZVs with the letters being the matrices r τ 0,n (x k ) appearing in the elliptic KZB equation of Z τ 0,n [80] r 0,n (Φ τ (x k )) = w≥0 k 1 ,...,kw≥0 The matrices r τ 0,n (x k ) are proportional to s ij and therefore to α (cf. eq. (2.1)), such that eq. (5.57) is simply the α -expansion of Φ τ . When plugged into the associator equation (5.56), it yields the α -expansion of the genus-one integrals Z τ n from the α -expansion of the genus-zero integrals Z tree n+2 . To obtain the α -expansion of Z τ n up to the order o τ α,max , words r τ 0,n (x k 1 . . . x kw ) up to the maximal word length 0 ≤ w ≤ w max with is the minimal order in α of Z tree n+2 . However, the actual genus-one configuration-space integrals appearing in one-loop open-string amplitudes are the coefficients of the η-variables from the integrals Z τ n . Since the matrices r τ 0,n (x k i ) are homogeneous in these variables of degree k i −1 ≥ −1, a configuration-space integral which is given by an η-degree k ≥ 1−n coefficient of Z τ n receives at most non-trivial contributions from words r τ 0,n (x k 1 . . . x kw ) with word length 0 ≤ w ≤ w max satisfying w = (k 1 + · · · + k w ) − k . (5.60) To summarize, in order to calculate the α -expansion of an n-point, genus-one configurationspace integral appearing as an η-coefficient 19 of degree k of Z τ n up to the order o τ α,max , only the finitely many words r τ 0,n (x k 1 . . . x kw ) with contribute to the corresponding η-coefficients of the elliptic KZB associator (5.57) and, thus, have to be included in the associator eq. (5.56). Upon rewriting the eMZVs in eq. (5.57) in terms of iterated Eisenstein integrals [19], we have checked eq. (5.56) to reproduce the α -expansion generated by eq. (3.23) for a wide range of orders in α and η j . Note that the results of eq. (5.56) for the integrals Z τ n |s ij relevant to one-loop open-string amplitudes no longer depend on s 01 , which is why the conditions eq. (2.3) on its real part to not pose any restrictions on the physical applications. . . x kw ) with word length 0 ≤ w ≤ 3 have to be considered. However, the condition (5.61) only selects the words which are explicitly written down in the following to contribute non-trivially:

Two-point example
Therefore, denoting the words written down above and the corresponding higher-order terms O(s 4 ij ) which give the order η −1 of the elliptic KZB associator by r 0,2 (Φ τ (x k )) | η −1 , we obtain the equation for the configuration-space integral in agreement with eq. (2.39) and [1].

Two organization schemes for α -expansions
The other expansion method for the genus-one Z-integrals Z τ n put forward in ref. [1] is reviewed in subsection 3.2 and consists of solving the differential eq. ω(k 1 , k 2 , . . . , k w |τ ) (5.67) is not obvious from the first glance at these types of series but guaranteed by the arguments in refs. [1,2] and the previous sections. The initial value Z i∞ n on the left-hand side is related to Z tree n+2 on the right-hand side by an s ij -and η j dependent (n−1)!×(n−1)! matrix described in ref. [1], see eq. (3.24) for the two-point example.
The discussion of the previous subsections yields a streamlined way of showing directly that both sides of eq. (5.67) obey the same differential equation in τ . The differential eq. (3.18) (with s ij in the place of s ij ) holds for the left-hand side by construction, and the analogous equation for P n C τ 1,n = e s 01 ω(1,0|τ ) Z τ n |s ij on the right-hand side can be inferred from properties of the augmented Z-integrals: according to the calculation (5.39), this differential equation for P n C τ 1,n can be written as where the identities (5.40) together with eq. (5.43) can be used to relate the matrix representations of different sizes P n r 0,n (x 1 )U n = −s 01 1 (n−1)!×(n−1)! , P n r 0,n ( 0 )U n = r n ( 0 )|s ij + 2ζ 2 s 01 , (5.69) P n (r 0,n ( k ) + r 0,n (x k−1 )) U n = r n ( k )|s ij , k ≥ 4 .
As a consequence, eq. (5.45) can be simplified to 2πi∂ τ e s 01 ω(1,0|τ ) Z τ n |s ij = 2πi∂ τ P n C τ 1,n (5.70) which is equivalent to eq. (3.18) for the Mandelstam variabless ij defined in eq. (5.32) after employing This concludes the direct proof that both sides of eq. (5.67) obey the same differential equation in τ . A direct comparison of the respective initial values as τ → i∞ may be challenging, but the consistency in this limit is guaranteed since both sides have been derived in refs. [1,2] and the previous sections. Note that the combination r 0,n ( k ) + r 0,n (x k−1 ) in the third line of eq. (5.69) also arises when adapting depth-three relations eq. (4.67) in the derivation algebra to the twice punctured torus.

Conclusion
In refs. [1,2] two different constructions for the α -expansion of configuration-space integrals in one-loop open-string amplitudes have been put forward. In both references the (elliptic) multiple zeta values in the α -expansions are derived from different types of differential equations. Here we have connected these two approaches within a more general framework and, in particular, shown that • both approaches and the definitions therein can be traced back to one class of iterated integrals, called augmented (genus-one) Z-integrals, a vector of n-point basis integrals Z τ 0,n is defined in eq. (4.3). Besides the usual fixed puncture z 1 = 0 on the torus C/(Z+τ Z) appearing in the definition of the genus-one Z-integrals of ref. [1], the integrals Z τ 0,n are augmented by a second unintegrated puncture z 0 with z 1 = 0 < z 0 < 1 as in ref. [2]. Thus -apart from Mandelstam invariants s ij -the augmented integrals depend on two parameters: the modular parameter τ of the torus and the additional puncture z 0 .
• differentiation of Z τ 0,n with respect to the two parameters τ and z 0 leads to a homogeneous linear system of two partial differential equations -an elliptic KZB system (4.52) on the twice-punctured torus • the genus-one Selberg integrals from ref. [2] are linear combinations of the components of the augmented Z-integrals Z τ 0,n ; they can be recovered according to the discussion in appendix D.
• the genus-one Z-integrals from ref. [1] -and hence the configuration-space integrals in n-point, planar, one-loop open-string amplitudes -are recovered as regularized boundary values of the augmented integrals in Z τ 0,n as z 0 → 1. Correspondingly, the two differential equations in the system (4.52) can be solved independently for the string integrals in the limit of z 0 → 1 via integration w.r.t. τ or z 0 . The respective initial values at τ → i∞ and z 0 → 0 are reduced to (n+2)-point genus-zero integrals whose α -expansion in terms of multiple zeta values is known from several all-multiplicity methods, see e.g. [35,11,14,44]. As summarized in subsection 5.4, this yields the two approaches in refs. [1,2] to calculate the α -expansion of one-loop open-string amplitudes in terms of (elliptic) multiple zeta values and iterated Eisenstein integrals.
• calculating the α -expansion using the integrals Z τ 0,n involves elementary operations only: differentiation in formal expansion variables η j and matrix algebra, with the matrices being determined by the elliptic KZB system (4.52). The entries of the corresponding n!×n! matrix representation determine the coefficients in the α -expansion and are explicitly given for an arbitrary number of points n in eqs. (4.21) and (4.44).
• the operators appearing in an elliptic KZB system of the form (4.52) satisfy the commutation relations (4.61) which serve as consistency checks for our explicit matrix representations.
Our construction of particular integral representations Z τ 0,n for the solution of an elliptic KZB system on the twice-punctured curve leads to the question as how it may be embedded into the existing Mathematics literature about similar systems. In particular, its connection to ref. [67] and the representation of the algebra generators therein appearing in the differential equations as nested commutators, should be clarified. the edge pointing away from i is given by the corresponding sum 1 ,a i (η a i a i+1 ...ap ) , where we use the convention η a i a i+1 ...ap = η a i,i+1...p for sums of η-variables associated to a sequence (a i , a i+1 , . . . , a p ) in graphs. It should always be clear from the context that whether edges without weights refer to the genus-zero notation or the genus-one notation where the weights are only implicit. In particular, in this and the next section, we exclusively discuss the genus-one case.
• The same accumulation of the η-variables is used for directed tree graphs, if it is not denoted explicitly: the weight of the edge pointing away from the vertex i is the sum of all the η-variables associated to the edges pointing towards i via a chain of Kronecker-Eisenstein series. For example The following identities, which are proven in appendix A.1, are particularly useful: for finite, disjoint sequences A = (a 1 , . . . , a p ), B = (b 1 , . . . , b q ), C = (c 1 , . . . , c m ) and D = (d 1 , . . . , d l ) as well as distinct labels r, r 0 , r 1 not contained in any of the sequences A, B, C, D, we find • the shuffle product of two chains beginning at the same point r where again by our convention η r = −η A = −η a 1 ...ap .

A.1 Shuffle and concatenation identities
The first identity is the concatenation (A.6) of two chains, where one has a shifted η-variable (A, r, B) .  (A, r, B) .
The second identity is the shuffle relation (A.8), which can be proven by induction in the length of the sequence A (and by the symmetry in A and B). Thus, let us assume that A = ∅ or B = ∅, then it is trivially satisfied according to the definition (4.2), i.e. ϕ τ (r) = 1. For A = (a 1 ) and B = (b 1 ), we simply find the Fay identity (2.24) for the Kronecker-Eisenstein series ϕ τ (r, a 1 )ϕ τ (r, b 1 ) = Ω r,a 1 (η a 1 )Ω r,b 1 Now, let us assume that it holds for (a 2 , . . . , a p ) and B = (b 1 , . . . , b q ), as well as for A = (a 1 , . . . , a p ) and (b 2 , . . . , b q ) and use the Fay identity for the induction step to show the identity for A and B a 1 , a 2 , . . . , a p )ϕ τ (r, a 1 , B) a 1 , a 2 , . . . , a p )ϕ τ (a 1 , B) a 1 , (a 2 , . . . , a p where we have used the concatenation property in the intermediate step. The next identity (A.10), is similar to the one before, but we have to be more careful with the shifts in the η-variables. For C = ∅ or D = ∅, it is trivial. Thus, let C = (c 1 , . . . , c m ) and D = (d 1 , . . . , d l ) with m, l = 0. For m = l = 1, it is simply the Fay identity. For m, l ≥ 2, we can iteratively apply the Fay identity: where according to eq. (A.2), To finish, let us proof the identity (A.14), for A = (a 1 , a 2 , . . . , a p ) and B = (b 1 , b 2 , . . . , b q ), where we already make use of the notation in eq. (4.19) for subsequences. It is trivially satisfied for A = ∅, thus, let us assume that it holds for A = (a 1 , . . . , a p ) and show it for A 0 = (a 0 , A) = (a 0 , a 1 , . . . , a p ). We do this in two steps. First, the following combinatorial identity is proven It is trivially satisfied for A = ∅ and for A = (a 1 ), it takes the form (−1)(a 0 ) (a 1 ) + (a 0 , a 1 ) = −(a 1 , a 0 ) .

B Derivation of the n-point z 0 -derivative
In this section, we derive the n-point formula for the z 0 -derivative. The starting point is eq. (4.16), where A = (a 1 , . . . , a p ) and B = (b 1 , . . . , b q ) are disjoint sequences without repetitions such that A ∪ B = {2, 3, . . . , n}, and the proof is split into three parts. Moreover, we will write γ in the place of γ 12...n0 for the integration domain (4.5) throughout the appendices. First, we derive some preliminary identities which will be useful to rewrite the term Using these identities, we then give the proof of the closed formula (4.21). Throughout this section, we accompany the crucial identities by the graphical notation for chains of Kronecker-Eisenstein series in order to facilitate the readability of the proofs for the n-point z 0 -and τ -derivatives.

B.1 Preliminary identities
Instead of only investigating the term (B.2) with the factor f (1) a k ,b j , we consider the corresponding generating series and, thus, the product of chains In particular, we will use the following identities: for k, j = 0, i.e. a k = 1 = a 0 and b j = 0 = b 0 , which can be depicted as while for k = 0 and j = 0 and for k = 0 and j = 0 The procedure to rewrite the Ω a k ,b j in eq. (B.2) is the following: first, we move the indices a k and b j in ϕ τ (1, A)ϕ τ (0, B) next to 1 and 0, respectively, by means of eq. (A.14), which yields for k = 0 i.e. and for j = 0 i.e. As a consequence, for k, j = 0 i.e. for k = 0, j = 0 )Ω a k ,0 (ξ) (B. 16) i.e. and for k = 0, j = 0 i.e.  20) i.e. The last sum, with the factor Ω 1,0 (ξ)Ω 1,a k (η A i ,p+1 − ξ)Ω 0,b j (η B l,q+1 + ξ), can be rewritten in terms of the original chains, using eqs. (B.10) and (B.12) in the reverse direction, leaving a shift of ∓ξ in the variables η a k and η b j , respectively, The remaining two sums in eq. (B.20) can either be written in terms of products of two chains, the first starting at 1 and the second at 0. This will yield the closed formula for the derivatives of Z τ 0,n . Or, they can be expressed in terms of the S-map. Both formulae are derived in the next two subsections.
To summarize, we have so far for k, j = 0 the identity i.e.
i.e. and for k = 0, j = 0 For k, j = 1, the first sum in eq. (B.23) is given by and similarly for the second sum The sum in in eq. (B.25) for k = 0, j = 0 can be rewritten as and, similarly, the one in eq. (B.27) for k = 0, j = 0 as follows Plugging eqs. (B.29) and (B.30) into (B.23), we can conclude that for k, j = 1 the corresponding graphical equation is obtained from eq. (B.24) by simply folding back any branch fork to a sum of single chains using the shuffle product, while all the η-variables and the corresponding shifts ±ξ stay the same (and will not be depicted in the following equation for notational simplicity) and similarly using eq. (B.31) in eq. (B.25) yields for k = 0, j = 0 which is also obtained from the graphs in eq. (B.26) by folding back any branches to a single chain using the shuffle product  A (b j ,B l,j B j+1,q+1 ) The non-trivial extraction of the ξ 0 -term occurs in the term proportional to Ω 10 (ξ) in eqs. (B.33) to (B.37), i.e. Ω 1,0 (ξ)ϕ τ (1, A)ϕ τ (0, B) with some shifts in the η-variables, and can be treated using the expansion in ξ of the single factors to project out differential operators from the chains. The extraction in the remaining terms can be done by (expanding factors of the form Ω ij (η + ξ) around ξ and) simply setting ξ = 0. For a single factor of the Kronecker-Eisenstein series, we find for k = 0 For a whole product, i.e. a chain, this procedure exactly reproduces the product rule due to the cross-terms and leads to for k, j = 0. Putting everything together, one indeed arrives at the closed formula eq. (4.21) for ∂ 0 Z τ 0,n ((1, A), (0, B)).

B.3 S-map formula
Alternatively, the derivatives of Z τ 0,n ((1, A), (0, B)) can be expressed in terms of the S-map as in eq. (4.26). For this purpose, we proceed from eq. (B.23) for k, j = 0 again using the shuffle identity (A.8) and the reflection property (A.12), but slightly differently than in the previous subsection: we want to keep a factor of Ω a k ,b j (η B l,q+1 + ξ) or Ω b j ,a k (η A i ,p+1 − ξ) as a bridge between the chains of A and B. For the first sum in eq. (B.23), the calculation amounts to where we applied the shuffle identity (A.8) and eq. (A.14) in the reverse direction in the first and third equality, respectively. The same calculation leads to a similar result for the second sum in eq. (B.23) ) Similarly, we find for the sum in (B.25) for k = 0, j = 0 and for the sum in eq. (B.27) with k = 0, j = 0 Thus, the identity (B.23), valid for k, j = 0 can be rewritten as follows Similarly, the identity (B.25) for k = 0, j = 0 is given by i.e. (B.52) Note that these three identities indeed all have a factor of Ω a k ,b j , which is the backbone in the formulation in terms of the S-map. This can be seen by summing these identities over all Comparing with eq. (4.23), we see that the second and third sum explicitly involve the definition of the S-map, except for the shift ∓ξ in the variables η a k and η b j . In order to obtain the S-map representation of the z 0 -derivative eq. (B.39) we simply need to extract the ξ 0 part of eq. (B.53), where we immediately recover the S-map formula (4.26).

C Derivation of the n-point τ -derivative
In this section, we determine the action of 2πi∂ τ on the integrals Z τ 0,n in eq. (4.6) to derive the corresponding formulae (4.42) and (4.44). The techniques in this appendix generalize those in [1,79], where the τ -derivatives of Z τ -integrals without augmentation were studied 20 . First, we recall that the τ -derivative of the Koba-Nielsen factor is given by eq. (4.40). Second, the action (up to integration by parts) on a chain with C = (c 1 , c 2 , . . . , c m ) can be expressed using the mixed heat equation for real z i , z j as follows: 20 See in particular section 4 and appendix A of [1] as well as section 4 and appendix E of [79].
We again use a step function θ j≥k which is taken to be 1 for j ≥ k and zero for j < k. Therefore, denoting (1, A) = (a 0 , a 1 , . . . , a p ) and (0, which implies eq. (4.41). As for eq. (B.1) in the calculation of the z 0 -derivative, this equation is the starting point to determine the τ -derivative of Z τ 0,n ((1, A), (0, B)). In the following two subsections, we give the corresponding formula in terms of the S-map and a closed expression.

D Recovering genus-one Selberg integrals
In this appendix, we relate the language used in section 4 to the genus-one Selberg integrals (3.32) employed in ref. [2].

D.1 Recovering genus-one Selberg integrals
Let us begin with defining the generating series of genus-one Selberg integrals with expansion variables ξ : where the i satisfy the admissibility condition (3.33), i.e. i ∈ {0, 1, +1, +2, . . . , n}. The integrals T τ 0,n are related to the augmented Z-integrals Z τ 0,n by Fay identities. This connection can be understood from the graphical approach in section 4.1 since the integrand Ω i 2 2 (ξ 2 ) . . . Ω inn (ξ n ) of eq. (D.1) with admissible i agrees precisely with the one in eq. (4.8). Once the ξ are identified as suitable linear combinations of η 2 , η 3 , . . . , η n to be denoted by η C ( i) , one can use Fay identities to expand are fixed from the discussion in section 4.1.1: the labels i = (i 2 , i 3 , . . . , i n ) of the integrand Ω i 2 2 (ξ 2 ) . . . Ω inn (ξ n ) in eq. (D.1) define a graph which in turn identifies ξ = η C ( i) with a linear combination of η 2 , η 3 , . . . , η n through the weights of its edges as explained below eq. (4.8).
In the same way as the Z τ 0,n ((1, A), (0, B)) are gathered in the n!-component vector Z τ 0,n in eq. such that the full basis transformation is given by In ref. [2], it was shown that the differential equation of the vector of all admissible genus-one Selberg integrals (3.32) of a certain weight w = k 2 + k 3 + · · · + k n , i.e. This equation expresses the relation between the matrices r 0,n (x k ) from section 4.2 and the submatrices r E 0,n (x k,w ) of r E 0,n (x k ) appearing in the KZB equation in ref. [2].

D.2.1 Two-point example
Let us illustrate and check the formula (D.19) for the two-point example and the block-matrices given in eqs. (3.42) and (3.43). The latter encode the matrices r E 0,n (x k,w ) for k = 0, 1 at weight w = 0

1,n
In this section, we address the claim above eq. (5.34). We argue that subleading terms appear in the limit z 0 → 1 of Z τ 0,n ((1, A), (0, B)) and estimate their order in 1−z 0 by first giving the two-point example and afterwards generalizing to n points.
Still, this example illustrates that the analysis of vector entries with additional suppression by powers of w 0 does not play any role for the above conclusion: all components of eq. (E.9) vanish when the limit w 0 → 0 is performed in presence of the regulating factor (−2πi(1−z 0 )) s 01 .