Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes

Within the program of studying iterated integrals on Riemann surfaces of various genera, the genus-zero case, which leads to multiple zeta values (MZVs) [1–3], is the starting point and takes the most prominent rôle. During the last years, however, the genus-one situation has received more attention: various iterated integrals on the elliptic curve as well as associated periods and elliptic associators have been investigated [4–7].

The simplest genus-one generalizations of MZVs are elliptic multiple zeta values (eMZVs), which arise from iterated integrals on the once-punctured elliptic curve, that is the elliptic curve where the origin is removed [8]. In this article, the notion of eMZVs is extended to twisted elliptic multiple zeta values (teMZVs), which are iterated integrals on a multiply-punctured elliptic curve. While for eMZVs it is sufficient to remove the origin, teMZVs arise when a lattice with rational coordinates as visualized in figure 1 is removed from the elliptic curve.

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The iterated integrals to be considered are performed over a path parallel to the real axis and are therefore a generalization of Enriquez' A-cycle eMZVs. If no lattice is removed, we obtain A-cycle eMZVs by definition⁴. A slight technical difficulty, which was absent for eMZVs, is that the integrands giving rise to teMZVs might have additional poles along the path of integration. We address the problem by suggesting a rather natural regularization scheme, which essentially amounts to integrating over an infinitesimal deformation of the real axis.

A crucial tool in the study of eMZVs was the existence of a certain first-order linear differential equation, expressing eMZVs as special linear combinations of iterated integrals of Eisenstein series and MZVs [7, 8, 10, 11]. In particular, this is an efficient way to compute their q-expansion, which is instrumental for finding (as well as excluding) linear relations among eMZVs. Even more so, since iterated integrals of Eisenstein series are linearly independent [12], the differential equation amounts to a decomposition of eMZVs into basic constituents, which reduces the study of relations among eMZVs to solving linear systems of equations. One of the main results of this article is the generalization of this differential equation to teMZVs. We find that the occurrence of the classical Eisenstein series (for $\newcommand{\SL}{\mathrm{SL}} \newcommand{\ZZ}{{\mathbb Z}} \SL_2(\ZZ)$) in the differential equation for eMZVs naturally generalizes to an occurrence of certain weighting functions $f^{(n)}(s+r\tau,\tau)$, where $\newcommand{\ZQ}{{\mathbb Q}} r, s \in \ZQ$, and the latter are also known to be modular forms for congruence subgroups of $\newcommand{\SL}{\mathrm{SL}} \newcommand{\ZZ}{{\mathbb Z}} \SL_2(\ZZ)$. Likewise, one can again identify a procedure delivering the boundary data for teMZVs at the cusp ${\rm i}\infty$ of the modular parameter τ, and relating them to integrals over genus-zero Riemann surfaces in a natural way. While for eMZVs this procedure leads to MZVs, in the case of teMZVs we obtain cyclotomic MZVs [3, 13–15]. A further parallel to eMZVs is the existence of shuffle and Fay relations [10, 16].

Scattering amplitudes in open-superstring theories have been recently noticed as a rewarding setup where iterated integrals on Riemann surfaces appear naturally. Generalizing the ubiquity of MZVs in tree-level amplitudes⁵, one-loop scattering amplitudes (corresponding to genus-one surfaces) provide a natural testing ground for eMZVs [30]. However, the analysis in [30] was focused on the planar sector of the one-loop amplitude where the integrations are performed over a single boundary of a genus-one surface with cylinder topology.

In this article, teMZVs will be identified as the suitable language for the calculation of the non-planar part of the open-string one-loop amplitude: the extension of the iterated integrals to both boundaries of the cylinder leads to the class of teMZVs with twist $\newcommand{\tauh}{\frac{\tau}{2}} \tauh$. We will employ these teMZVs to calculate non-planar contributions to the low-energy expansion⁶ of the four-point one-loop open-string amplitude. Explicit results will be given up to the third subleading low-energy order which are checked to match the expressions available in the literature at the first subleading low-energy order [31] and at the cusp [32]. As in the planar case, their efficient computation crucially relies on the differential equation satisfied by teMZVs.

Interestingly, our final result for the non-planar part of string scattering amplitudes in the cases considered can be expressed in terms of eMZVs alone, i.e. teMZVs with twist $\newcommand{\tauh}{\frac{\tau}{2}} \tauh$ cancel out. This observation relies on providing explicit formulas for certain linear combinations of teMZVs with twist $\newcommand{\tauh}{\frac{\tau}{2}} \tauh$ in terms of eMZVs, which can in turn be checked using their differential equation. We will provide physics arguments bolstering the conjecture that this feature will persist to all orders in the low-energy expansion, and it would be very interesting to find a mathematical explanation for this effect.

Finally, we expect the teMZVs defined here to be closely related to the monodromy of the universal twisted elliptic KZB equation to be studied in upcoming work of Calaque and Gonzalez [9]. A particularly important aspect of their work is the definition of a twisted version of the derivation algebra, the untwisted version of which [5, 33–35] already appeared in the study of eMZVs [10, 11, 36]. Similar to the situation for eMZVs, this twisted derivation algebra might be capable of encoding the number of indecomposable teMZVs of a given weight and length.

In section 2 we introduce teMZVs, and discuss the expansion of their constituents with respect to the modular parameter of the elliptic curve. Thereby we set the stage for section 3, where a differential equation for teMZVs w.r.t. τ as well as a procedure to extract their $\tau \rightarrow {\rm i} \infty$ limit is presented. In section 4, the formalism is applied to the calculation of the non-planar contribution to the open-string one-loop scattering amplitude and the rôle of teMZVs therein is discussed. After concluding and pointing out a couple of open problems in section 5 we provide various appendices containing collections of definitions for the numerous objects appearing as well as several detailed calculations omitted in the main text.

Elliptic multiple zeta values can be represented as iterated integrals on the multiply punctured elliptic curve $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\ZC}{{\mathbb C}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\e}{{\rm e}} \ZC / (\ZZ+\ZZ\tau) \setminus \lbrace b_1 , \dots , b_\ell \rbrace$ with parameter τ in the upper half plane $\newcommand{\ZH}{{\mathbb H}} \ZH$, where we denote $\newcommand{\e}{{\rm e}} q = \exp(2 \pi {\rm i} \tau)$. Starting from $\newcommand{\Ga}{\Gamma} \newcommand{\GL}{\Gamma} \GL(;z) = 1$, elliptic iterated integrals are defined recursively via

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\GLargz}[2]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};z\right)} \newcommand{\GL}{\Gamma} \newcommand{\dd}{\mathrm{d}} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \GLargz{n_1 &n_2 &\ldots &n_\ell}{b_1 &b_2 &\ldots &b_\ell} = \int^z_0 \dd t \, f^{(n_1)}(t-b_1) \, \GLarg{n_2 &\ldots &n_\ell}{b_2 &\ldots &b_\ell}{t}, \qquad z \in [0,1], \label{eqn:defGell} \nonumber \end{align} \tag{ 2.1 }$$

where the interval of integration is $[0, z]$. As will be discussed in section 2.1, regularization prescriptions have to be specified, if n_i  =  1 and $b_i\in[0, z]$.

The weighting functions $f^{(n)}(z, \tau)$ arise as expansion coefficients of the doubly-periodic completion of the Eisenstein–Kronecker series, starting with

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \renewcommand{\Im}{{\rm Im\,}} \newcommand{\re}{{\rm Re}} \displaystyle f^{(0)}(z,\tau)=1 \, , \qquad f^{(1)}(z,\tau)= \frac{\theta_1'(z,\tau)}{\theta_1(z,\tau)} + 2\pi {\rm i} \frac{\Im(z) }{\Im (\tau)} \, , \label{eqn:expl} \nonumber \end{align} \tag{ 2.2 }$$

see appendix B for details and conventions. They are doubly-periodic functions of alternating parity

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle f^{(n)}(z+1,\tau)= f^{(n)}(z+\tau,\tau)= f^{(n)}(z,\tau)\, , \qquad f^{(n)}(-z,\tau)=(-1)^{n}f^{(n)}(z,\tau) \, , \label{eqn:fparity1} \nonumber \end{align} \tag{ 2.3 }$$

and the function $f^{(1)}(z-b_i, \tau)$ in equation (2.2) acquires a pole at z  =  b_i which requires regularization of equation (2.1). Throughout the article, we will frequently omit noting the τ-dependence of both weighting functions f⁽ⁿ⁾ and elliptic iterated integrals equation (2.1).

In [10, 30], the main focus was on elliptic multiple zeta values, whose shifting parameters b_i–referred to as twists–have been limited to b_i  =  0. Correspondingly, the elliptic curve in question has a single puncture only: $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\ZC}{{\mathbb C}} \newcommand{\ZZ}{{\mathbb Z}} E_\tau^\times = \ZC / (\ZZ+\ZZ\tau) \setminus \{0\}$. Evaluating this subclass of elliptic iterated integrals at z  =  1 leads to the definition of Enriquez' A-cycle elliptic multiple zeta values or eMZVs for short:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\omm}{\omega} \newcommand{\GL}{\Gamma} \newcommand{\dd}{\mathrm{d}} \newcommand{\Ga}{\Gamma} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \label{eqn:defeMZV} \omm(n_1,n_2,\ldots,n_\ell) &= \int_{0 \leqslant z_i \leqslant z_{i+1} \leqslant 1} f^{(n_1)}(z_1) \dd z_1\, f^{(n_2)}(z_2)\dd z_2\, \ldots f^{(n_\ell)}(z_\ell) \dd z_\ell \nonumber \\ &= \GLarg{n_\ell &\ldots & n_2 & n_1}{0 &\ldots & 0 & 0}{1} = \GL(n_\ell, \ldots, n_2 ,n_1;1) \, .\nonumber \end{align} \tag{ 2.4 }$$

The quantities $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\e}{{\rm e}} w=\sum_{i=1}^\ell n_i$, and the number $\newcommand{\e}{{\rm e}} \ell$ of integrations in equations (2.1) and (2.4) are referred to as weight and length of the elliptic iterated integral and of the corresponding eMZV, respectively.

Allowing for rational values s_i and r_i in $b_i=s_i+r_i\tau$, leads to twisted elliptic multiple zeta values or teMZVs:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\GL}{\Gamma} \newcommand{\nbeta}{b} \newcommand{\dd}{\mathrm{d}} \newcommand{\Ga}{\Gamma} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \label{eqn:defteMZV} \omwb{n_1,n_2,\ldots,n_\ell}{\nbeta_1,\nbeta_2,\ldots,\nbeta_\ell} &= \int_{0\leqslant z_i \leqslant z_{i+1} \leqslant 1} f^{(n_1)}(z_1-\nbeta_1) \dd z_1\, f^{(n_2)}(z_2-\nbeta_2)\dd z_2\, \ldots f^{(n_\ell)}(z_\ell-\nbeta_\ell) \dd z_\ell \nonumber \\ &=\quad \GLarg{n_\ell & n_{\ell-1} & \ldots & n_1}{\nbeta_\ell & \nbeta_{\ell-1} & \ldots & \nbeta_1}{1} , \nonumber \end{align} \tag{ 2.5 }$$

where the notions of weight and length carry over from equation (2.4) directly. Taking the double-periodicity (2.3) of the weighting functions f⁽ⁿ⁾ into account, one can limit the attention to the fundamental domain of the elliptic curve with $r_i, s_i \in [0, 1)$, i.e. the shaded region in figure 2.

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In this article we are going to limit our attention to twists $\newcommand{\ZQ}{{\mathbb Q}} \ZQ+\ZQ\tau$, that is $\newcommand{\ZQ}{{\mathbb Q}} r_i, s_i\in\ZQ$. In order to classify those, let us introduce

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\La}{\Lambda} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle \Lambda_N= \Big\{0, \frac 1N, \frac 2N,\ldots, \frac{N-1}{N} \Big\} , \ \ \ \ \ \ \Lambda_N^{\times} = \Lambda_N \setminus \{0\} \ . \label{deflambda} \nonumber \end{align} \tag{ 2.6 }$$

If $\newcommand{\La}{\Lambda} b_i\in\Lambda_N^\times$, the twist is referred to as proper rational. Correspondingly, all other twists—that is those with $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\La}{\Lambda} b_i\in(\Lambda_N+\Lambda_N\tau)\setminus \Lambda_N^\times$ as visualized in figure 2—are called generic twists. While in the latter situation divergences occur at endpoints only and can be addressed using the methods in [8], the presence of a proper rational twist requires more work as discussed in section 2.1.

Twisted eMZVs based on proper rational twists do not make an appearance in the open-string one-loop amplitude. However, they are interesting from a number-theoretic point of view because their constant terms give rise to cyclotomic generalizations of MZVs or 'cyclotomic MZVs' for short [3, 13–15]. The set of (generic) twists $b_i\in\lbrace 0, \tau/2\rbrace$ turns out to lead to teMZVs relevant for the non-planar open-string amplitude, which we are going to discuss in section 4.

2.1. Regularization

In order to regularize the divergences in equation (2.5) caused by twists $\newcommand{\La}{\Lambda} \newcommand{\e}{{\rm e}} b_1, \ldots, b_\ell \in \Lambda^\times_N$, we propose to replace the straight line $[0, 1]$ by the domain of integration $[0, 1]_{\varepsilon}$ in the right panel of figure 3.

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Here, $\varepsilon>0$ is an additional real parameter, which determines the radii of the semicircles around the proper rational twists in figure 3. One then defines regularized values of teMZVs

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\dd}{\mathrm{d}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \label{eqn:rigorous} \omwb{n_1,\,n_2,\,\ldots,\,n_\ell}{b_1,\,b_2,\,\ldots,\,b_\ell} = \lim_{\varepsilon \to 0} \int_{[0,1]_{\varepsilon}}f^{(n_1)}(z_1-b_1)\,\dd z_1\,f^{(n_2)}(z_2-b_2)\,\dd z_2\ldots f^{(n_\ell)}(z_\ell-b_\ell)\,\dd z_\ell , \nonumber \end{align} \tag{ 2.7 }$$

which agree with equation (2.5) if all twists are generic. The existence of the limit in equation (2.7) requires some explanation because $f^{(1)}(z-b_i)$ has a pole at z  =  b_i. For a single proper rational twist $\newcommand{\La}{\Lambda} b \in \Lambda^\times_N$, the path $[0, 1]_{\varepsilon}$ can be written as the composition of a straight line from 0 to $b-\varepsilon$, followed by a semicircle from $b-\varepsilon$ to $b+\varepsilon$ above b, and then followed by a straight line from $b+\varepsilon$ to 1:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\dd}{\mathrm{d}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \label{eqn:om1} \omwb{1}{b}=\lim_{\varepsilon \to 0}\int_{[0,b-\varepsilon]}\dd z \ f^{(1)}(z-b)+\int_{[b-\varepsilon,b+\varepsilon]}\dd z \ f^{(1)}(z-b)+\int_{[b+\varepsilon,1]}\dd z \ f^{(1)}(z-b) \ . \nonumber \end{align} \tag{ 2.8 }$$

Clearly, the contribution to equation (2.8) coming from the non-holomorphic part $\newcommand{\re}{{\rm Re}} \renewcommand{\Im}{{\rm Im\,}} 2\pi {\rm i}\frac{\Im(z)}{\Im(\tau)}$ of f⁽¹⁾ in equation (2.2) vanishes in the limit $\varepsilon \to 0$. The contribution coming from the closed one-form $\newcommand{\dd}{\mathrm{d}} \frac{\theta_1'(z, \tau)}{\theta_1(z, \tau)}\dd z$ in turn is independent of ε by Stokes' theorem, since the paths $[0, 1]_{\varepsilon}$ belong to the same homotopy class. Computing the right hand side of equation (2.8), we find that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\be}{\beta} \displaystyle \omwb{1}{b}=\log\left(\frac{\theta_1(1-b,\tau)}{\theta_1(b,\tau)}\right)-{\rm i} \pi=-{\rm i} \pi , \nonumber \end{align} \tag{ 2.9 }$$

where the first term comes from the first and third integral in equation (2.8) and vanishes by reflection and periodicity of the $\theta_1$ function. The contribution of $-{\rm i}\pi$ is due to the second integral in equation (2.8), by the residue theorem.

The higher-length case is handled similarly. First we note that on the semicircles we have additional contributions from the non-holomorphic parts of the weighting functions $f^{(n_i)}(z_i - b_i)$ (see equation (2.17) below), given by powers of $\newcommand{\re}{{\rm Re}} \renewcommand{\Im}{{\rm Im\,}} 2\pi {\rm i}\frac{\Im(z_i-b_i)}{\Im(\tau)}$. These additional contributions on the semicircle are bounded by $\newcommand{\re}{{\rm Re}} \renewcommand{\Im}{{\rm Im\,}} \Im(z_i)\leqslant \varepsilon$, and the accompanying meromorphic functions have at most a simple pole at b_i. Hence, the overall integrand on the semicircle is finite as $\varepsilon \to 0$. Subsequently, we may use the composition of paths formula for iterated integrals (see equation (C.3)) to check that the contributions from the non-holomorphic parts on the semicircles are in fact of order $\mathcal{O}(\varepsilon)$ and therefore do not contribute in the limit $\varepsilon \rightarrow 0$. Thus we are left with integrals over meromorphic functions of the z_i, which do not depend on $\newcommand{\ve}{\varepsilon} \ve$ by homotopy of all paths $\newcommand{\ve}{\varepsilon} [0, 1]_{\ve}$.

The upshot is then that, up to terms which vanish in the limit $\varepsilon \to 0$, the right hand side of equation (2.7) is independent of ε, thus convergent. An example at length two can be found in appendix C.

2.2. General properties of elliptic iterated integrals and (t)eMZVs

In general, iterated integrals of the form (2.1) satisfy shuffle relations. In terms of combined letters $B_i = {n_i \atop b_i }$, the shuffle relation for elliptic iterated integrals reads

where denotes the shuffle product [37]. Naturally, the shuffle relation equation (2.10) straightforwardly carries over to eMZVs,

and teMZVs

where the $B_i, C_i$ are combined letters as defined above.

Taking into account the parity property (2.3) of the weighting functions $f^{(n_i)}$ and the definition of elliptic iterated integrals, one finds the reflection identity

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\GLargz}[2]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};z\right)} \newcommand{\GL}{\Gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \GLargz{n_1 &n_2 &\ldots &n_\ell}{b_1 &b_2 &\ldots &b_\ell} =(-1)^{n_1+n_2+\ldots+n_\ell} \GLargz{n_\ell &\ldots &n_2 &n_1}{z-b_\ell &\ldots &z-b_2 &z-b_1}, \label{eqn:reflection} \nonumber \end{align} \tag{ 2.13 }$$

which is, however, valid only if the combined letter $\newcommand{\be}{\beta} \newcommand{\e}{{\rm e}} B_i = \begin{array}{@{}c@{}} 1 \\ b_i \end{array}$ with b_i a proper rational twist does not occur. The need to exclude such instances of B_i stems from the regularization of section 2.1 which does not preserve the reflection property.

Again, as in the case of the shuffle relation, there is an echo of the reflection identity for eMZVs and teMZVs:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \label{eqn:reflectioneMZV} \omm(n_1,\,n_{2} ,\,\ldots,\,n_{\ell-1},\,n_\ell)&= (-1)^{n_1+n_2+\ldots+n_\ell}\omm(n_\ell,\,n_{\ell-1},\,\ldots,\,n_2,\,n_1)\nonumber \\ \omwb{n_1,\,n_2,\,\ldots,\,n_\ell}{\nbeta_1,\,\nbeta_2,\,\ldots,\,\nbeta_\ell} &= (-1)^{n_1+n_2+\ldots+n_\ell} \omwb{n_\ell,\,n_{\ell-1},\,\ldots,\,n_1}{\tilde\nbeta_\ell,\,\tilde\nbeta_{\ell-1},\,\ldots,\,\tilde\nbeta_1}, \nonumber \end{align} \tag{ 2.14 }$$

where $\tilde{b}_i$ is the representative of  −b_i in the fundamental domain of the elliptic curve and letters $\newcommand{\be}{\beta} \newcommand{\e}{{\rm e}} B_i = \begin{array}{@{}c@{}} 1 \\ b_i \end{array}$ with b_i a proper rational twist are again excluded.

2.3. q-expansion of teMZVs

In contrast to usual MZVs, which are just numbers, eMZVs and teMZVs are functions of the modular parameter τ and can be expanded in its exponentiated cousin $q= {\rm e}^{2\pi {\rm i} \tau}$. The q-expansions of eMZVs and teMZVs rely on the available q-expansions of the weighting functions f⁽ⁿ⁾. The discussion below will simplify considerably, if we consider in addition a class of meromorphic weighting functions g⁽ⁿ⁾.

While the weighting functions f⁽ⁿ⁾ appear as expansion coefficients of the doubly-periodic completion $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \Omega(z, \alpha, \tau)$ of the Eisenstein–Kronecker series $\newcommand{\al}{\alpha} F(z, \alpha, \tau)$ (see equation (B.1)) [6]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Om}{\Omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \renewcommand{\Im}{{\rm Im\,}} \displaystyle \label{alt51} \Omega(z,\alpha,\tau)= \exp \bigg(2\pi {\rm i} \alpha\, \frac{\Im(z)}{\Im(\tau)} \bigg) F(z,\alpha,\tau)= \sum_{n=0}^{\infty}f^{(n)}(z,\tau)\alpha^{n-1} \, , \nonumber \end{align} \tag{ 2.15 }$$

the functions g⁽ⁿ⁾ are the expansion coefficients of the Eisenstein–Kronecker series [38, 39]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle F(z,\alpha,\tau) = \sum_{n =0}^{\infty} g^{(n)}(z,\tau) \alpha^{n-1}. \label{eqn:exp_F} \nonumber \end{align} \tag{ 2.16 }$$

The set of meromorphic functions $g^{(n)}(z, \tau)$ starts with g⁽⁰⁾  =  1 and $g^{(1)}(z, \tau)=\frac{\theta_1'(z, \tau)}{\theta_1(z, \tau)}$ and can be related to their doubly-periodic but non-holomorphic⁷ completions via equation (2.15):

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \renewcommand{\Im}{{\rm Im\,}} \newcommand{\re}{{\rm Re}} \displaystyle f^{(n)}(z,\tau) = \sum_{j=0}^{n} \frac{1}{(n-j)!} \left(2 \pi {\rm i} \frac{\Im(z)}{\Im(\tau)} \right)^{n-j} g^{(\,j)}(z,\tau) \; . \label{comp} \nonumber \end{align} \tag{ 2.17 }$$

Quasi-periodicity and the reflection property of $\newcommand{\al}{\alpha} F(z, \alpha, \tau)$ (see equations (B.5) and (B.6)) imply the following properties of the $g^{(n)}(z, \tau)$,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle g^{(n)}(z) = g^{(n)}(z + 1) , \quad g^{(n)}(z + \tau) = \sum_{j=0}^n \frac{(-2 \pi {\rm i})^{\,j}}{j!} g^{(n-j)}(z) , \quad g^{(n)}(-z) = (-1)^n g^{(n)}(z), \label{eqn:props_gn} \nonumber \end{align} \tag{ 2.18 }$$

and their Fourier expansions are given by [6, 30, 39]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\co}{~,\quad} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle g^{(1)}(z,\tau) &= \pi \, \cot(\pi z) - 2 {\rm i} (2 \pi {\rm i}) \sum_{n,m=1}^\infty \sin(2 \pi m z) \, q^{m n} \nonumber \\ g^{(2k)}(z,\tau) & = -2 \zeta_{2k} -2 \frac{(2\pi {\rm i})^{2k}}{(2k-1)!} \sum_{n,m=1}^\infty \cos(2 \pi m z) \, n^{2k-1} q^{m n} \, , & k > 0 \, \nonumber \\ g^{(2k+1)}(z,\tau) &= -2i \frac{(2\pi {\rm i})^{2k+1}}{(2k)!} \sum_{n,m=1}^\infty \sin(2 \pi m z) \, n^{2k} q^{m n} \, , & k > 0 \, . \nonumber \end{align} \tag{ 2.19 }$$

For real values of z one finds from equations (2.15) and (2.17) that $f^{(n)}(z)=g^{(n)}(z)$ and their q-expansions agree. In particular, they can be employed to find q-expansions for eMZVs

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle \omm(n_1,\,n_2,\,\ldots,\,n_\ell)=\om_0(n_1,\,n_2,\,\ldots,\,n_\ell)+\sum_{k=1}^{\infty}c_k(n_1,\,n_2,\,\ldots,\,n_\ell) q^k \ . \label{eqn:qexp} \nonumber \end{align} \tag{ 2.20 }$$

The q-independent quantity $\newcommand{\om}{\omega} \omega_0$ in equation (2.20) is called the constant term of the eMZV ω and is known to be a $\newcommand{\ZQ}{{\mathbb Q}} \ZQ[(2\pi {\rm i}){}^{-1}]$-linear combination of MZVs (see [10, 11, 30]).

In order to describe the q-dependence of teMZVs in a similar manner, we consider the twist $\newcommand{\La}{\Lambda} b=s+r\tau \in \Lambda_N+\Lambda_N \tau$ in the weighting function f⁽ⁿ⁾(z  −  b) equation (2.17) for real values of z:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle f^{(n)}(z-s-r\tau,\tau) = \sum_{j=0}^{n} \frac{(- 2 \pi {\rm i} r)^{n-j} }{(n-j)!} g^{(\,j)}(z-s-r\tau,\tau) \; , \quad z \in {\mathbb R}\; . \label{eqn:f_via_g} \nonumber \end{align} \tag{ 2.21 }$$

Employing equation (2.19), the functions $g^{(\,j)}(z-b, \tau)$, can be expanded in non-negative rational powers of q,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\co}{~,\quad} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \displaystyle &g^{(2k+1)}(z - s - r \tau ,\tau) = \delta_{k,0} \pi \, \cot(\pi(z-s-r\tau)) + \frac{(2\pi {\rm i})^{2k} }{(2k)!} \sum_{m,n=1}^{\infty} n^{2k} q^{mn} \nonumber \\ &\ \ \ \times \Big\{\cos(2\pi m(z-s)) (q^{mr}-q^{-mr}) -{\rm i} \sin(2\pi m(z-s)) (q^{mr}+q^{-mr}) \Big\} , &k \geqslant 0 \, \nonumber \\ & g^{(2k)}(z - s - r \tau ,\tau) = -2 \zeta_{2k} - \frac{(2\pi {\rm i})^{2k-1} }{(2k-1)!} \sum_{m,n=1}^{\infty} n^{2k-1} q^{mn} \nonumber \\ &\ \ \ \times \Big\{\cos(2\pi m(z-s)) (q^{mr}+q^{-mr}) -{\rm i} \sin(2\pi m(z-s)) (q^{mr}-q^{-mr}) \Big\} , &k > 0 . \nonumber \end{align} \tag{ 2.22 }$$

If 0  <  r  <  1, i.e. if b is a generic twist, then the cotangent term in g⁽¹⁾ may be rewritten as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\co}{~,\quad} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle \pi \cot(\pi(z-s-r\tau)) = {\rm i} \pi (1+q^r {\rm e}^{2\pi {\rm i}(s-z)}) \sum_{n=0}^{\infty}(q^r {\rm e}^{2\pi {\rm i} (s-z)})^n . \nonumber \end{align} \tag{ 2.23 }$$

On these grounds, $f^{(n)}(z-s-r\tau)$ can be expanded in powers of q^r and q^1−r such that every teMZV admits an expansion in q^p,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle \omwb{n_1,\,n_2,\,\ldots,\,n_\ell}{\nbeta_1,\,\nbeta_2,\,\ldots,\,\nbeta_\ell} = \omwbc{n_1,\,n_2,\,\ldots,\,n_\ell}{\nbeta_1,\,\nbeta_2,\,\ldots,\,\nbeta_\ell} + \sum_{k=1}^{\infty} c_{k} \left(\begin{array}{@{}c@{}} n_1,n_2,\ldots,n_\ell \nonumber \\ \nbeta_1,\nbeta_2,\ldots,\nbeta_\ell \end{array}\right) (q^{\,p})^k , \label{eqn:np_qexp} \nonumber \end{align} \tag{ 2.24 }$$

where $\newcommand{\ZQ}{{\mathbb Q}} 1/p\in \ZQ$ is the least common denominator of all occurring r_i. The q-independent quantity $\newcommand{\om}{\omega} \omega_0$ in equation (2.24) is called the constant term of the teMZV, which we are going to study in section 3. Depending on the set of twists b_i, different classes of objects appear as constant terms: while MZVs cover constant terms for generic twists, proper rational twists lead to cyclotomic MZVs [3, 13–15]. We will refer to teMZVs for which $\newcommand{\be}{\beta} \newcommand{\nbeta}{b} \newcommand{\e}{{\rm e}} c_{k} \left(\begin{array}{@{}c@{}} n_1, n_2, \ldots, n_\ell \\ \nbeta_1 , \nbeta_2, \ldots, \nbeta_\ell \end{array}\right)=0$ for all $\newcommand{\ZN}{{\mathbb N}} k \in \ZN^+$ as constant.

The goal of this section is to set up an initial value problem for teMZVs equation (2.5) and to obtain their q-expansion without performing any integral over their trigonometric constituents in equation (2.22). The differential equation to be derived in this section will prove to be the main tool to allow the efficient computation and comparison of teMZVs in a convenient representation as iterated integrals. In particular, this representation will prove useful in the context of calculating non-planar contributions to the one-loop amplitude in section 4.

Following the strategy for computing the usual eMZV's q-expansion in [8, 10], in a first step we derive a first-order differential equation in τ for teMZVs. In the second step, a boundary value at the cusp $\tau \rightarrow {\rm i} \infty$ will be determined to identify a unique solution to the differential equation. Since the action of $\newcommand{\pd}{\partial} \pd_\tau$ reduces the length of teMZVs, one can derive the q-expansion for teMZVs recursively.

For eMZVs, classical Eisenstein series and MZVs are the building blocks for the τ-derivative and constant term respectively [8, 10, 11]. Similarly, we will show that the weighting functions $f^{(n)}(b, \tau)$ evaluated at lattice points $\newcommand{\La}{\Lambda} b\in\Lambda_N+\Lambda_N\tau$ and cyclotomic MZVs are suitable generalizations thereof for teMZVs.

After deriving the differential equation in section 3.1, the constant term will be elaborated on in section 3.2 for generic twists and modifications when including proper rational twists are discussed in section 3.3.

3.1. Differential equation

We begin by defining a generating series for $\newcommand{\te}{\textrm} \newcommand{\temzv}{{\rm teMZV}} \temzv$s of length $\newcommand{\e}{{\rm e}} \ell$,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\dd}{\mathrm{d}} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \Tgen{\alpha_1, \alpha_2, \dots, \alpha_\ell}{\nbeta_1, \nbeta_2, \dots, \nbeta_\ell} & = \int_{0\leqslant z_i \leqslant z_{i+1} \leqslant 1} \Omega(z_1 - \nbeta_1,\alpha_1,\tau)\,\dd z_1\, \Omega(z_2 - \nbeta_2,\alpha_2,\tau)\,\dd z_2 \ldots \Omega(z_\ell - \nbeta_\ell,\alpha_\ell,\tau)\,\dd z_\ell \nonumber \\ & = \sum_{n_1,n_2,\ldots,n_\ell=0 }^{\infty} \alpha_1^{n_1-1} \alpha_2^{n_2-1} \dots \alpha_\ell^{n_\ell-1} \omwb{n_1, n_2, \dots, n_\ell}{\nbeta_1, \nbeta_2, \dots, \nbeta_\ell} \; , \label{eqn:generating_T} \nonumber \end{align} \tag{ 3.1 }$$

generalizing a construction of [8]. For simplicity, we will assume in this subsection that the twists $b_i=s_i+r_i\tau$ are generic, i.e. $r_i \in (0, 1)$. The case of proper rational twists is discussed in appendix F.

First, since the domain of integration in equation (3.1) is the interval $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\ZR}{{\mathbb R}} [0, 1] \subset \ZR$, it is natural to restrict the function $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} z\mapsto \Omega(z-b, \alpha, \tau)$ to real arguments of z. With this restriction, the following differential equation is then a consequence of the mixed heat equation (B.4)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \newcommand{\pd}{\partial} \displaystyle \pd_\tau \Omega(z - s - r \tau,\alpha,\tau) & = \exp(-2 \pi {\rm i} r \alpha) \pd_\tau F(z - s - r \tau,\alpha,\tau) \nonumber \\ & = \exp(- 2 \pi {\rm i} r \alpha) \Big(- r \pd_{z} + \frac{1}{2 \pi {\rm i}} \pd_{z} \pd_{\alpha} \Big) F(z - s - r \tau,\alpha,\tau) \label{eqn:mixed_heat_type_eq}\nonumber \\ & = \frac{1}{2 \pi {\rm i}} \pd_{z} \pd_{\alpha} \Omega(z - s - r \tau,\alpha,\tau).\nonumber \end{align} \tag{ 3.2 }$$

Here the partial derivative $\newcommand{\pd}{\partial} \pd_\tau$ is understood to act on all occurrences of the variable τ. Furthermore, we have used that $\newcommand{\pd}{\partial} \pd_z r=\pd_\tau r=0$, since the twist $b=s+r\tau$ is fixed, and therefore neither depends on z nor τ. Note that in going from the first to the second line in equation (3.2), the term $r \partial_z$ appears by taking the occurrence of τ in the first argument of the Kronecker series into account. However, this additional term gets neatly absorbed when returning to the doubly-periodic completion $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \Omega(z-s-r\tau, \alpha, \tau)$ in the last line.

The τ-derivative of the generating function in equation (3.1) reads

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Tgen}[2]{T\left[\begin{array}{l}#1\nonumber \\#2\end{array}\right]} \newcommand{\nbeta}{b} \newcommand{\dd}{\mathrm{d}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \newcommand{\pd}{\partial} \displaystyle \label{eqn:taugen} 2 \pi {\rm i} \frac{\pd}{\pd \tau} \Tgen{\alpha_1, \alpha_2, \dots, \alpha_\ell}{\nbeta_1, \nbeta_2, \dots, \nbeta_\ell} &= \int_{0\leqslant z_i \leqslant z_{i+1} \leqslant 1} \dd z_1 \, \dd z_2 \, \ldots \, \dd z_\ell \sum_{i=1}^\ell \pd_{z_i} \pd_{\alpha_i} \Omega(z_i - \nbeta_i,\alpha_i) \prod_{j \neq i}^{\ell} \Omega(z_j - \nbeta_j,\alpha_j) \nonumber \\ &= \pd_{\alpha_\ell} \Omega(-\nbeta_\ell,\alpha_\ell) \Tgen{\!\!\alpha_1, \dots, \alpha_{\ell-1}}{\nbeta_1, \dots, \nbeta_{\ell-1\!\!}} - \pd_{\alpha_1} \Omega(-\nbeta_1,\alpha_1) \Tgen{\alpha_2, \dots, \alpha_\ell}{\nbeta_2, \dots, \nbeta_\ell} \nonumber \\ & \quad + \sum_{i=2}^{\ell} \Big(\Tgen{\!\alpha_1, \dots ,\alpha_{i-2}, \alpha_{i-1} + \alpha_i , \, \alpha_{i+1\!\!}, \dots ,\alpha_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, ~~\quad \nbeta_{i}, \,~~\quad \nbeta_{i+1}, \dots ,\nbeta_\ell} \pd_{\alpha_{i-1}} \Omega(\nbeta_i-\nbeta_{i-1},\alpha_{i-1}) \nonumber \\ & - \Tgen{\!\!\alpha_1, \dots, \alpha_{i-2}, \alpha_{i-1} + \alpha_i, \, \alpha_{i+1}, \dots, \alpha_\ell}{\nbeta_1, \dots ,\nbeta_{i-2} , ~\quad \nbeta_{i-1\!\!} ,\quad ~~ \, \nbeta_{i+1}, \dots ,\nbeta_\ell} \pd_{\alpha_i} \Omega(\nbeta_{i-1}-\nbeta_i,\alpha_i) \Big) , \nonumber \end{align} \tag{ 3.3 }$$

where we used equation (3.2) in the first line and suppressed the dependence of $\newcommand{\TL}{T} \TL$ and Ω on the modular parameter τ. In the second equality, the number of integrations is reduced by evaluating $\newcommand{\pd}{\partial} \newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \newcommand{\dd}{\mathrm{d}} \newcommand{\nbeta}{b} \int \dd z_i \, \pd_{z_i} \pd_{\alpha_i} \Omega(z_i - \nbeta_i, \alpha_i)$ via boundary terms $\newcommand{\pd}{\partial} \newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \newcommand{\nbeta}{b} \pd_{\alpha_i} \Omega(z_i - \nbeta_i, \alpha_i)|_{z_{i-1}}^{z_{i+1}}$ with z₀  =  0 and $\newcommand{\e}{{\rm e}} z_{\ell+1}=1$. The resulting products of the form $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \Omega(z_i-b_{i-1}, \alpha_{i-1}) \Omega(z_i-b_i, \alpha_i)$ are rewritten using the Fay identity equation (B.3) such that each integration variable z_i appears in at most one factor of Ω. The details of the computation can be found in appendix E.

Upon expanding Ω and $\newcommand{\TL}{T} \TL$ in equation (3.3) in $\newcommand{\al}{\alpha} \alpha_i$, one can compare the coefficients of the monomials $\newcommand{\al}{\alpha} \newcommand{\e}{{\rm e}} \alpha_1^{m_1} \ldots \alpha_\ell^{m_\ell}$. The coefficient of each monomial is a linear combination of some f⁽ⁿ⁾ multiplied by $\newcommand{\te}{\textrm} \newcommand{\temzv}{{\rm teMZV}} \temzv$s of length $\newcommand{\e}{{\rm e}} \ell-1$. Working out the details yields the following differential equation for $\newcommand{\te}{\textrm} \newcommand{\temzv}{{\rm teMZV}} \temzv$s ($\newcommand{\e}{{\rm e}} \ell \geqslant2$), and using the shorthand

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle h^{(n)}(u,\tau) = (n-1)\,f^{(n)} (u,\tau). \label{eqn:defh} \nonumber \end{align} \tag{ 3.4 }$$

for $\newcommand{\ZC}{{\mathbb C}} \newcommand{\ZZ}{{\mathbb Z}} u\in\ZC/(\ZZ+\ZZ\tau)$, we find

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\pd}{\partial} \displaystyle \label{eqn:diff_eq_nemzv} & 2 \pi {\rm i} \pd_\tau \omwb{n_1, \dots , n_\ell}{\nbeta_1, \dots , \nbeta_\ell} = h^{(n_\ell+1)}(-\nbeta_\ell) \omwb{n_1, \dots , n_{\ell-1}}{\nbeta_1, \dots , \nbeta_{\ell-1}} - h^{(n_1+1)}(-\nbeta_1)\omwb{n_2, \dots , n_{\ell}}{\nbeta_2, \dots , \nbeta_{\ell}} \nonumber \\ & \quad + \sum_{i=2}^\ell \Bigg[ \theta_{n_{i} \geqslant 1} \sum_{k=0}^{n_{i-1}+1} \left(\begin{array}{@{}c@{}}{n_i + k - 1} \nonumber \\{k}\end{array}\right) h^{(n_{i-1}-k+1)}(\nbeta_i - \nbeta_{i-1}) \omwb{n_1, \dots, n_{i-2}, n_i+k ,\, n_{i+1}, \dots, n_\ell}{\nbeta_1, \dots, \nbeta_{i-2}, ~~~\nbeta_{i},~~~ \,\nbeta_{i+1}, \dots, \nbeta_\ell} \nonumber \\ & \quad - \theta_{n_{i-1} \geqslant 1} \sum_{k=0}^{n_{i}+1} \left(\begin{array}{@{}c@{}}{n_{i-1} + k -1} \nonumber \\{k}\end{array}\right) h^{(n_{i}-k+1)}(\nbeta_{i-1} - \nbeta_{i}) \omwb{n_1, \dots, n_{i-2}, n_{i-1}+k ,\, n_{i+1}, \dots, n_\ell}{\nbeta_1, \dots, \nbeta_{i-2}, ~~\nbeta_{i-1},~~ \nbeta_{i+1}, \dots, \nbeta_\ell} \nonumber \\ & \quad + (-1)^{n_{i}+1} \theta_{n_{i-1} \geqslant 1} \theta_{n_i \geqslant 1} h^{(n_{i-1} + n_i + 1)}(\nbeta_{i}-\nbeta_{i-1}) \omwb{n_1, \dots ,n_{i-2}, 0, n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots, \nbeta_{i-2},~~ 0,~~ \nbeta_{i+1}, \dots, \nbeta_\ell} \Bigg], \nonumber \end{align} \tag{ 3.5 }$$

where we have introduced $\newcommand{\de}{\delta} \theta_{n \geqslant 1} = 1- \delta_{n, 0}$ for non-negative n to indicate that some of the contributions in the last three lines vanish for n_i  =  0. For vanishing twists $\newcommand{\nbeta}{b} \nbeta_i=0$, equation (3.5) reduces to the differential equation for eMZVs stated in equation (2.47) of [10] since the weighting functions f⁽ⁿ⁾ are related to holomorphic Eisenstein series (with $\newcommand{\GGs}{G} \newcommand{\GG}[1]{\GGs_{#1}} \GG{0}(\tau)=-1$) via

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GG}[1]{\GGs_{#1}} \newcommand{\zm}{\zeta} \newcommand{\GGs}{G} \newcommand{\te}{\textrm} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle -\lim_{z \to 0}f^{(k)}(z,\tau)= \GG{k}(\tau) = \left\{\begin{array}{@{}rl@{}} 2\zm_k+\frac{2 (2\pi {\rm i})^k}{(k-1)!} \sum_{m,n=1}^{\infty} m^{k-1} q^{mn} &: \ k \geqslant 2 \ \te{even} \nonumber \\ -1 &: \ k=0 \ \nonumber \\ 0 &: \ k \neq 1 \ \te{odd} \end{array} \right.\, , \label{eqn:h0Eisen} \nonumber \end{align} \tag{ 3.6 }$$

where the limit is understood to be taken along the real axis. In other words, the functions $h^{(n)}(b, \tau)$ occurring in the differential equation (3.5) (which are modular forms for congruence subgroups of $\newcommand{\SL}{\mathrm{SL}} \newcommand{\ZZ}{{\mathbb Z}} \SL_2(\ZZ)$) take the rôle of Eisenstein series in the differential equation for eMZVs. Also, note that the exceptional case $f^{(1)}(z, \tau)$ (which has a pole at z  =  0) does not appear in equation (3.3) since it is accompanied by $\newcommand{\al}{\alpha} \alpha^0=1$ in the generating series $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \Omega(z, \alpha, \tau)$ and is therefore annihilated upon application of the α-derivative.

3.2. Constant terms for generic twists

In this subsection we are going to extend the constant-term procedure for eMZVs studied in [10, 11] to a procedure delivering the constant terms for teMZVs. Calculating the constant term for $\newcommand{\te}{\textrm} \newcommand{\temzv}{{\rm teMZV}} \temzv$s amounts to the computation of the limit $\tau \to {\rm i}\infty$ of equation (2.5). This limit will figure as the initial value for the differential equation (3.5) discussed in the previous subsection.

In order to make the bookkeeping more efficient, it is convenient to consider a suitable generating series of teMZVs, which is a generalization of the A-part of Enriquez' elliptic KZB associator [7] to the realm of teMZVs: More precisely, for every $N\geqslant 1$ we will consider a formal power series in nested commutators

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\adrrr}{{\rm ad}} \displaystyle \adrrr^n_x(y)= \underbrace{[x, \ldots [x,[x,}_{n \ {\rm times}} y]] \ldots ], \quad n \geqslant 0\, , \label{defcommut} \nonumber \end{align} \tag{ 3.7 }$$

of the non-commutative variables $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\La}{\Lambda} y, \{x_{b_i}\}_{b_i \in (\Lambda_N + \Lambda_N \tau) \setminus \Lambda_N^{\times}}$ as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\dd}{\mathrm{d}} \newcommand{\te}{\textrm} \newcommand{\La}{\Lambda} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\adrrr}{{\rm ad}} \displaystyle A^{\rm twist}_{(\Lambda_N+\Lambda_N\tau)\setminus\Lambda_N^\times}(\tau)&= \sum_{\ell \geqslant 0}(-1)^\ell \hspace{-20pt} \sum\limits_{n_1,n_2,\ldots,n_\ell\geqslant 0 \atop {b_1,b_2,\ldots,b_\ell \in (\Lambda_N+\Lambda_N\tau) \setminus \Lambda_N^{\times}}} \hspace{-20pt} \omwb{n_1,n_2,\ldots,n_\ell}{\nbeta_1,\nbeta_2,\ldots,\nbeta_\ell}\adrrr^{n_\ell}_{x_{b_\ell}}(y)\ldots \adrrr^{n_2}_{x_{b_2}}(y) \, \adrrr^{n_1}_{x_{b_1}}(y) \nonumber \\ &=\tilde {{\mathcal P}} \, \exp \bigg(- \int^1_0 \dd z \sum\limits_{k=0}^{\infty} \hspace{5pt} \sum\limits_{b\in(\Lambda_N+\Lambda_N\tau) \setminus \Lambda_N^{\times} } \hspace{5pt} f^{(k)} (z-b,\tau) \, \te{ad}_{x_{b}}^{k}(y) \bigg), \label{eqn:genfunction} \nonumber \end{align} \tag{ 3.8 }$$

where $\newcommand{\La}{\Lambda} \Lambda_N$ and $\newcommand{\La}{\Lambda} \Lambda_N^\times$ were defined in equation (2.6), and $\newcommand{\e}{{\rm e}} \tilde{\mathcal{P}} \, \exp(\ldots)$ denotes the path-ordered exponential with reverted order of multiplication for the non-commutative variables in comparison to the order of the integration variables z.⁸ We note that there is no loss of generality in studying the lattice $\newcommand{\La}{\Lambda} \Lambda_N+\Lambda_N\tau$ rather than $\newcommand{\La}{\Lambda} \Lambda_N+\Lambda_M\tau$ with $M\neq N$: the latter can be embedded into the lattice $\newcommand{\La}{\Lambda} \Lambda_{N'}+\Lambda_{N'}\tau$ with $N'$ the least common multiple of M and N. Also, proper rational twists $\newcommand{\La}{\Lambda} b \in \Lambda_N^{\times}$ have been excluded from the summation range for b in equation (3.8) in order to relegate a discussion of the additional ingredients required in these cases to section 3.3.

The series (3.8) combines different instances of the generating series $\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\nbeta}{b} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\\#2\end{array}\right]} \newcommand{\Tgenb}[2]{T\left[\begin{array}{cccc}#1\\#2\end{array}\right]} \newcommand{\e}{{\rm e}} \Tgenb{\alpha_1& \alpha_2& \dots& \alpha_\ell}{\nbeta_1& \nbeta_2& \dots& \nbeta_\ell}$ in equation (3.1),

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Tgenb}[2]{T\left[\begin{array}{cccc}#1\nonumber \\#2\end{array}\right]} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\La}{\Lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle A^{\rm twist}_{(\Lambda_N+\Lambda_N\tau)\setminus\Lambda_N^\times}(\tau) \longleftrightarrow \sum_{\ell \geqslant 0}(-1)^\ell\sum_{b_1,\ldots,b_\ell\in (\Lambda_N+\Lambda_N\tau) \setminus \Lambda_N^{\times}} \Tgenb{\alpha_\ell& \alpha_{\ell-1}& \dots& \alpha_1}{b_\ell& b_{\ell-1}& \dots & b_1}, \nonumber \end{align} \tag{ 3.9 }$$

summing over all values of the length $\newcommand{\e}{{\rm e}} \ell \geqslant 0$ and the generic twists $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\La}{\Lambda} b_i \in (\Lambda_N+\Lambda_N\tau) \setminus \Lambda_N^{\times}$. The non-commutative product of $\newcommand{\adrrr}{{\rm ad}} \adrrr_{x_{b_i}}^{k_i}(y)$ corresponds to commutative variables $\newcommand{\al}{\alpha} \alpha_i^{k_i-1}$ in equation (3.1), which accompany individual teMZVs in the respective generating series. While the organization via $\newcommand{\al}{\alpha} \alpha_i^{k_i-1}$ is better suited for the study of the differential equation of teMZVs, the non-commutative variables $\newcommand{\adrrr}{{\rm ad}} \adrrr_{x_{b_i}}^{k_i}(y)$ in equation (3.7) are well adapted to the subsequent analysis of their constant terms⁹.

3.2.1. Degeneration of weighting functions.

In order to compute $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\La}{\Lambda} \lim_{\tau \to {\rm i}\infty}A^{\rm twist}_{(\Lambda_N+\Lambda_N\tau)\setminus\Lambda_N^\times}(\tau)$, we need to study the degeneration of the weighting functions $f^{(k)}(z-b, \tau)$ as $\tau \to {\rm i}\infty$ (or equivalently $q\to0$). Conveniently, the limit is expressed in the variables

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\co}{~,\quad} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle w = {\rm e}^{2\pi {\rm i} z} \co \dd z = \frac{1}{2\pi {\rm i} } \, \frac{\dd w }{w} . \label{eqn:drin12} \nonumber \end{align} \tag{ 3.10 }$$

Using the q-expansions equations (2.22) and (2.23) together with equation (2.21) we obtain, for generic twists and k  >  1,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \lim_{\tau \to {\rm i}\infty} f^{(k)}(z-s-r\tau,\tau) \, \dd z &= \bigg(\frac{\pi {\rm i} (-2 \pi {\rm i} r)^{k-1} }{(k-1)!} - 2 \sum_{m=0}^{\lfloor \frac k2\rfloor } \frac{(-2 \pi {\rm i} r)^{k-2m}}{(k-2m)!} \zeta_{2m} \bigg) \frac{1}{2 \pi {\rm i}} \frac{\dd w}{w}\nonumber \\ &= -\frac{\dd w}{w}\sum_{m=0}^k\frac{B_m(-2\pi {\rm i})^{m-1}}{m!}\frac{(-2\pi {\rm i}r)^{k-m}}{(k-m)!},\qquad \ k >1 . \label{eqn:drin11a} \nonumber \end{align} \tag{ 3.11 }$$

Here, we have used $\newcommand{\ze}{\zeta} \newcommand{\z}{\zeta} \zeta_{2m}=-\frac{B_{2m}(2\pi {\rm i}){}^{2m}}{2(2m)!}$, where B_k denotes the kth Bernoulli number (such that $B_1=-\frac 12$). While f⁽⁰⁾  =  1, the case of f⁽¹⁾(z  −  b) is special and we find

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \lim_{\tau \to {\rm i}\infty}f^{(1)}(z-s-r\tau,\tau) \, \dd z = \left\{\begin{array}{@{}cl@{}} \left(\frac 12-r\right) \, \frac{\dd w }{w} &: \ r \neq 0 \nonumber \\ \nonumber \\ -\frac{1}{2} \, \frac{\dd w }{w} + \frac{\dd w }{w-1} &: \ r=s=0. \end{array} \right. \label{eqn:drin11b} \nonumber \end{align} \tag{ 3.12 }$$

Combining equations (3.11) and (3.12) allows to rewrite the exponent of equation (3.8) as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\te}{\textrm} \newcommand{\La}{\Lambda} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle \lim_{\tau \to {\rm i}\infty}- \dd z \ \sum_{k=0}^{\infty} \bigg(\sum_{b \in (\Lambda_N+\Lambda_N\tau) \setminus \Lambda_N^\times} f^{(k)}(z-b,\tau) \, \te{ad}_{x_b}^{k}(y) \bigg) = \tilde{y}_N \, \frac{\dd w}{w} + t \, \frac{\dd w }{w-1} , \label{eqn:drin23} \nonumber \end{align} \tag{ 3.13 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\La}{\Lambda} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\adrrr}{{\rm ad}} \displaystyle \tilde{y}_N = -\frac{\adrrr_{x_0}}{{\rm e}^{2\pi {\rm i}\,\adrrr_{x_0}}-1}(y)+\sum_{b \in (\Lambda_N+\Lambda_N\tau) \setminus \Lambda_N}\frac{\adrrr_{x_b}{\rm e}^{-2\pi {\rm i}r\,\adrrr_{x_b}}}{{\rm e}^{-2\pi {\rm i}\,\adrrr_{x_b}}-1}(y) , \ \ \ \ \ \ t = [y,x_0]. \label{eqn:drin24} \nonumber \end{align} \tag{ 3.14 }$$

These definitions of $\tilde{y}_N$ and t are tailored to track the appearance of the forms $\newcommand{\dd}{\mathrm{d}} \frac{\dd w }{w}$ and $\newcommand{\dd}{\mathrm{d}} \frac{\dd w }{w-1}$ in the degeneration limits equations (3.11) and (3.12) of $f^{(k)}(z-b, \tau)$. In absence of twists, for instance, the first contribution to $\tilde{y}_N$ in equation (3.14) stems from setting r  =  0 in equation (3.11) and identifying the generating series of $\newcommand{\adrrr}{{\rm ad}} \frac{B_k(-2\pi {\rm i}){}^{k-1}}{k!}\adrrr^k_{x_0}(y)$:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\te}{\textrm} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\adrrr}{{\rm ad}} \displaystyle \lim_{\tau \to {\rm i}\infty}- \dd z \ \sum_{k=0}^{\infty} \bigg(\,f^{(k)}(z,\tau) \, \te{ad}_{x_0}^{k}(y) \bigg)&=-\frac{\dd w}{w}\sum_{k=0}^{\infty}\frac{B_k(2\pi {\rm i})^{k-1}}{k!}\adrrr^k_{x_0}(y)-\frac{\dd w}{w-1}\adrrr_{x_0}(y) \nonumber \\ &=-\frac{\adrrr_{x_0}}{{\rm e}^{2\pi {\rm i}\adrrr_{x_0}}-1}(y)\frac{\dd w}{w}-\frac{\dd w}{w-1}\adrrr_{x_0}(y) \, . \label{explain} \nonumber \end{align} \tag{ 3.15 }$$

In the generalization to non-zero twists $r \neq 0$, the second contribution to $\tilde{y}_N$ in equation (3.14) arises as the generating series of $\newcommand{\adrrr}{{\rm ad}} \newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \sum_{m=0}^k\frac{B_m(-2\pi {\rm i}){}^{m-1}}{m!}\frac{(-2\pi {\rm i}r){}^{k-m}}{(k-m)!} \adrrr^k_{x_b}(y)$. Note in particular that the dependence on r gives rise to the factor of $\newcommand{\adrrr}{{\rm ad}} {\rm e}^{-2\pi {\rm i}r\adrrr_{x_b}}$ in the numerator.

By comparing the degeneration behaviour in equation (3.13) with equation (3.8), we deduce that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\La}{\Lambda} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle \label{eqn:drin16} \lim_{\tau \to {\rm i}\infty}A^{\rm twist}_{(\Lambda_N+\Lambda_N\tau)\setminus\Lambda_N^\times}(\tau)=\tilde {{\mathcal P}} \, \exp \bigg(\int_{C_0^{2\pi}(1)} \bigg[ \frac{\tilde{y}_N }{w} + \frac{t }{w-1} \bigg] \, \dd w \bigg) \, , \nonumber \end{align} \tag{ 3.16 }$$

where the unit circle $w \in C_0^{2\pi}(1)$ arises from the path of integration $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\ZC}{{\mathbb C}} [0, 1]\subset \ZC$ under the change of variables equation (3.10). Strictly speaking, equation (3.16) requires regularization, due to divergences at w  =  1. They are treated in analogy to eMZVs as described in [8, 11, 30] and cause modifications to be pointed out in the subsequent discussion.

3.2.2. Deforming the integration contour.

We have expressed $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\La}{\Lambda} \lim_{\tau \to {\rm i}\infty}A^{\rm twist}_{(\Lambda_N+\Lambda_N\tau)\setminus\Lambda_N^\times}(\tau)$ as a generating series of iterated integrals of explicit differential forms along the unit circle $C_0^{2\pi}(1)$, see the left panel of figure 4 below. Although the functions $\lim_{\tau \to {\rm i}\infty}f^{(k)}(z-b, \tau)$ in equations (3.11) and (3.12) were restricted to real values of z, they can be extended to the doubly-punctured complex plane $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\ZC}{{\mathbb C}} z \in \ZC \setminus \{0, 1\}$ (with r kept constant). In this way, the integrand in equation (3.13) is holomorphic on $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\ZC}{{\mathbb C}} \ZC \setminus \{0, 1\}$, and the resulting path-ordered exponential equation (3.16) is homotopy-invariant. Therefore one can replace the unit circle by a contour homotopic to it, visualized in the right panel of figure 4.

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This deformed contour can in turn be viewed as the composition of straight paths $P_1, P_1^{-1}$ connecting the points $w=0, 1$ along with a circle $C_0^{2\pi}(\varepsilon)$ of infinitesimal radius around the origin, as shown in figure 4. Moreover, the regularization alluded to above manifests itself in figure 4: both paths $C_0^{2\pi}(1)$ as well as the composition $P_1^{-1} C_0^{2\pi}(\varepsilon) P_1 \hat C_{\pi}^0(\varepsilon)$ have to leave w  =  1 with velocity  −1 and arrive back at w  =  1 with the same velocity. More precisely, both paths, which really are smooth functions $\newcommand{\ZC}{{\mathbb C}} [0, 1] \rightarrow \ZC^{\times}$ must have a derivative equal to $\newcommand{\pd}{\partial} \newcommand{\ZC}{{\mathbb C}} -\frac{\pd}{\pd w} \in T_1(\ZC^{\times})$, where T₁ denotes the tangent space at 1. This is also the reason for the semicircle $\hat{C}_{\pi}^0(\varepsilon)$.

The virtue of deforming the path of integration is that equation (3.16) can now be computed rather explicitly. First, in view of the reversal operation contained in the definition of $\tilde {{\mathcal P}}$, the composition of paths α and β translates into a concatenation of the non-commutative series with reversed order,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\om}{\omega} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \tilde{{\mathcal P}} \exp\left(\int_{\alpha\beta} \omega \right) = \tilde{{\mathcal P}} \exp\left(\int_{\beta} \omega \right) \tilde{{\mathcal P}} \exp\left(\int_{\alpha} \omega \right) \nonumber \end{align} \tag{ 3.17 }$$

regardless of the differential form ω. Hence, the equality of homotopy classes of paths (relative to the tangent vector $\newcommand{\pd}{\partial} -\frac{\pd}{\pd w}$ at 1) $[C_0^{2\pi}(1)]=[P_1^{-1} C_0^{2\pi}(\varepsilon)P_1 \hat C_{\pi}^0(\varepsilon)]$ allows to rewrite equation (3.16) as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\La}{\Lambda} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle \lim_{\tau \rightarrow {\rm i} \infty} A^{\rm twist}_{(\Lambda_N+\Lambda_N\tau)\setminus\Lambda_N^\times}(\tau) ={\rm e}^{{\rm i} \pi t}\Phi(\tilde{y}_N,t){\rm e}^{2\pi {\rm i}\tilde{y}_N}\Phi(\tilde{y}_N,t)^{-1}, \label{eqn:drin17} \nonumber \end{align} \tag{ 3.18 }$$

where $\Phi(\tilde{y}_N, t)$ denotes the Drinfeld associator [40]. In deducing equation (3.18), we have used the identities

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\be}{\beta} \displaystyle \Phi(\tilde{y}_N,t) = \tilde{{\mathcal P}} \exp\bigg(\, \int_{P_1} \Big[ \frac{\tilde{y}_N }{w}+ \frac{t }{w-1} \Big] \, \dd w \bigg) , \nonumber \end{align} \tag{ 3.19 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\be}{\beta} \displaystyle {\rm e}^{2\pi {\rm i} \tilde{y}_N } = \tilde{{\mathcal P}} \exp\bigg(\, \int_{C_0^{2\pi}(\varepsilon)} \Big[ \frac{\tilde{y}_N }{w}+ \frac{t }{w-1} \Big] \, \dd w \bigg) , \nonumber \end{align} \tag{ 3.20 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \label{drin18} {\rm e}^{{\rm i} \pi t } = \tilde{{\mathcal P}} \exp\bigg(\, \int_{\hat C_{\pi}^0(\varepsilon)} \Big[ \frac{\tilde{y}_N }{w}+ \frac{t }{w-1} \Big] \, \dd w \bigg) . \nonumber \end{align} \tag{ 3.21 }$$

Since the coefficients of Φ are $\newcommand{\ZQ}{{\mathbb Q}} \ZQ$-linear combinations of MZVs [41], an implementation of equation (3.18) using a standard computer algebra system can be used to explicitly write the constant terms of teMZVs for generic twists as $\newcommand{\ZQ}{{\mathbb Q}} \ZQ[(2\pi {\rm i}){}^{-1}]$-linear combinations of MZVs. Equation (3.18) is a generalization of a similar formalism for eMZVs, which has been established in [7] (see also [10]). Examples for constant terms of teMZVs computed via equation (3.18) are gathered in appendix G.1.

3.3. Constant terms for all twists

So far, we have only considered the constant terms of teMZVs with generic twists. The presence of proper rational twists $\newcommand{\La}{\Lambda} b \in \Lambda^\times_N$ requires a separate discussion due to the additional features of the corresponding weighting function $f^{(1)}(z-b, \tau)$:

• its simple pole in the interior of the domain of integration requires the regularization procedure of section 2.1
• its $\tau \to {\rm i}\infty$ limit introduces singularities $\newcommand{\ze}{\zeta} \newcommand{\z}{\zeta} \newcommand{\dd}{\mathrm{d}} \frac{\dd w}{w-\zeta}$ with ζ denoting a root of unity

In order to facilitate the computation of the constant terms of teMZVs including proper rational twists, we again introduce a generating series for bookkeeping purposes

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\dd}{\mathrm{d}} \newcommand{\te}{\textrm} \newcommand{\La}{\Lambda} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\adrrr}{{\rm ad}} \displaystyle A^{\rm twist}_{\Lambda_N+\Lambda_N\tau}(\tau) &=\sum_{\ell \geqslant 0}(-1)^\ell \hspace{-20pt} \sum\limits_{{n_1,n_2,\ldots,n_\ell\geqslant 0} \atop {b_1,b_2,\ldots,b_\ell \in \Lambda_N+\Lambda_N\tau}} \hspace{-20pt} \omwb{n_1,n_2,\ldots,n_\ell}{\nbeta_1,\nbeta_2,\ldots,\nbeta_\ell}\adrrr^{n_\ell}_{x_{b_\ell}}(y)\ldots \adrrr^{n_2}_{x_{b_2}}(y) \, \adrrr^{n_1}_{x_{b_1}}(y) \nonumber \\ &=\lim_{\varepsilon \to 0}\tilde {{\mathcal P}} \, \exp \bigg(- \int_{[0,1]_{\varepsilon}} \dd z \sum_{k=0}^{\infty} \hspace{5pt}\sum_{b\in\Lambda_N+\Lambda_N\tau} f^{(k)} (z-b,\tau) \, \te{ad}_{x_{b}}^{k}(y) \bigg) , \label{eqn:genfunction2} \nonumber \end{align} \tag{ 3.22 }$$

which generalizes equation (3.8) to the case of arbitrary twists.

3.3.1. Degeneration of weighting functions.

As in the above situation we need to determine the degeneration limit of the weighting functions $f^{(k)}(z-b, \tau)$ as $\tau \to {\rm i}\infty$. The only case, where this degeneration limit differs from the results of the previous subsection (see equations (3.11) and (3.12)) is k  =  1 and r  =  0:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \lim_{\tau \to {\rm i}\infty} f^{(1)}(z-s,\tau) \, \dd z = -\frac{1}{2} \frac{\dd w}{w} + \frac{\dd w}{w-{\rm e}^{2\pi {\rm i}s}} . \label{eqn:drin20} \nonumber \end{align} \tag{ 3.23 }$$

Note the occurrence of the root of unity ${\rm e}^{2\pi {\rm i}s}$. Denoting the set of Nth roots of unity by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\La}{\Lambda} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\be}{\beta} \newcommand{\ve}{\varepsilon} \displaystyle \mu_N = \big\lbrace {\rm e}^{2\pi {\rm i}s} \, \vert \, s \in \Lambda_N \big\rbrace , \nonumber \end{align} \tag{ 3.24 }$$

the generalization of equation (3.13) to twists $\newcommand{\La}{\Lambda} b \in \Lambda_N+\Lambda_N\tau$ reads

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\te}{\textrm} \newcommand{\La}{\Lambda} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \lim_{\tau \to {\rm i}\infty}- \dd z \sum_{k=0}^{\infty} \; \sum\limits_{b \in \Lambda_N{+}\Lambda_N\tau} f^{(k)}(z-b,\tau) \, \te{ad}_{x_b}^{k}(y) = \tilde{y}_N \, \frac{\dd w}{w} + \sum_{\zeta \in \mu_N}t_{\zeta} \, \frac{\dd w }{w-\zeta} , \label{eqn:drin23b} \nonumber \end{align} \tag{ 3.25 }$$

where¹⁰

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\La}{\Lambda} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \newcommand{\adrrr}{{\rm ad}} \displaystyle \tilde{y}_N &= -\sum_{b \in \Lambda_N}\frac{\adrrr_{x_b}}{{\rm e}^{2\pi {\rm i}\adrrr_{x_b}}-1}(y)+ \hspace{-2pt} \sum\limits_{b \in (\Lambda_N+\Lambda_N\tau) \setminus \Lambda_N} \frac{\adrrr_{x_b}{\rm e}^{-2\pi {\rm i}r\adrrr_{x_b}}}{{\rm e}^{-2\pi {\rm i}\adrrr_{x_b}}-1}(y) \nonumber \\ t_{\zeta} &= [y,x_{s}] , \ \ \ \ {{\rm for}} \ \zeta = {\rm e}^{2\pi {\rm i} s} \in \mu_N . \label{eqn:drin24b} \nonumber \end{align} \tag{ 3.26 }$$

Equation (3.26) is the generalization of equation (3.14) to arbitrary twists in the lattice $\newcommand{\La}{\Lambda} \Lambda_N+\Lambda_N\tau$, and can be proved along the lines of the previous subsection. In particular, the expression for $\tilde{y}_N$ follows by repeating the steps which have been detailed around equation (3.15).

3.3.2. Deforming the integration contour.

Now the image of the integration contour $[0, 1]_{\varepsilon}$ under the transformation $z \mapsto w={\rm e}^{2\pi {\rm i}z}$ is the unit circle around 0, which is dented at the roots of unity ${\rm e}^{2\pi {\rm i}s} \in \mu_N$ as pictured in figure 5 below. However, the point w  =  1 is special and will be taken care of by the regularization of section 3.2. Similar to the situation above, the dented unit circle is homotopic to $[P_1^{-1} C_0^{2\pi}(\varepsilon)P_1 \hat C_{\pi}^0(\varepsilon)]$ as depicted in figure 5 for twists in $\newcommand{\La}{\Lambda} \Lambda_3$.

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Hence, equation (3.18) can be generalized to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\La}{\Lambda} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \lim_{\tau \rightarrow {\rm i} \infty} A^{\rm twist}_{\Lambda_N+\Lambda_N\tau}(\tau) ={\rm e}^{{\rm i} \pi t_1}\Phi_N(\tilde{y}_N,(t_{\zeta})_{\zeta \in \mu_N}){\rm e}^{2\pi {\rm i}\tilde{y}_N} \Phi_N(\tilde{y}_N,(t_{\zeta})_{\zeta \in \mu_N})^{-1} , \label{eqn:drin29} \nonumber \end{align} \tag{ 3.27 }$$

where $\newcommand{\ze}{\zeta} \newcommand{\z}{\zeta} \Phi_N(e_0, (e_{\zeta})_{\zeta \in \mu_N})$ is the cyclotomic version of the Drinfeld associator [42], defined by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \Phi_N(e_0,(e_{\zeta})_{\zeta \in \mu_N}) = \tilde {{\mathcal P}} \, \exp \bigg(\int^1_0 \bigg[ \frac{e_0}{w}+\sum_{\zeta \in \mu_N}\frac{e_{\zeta}}{w-\zeta} \bigg] \, \dd w \bigg) . \nonumber \end{align} \tag{ 3.28 }$$

Since $\Phi_N$ is the generating series of N-cyclotomic MZVs [3, 13–15], the constant terms of teMZVs for arbitrary twists in the lattice $\newcommand{\La}{\Lambda} \Lambda_N+\Lambda_N\tau$ are $\newcommand{\ZQ}{{\mathbb Q}} \ZQ[(2\pi {\rm i}){}^{-1}]$-linear combinations of cyclotomic MZVs. Definitions and properties of cyclotomic MZVs are collected in appendix D, and examples for constant terms of teMZVs with proper rational twists can be found in appendix G.2.

As exemplified by $\newcommand{\be}{\beta} \newcommand{\om}{\omega} \newcommand{\omm}{\omega} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\\#2\end{array}\right)} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\\#2\end{array}\right)} \omwbc{1}{\oneh} = -{\rm i} \pi$, it is the regularization of divergences occurring for proper rational twists, which spoils the validity of the reflection property equation (2.14) for letters $\newcommand{\be}{\beta} \newcommand{\e}{{\rm e}} B = \begin{array}{@{}c@{}} 1 \\ b \end{array}$ with $\newcommand{\La}{\Lambda} b\in\Lambda_N^\times$. It would be interesting to identify an alternative regularization scheme where equation (2.14) is preserved.

This section is devoted to the discussion of the appearance of teMZVs in a physics context—in the low-energy expansion of scattering amplitudes in string theory [43–47]. In general, string amplitudes at lower loop orders¹¹ $g\leqslant 2$, possibly also at $g=3, 4$, can be represented by integrals over the moduli space of punctured Riemann surfaces of genus g. For one-loop scattering of open strings, the Riemann surfaces or worldsheets of interest are the cylinder and the Mœbius strip. The punctures—the insertion points of vertex operators for external states—are then integrated over the boundary components of the worldsheets. These boundary integrals are weighted by traces over Lie-algebra generators t^a associated with the gauge degrees of freedom of the open-string states: Each boundary component contributes a separate trace factor, in each of which the order of multiplication matches the ordering of the associated punctures.

A convenient parametrization of the one-loop open-string topologies—the cylinder and the Mœbius strip—can be obtained starting from the torus by restricting the modular parameter to $\tau_C = {\rm i} t$ and to $\tau_M = {\rm i} t+\frac{1}{2}$, respectively, where $\newcommand{\ZR}{{\mathbb R}} t \in \ZR_+$. In both cases, the boundary is parametrized via $z\in {\mathbb C}/({\mathbb Z} + {\mathbb Z} \tau)$ with $\newcommand{\re}{{\rm Re}} \renewcommand{\Re}{{\rm Re\,}} \Re(z) \in [0, 1]$ and $\newcommand{\re}{{\rm Re}} \renewcommand{\Im}{{\rm Im\,}} \Im(z)=0$ or $\newcommand{\re}{{\rm Re}} \renewcommand{\Im}{{\rm Im\,}} \Im(z)=\frac{t}{2}$ which is sometimes referred to as the closed-string channel.

In this setup, elliptic iterated integrals equation (2.1) appear naturally when integrating over moduli spaces of cylinder- and Mœbius-strip punctures. This is yet another example, where the iterated integrals on the boundary of open-string worldsheets yield special values of polylogarithms tailored to the corresponding Riemann surfaces, generalizing the ubiquity of MZVs at genus zero.

Cylindrical worldsheets with all insertions on the same boundary are referred to as planar cylinders. For these contributions to the amplitude, all integrals over the punctures were shown to boil down to eMZVs in [30]. Moreover, the only difference between integrals over punctures on the Mœbius strip and those on the planar cylinder is the value of the modular parameter τ [49]: therefore one can straightforwardly convert the contributions from the planar cylinder to those of the Mœbius strip by

• replacing $q_C = {\rm e}^{2\pi {\rm i} \tau_C} = {\rm e}^{-2\pi t}$ in the Fourier expansion of the eMZVs in the planar-cylinder contribution by $q_M = {\rm e}^{2\pi i \tau_M} = {\rm e}^{-2\pi t +{\rm i}\pi} = -q_C$; this results in alternating relative signs between the coefficients in the q_C-expansion of the Mœbius-strip contributions and the cylinder contributions.
• inserting a factor of $\pm \frac{32}{N_G}$ for the Mœbius strip to account for its single boundary of doubled length compared to individual boundary components of the cylinder [49]. The '+' sign is for gauge groups USp(N_G) and '−' for SO(N_G).

Hence, for a gauge group $SO(32)$, the constant term in the q-expansion w.r.t. q_C which would give rise to a UV divergence upon integration over $\newcommand{\ZR}{{\mathbb R}} t \in \ZR_+$ cancels between the cylinder and the Mœbius strip [49].

The double-trace contributions, on the other hand, stem entirely from the cylinder topology with punctures on two different boundaries— non-planar cylinder diagrams. We will see that the integrals over the punctures boil down to teMZVs with purely imaginary modular parameter and twists $\newcommand{\nbeta}{b} \nbeta \in \{0, \frac{\tau_C}{2}\}=\{0, \frac{{\rm i}t}{2}\}$.

In the planar case mentioned above the link between eMZVs and the worldsheet integral over the cylinder boundary was established as follows: Punctures on the same boundary enter through the genus-one Green function at real arguments which can be written as an integral over the weighting function f⁽¹⁾(x) with real argument $x \in (0, 1)$ [30]. Consequently, we will show that the new ingredient in the case of the non-planar cylinder, i.e. the genus-one Green function for two insertions on different boundaries, is related to integrals over $\newcommand{\tauh}{\frac{\tau}{2}} f^{(1)}(x - \tauh)$. Hence, the non-planar contributions may be expressed as iterated integrals on $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\tauh}{\frac{\tau}{2}} E_\tau^\times \setminus \left\{\tauh \right\}$ described above, which eventually lead to the teMZVs introduced in section 2.

In order to simplify the final formulas, we employ the differential equation of section 3 to build in all the relations among the teMZVs we encounter (see appendix G.3). In fact, as will be detailed in section 4.3, these relations ultimately reduce all instances of teMZVs to eMZVs alone. Still, teMZVs are an essential tool for intermediate steps and to render the subsequent computations completely algorithmic.

4.1. The four-point integrals

We will illustrate the emergence of teMZVs through the non-planar contribution to the four-point one-loop amplitude of the open superstring. Its dependence on the external polarizations enters through a prefactor K universal to all worldsheet topologies [50] and is irrelevant for the subsequent discussion. Then, setting q_C  =  q and q_M  =  −q as discussed above, the complete expression for the one-loop open-string four-point amplitude reads [49]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle {{\mathcal A}}_4^{\rm 1-loop} =&~ K\int^1_{0} \frac{\dd q }{q} \Big\{{\rm Tr}(t^1 t^2 t^3 t^4) \, \big[ N_G \, I_{1234}(q) - 32 \,I_{1234}(-q) \big] \nonumber \\ & + {\rm Tr}(t^1 t^2)\, {\rm Tr}(t^3 t^4) I_{12|34}(q) + {\rm cyc}(2,3,4) \Big\} \; , \label{eqn:singtrace} \nonumber \end{align} \tag{ 4.1 }$$

where t^a are traceless generators of the gauge group SO(N_G) and the traces are taken in its fundamental representation. The accompanying integrals are given by¹²

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \displaystyle I_{1234}(q) = \int^{1}_0 \dd x_4 \int^{x_4}_0 \dd x_3 \int^{x_3}_0 \dd x_2 \int^{x_2}_0 \dd x_1 \,\delta(x_1) \prod_{i<j}^4 \exp\bigg[ \frac{1}{2} s_{ij} G(x_{ij},\tau) \bigg] \; , \label{eqn:itintA} \nonumber \end{align} \tag{ 4.2 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \displaystyle I_{12|34}(q) &= \int^{1}_0 \dd x_4 \int^{1}_0 \dd x_3 \int^{1}_0 \dd x_2 \int^{1}_0 \dd x_1 \,\delta(x_1) \label{eqn:itintB} \nonumber \\ & \ \ \ \times \exp\bigg[ \frac{1}{2} s_{12} G(x_{12},\tau)+ \frac{1}{2} s_{34} G(x_{34},\tau)+\frac{1}{2} \sum_{i=1,2 \atop {j=3,4}} s_{ij} G(x_{ij}-\tauh,\tau) \bigg] , \nonumber \end{align} \tag{ 4.3 }$$

where the insertion points $z_{1, 2}= x_{1, 2}$ and $\newcommand{\tauh}{\frac{\tau}{2}} z_{3, 4}= x_{3, 4} + \tauh$ of the vertex operators are parametrized by real integration variables x_i with $x_{ij} = x_i - x_j$, see figure 6. Translation invariance on a genus-one surface has been used to fix x₁  =  0 through the above delta function. The genus-one Green functions¹³ [32]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle G(z,\tau) = \log \left| \frac{\theta_1(z,\tau)}{\theta_1'(0,\tau)} \right|^2 - \frac{2 \pi}{{{\rm Im}}(\tau)} {{\rm Im}}(z)^2 \label{eqn:1l_prop} \nonumber \end{align} \tag{ 4.4 }$$

depend on the differences of punctures z_i and their second argument τ will often be suppressed. Considering the parametrization of the cylinder visualized in figure 6, vertex insertions on different boundaries give rise to arguments $\newcommand{\tauh}{\frac{\tau}{2}} x_{ij} -\tauh$ as for instance seen in the contribution $\newcommand{\tauh}{\frac{\tau}{2}} G(x_{13}-\tauh, \tau)$ to the exponent of equation (4.3).

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Given the relations between the dimensionless Mandelstam invariants¹⁴,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle s_{34}=s_{12} , \ \ \ \ \ \ s_{14} = s_{23} , \ \ \ \ \ \ s_{13}=s_{24}=-s_{12}-s_{23} \, , \label{eqn:mand} \nonumber \end{align} \tag{ 4.5 }$$

the integrands of equations (4.2) and (4.3) are unchanged if the Green function equation (4.4) is shifted by a z-independent function. This feature will be made use of in the following subsections.

Configurations with three punctures on the same boundary lead to color factors such as ${\rm Tr}(t^1 t^2 t^3) {\rm Tr}(t^4)$ which vanish for traceless SO(N_G) generators considered in equation (4.1). Nevertheless, the accompanying integral

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \displaystyle I_{123|4}(q) &= \int^{1}_0 \dd x_4 \int^{1}_0 \dd x_3 \int^{x_3}_0 \dd x_2 \int_{0}^{x_2} \dd x_1 \,\delta(x_1) \label{eqn:itintC} \nonumber \\ & \ \ \ \times \exp\bigg[ \frac{1}{2}\sum_{1\leqslant i<j}^3 s_{ij} G(x_{ij},\tau) + \frac{1}{2} \sum_{j=1}^3 s_{j4} G(x_{j4}-\tauh,\tau) \bigg] \nonumber \end{align} \tag{ 4.6 }$$

plays an important rôle for one-loop monodromy relations [31, 53]. It will be demonstrated in appendix H that equation (4.6) may be expanded in terms of teMZVs using the same techniques as will be applied to the integral I_12|34(q) in equation (4.3) along with two punctures on each boundary. For both non-planar integrals I_12|34(q) and I_123|4(q), our results up to and including the order of $s_{ij}^{3}$ can be simplified to ultimately yield combinations of eMZVs, i.e. as mentioned earlier, all of their twisted counterparts are found to drop out at the orders considered.

4.1.1. Analytic versus non-analytic momentum dependence.

The one-loop four-point amplitude is a non-analytic function of the Mandelstam invariants equation (4.5): From the integration over q in equation (4.1), the region with $t\to\infty$ or $q={\rm e}^{-2\pi t}\to 0$ leads to branch cuts well-known from the Feynman integrals in the field-theory limit [50]. Moreover, the non-planar contribution I_12|34(q) additionally integrates to kinematic poles in s₁₂, reflecting the exchange of closed-string states between the cylinder boundaries [44]. Since both the poles and the branch cuts stem from the integration over q, the integrals equations (4.2) and (4.3) over the punctures by themselves do not reflect the singularity structure of the one-loop amplitude.

At fixed values of q it is possible to separate the overall amplitude into analytic and non-analytic parts. For the four-point closed-string one-loop amplitude, a careful procedure to isolate the logarithmic dependence on s_ij has been developed in [54]: this method allows for a focused study of the analytic sector where modular graph functions take the rôle of eMZVs [55, 56].

While the integrals in the expansion of equation (4.2) have been performed at fixed value of q, integrating for instance equation (4.7) below and its counterpart from the Mœbius strip over q introduces divergences, also for the gauge group $SO(32)$. The choice of regularization scheme for these divergences (see [31] for an example at the first subleading order in $\newcommand{\al}{\alpha} \alpha'$) reflects a particular way of splitting the analytic and non-analytic parts of the final expression for the amplitude after integrating over q.

4.1.2. The low-energy expansion in the single-trace sector.

In the following, we will study the analytic part of the non-planar one-loop four-point amplitude by Taylor-expansion of equation (4.3) in s_ij and thereby in $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$, probing the low-energy behaviour. The analogous low-energy expansion for the single-trace integral equation (4.2) has been performed in [30],

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ap}{\alpha'} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle &I_{1234}(q) = \omega(0,0,0) \, - \, 2\omega(0,1,0,0) \, (s_{12}+s_{23}) \, + \, 2 \omega(0,1,1,0,0)\, \big(s_{12}^2 + s_{23}^2 \big) \label{4ptja} \nonumber \\ &\, - \, 2 \omega(0,1,0,1,0) \,s_{12}s_{23} \, + \, \beta_5 \, (s_{12}^3+2 s_{12}^2 s_{23} + 2s_{12} s_{23}^2+s_{23}^3) \, + \, \beta_{2,3} \, s_{12} s_{23}(s_{12}+s_{23}) \, + \, {{\mathcal O}}(\ap^4) \nonumber \end{align} \tag{ 4.7 }$$

with the following combinations of eMZVs at order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^3$:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \beta_5 = \frac{4}{3} \, \big[ \omm(0,0,1,0,0,2)+\omm(0,1,1,0,1,0) - \omm(2,0,1,0,0,0) - \zeta_2 \omm(0,1,0,0) \big] \nonumber \end{align} \tag{ 4.8 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \beta_{2,3} =\frac{\zeta_3}{12}+\frac{8\,\zeta_2}{3}\omm(0,1,0,0)-\frac{5}{18}\omm(0,3,0,0) \, . \nonumber \end{align} \tag{ 4.9 }$$

It was explained in the reference that the dependence of the single-trace integral equation (4.7) on q is captured by eMZVs at any order in $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$. Note that the contributions of the planar cylinder and the Mœbius strip to equation (4.1) are obtained by integrating equation (4.7) over arguments $q \rightarrow {\rm e}^{-2\pi t}$ and $q \rightarrow - {\rm e}^{-2\pi t}$, respectively, with $\newcommand{\ZR}{{\mathbb R}} t \in \ZR_+$.

In analogy with equation (4.7), we will determine the $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$-expansion of the non-planar integral equation (4.3) in the framework of teMZVs. Since the main emphasis of this article is to exemplify the use of teMZVs in the calculation of non-planar one-loop amplitudes, a detailed analysis of the singularity structure after integration over q is left for the future.

4.2. The genus-one Green function as an elliptic iterated integral

The link between open-string amplitudes and the framework of elliptic iterated integrals is the holomorphic derivative¹⁵

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\pd}{\partial} \displaystyle \pd_z G(z,\tau) = f^{(1)}(z,\tau) , \ \ \ \ \ \ z\in {\mathbb C} \setminus (\mathbb{Z} + \mathbb{Z} \tau) \, , \label{eqn:del1l_prop} \nonumber \end{align} \tag{ 4.10 }$$

of the genus-one Green function equation (4.4) in the integrals equations (4.2) and (4.3). It allows to rewrite the exponent in the non-planar integral equation (4.3) into the form

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle &\frac{1}{2} s_{12} G(x_{12},\tau)+ \frac{1}{2} s_{34} G(x_{34},\tau)+ \sum_{i=1,2 \atop {j=3,4}} \frac{1}{2} s_{ij} G \big(x_{ij}-\tauh,\tau \big) \nonumber \\ &\ \ \ \ \ = s_{12} P(x_{12}) + s_{34} P(x_{34}) + \sum_{i=1,2 \atop {j=3,4}} s_{ij} Q(x_{ij}) \; , \label{eqn:ellipticKN} \nonumber \end{align} \tag{ 4.11 }$$

where the entire dependence on the real parts¹⁶ $\newcommand{\ZR}{{\mathbb R}} x_i \in \ZR$ of the punctures $z_{1, 2}= x_{1, 2}$ and $\newcommand{\tauh}{\frac{\tau}{2}} z_{3, 4}= x_{3, 4} + \tauh$ is captured via elliptic iterated integrals equation (2.1):

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{l}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\GL}{\Gamma} \newcommand{\dd}{\mathrm{d}} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle P(x) = \int^x_0 \dd y \ f^{(1)}(y) = \GLarg{\!\!1}{\!\!0 }{x} \label{eqn:defPP} \nonumber \end{align} \tag{ 4.12 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\GL}{\Gamma} \newcommand{\dd}{\mathrm{d}} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle Q(x) &= c(q)+ \int^x_0 \dd y \ f^{(1)} \left(y- \tauh \right) = c(q) + \GLarg{\!\!1}{\!\!\tau/2 }{x} \; . \label{eqn:defQQ} \nonumber \end{align} \tag{ 4.13 }$$

The appearance of the x_i-independent quantity

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle c(q) = -\frac{{\rm i} \pi }{2} - \frac{1}{8} \log(q) + 2 \sum_{m=1}^{\infty} \frac{1}{m} \Big[ \frac{q^m}{1-q^m} - \frac{q^{m/2} }{1-q^m} \Big] \label{eqn:defcq} \nonumber \end{align} \tag{ 4.14 }$$

is special to the non-planar cylinder and caused by the different properties of the Green functions G(x_ij) and $\newcommand{\tauh}{\frac{\tau}{2}} G(x_{ij}-\tauh)$ connecting punctures on the same and different boundaries of the cylinder, respectively.

In passing to the right hand side of equation (4.11) we have used momentum conservation $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \sum_{i<j}^4 s_{ij}=0$ to discard

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \frac{1}{2}G(x) -P(x) = \frac{{\rm i} \pi }{2} - \log(2 \pi) = \frac{1}{2}G\big(x- \tauh \big) - Q(x) . \label{eqn:newlabel} \nonumber \end{align} \tag{ 4.15 }$$

A detailed explanation of the relations above and the underlying regularization will be given in the following two subsections.

4.2.1. ${G(x)}$ versus ${P(x)}$.

In the case of punctures on the same boundary of the cylinder, the Green functions with arguments $x\in [0, 1]$ and $\newcommand{\ZR}{{\mathbb R}} \tau \in {\rm i}\ZR_+$ reduce to $\frac{1}{2}G(x, \tau)=\log \frac{\theta_1(x, \tau)}{\theta_1'(0, \tau)}$. Up to an additive constant, this expression can be recovered by the following integration with a regulator $\varepsilon>0$ in the lower limit:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \int_\varepsilon^{x} \dd y\, f^{(1)}(y) &= \log(\theta_1(x)) - \log(\theta_1(\varepsilon)) \nonumber \\ &= \log(\theta_1(x)) - \log(\theta_1'(0)) - \log(\varepsilon) + \mathcal{O}(\varepsilon) \label{eqn:Pbyint} \nonumber \\ &= \frac{1}{2}G(x) - \log(-2\pi {\rm i} \varepsilon) + \log(-2 \pi {\rm i}) + \mathcal{O}(\varepsilon) \nonumber \; . \nonumber \end{align} \tag{ 4.16 }$$

The regularization scheme for the limit $\varepsilon\rightarrow 0$ has to be chosen consistently with the treatment of divergent eMZVs: Following the conventions of [30], the regularized value of an eMZV is defined to be the constant term in an expansion¹⁷ w.r.t. $\log(-2 \pi {\rm i} \varepsilon)$, and we choose the principal branch of the logarithm, such that $\log(-{\rm i})= -\frac{{\rm i} \pi}{2}$. Hence, $\log(-2 \pi {\rm i} \varepsilon)$ is formally set to zero in equation (4.16), and we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle P(x) = \lim_{\varepsilon\rightarrow 0} {\rm Reg} \int_\varepsilon^{x} \dd y\, f^{(1)}(y) = \frac{1}{2}G(x) + \log(-2 \pi {\rm i}) = \frac{1}{2}G(x) - \frac{{\rm i} \pi }{2} + \log(2 \pi) \; , \label{eqn:PbyintA} \nonumber \end{align} \tag{ 4.17 }$$

reproducing the first equality in equation (4.15).

4.2.2. ${ \newcommand{\tauh}{\frac{\tau}{2}} G(x-\tauh)}$ versus ${Q(x)}$.

Pairs of punctures on different boundaries of the cylinder lead to arguments $\newcommand{\tauh}{\frac{\tau}{2}} x-\tauh$ of the Green function equation (4.4), where $x \in [0, 1]$. In these cases, the Green function may be related to the integral

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \int_{0}^{x} \dd y \, f^{(1)}\left(y- \tauh \right) &= \log \big(\theta_1\big(x - \tauh \big) \big) -{\rm i} \pi x - \log \big(\theta_1 \big(-\tauh\big) \big) \nonumber \\ &= \log \big| \theta_1 \big(x - \tauh \big) \big| - \frac{{\rm i} \pi }{2} - \log \big(-\theta_1 \big(\tauh \big) \big) \label{eqn:1l_prop_diffb} \nonumber \\ &= \frac{1}{2}G\big(x-\tauh \big) + \log \theta_1'(0) -\frac{1}{8} \log(q) - \frac{{\rm i} \pi }{2} - \log \big(-{\rm i} q^{-1/8} \theta_4(0) \big) \nonumber \\ &= \frac{1}{2} G \big(x-\tauh \big) + \log\left(\frac{\theta_1'(0) }{\theta_4(0) } \right) = \frac{1}{2} G\big(x-\tauh \big)- \frac{1}{2} G\big(\tauh \big) \; , \nonumber \end{align} \tag{ 4.18 }$$

without any need for regularization. In passing to the second line, we have chosen the principal branch of the logarithm to relate $\log(\theta_1(x - \tau/2)) -{\rm i} \pi x= \log| \theta_1(x - \tau/2) | - \frac{{\rm i} \pi }{2 }$, see appendix A for our conventions and several identities for the theta functions. In the next step, we have introduced the Green function via $\log| \theta_1(x - \tau/2) | = \frac{1}{2}G(x-\tau/2) + \log \theta_1'(0) -\frac{1}{8} \log(q)$ and used the theta-function identity $\theta_1(\tau/2) = {\rm i} q^{-1/8} \theta_4(0)$, see equation (A.3).

Using the infinite-product representations equation (A.1) of the theta functions, the result of equation (4.18) can be rewritten as [57]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle & \int_{0}^{x} \dd y \, f^{(1)}\left(y- \tauh \right) - \frac{1}{2} G(x- \tauh) = \log\left(\frac{\theta_1'(0)}{\theta_4(0)} \right) \nonumber \\ & \ \ \ \ = \log(2 \pi q^{1/8}) + 2 \sum_{n=1}^\infty \left[ \log(1 - q^n) - \log(1 - q^{n-\oneh}) \right] \nonumber \\ & \ \ \ \ = \log(2 \pi q^{1/8}) + 2 \sum_{m=1}^\infty \frac{1}{m} \Big[ \frac{q^{m/2} }{1-q^m} - \frac{q^m}{1-q^m} \Big] \label{moresteps} \nonumber \\ & \ \ \ \ = -\frac{{\rm i} \pi }{2} + \log(2\pi) - c(q) . \nonumber \end{align} \tag{ 4.19 }$$

In passing to the third line, we have rearranged the infinite sums¹⁸ to identify the quantity $c(q)$ in equation (4.14). In this way, one arrives at

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle Q(x) = c(q)+\int_0^{x} \dd y\, f^{(1)}\big(y-\tauh \big) = \frac{1}{2}G \big(x-\tauh \big) - \frac{{\rm i} \pi }{2} + \log(2 \pi) \; , \label{gibmireinQ} \nonumber \end{align} \tag{ 4.20 }$$

and the second equality in equation (4.15) is confirmed.

4.3. Non-planar contribution to the four-point amplitude

Using the identities discussed in the previous subsections, the integral equation (4.3) relevant to the non-planar cylinder can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle I_{12|34}(q) = \int_{12}^{34} \exp \bigg[ s_{12} P(x_{12}) + s_{34} P(x_{34}) + \sum_{i=1,2 \atop {j=3,4}} s_{ij} Q(x_{ij}) \bigg] , \label{eqn:newrep} \nonumber \end{align} \tag{ 4.21 }$$

where $P(x)$ and $Q(x)$ are given by the elliptic iterated integrals equations (4.12) and (4.13) and we have introduced the following shorthand for the integration measure:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \displaystyle \int_{12}^{34} = \int_{0}^{1} \dd x_4 \int_{0}^{1} \dd x_3 \int_{0}^{1} \dd x_2 \int_{0}^{1} \dd x_1\, \delta(x_1) \; . \label{eqn:int_bdy_np} \nonumber \end{align} \tag{ 4.22 }$$

We will now investigate the Taylor-expansion of equation (4.21) in the dimensionless Mandelstam invariants equation (4.5) and thus in $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$ by expanding the exponentials ${\rm e}^{s_{ij} P(x_{ij})}$ and ${\rm e}^{s_{ij} Q(x_{ij})}$ in the integrand:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\GL}{\Gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle I_{12|34}(q) = {\rm e}^{-2s_{12}c(q)} \int_{12}^{34} \sum_{n_{ij}=0}^{\infty} \frac{\left (s_{12} \GLarg{1}{0 }{x_{12}} \right)^{n_{12}} \left(s_{34} \GLarg{1}{0 }{x_{34}}\right)^{n_{34}} }{n_{12}! \, n_{34}!} \prod_{i=1,2 \atop {j=3,4}} \frac{\left(s_{ij} \GLarg{1}{\tau/2 }{x_{ij}} \right)^{n_{ij}} }{n_{ij}!}\ . \label{eqn:newnewrep} \nonumber \end{align} \tag{ 4.23 }$$

The punctures x_i only enter via elliptic iterated integrals, and Fay relations among the weighting functions f⁽ⁿ⁾ guarantee that the individual integrations over x_i can always be performed in terms of further elliptic iterated integrals, see appendix I. Hence, each order in $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$ can be expressed in terms of teMZVs and the quantity $c(q)$ in equation (4.14), where the latter will also be related to teMZVs in equation (4.49). Moreover, the explicit results up to the third order can in fact be expressed in terms of (untwisted) eMZVs only, without the need to involve their twisted counterparts. Whether this behaviour persists at any order in $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$ will be discussed in section 4.3.5.

4.3.1. Structure of the leading orders ${ \newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}}$.

As a first step towards an expansion in terms of teMZVs, we classify the inequivalent integrals w.r.t. the cycle structure of ${\rm Tr}(t^1 t^2)\, {\rm Tr}(t^3 t^4)$ which occur at the orders $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}$ of equation (4.23): We will use the shorthand $P_{ij}=P(x_{ij})$ and $Q_{ij}=Q(x_{ij})$ for the two integrals at order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{1}$,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\co}{~,\quad} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d_1^1= \int_{12}^{34} P_{12} \co d_2^1= \int_{12}^{34} Q_{13} , \label{npl9} \nonumber \end{align} \tag{ 4.24 }$$

the six integrals at order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{2}$,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\co}{~,\quad} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d_1^2 &= \frac{1}{2} \int_{12}^{34} P_{12}^2 \co d^2_3 = \int_{12}^{34} P_{12} Q_{13} \co d^2_5 = \int_{12}^{34} Q_{13} Q_{14} \label{npl10} \nonumber \\ d_2^2 &= \frac{1}{2} \int_{12}^{34}Q_{13}^2 \co d^2_4 = \int_{12}^{34} P_{12} P_{34} \co d^2_6 = \int_{12}^{34} Q_{13} Q_{24} , \nonumber \end{align} \tag{ 4.25 }$$

and the twelve integrals at order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{3}$:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\co}{\,,\quad} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d_1^3 &= \frac{1}{6} \int_{12}^{34} P_{12}^3 \co \ \ \ \ \, \, d_5^3= \frac{1}{2} \int_{12}^{34} P_{12}^2 P_{34}\! \co \ \ \ \ \, \, d_9^3= \int_{12}^{34} P_{12} Q_{13} Q_{24} \ \nonumber \\ d_2^3 &= \frac{1}{6} \int_{12}^{34} Q_{13}^3 \co \ \ \ \ \, \, d_6^3= \frac{1}{2} \int_{12}^{34} Q_{13}^2 Q_{14} \co \ \ \ \ \, \, d_{10}^3= \int_{12}^{34} P_{12} Q_{13} Q_{14} \ \label{npl11} \nonumber \\ d_3^3 &= \frac{1}{2} \int_{12}^{34} P_{12}^2 Q_{13} \co \hspace{0.1mm} d_7^3= \frac{1}{2} \int_{12}^{34} Q_{13}^2 Q_{24} , \co \ \ \ \ \, \, d_{11}^3= \int_{12}^{34} P_{34} Q_{13} Q_{14} \ \nonumber \\ d_4^3 &= \frac{1}{2} \int_{12}^{34} P_{12} Q_{13}^2 \co \hspace{0.1mm} d_8^3= \int_{12}^{34} P_{12} P_{34} Q_{13} \co \ \ \ \ \, \, d_{12}^3= \int_{12}^{34} Q_{13} Q_{14} Q_{23} \ . \nonumber \end{align} \tag{ 4.26 }$$

In fact, some of the above $d_i^{\,j}$ can be related via cyclicity and reflection properties of the five-point open-string worldsheet setup where the integration measure equation (4.22) is generalized to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \displaystyle \int_{123}^{45} = \int_{0}^{1} \dd x_5 \int_{0}^{1} \dd x_4 \int_{0}^{1} \dd x_3 \int_{0}^{x_3} \dd x_2 \int_{0}^{x_2} \dd x_1 \,\delta(x_1) \ . \label{eqn:int_bdy_np5} \nonumber \end{align} \tag{ 4.27 }$$

Using $\newcommand{\pd}{\partial} \pd_i Q_{ij}=-\pd_j Q_{ij}$ and the vanishing of $\newcommand{\pd}{\partial} \newcommand{\dd}{\mathrm{d}} \int^1_0 \dd x_j \ \pd_j Q_{ij}$ by double periodicity of the Green function, we find one relation

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle 0 = \int_{123}^{45} Q_{25} \, \partial Q_{14} = \int_{123}^{45} Q_{25} \, (Q_{24}-Q_{34}) \ \ \ \Rightarrow \ \ \ d_5^2 = d_6^2 \label{npl22} \nonumber \end{align} \tag{ 4.28 }$$

among the integrals equation (4.25) at order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^2$. The same methods yield the two relations

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\pd}{\partial} \displaystyle \left. \begin{array}{@{}l@{}} 0= \int_{123}^{45} Q_{25}^2 \pd Q_{14} \nonumber \\ 0 = \int_{123}^{45} P_{23} Q_{25} \pd Q_{14} \end{array} \right \} \ \ \ \Rightarrow \ \ \ \left\{\begin{array}{@{}l@{}} d_7^3 = d_6^3 \nonumber \\ d_9^3 = d_{10}^3 \end{array} \right. \label{npl0} \nonumber \end{align} \tag{ 4.29 }$$

among the integrals equation (4.26) at order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^3$.

4.3.2. teMZVs at orders ${ \newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}}$.

As a next step, we exploit the representations equations (4.12) and (4.13) of P_ij and Q_ij to express the above $d^i_j$ in terms of teMZVs: The two instances at order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$ yield

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{l}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\dd}{\mathrm{d}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d_{1}^1&= \int_0^1 \dd x_2 \int_0^{x_2} \dd y \; f^{(1)}(y) = \omwb{\!\!1,0\!\!}{\!\!0,0\!\!} \label{eqn:d11} \nonumber \end{align} \tag{ 4.30 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d_{2}^1 &= c(q) + \int_0^{1} \dd x_3 \int_0^{x_3} \dd y \ f^{(1)}\big(y - \tauh\big) \nonumber \\ &= c(q) + \omwb{\!\!1,0\!\!}{\!\!\tauh,0\!\!} \; , \label{eqn:d12} \nonumber \end{align} \tag{ 4.31 }$$

and we can similarly convert the five independent integrals at order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^2$ to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^2_1 & = \omwb{1 , 1 , 0 }{0 , 0 , 0} \label{eqn:d21} \nonumber \end{align} \tag{ 4.32 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^2_2 & = \frac{1}{2} c(q)^2 + c(q) \omwb{1 , 0 }{\tauh, 0 } + \omwb{1 , 1 , 0 }{\tauh,\tauh, 0} \label{eqn:d22} \nonumber \end{align} \tag{ 4.33 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^2_{3} &= d^1_1 d^1_2= \omwb{1 , 0 }{0 , 0 } \left(c(q) + \omwb{1 , 0 }{\tauh,0 } \right) \label{eqn:d23} \nonumber \end{align} \tag{ 4.34 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^2_4 & =(d_1^1)^2= \omwb{1 , 0 }{0 , 0}^2 \label{eqn:d24} \nonumber \end{align} \tag{ 4.35 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^2_5 & = (d_2^1)^2= \left(c(q) + \omwb{1 , 0 }{\tauh,0 } \right)^2 \label{eqn:d25} \nonumber \end{align} \tag{ 4.36 }$$

and the ten independent integrals at order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^3$ to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^{3}_{1} & = \omwb{1 , 1 , 1 , 0 }{0 , 0 , 0, 0} \label{eqn:d31} \nonumber \end{align} \tag{ 4.37 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^{3}_{2} & = \frac{1}{6} c(q)^3 + \frac{1}{2} c(q)^2 \omwb{1 , 0}{\tauh, 0} + c(q) \omwb{1 , 1 , 0}{\tauh, \tauh, 0} + \omwb{1 , 1 , 1 , 0}{\tauh, \tauh, \tauh, 0} \label{eqn:d32} \nonumber \end{align} \tag{ 4.38 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^{3}_{3} & = d^{2}_{1} d^{1}_{2} = \omwb{1 , 1 , 0 }{0 , 0 , 0 } \left(c(q) + \omwb{1 , 0}{\tauh, 0} \right) \label{eqn:d33} \nonumber \end{align} \tag{ 4.39 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^{3}_{4} &= d^1_{1} d^2_{2} = \omwb{1 , 0}{0 , 0} \left(\frac{1}{2} c(q)^2 + c(q) \omwb{1 , 0}{\tauh, 0} + \omwb{1 , 1 , 0}{\tauh, \tauh, 0} \right) \label{eqn:d34} \nonumber \end{align} \tag{ 4.40 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^{3}_{5} & = d^1_{1} d^{2}_{1} = \omwb{1 , 1 , 0 }{0 , 0 , 0} \omwb{1 , 0 }{0 , 0} \label{eqn:d35} \nonumber \end{align} \tag{ 4.41 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^{3}_{7} & = d^1_{2} d^2_{2} = \left(c(q) + \omwb{1,0}{\tauh,0} \right) \left(\frac{1}{2} c(q)^2 + c(q) \omwb{1 , 0}{\tauh, 0} + \omwb{1 , 1 , 0}{\tauh, \tauh, 0} \right) \label{eqn:d37} \nonumber \end{align} \tag{ 4.42 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^3_{8} & = d^1_{1} d^2_{3} = (d^1_{1})^2 d^1_{2} = \omwb{1 , 0 }{0 , 0 } ^2 \left(c(q) + \omwb{1 , 0 }{\tauh, 0 } \right) \label{eqn:d38} \nonumber \end{align} \tag{ 4.43 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^3_{9} & = d^1_{2} d^2_{3} = d^1_{1} (d^1_{2})^2 = \omwb{1 , 0 }{0 , 0 } \left(c(q) + \omwb{1 , 0 }{\tauh, 0 } \right)^2 \label{eqn:d39} \nonumber \end{align} \tag{ 4.44 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^3_{11} & = \omwb{1 , 0 }{0 , 0 } \left(c(q) + \omwb{1 , 0 }{\tauh, 0 } \right)^2 +\frac{1}{3} \,\omwb{0 , 3 , 0 , 0 }{0 , 0 , 0 , 0} \label{eqn:d311} \nonumber \end{align} \tag{ 4.45 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d^3_{12} & = d^1_{2} d^2_{5} = (d^1_{2})^3 = \left(c(q) + \omwb{1 , 0 }{\tauh, 0 } \right)^3 \ . \label{eqn:d312} \nonumber \end{align} \tag{ 4.46 }$$

Note that $d_6^2, d_6^3$ and $d_{10}^3$ are determined by equations (4.28) and (4.29), and the derivation of the more involved integral $d_{11}^3$ is detailed in appendix I.

4.3.3. Assembling the orders ${ \newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}}$.

Once we apply momentum conservation equation (4.5) to the integral equation (4.23), its leading orders in the $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$-expansion simplify to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ap}{\alpha'} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle I_{12|34}(q) = 1\ &+\ 2 s_{12}\, (d_1^1 - d_2^1) \ +\ s_{12}^2 \,(2 d_1^2 + 2 d_2^2 - 4 d_3^2 + d_4^2 + d_5^2) \ + \ s_{13}s_{23} \, (2 d_5^2-4 d_2^2) \nonumber \\ & + \ 2s_{12}^3\, (d_1^3 - d_2^3 - 2 d_3^3 + 2 d_4^3 + d_5^3 - d_7^3 - d_8^3 + d_9^3) \label{npl12} \nonumber \\ & - \ 2s_{12}s_{13}s_{23}\,(d_{12}^3 - 3 d_2^3 + 4 d_4^3 - d_7^3 - 2 d_{11}^3) + \ {{\mathcal O}}(\ap^4) \ . \nonumber \end{align} \tag{ 4.47 }$$

The q-dependence of the relevant teMZVs can be determined by solving the initial-value problem set up in section 3, yielding for instance

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle \omwb{1,0}{0,0} &= - \frac{{\rm i} \pi}{2} + 2\sum_{n,m=1}^\infty \frac{q^{m n}}{m} , \ \ \ \ \ \ \omwb{1,0}{\tauh,0} = 2 \sum_{n,m=1}^\infty \frac{q^{m(n-\oneh)}}{m} \; , \label{eqn:iti_ex_d11} \nonumber \end{align} \tag{ 4.48 }$$

and further examples can be found in appendix G.4. By comparing the q-expansion with the expression equation (4.14) for $c(q)$, we infer that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle c(q) = \omwb{1,0}{0,0} - \omwb{1,0}{\tauh,0} - \frac{1}{8} \log(q) , \label{eqn:cversuslogq} \nonumber \end{align} \tag{ 4.49 }$$

which identifies the prefactor ${\rm e}^{-2s_{12}c(q)}$ in equation (4.23) as $q^{s_{12}/4}$ multiplied by a series in teMZVs $\newcommand{\be}{\beta} \newcommand{\om}{\omega} \newcommand{\omm}{\omega} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\\#2\end{array}\right)} \omwb{1, 0}{0, 0}$ and $\newcommand{\be}{\beta} \newcommand{\om}{\omega} \newcommand{\omm}{\omega} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\\#2\end{array}\right)} \omwb{1, 0}{\tauh, 0}$. Moreover, equation (4.49) simplifies the order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^1$ of equation (4.47) to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle I_{12|34}(q) \, \big|_{s_{12}} = 2(d_1^1 - d_2^1) = 2 \Big\{\omwb{1,0}{0,0} - \omwb{1,0}{\tauh,0} - c(q) \Big\} = \frac{1}{4} \log(q) , \label{eqn:orderone} \nonumber \end{align} \tag{ 4.50 }$$

in agreement with [31]. Also at higher orders of equation (4.47), we convert any appearance of $c(q)$ into $-\frac{1}{8}\log(q)$ via equation (4.49) and obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle I_{12|34}(q) \, \big|_{s_{12}^2} &= \frac{(\log(q)^2) }{32} + 2 \omwb{1,1 ,0}{0,0,0} - \omwb{1,0}{0,0}^2 + 2 \omwb{1,1,0}{\tauh,\tauh ,0} - \omwb{1,0}{\tauh,0}^2 \nonumber \\ &= \frac{(\log(q)^2) }{32} + \frac{7\zeta_2 }{6} + 2\,\omega(0,0,2) \nonumber \end{align} \tag{ 4.51 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle I_{12|34}(q) \, \big|_{s_{13} s_{23} } & = 2 \omwb{1,0}{\tauh,0}^2 - 4 \omwb{1,1,0}{\tauh,\tauh,0} = -2 \, \omega(0,0,2) - \frac{2 \zeta_2 }{3} \nonumber \end{align} \tag{ 4.52 }$$

via teMZV relations equations (G.11) and (G.12) as well as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle I_{12|34}(q) \, \big|_{s_{12}^3} &= \frac{1}{3!} \left(\frac{\log(q) }{4} \right)^3 + \frac{1 }{4} \log(q) \, \left(\frac{7\zeta_2 }{6} + 2\,\omega(0,0,2) \right) \nonumber \\ & \ \ + \frac{2}{3} \omwb{1,0}{0,0}^3 - 2 \omwb{1,0}{0,0} \omwb{1 ,1,0}{0,0,0} + 2 \omwb{1,1,1,0}{0,0,0,0} \nonumber \\ & \ \ - \frac{2}{3} \omwb{1,0}{\tauh,0}^3 + 2 \omwb{1,0}{\tauh,0} \omwb{1 ,1,0}{\tauh,\tauh,0} - 2 \omwb{1,1,1,0}{\tauh,\tauh,\tauh,0} \nonumber \\ &= \frac{1}{3!} \left(\frac{\log(q) }{4} \right)^3 + \frac{1}{4} \log(q) \, \left(\frac{7\zeta_2 }{6} + 2\,\omega(0,0,2) \right) - 4 \, \zeta_2 \, \omega(0,1,0,0) \nonumber \end{align} \tag{ 4.53 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle I_{12|34}(q) \, \big|_{s_{12}s_{23} s_{13}} &= \frac{1 }{4} \log(q) \, \left(-2 \, \omega(0,0,2) - \frac{2 \zeta_2 }{3} \right) + \frac{4}{3} \, \omega(0,3,0,0) \nonumber \\ & \ \ \ +2 \omwb{1,0}{\tauh,0}^3 -6 \omwb{1,0}{\tauh,0} \omwb{1 ,1,0}{\tauh,\tauh,0} +6 \omwb{1,1,1,0}{\tauh,\tauh,\tauh,0} \nonumber \\ &= \frac{1 }{4} \log(q) \, \left(-2 \, \omega(0,0,2) - \frac{2 \zeta_2 }{3} \right) + \frac{5}{3}\,\omega(0,3,0,0) + 4 \, \zeta_2 \, \omega(0,1,0,0) - \frac{1}{2} \zeta_3 \nonumber \end{align} \tag{ 4.54 }$$

via teMZV relations equations (G.13) and (G.14). As will be discussed shortly, the cancellation of teMZVs with nonzero twist manifests the absence of unphysical poles in the string amplitude after integration over q.

4.3.4. Summary of the orders ${ \newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}}$.

As is clear by comparing equation (4.23) with equation (4.49), any appearance of $\log(q)$ can be traced back to the expansion of $q^{s_{12}/4}$. Hence, the above orders $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}$ can be summarized as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ap}{\alpha'} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle I_{12|34}(q) & = q^{s_{12}/4} \bigg\{1+ s_{12}^2 \Big(\frac{7\zeta_2 }{6} + 2\,\omega(0,0,2) \Big) -2s_{13}s_{23} \Big(\frac{\zeta_2 }{3} + \omega(0,0,2) \Big) \label{finalqtononzero} \nonumber \\ & \ \ - 4 \, \zeta_2 \, \omega(0,1,0,0) \, s_{12}^3 + s_{12}s_{13}s_{23} \Big(\frac{5}{3}\,\omega(0,3,0,0) + 4 \, \zeta_2 \, \omega(0,1,0,0) - \frac{1}{2} \zeta_3 \Big) + {{\mathcal O}}(\ap^4) \bigg\}\; , \nonumber \end{align} \tag{ 4.55 }$$

and the integral I_123|4(q) in equation (4.6) admits a similar low-energy expansion in terms of eMZVs only, see equation (H.26). While the q-expansions of the above eMZVs are listed in appendix G.4, their constant terms yield

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ap}{\alpha'} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle I_{12|34}(q) = q^{s_{12}/4} \Big\{1+ \frac{1}{2} \zeta_2 s_{12}^2 - \frac{1 }{2} \zeta_3 s_{12}^3 + {{\mathcal O}}(q,\ap^4) \Big\}\; , \label{qtozeroA} \nonumber \end{align} \tag{ 4.56 }$$

in agreement with the all-order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$ expression given in [31]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle I_{12|34}(q) = \frac{2^{2s_{12}} q^{s_{12}/4 }}{\pi} \left\{\left(\frac{\Gamma(\frac{1}{2} + \frac{s_{12}}{2})}{\Gamma(1+ \frac{s_{12}}{2})} \right)^2 + {{\mathcal O}}(q) \right\}\ . \label{qtozeroB} \nonumber \end{align} \tag{ 4.57 }$$

With the above strategy and the techniques exemplified in appendix I, there is no limitation in obtaining higher orders of the $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$-expansion equation (4.55). Similarly, non-planar open-string amplitudes with five and more external legs can be expanded along the same lines¹⁹ because their integrands depend on the punctures and τ through products of f⁽ⁿ⁾ and Eisenstein series [30, 58, 59].

4.3.5. Higher orders in ${ \newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap}$ and eMZVs.

The exclusive appearance of eMZVs at the leading orders of the $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$-expansion equation (4.55) illustrates a general property of the non-planar integral I_12|34(q): apart from the prefactor $q^{s_{12}/4}$, its q-expansion comprises integer powers only. This property is shared by eMZVs but not by typical teMZVs involving twists $\newcommand{\tauh}{\frac{\tau}{2}} \tauh$.

The absence of half-odd integer powers $\newcommand{\oneh}{\frac{1}{2}} q^{n+\oneh}$ with $\newcommand{\ZN}{{\mathbb N}} n \in \ZN_0$ can be explained from physical constraints on the pole structure of the open-string amplitude: Performing the q-integration in the amplitude prescription equation (4.1) over expansions of the schematic form

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle I_{12|34}(q) = q^{s_{12}/4 } \sum_{n=0}^{\infty}(a_n q^n + c_n q^{n+\oneh}) \label{justify1} \nonumber \end{align} \tag{ 4.58 }$$

yields kinematic poles at s₁₂  =  −4n in case of integer powers of q and at s₁₂  =  −4n  −  2 in case of half-odd integer powers, respectively, with $\newcommand{\ZN}{{\mathbb N}} n\in\ZN_0$:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle {\int^{1}\limits_{0}} \frac{\dd q}{q} \; I_{12|34}(q) = \sum_{n=0}^{\infty}\bigg\{\frac{4a_n}{s_{12} + 4n} + \frac{4c_n}{s_{12}+ 4n+2} \bigg\} \; . \end{align} \tag{ 4.59 }$$

The expansion coefficients a_n and c_n are understood to be formal power series in the Mandelstam variables s_ij accompanied by $\newcommand{\ZQ}{{\mathbb Q}} \ZQ[(2\pi {\rm i}){}^{-1}]$-linear combinations of MZVs. The singular values of Mandelstam variables $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} s_{12}=\ap(k_1+k_2){}^2$ in scattering amplitudes with external momenta k_i correspond to internal masses $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} -\ap m^2$. As a general property of non-planar one-loop open-string string amplitudes, the kinematic poles arising from the integration over q reveal the appearance of closed-string modes among the internal states [44].

In particular, the poles in equation (4.59) with residues proportional to a_n and c_n signal the exchange of closed-string states with masses $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} m^2 = \frac{4n}{\ap}$ and $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} m^2 = \frac{4n+2}{\ap}$, respectively, with $\newcommand{\ZN}{{\mathbb N}} n \in \ZN_0$. However, the closed-superstring spectrum only comprises masses $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} m^2 = \frac{4n}{\ap}$, whereas states with $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} m^2 = \frac{4n+2}{\ap}$ cannot be found in GSO projected string theories [43, 44, 46, 47]. Hence, the pole structure of $\frac{4c_n}{s_{12}+ 4n+2}$ due to half-odd integer powers of q would signal the propagation of unphysical states and violate unitarity if some of the c_n were nonzero.

However, it is not immediately clear if an integer-power q-expansion equation (4.58) with c_n  =  0 is necessarily expressible in terms of eMZVs. Given that $q^{-s_{12}/4 }I_{12|34}(q)$ was argued to comprise teMZVs with twists $\newcommand{\tauh}{\frac{\tau}{2}} \in \{0, \tauh\}$, this leads to the following, purely mathematical question: If a linear combination of teMZVs with twists $\newcommand{\tauh}{\frac{\tau}{2}} \in \{0, \tauh\}$ is such that its q-expansion has integer exponents only, can it be written as a linear combination of eMZVs (i.e. teMZVs with vanishing twists) only?

We expect that an answer to this question will necessitate a closer study of the decomposition of teMZVs into iterated τ-integrals of the functions h⁽ⁿ⁾(b) in equation (3.4), with $\newcommand{\tauh}{\frac{\tau}{2}} b\in \{0, \tauh\}$. Given that h⁽ⁿ⁾(b) are modular forms for congruence subgroups of $\newcommand{\SL}{\mathrm{SL}} \newcommand{\ZZ}{{\mathbb Z}} \SL_2(\ZZ)$, this decomposition of teMZVs generalizes the decomposition of eMZVs into linear combinations of iterated Eisenstein integrals [10, 36]. In particular, a natural first step would be to prove linear independence of iterated τ-integrals comprised of the integrands h⁽ⁿ⁾(b) with $\newcommand{\tauh}{\frac{\tau}{2}} b \in \{0, \tauh\}$, which would generalize results of [12, 60], and can presumably be proved along similar lines.

In any case, based on the arguments presented in this subsection, we conjecture that all orders in the $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$-expansion equation (4.55) are furnished by eMZVs. The same is expected to hold for all non-planar n-point amplitudes at one loop prior to integration over q. It is conceivable that this can be derived from the one-loop monodromy relations [31, 53], and it would be interesting to work out a rigorous proof.

In this article, teMZVs have been introduced as iterated integrals on an elliptic curve with multiple punctures on a lattice $\newcommand{\ZQ}{{\mathbb Q}} \ZQ+ \ZQ \tau$. Our main result is the identification of an initial-value problem satisfied by teMZVs, which expresses them in terms of linear combinations of iterated τ-integrals of the weighting functions f⁽ⁿ⁾ with coefficients given by cyclotomic MZVs.

As an application of teMZVs in physics we have studied one-loop scattering amplitudes of open-string states. In the non-planar sector of the four-point amplitude, the low-energy expansion can be computed by integrals over the two boundaries of a cylinder. A systematic procedure is established, which allows to evaluate these integrals in terms of teMZVs. There is no conceptual bottleneck in extending the procedure to one-loop amplitudes with an arbitrary number of external states. Having calculated the non-planar amplitude up to the third subleading low-energy order, we find that the results can ultimately be simplified to eMZVs.

The results of this article trigger a variety of questions: From a mathematical perspective, the differential equation of teMZVs could serve as a starting point to classify their relations and to understand the underlying algebraic principles. In the untwisted case, a crucial rôle was played by a certain derivation algebra. We expect that a suitable twisted analogue of the derivation algebra [9] will likewise control the algebraic structure of teMZVs.

In a physics context, the methods of this article allow to compute higher orders in the low-energy expansion of non-planar one-loop open-string amplitudes and to investigate its structure. More interestingly, higher-loop open-string amplitudes should require an extension of elliptic iterated integrals to Riemann surfaces of higher genus and a suitable generalization of teMZVs to accommodate multiple boundaries.

We would like to thank Claire Glanois, Martin Gonzalez, David Meidinger and Federico Zerbini for helpful discussions, and the Kolleg Mathematik und Physik Berlin for support in various ways. JB and OS would like to thank Universität Hamburg for hospitality. The work of JB and GR is supported in part by the SFB 647 'Raum–Zeit–Materie. Analytische und Geometrische Strukturen'. GR is furthermore supported by the International Max Planck Research School for Mathematical and Physical Aspects of Gravitation, Cosmology and Quantum Field Theory. This paper was written while NM was a Ph.D. student at Universität Hamburg under the supervision of Ulf Kühn. The research of OS was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation and Science.

For later use we note as well the expansion of Jacobi theta functions (and derivatives thereof) as an expansion in the parameter $q={\rm e}^{2\pi {\rm i} \tau}$ [57]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \theta_1(z,\tau) & = 2 q^{1/8} {{\rm ~sin}}(\pi z) \prod_{n=1}^{\infty} (1-q^n)(1 - 2q^n {{\rm ~cos}}(2\pi z) + q^{2n})\nonumber \\ \theta_1'(0,\tau)& = 2 \pi q^{1/8} \prod_{n=1}^{\infty} (1-q^n)^3 \nonumber \\ \theta_4(z,\tau) & = \prod_{n=1}^{\infty} (1-q^n)(1 - 2q^{n-1/2} {{\rm ~cos}}(2\pi z) + q^{2n-1}) \; ,\nonumber \label{eqn:conv_theta} \nonumber \end{align} \tag{ A.1 }$$

which all turn out to be positive given $z \in [ 0, 1 ]$ and $q \in [ 0, 1 ]$. Furthermore, a periodicity property useful for calculating equation (4.18) as well as a relation between $\theta_1$ and $\theta_4$ reads

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\be}{\beta} \displaystyle \theta_1(z+m+n \tau,\tau) = (-1)^{n+m} q^{-n^2/2} {\rm e}^{-2\pi {\rm i} n z}\,\theta_1(z,\tau) \nonumber \end{align} \tag{ A.2 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \theta_4(z,\tau) = -{\rm i} {\rm e}^{{\rm i}\pi z}q^{1/8}\,\theta_1(z+\tauh,\tau). \label{nonameeq} \nonumber \end{align} \tag{ A.3 }$$

The weighting functions $f^{(n)}(z, \tau)$ arise as expansion coefficients of the doubly-periodic completion $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \Omega(z, \alpha, \tau)$ of the Eisenstein–Kronecker series $\newcommand{\al}{\alpha} F(z, \alpha, \tau)$,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Om}{\Omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \renewcommand{\Im}{{\rm Im\,}} \displaystyle \Omega(z,\alpha,\tau)= \exp \bigg(2\pi {\rm i} \alpha\, \frac{\Im(z)}{\Im(\tau)} \bigg) F(z,\alpha,\tau)= \sum_{n=0}^{\infty}f^{(n)}(z,\tau)\alpha^{n-1} \, , \label{alt5} \nonumber \end{align} \tag{ B.1 }$$

which, in turn is given by [6, 61]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle F(z,\alpha,\tau) = \frac{\theta_1'(0,\tau)\theta_1(z+\alpha,\tau)}{\theta_1(z,\tau)\theta_1(\alpha,\tau)} \ . \label{alt1} \nonumber \end{align} \tag{ B.2 }$$

The odd Jacobi theta function $\theta_1$ is defined in appendix A, and the derivative with respect to the first argument is denoted by a tick. For real z, the expressions in equations (B.1) and (B.2) agree, and the lowest-order examples of f⁽ⁿ⁾ are spelt out in equation (2.2). In fact, f⁽¹⁾ is the only weighting function with a simple pole on the lattice $\newcommand{\ZZ}{{\mathbb Z}} \ZZ+\ZZ\tau$, while all f⁽ⁿ⁾ with $n\neq 1$ are non-singular on the entire elliptic curve.

Both the Eisenstein–Kronecker series $\newcommand{\al}{\alpha} F(z, \alpha, \tau)$ and its doubly periodic completion $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \Omega(z, \alpha, \tau)$ satisfy the Fay identity

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \Omega(z_1,\alpha_1,\tau) \Omega(z_2,\alpha_2,\tau) & = \Omega(z_1,\alpha_1 + \alpha_2,\tau) \Omega(z_2 - z_1,\alpha_2,\tau)\nonumber \\ & \quad + \Omega(z_2,\alpha_1 + \alpha_2,\tau) \Omega(z_1 - z_2,\alpha_1,\tau) . \label{fayfay_Om} \nonumber \end{align} \tag{ B.3 }$$

Furthermore, the Eisenstein–Kronecker series satisfies the mixed heat equation

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \newcommand{\pd}{\partial} \displaystyle 2\pi {\rm i}\,\pd_\tau F(z,\alpha,\tau) = \pd_\alpha \pd_z F(z,\alpha,\tau) \; . \label{mixedheat} \nonumber \end{align} \tag{ B.4 }$$

Both the Fay identity equation (B.3) and the mixed heat equation (B.4) are relevant in the calculations of section 3. Starting from the quasi-periodicity of the Eisenstein–Kronecker series,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle F(z+1,\alpha,\tau) = F(z,\alpha,\tau) \; , \qquad F(z+\tau,\alpha,\tau) = \exp(-2 \pi {\rm i} \alpha) F(z,\alpha,\tau) \; , \label{eqn:period_F} \nonumber \end{align} \tag{ B.5 }$$

and its reflection property

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle F(-z,-\alpha,\tau)=-F(z,\alpha,\tau) \; , \label{eqn:period_F1} \nonumber \end{align} \tag{ B.6 }$$

it is straightforward to derive properties equation (2.3) of the weighting functions f⁽ⁿ⁾.

We illustrate the definition of teMZVs in the case of proper rational twists $\newcommand{\La}{\Lambda} b \in \Lambda_N^{\times} = \{\frac{1}{N}, \ldots, \frac{N-1}{N}\}$, via an explicit computation of $\newcommand{\be}{\beta} \newcommand{\om}{\omega} \newcommand{\omm}{\omega} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\\#2\end{array}\right)} \omwb{0 , 1}{0 , \oneh}$. Our starting point is the definition equation (2.7) of teMZVs with twists $\newcommand{\oneh}{\frac{1}{2}} b_i \in \left\{0 , \oneh \right\}$ through the integral

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\dd}{\mathrm{d}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \omwb{n_1 , n_2 , \dots , n_{\ell}}{b_1 , b_2 , \dots , b_\ell} = \lim_{\varepsilon \rightarrow 0} \int_{\alpha_1 \delta_{\varepsilon} \alpha_2} f^{(n_1)}(z-b_1)\dd z_1 \, f^{(n_2)}(z-b_2)\dd z_2 \ldots f^{(n_\ell)}(z-b_\ell)\dd z_\ell \; . \label{eqn:def_real_twist} \nonumber \end{align} \tag{ C.1 }$$

We choose the parametrization of the individual path segments depicted in figure 3 as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \alpha_1(t) &= \left(\frac{1}{2} - \varepsilon \right) t \nonumber \\ \delta_{\varepsilon}(t) &= \frac{1}{2} - \varepsilon \exp(-{\rm i} \pi t) \nonumber \\ \alpha_2(t) &= \frac{1}{2} + \varepsilon + \left(\frac{1}{2} - \varepsilon \right)t \label{eqn:path_para} \nonumber \end{align} \tag{ C.2 }$$

with $t \in (0, 1)$ in each case. Then we may compute the iterated integral using the composition of paths formula, for the smooth one-forms $\newcommand{\om}{\omega} \newcommand{\dd}{\mathrm{d}} \omega_i = f^{(n_i)}(z-b_i) \dd z$ (see [62], proposition 2.9)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \label{eqn:comppath} \int_{\alpha\beta} \omega_1 \omega_2 \ldots\omega_\ell =\sum_{k=0}^\ell \int_{\alpha} \omega_1\omega_2\ldots\omega_k \int_{\beta} \omega_{k+1}\ldots\omega_\ell \, , \nonumber \end{align} \tag{ C.3 }$$

where the paths $\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \alpha, \beta$ are such that $\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \alpha(1)=\beta(0)$ and the empty integral is defined to be one.

As the forms $\newcommand{\om}{\omega} \newcommand{\dd}{\mathrm{d}} \omega_i = f^{(n_i)}(z-b_i) \dd z$ admit an expansion in q we may treat the q⁰ term separately from the rest, assuming the q-expansion can be exchanged with the integration. Then, as the coefficients of the jth power q^j for $j \neq 0$ are well defined on the real line, we may exchange the limit $\varepsilon \rightarrow 0$ with the integration and compute this part of the integral over the much more mundane path $\newcommand{\ga}{\gamma} \gamma(t) = t$. Specifically, the q dependent part is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle I_{q} = -2 {\rm i} (2 \pi {\rm i}) \int_{0 < t_1 < t_2 < 1} \dd t_1 \dd t_2 \sum_{n,m = 1}^{\infty} q^{m n} \sin\left(2 \pi m \left(t_2-\frac{1}{2} \right) \right) = - 2 \sum_{n,m = 1}^{\infty} \frac{(-1)^m q^{m n}}{m} \; . \label{eqn:qn_terms} \nonumber \end{align} \tag{ C.4 }$$

We note that this can be reproduced from the differential equation (3.5) for real twists

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\omm}{\omega} \newcommand{\dd}{\mathrm{d}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle \omwb{0 , 1}{0 , \oneh} &= \omwbc{0 , 1}{0 , \oneh} + \int_0^q \frac{\dd \log(q_1)}{-4 \pi^2} \left[ \omwb{2}{\oneh} - f^{(2)}\left(\frac{1}{2}, q_1\right) \right]\nonumber \\ &= \omwbc{0 , 1}{0 , \oneh} + \int_0^q \frac{\dd \log(q_1)}{-4 \pi^2} \left[ 8 \pi^2 \sum_{m,n=1}^\infty (-1)^m n \, q_1^{m n} \right] \; , \label{eqn:de_w010h} \nonumber \end{align} \tag{ C.5 }$$

where the integration constant $\newcommand{\be}{\beta} \newcommand{\om}{\omega} \newcommand{\omm}{\omega} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\\#2\end{array}\right)} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\\#2\end{array}\right)} \omwbc{0 , 1}{0 , \oneh}$ remains to be determined.

Application of equation (C.3) for the constant term $f_0^{(n)}(z-b)$ of the q expansion of f⁽ⁿ⁾(z  −  b) yields, bearing in mind that f⁽⁰⁾(z)  =  1,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle I_0 &= \int_{\alpha_1 \delta_{\varepsilon} \alpha_2} \dd z_1 \, f^{(1)}_0\left(z_2 - \frac{1}{2} \right) \dd z_2 \nonumber \\ &= \int_{\alpha_1} \dd z_1 \, f^{(1)}_0\left(z_2 - \frac{1}{2} \right)\dd z_2 + \int_{\delta_{\varepsilon}} \dd z_1 \, f^{(1)}_0\left(z_2 - \frac{1}{2} \right) \dd z_2 + \int_{\alpha_2}\dd z_1 \, f^{(1)}_0\left(z_2 - \frac{1}{2} \right) \dd z_2 \nonumber \\ & \quad + \int_{\alpha_1} \dd w \int_{\delta_{\varepsilon}} f^{(1)}_0\left(z - \frac{1}{2} \right) \dd z + \int_{\alpha_1} \dd w \int_{\alpha_2} f^{(1)}_0\left(z - \frac{1}{2} \right) \dd z + \int_{\delta_{\varepsilon}} \dd w \int_{\alpha_2} f^{(1)}_0\left(z - \frac{1}{2} \right) \dd z \; . \label{eqn:comppath_q0} \nonumber \end{align} \tag{ C.6 }$$

The individual integrals are given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\co}{~,\quad} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \int_{\alpha_1} \dd z_1\, f^{(1)}_0\left(z_2-\frac{1}{2}\right) \dd z_2 &= \int_{0 < t_1 < t_2 < 1 } \dd t_1 \dd t_2 \, \left(\frac{1}{2} - \varepsilon\right)^2 \pi \cot \left(\pi \left[ \left(\frac{1}{2}-\varepsilon \right)t_2 -\frac{1}{2} \right] \right) \nonumber \\ &=\frac{\log(2)}{2} + \frac{\log(\pi \varepsilon) }{2} + \mathcal{O}(\varepsilon) \label{eqn:i1} \nonumber \end{align} \tag{ C.7 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\co}{~,\quad} \newcommand{\la}{\lambda} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \displaystyle \int_{\delta_{\varepsilon}} \dd z_1 \, f^{(1)}_0 \left(z_2-\frac{1}{2} \right) \dd z_2 &= \int_{0 < t_1 < t_2 < 1 } \dd t_1 \dd t_2 \, ({\rm i} \pi)^2 \varepsilon^2 {\rm e}^{-{\rm i} \pi (t_1+t_2)} \pi \cot(- \pi \varepsilon {\rm e}^{-{\rm i} \pi t_2}) \nonumber \\ &= \mathcal{O}(\varepsilon) \label{eqn:i2} \nonumber \end{align} \tag{ C.8 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\co}{~,\quad} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \int_{\alpha_2} \dd z_1 \, f^{(1)}_0\left(z_2-\frac{1}{2} \right) \dd z_2 &= \int_{0 < t_1 < t_2 < 1 } \dd t_1 \dd t_2 \, \left(\frac{1}{2} - \varepsilon\right)^2 \pi \cot\left(\pi \left[ \left(\frac{1}{2}-\varepsilon \right)t_2 + \varepsilon \right] \right) \nonumber \\ &= \frac{\log(2)}{2} + \varepsilon \log(\pi \varepsilon) + \mathcal{O}(\varepsilon) \label{eqn:i3} \nonumber \end{align} \tag{ C.9 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \int_{\alpha_1} \dd w \int_{\delta_{\varepsilon}} f^{(1)}_0\left(z-\frac{1}{2} \right) \dd z = {\rm i} \pi \left(\varepsilon - \frac{1}{2} \right) \label{eqn:i4} \nonumber \end{align} \tag{ C.10 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \int_{\alpha_1} \dd w \int_{\alpha_2} f^{(1)}_0\left(z-\frac{1}{2} \right) \dd z = \left(\varepsilon - \frac{1}{2} \right) \log(\sin(\pi \varepsilon)) \label{eqn:i5} \nonumber \end{align} \tag{ C.11 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \int_{\delta_{\varepsilon}} \dd w \int_{\alpha_2} f^{(1)}_0\left(z-\frac{1}{2}\right)\dd z = -2 \varepsilon \log(\sin(\pi \varepsilon)) \, . \label{eqn:i6} \nonumber \end{align} \tag{ C.12 }$$

Note that due to $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \lim_{\varepsilon \rightarrow 0} \varepsilon \log(\sin(\pi \varepsilon)) = 0$ the only singular contributions stem from the integrals in equations (C.7) and (C.11), which cancel in their sum. Then, we arrive at

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \lim_{\varepsilon \rightarrow 0} I_0 = - \frac{{\rm i} \pi}{2} + \log(2) \, , \label{eqn:c_w010h} \nonumber \end{align} \tag{ C.13 }$$

as predicted by the constant-term procedure equation (3.27), see equation (G.7). Finally, upon combination with the q-series in equation (C.4), the desired teMZV is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle \omwb{0 , 1}{0 , \oneh} = \lim_{\varepsilon \rightarrow 0} (I_0 + I_q) = - \frac{{\rm i} \pi}{2} + \log(2) - 2 \sum_{n,m = 1}^{\infty} \frac{(-1)^m q^{m n}}{m} \; . \label{eqn:f_w010h} \nonumber \end{align} \tag{ C.14 }$$

Cyclotomic MZVs (also called 'multiple polylogarithms at roots of unity') are generalizations of MZVs. While MZVs are represented by nested sums of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\zm}{\zeta} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \zm_{n_1,n_2,\ldots,n_r}=\zm(n_1,n_2,\ldots,n_r)=\sum^{\infty}_{0<k_1<k_2<\ldots<k_r}\frac{1}{k_1^{n_1}k_2^{n_2}\ldots k_r^{n_r}}, \quad n_1,\ldots,n_{r-1} \geqslant 1, \, n_r \geqslant 2, \nonumber \end{align} \tag{ D.1 }$$

cyclotomic MZVs are represented by nested sums with additional 'coloring' given by Nth roots of unity $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \sigma_1, \ldots, \sigma_r \in \mu_N$:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\BZ}[2]{\zeta\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\co}{~,\quad} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \BZ{n_1,n_2,\ldots,n_r}{\sigma_1,\sigma_2,\ldots,\sigma_r} = \sum_{0<k_1<k_2<\ldots<k_r} \frac{\sigma_1^{k_1} \sigma_2^{k_2} \ldots \sigma_r^{k_r}} {k_1^{n_1} k_2^{n_2} \ldots k_r^{n_r}} \co n_1,\ldots,n_r \geqslant 1, \, n_r \geqslant 2 \ {\rm if} \ \sigma_r=1 \ . \label{cyclo11} \nonumber \end{align} \tag{ D.2 }$$

Likewise, the integral representation of MZVs

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \zeta(n_1,\ldots,n_r)=\int_{0 \leqslant t_i \leqslant t_{i+1} \leqslant 1}\frac{\dd t_1}{1-t_1}\frac{\dd t_2}{t_2}\ldots \frac{\dd t_{n_1}}{t_{n_1}}\frac{\dd t_{n_1+1}}{1-t_{n_1+1}}\ldots \frac{\dd t_w}{t_w} , \quad w=n_1+\ldots +n_r , \nonumber \end{align} \tag{ D.3 }$$

generalizes to an integral representation for cyclotomic MZVs

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\BZ}[2]{\zeta\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\dd}{\mathrm{d}} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \BZ{n_1,n_2,\ldots,n_r}{\sigma_1,\sigma_2,\ldots,\sigma_r} = \int_{0 \leqslant t_i \leqslant t_{i+1} \leqslant 1}\frac{\dd t_1}{\eta_1-t_1}\frac{\dd t_2}{t_2}\ldots \frac{\dd t_{n_1}}{t_{n_1}}\frac{\dd t_{n_1+1}}{\eta_2-t_{n_1+1}}\ldots \frac{\dd t_w}{t_w} , \nonumber \end{align} \tag{ D.4 }$$

where $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\e}{{\rm e}} \eta_i=(\sigma_i \sigma_{i+1}\cdots\sigma_r){}^{-1}$. The positive integer N, implicit in the definition of cyclotomic MZVs, is sometimes considered an additional datum, and one speaks of N-cyclotomic MZVs to emphasize the choice of N.

Cyclotomic MZVs have first been considered by Goncharov [3]. Suitable references for cyclotomic MZVs include [13–15]. For a detailed study of N-cyclotomic MZVs where $N=2, 3, 4, 6, 8$, see [63]. More recently, the case N  =  6 has generated further interest [64–66].

In this appendix, we give a detailed derivation of the differential equation (3.5) in the case $\newcommand{\e}{{\rm e}} b_1, b_\ell \neq 0$. The case b₁  =  0 or $\newcommand{\e}{{\rm e}} b_\ell=0$ is technically more complicated, since the iterated integrals involved need to be regularized according to [8]. However, using proposition 3.1 of [8], the arguments of this section go through for b₁  =  0 or $\newcommand{\e}{{\rm e}} b_\ell=0$ as well.

Using the mixed heat equation (3.2) for $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \Omega(z-b, \alpha)$, we may rewrite the τ-derivative of the generating function equation (3.1) of length-$\newcommand{\e}{{\rm e}} \ell$ teMZVs as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Tgenb}[2]{T\left[\begin{array}{cccc}#1\nonumber \\#2\end{array}\right]} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\nbeta}{b} \newcommand{\dd}{\mathrm{d}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \newcommand{\pd}{\partial} \displaystyle 2 &\pi {\rm i} \frac{\pd}{\pd \tau} \Tgen{\alpha_1, \alpha_2, \dots, \alpha_\ell}{\nbeta_1, \nbeta_2, \dots, \nbeta_\ell} = \int_{0\leqslant z_p \leqslant z_{p+1} \leqslant 1} \dd z_1 \, \dd z_2 \, \ldots \, \dd z_\ell \sum_{i=1}^\ell \pd_{\alpha_i} \pd_{z_i} \Omega(z_i - \nbeta_i,\alpha_i) \prod^\ell_{j \neq i} \Omega(z_j - \nbeta_j,\alpha_j) \nonumber \\ &= \int_{0\leqslant z_p \leqslant z_{p+1} \leqslant 1} \dd z_2 \, \ldots \, \dd z_\ell \, \pd_{\alpha_1} \Omega(z_2 - \nbeta_1,\alpha_1) \prod_{j =2}^\ell \Omega(z_j - \nbeta_j,\alpha_j) - \pd_{\alpha_1} \Omega(-\nbeta_1,\alpha_1) \Tgen{\alpha_2, \dots, \alpha_\ell}{\nbeta_2, \dots, \nbeta_\ell} \nonumber \\ & \quad + \pd_{\alpha_\ell} \Omega(-\nbeta_\ell,\alpha_\ell) \Tgen{\alpha_1, \dots, \alpha_{\ell-1}}{\nbeta_1, \dots, \nbeta_{\ell-1}} \!-\! \int_{0\leqslant z_p \leqslant z_{p+1} \leqslant 1} \dd z_1 \, \ldots \, \dd z_{\ell-1}\, \pd_{\alpha_\ell} \Omega(z_{\ell-1} - \nbeta_\ell,\alpha_\ell) \prod_{j=1}^{\ell-1} \Omega(z_j - \nbeta_j,\alpha_j) \nonumber \\ & \quad + \sum_{i=2}^{\ell-1} \int_{0\leqslant z_p \leqslant z_{p+1} \leqslant 1} \dd z_1 \, \ldots \, \dd z_{i-1} \,\dd z_{i+1} \, \ldots \, \dd z_{r} \, \pd_{\alpha_i} \Omega(z_{i} - \nbeta_i,\alpha_i) \Big|^{z_i=z_{i+1}}_{z_i = z_{i-1}} \, \prod_{j \neq i}^\ell \Omega(z_j - \nbeta_j,\alpha_j) \nonumber \\ &= \pd_{\alpha_\ell} \Omega(-\nbeta_\ell,\alpha_\ell) \Tgen{\alpha_1, \dots, \alpha_{\ell-1}}{\nbeta_1, \dots, \nbeta_{\ell-1}} - \pd_{\alpha_1} \Omega(-\nbeta_1,\alpha_1) \Tgenb{\alpha_2& \dots& \alpha_\ell}{\nbeta_2& \dots& \nbeta_\ell} \label{eqn:diff_eq_gen_T}\nonumber \\ & \quad + \int_{0\leqslant z_p \leqslant z_{p+1} \leqslant 1} \dd z_2 \,\dd z_3 \, \ldots \, \dd z_\ell \, \sum_{i=2}^{\ell} \, \prod_{j \neq i,1}^\ell \Omega(z_j - \nbeta_j,\alpha_j) \, (\pd_{\alpha_{i-1}} - \pd_{\alpha_{i}}) \Omega(z_i - \nbeta_{i-1},\alpha_{i-1}) \Omega(z_i - \nbeta_{i},\alpha_{i}) \nonumber \\ &= \pd_{\alpha_\ell} \Omega(-\nbeta_\ell,\alpha_\ell) \Tgen{\alpha_1, \dots, \alpha_{\ell-1}}{\nbeta_1, \dots, \nbeta_{\ell-1}} - \pd_{\alpha_1} \Omega(-\nbeta_1,\alpha_1) \Tgen{\alpha_2, \dots, \alpha_\ell}{\nbeta_2, \dots, \nbeta_\ell} \nonumber \\ & \quad + \sum_{i=2}^{\ell} \Big(\Tgen{\alpha_1, \dots, \alpha_{i-2}, \alpha_{i-1} + \alpha_i , \alpha_{i+1}, \dots, \alpha_\ell}{\nbeta_1, \dots, \nbeta_{i-2}, ~\quad \nbeta_{i},\quad ~ \nbeta_{i+1}, \dots, \nbeta_\ell} \pd_{\alpha_{i-1}} \Omega(\nbeta_{i}-\nbeta_{i-1},\alpha_{i-1}) \nonumber \\ & \qquad - \Tgen{\alpha_1, \dots, \alpha_{i-2}, \alpha_{i-1} + \alpha_i, \alpha_{i+1}, \dots, \alpha_\ell}{\nbeta_1, \dots ,\nbeta_{i-2} , ~~~\nbeta_{i-1},~~~ \nbeta_{i+1}, \dots, \nbeta_\ell} \pd_{\alpha_i} \Omega(\nbeta_{i-1} - \nbeta_{i},\alpha_i) \Big) \; , \nonumber \end{align} \tag{ E.1 }$$

where we mapped the integrals over $\newcommand{\pd}{\partial} \pd_{z_i}(\ldots)$ to boundary terms in the second equality and used the Fay identity (B.3) in the last equality to simplify

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\nbeta}{b} \newcommand{\Om}{\Omega} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \newcommand{\pd}{\partial} \displaystyle &(\pd_{\alpha_{i-1}} - \pd_{\alpha_{i}}) \Omega(z_i - \nbeta_{i-1},\alpha_{i-1}) \Omega(z_i - \nbeta_{i},\alpha_{i}) \nonumber \\ & = (\pd_{\alpha_{i-1}} - \pd_{\alpha_{i}})\Big[\Omega(z_i {-} \nbeta_{i},\alpha_{i-1}{+}\alpha_i) \Omega(\nbeta_i {-} \nbeta_{i-1},\alpha_{i-1}) + \Omega(z_i {-} \nbeta_{i-1},\alpha_{i-1}{+}\alpha_i) \pd_{\alpha_{i}} \Omega(\nbeta_{i-1} {-} \nbeta_{i},\alpha_{i})\Big] \nonumber \\ &= \Omega(z_i - \nbeta_{i},\alpha_{i-1}+\alpha_i) \pd_{\alpha_{i-1}} \Omega(\nbeta_i - \nbeta_{i-1},\alpha_{i-1}) - \Omega(z_i - \nbeta_{i-1},\alpha_{i-1}+\alpha_i) \pd_{\alpha_{i}} \Omega(\nbeta_{i-1} - \nbeta_{i},\alpha_{i}) \, . \nonumber \end{align} \tag{ E.2 }$$

From the above differential equation for the generating function one may deduce a differential equation for teMZVs,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \newcommand{\pd}{\partial} \displaystyle 2 \pi {\rm i} \pd_\tau \Tgen{\alpha_1, \alpha_2, \dots, \alpha_\ell}{\nbeta_1, \nbeta_2, \dots, \nbeta_\ell} &= 2 \pi {\rm i} \sum_{n_1,n_2,\ldots,n_\ell=0}^{\infty} \alpha_1^{n_1-1} \dots \alpha_\ell^{n_\ell-1} \pd_\tau \omwb{n_1, \dots , n_\ell}{\nbeta_1, \dots , \nbeta_\ell}\nonumber \\ &= \pd_{\alpha_\ell} \Omega(-\nbeta_\ell,\alpha_\ell) \Tgen{\alpha_1, \dots, \alpha_{\ell-1}}{\nbeta_1, \dots, \nbeta_{\ell-1}} - \pd_{\alpha_1} \Omega(-\nbeta_1,\alpha_1) \Tgen{\alpha_2, \dots, \alpha_\ell}{\nbeta_2, \dots, \nbeta_\ell} \nonumber \\ & \quad + \sum_{i=2}^{\ell} \Big(\Tgen{\alpha_1, \dots, \alpha_{i-2}, \alpha_{i-1} + \alpha_i , \alpha_{i+1}, \dots, \alpha_\ell}{\nbeta_1, \dots, \nbeta_{i-2}, ~\quad \nbeta_{i},\quad ~ \nbeta_{i+1}, \dots, \nbeta_\ell} \pd_{\alpha_{i-1}} \Omega(\nbeta_{i}-\nbeta_{i-1},\alpha_{i-1}) \nonumber \\ & \quad - \Tgen{\alpha_1, \dots, \alpha_{i-2}, \alpha_{i-1} + \alpha_i, \alpha_{i+1}, \dots, \alpha_\ell}{\nbeta_1, \dots ,\nbeta_{i-2} , ~~~\nbeta_{i-1} , ~~~\nbeta_{i+1}, \dots, \nbeta_\ell} \pd_{\alpha_i} \Omega(\nbeta_{i-1} - \nbeta_{i},\alpha_i) \Big) \ .\nonumber \label{eqn:diff_eq_again} \nonumber \end{align} \tag{ E.3 }$$

Eventually we want to equate coefficients to extract the τ-derivative of a particular $\newcommand{\te}{\textrm} \newcommand{\temzv}{{\rm teMZV}} \temzv$. For this purpose let us consider the terms in equation (E.3) separately. Recalling the definition equation (3.4) of h⁽ⁿ⁾ we may rewrite the first term on the right hand side of equation (E.3) as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \newcommand{\pd}{\partial} \displaystyle \pd_{\alpha_\ell} \Omega(-\nbeta_\ell,\alpha_\ell) \Tgen{\alpha_1, \dots, \alpha_{\ell-1}}{\nbeta_1, \dots, \nbeta_{\ell-1}} & = \sum_{n_\ell = 0}^\infty h^{(n_\ell)}(-\nbeta_\ell) \alpha_\ell^{n_\ell-2} \sum_{n_1,n_2,\ldots,n_{\ell-1}=0}^{\infty} \alpha_1^{n_1 - 1} \dots \alpha_{\ell-1}^{n_{\ell-1} - 1} \omwb{n_1, \dots , n_{\ell-1}}{\nbeta_1, \dots , \nbeta_{\ell-1}} \nonumber \\ &= \sum_{n_1,n_2,\ldots,n_\ell=0}^{\infty} \alpha_1^{n_1-1} \dots \alpha_{\ell-1}^{n_{\ell-1}-1} \alpha_\ell^{n_\ell-1} h^{(n_\ell+1)}(-\nbeta_\ell)\omwb{n_1, \dots , n_{\ell-1}}{\nbeta_1, \dots , \nbeta_{\ell-1}} \nonumber \\ & \quad + h^{(0)}(-\nbeta_\ell) \alpha_\ell^{-2} \sum_{n_1,n_2,\ldots,n_{\ell-1}=0}^{\infty} \alpha_1^{n_1-1} \dots \alpha_{\ell-1}^{n_{\ell-1}-1} \omwb{n_1, \dots , n_{\ell-1}}{\nbeta_1, \dots , \nbeta_{\ell-1}} \; , \label{eqn:diff_eq_bdy_term} \nonumber \end{align} \tag{ E.4 }$$

and similarly for the second term. The sum $\newcommand{\al}{\alpha} \newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\e}{{\rm e}} \sim \alpha_\ell^{-2}$ is canceled by contributions from the last two lines of equation (E.3), which we will now turn to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \newcommand{\pd}{\partial} \displaystyle &\Tgen{\alpha_1, \dots, \alpha_{i-2}, \alpha_{i-1} + \alpha_i , \alpha_{i+1}, \dots, \alpha_\ell}{\nbeta_1, \dots, \nbeta_{i-2}, \quad ~\nbeta_{i},\quad ~ \nbeta_{i+1}, \dots, \nbeta_\ell} \pd_{\alpha_{i-1}} \Omega(\nbeta_{i}-\nbeta_{i-1},\alpha_{i-1}) \nonumber \\ &= \sum_{n_1, \ldots, n_{i-2}, n_{i+1} , \ldots, n_{\ell} = 0}^{\infty} \alpha_1^{n_1-1} \dots \alpha_{i-2}^{n_{i-2}-1} \alpha_{i+1}^{n_{i+1}-1} \dots \alpha_\ell^{n_\ell-1} \nonumber \\ & \qquad \times \, \Big[ \underbrace{(\alpha_{i-1} + \alpha_i)^{-1} \omwb{n_1, \dots, n_{i-2}, 0 , \, n_{i+1}, \dots, n_\ell}{\nbeta_1, \dots, \nbeta_{i-2}, \nbeta_{i}, \, \nbeta_{i+1}, \dots, \nbeta_\ell} \sum_{j=0}^\infty h^{(\,j)}(\nbeta_{i}-\nbeta_{i-1}) \alpha_{i-1}^{\,j-2}}_{= \ B_{i,+}} \nonumber \\ & \qquad \qquad + \underbrace{\sum_{j,k = 0}^\infty \sum_{p=0}^{k} \left(\begin{array}{@{}c@{}}{k} \nonumber \\{\,p}\end{array}\right) \alpha_{i-1}^{k-p+j-2} \alpha_i^{\,p} \omwb{n_1, \dots, n_{i-2}, k+1 ,\, n_{i+1}, \dots, n_\ell}{\nbeta_1, \dots, \nbeta_{i-2}, \quad \nbeta_{i},\quad \nbeta_{i+1}, \dots, \nbeta_\ell} h^{(\,j)}(\nbeta_{i} - \nbeta_{i-1})}_{= \ C_{i,+}}\Big] \label{eqn:diff_eq_complicated_term} \nonumber \end{align} \tag{ E.5 }$$

with an analogous definition for B_i,− and C_i,− relevant to the last term of equation (E.3)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\om}{\omega} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle B_{i,-} &= (\alpha_{i-1} + \alpha_i)^{-1} \omwb{n_1, \dots ,n_{i-2}, 0 , \, n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, \nbeta_{i-1}, \, \nbeta_{i+1}, \dots ,\nbeta_\ell} \sum_{j=0}^\infty h^{(\,j)}(\nbeta_{i-1}-\nbeta_{i}) \alpha_{i}^{\,j-2} \nonumber \end{align} \tag{ E.6 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\om}{\omega} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle C_{i,-} &= \sum_{j,k = 0}^\infty \sum_{p=0}^{k} \left(\begin{array}{@{}c@{}}{k} \nonumber \\{\,p}\end{array}\right) \alpha_{i}^{k-p+j-2} \alpha_{i-1}^{\,p} \omwb{n_1, \dots ,n_{i-2}, k+1 , \, n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, \nbeta_{i-1}, \, \nbeta_{i+1}, \dots ,\nbeta_\ell} h^{(\,j)}(\nbeta_{i-1} - \nbeta_{i}) \ . \nonumber \end{align} \tag{ E.7 }$$

In the following the manipulations only affect pairs $\newcommand{\al}{\alpha} (\alpha_{i-1}, \alpha_i)$, hence we will suppress the summation over the other α's. Since $\newcommand{\nbeta}{b} h^{(0)}(\nbeta)=-1$ does not depend on $\newcommand{\nbeta}{b} \nbeta$, we can set it to zero for the iterated integral

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \omwb{n_1, \dots ,n_{i-2}, 0 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots, \nbeta_{i-2}, \nbeta_{i-1} , \nbeta_{i+1}, \dots ,\nbeta_\ell} = \omwb{n_1, \dots ,n_{i-2}, 0 , \, n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots, \nbeta_{i-2}, 0 , \nbeta_{i+1}, \dots, \nbeta_\ell} \; . \label{eqn:equality_n=0} \nonumber \end{align} \tag{ E.8 }$$

Hence, for the terms not expressible by the binomial law $B_{i, \pm}$, we can use h⁽⁰⁾  =  −1, h⁽¹⁾  =  0, $h^{(2)}(b) = h^{(2)}(-b)$ as well as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle (\alpha_{i-1}^{\,j} -(-1)^{\,j} \alpha_i^{\,j}) = (\alpha_{i-1} + \alpha_i) \sum_{a=0}^{\,j-1} (-1)^{\,j-1-a} \alpha_{i-1}^{a} \alpha_{i}^{\,j-1-a} \; , \; j>0 \label{eqn:random_sum_1} \nonumber \end{align} \tag{ E.9 }$$

to obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle & B_{i,+} - B_{i,-} \nonumber \\ &=(\alpha_{i-1} + \alpha_i)^{-1} \omwb{n_1, \dots ,n_{i-2}, 0 , \, n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, 0, \, \nbeta_{i+1}, \dots ,\nbeta_\ell} \sum_{j = 0}^\infty \left(h^{(\,j)}(\nbeta_{i}-\nbeta_{i-1}) \alpha_{i-1}^{\,j-2} - h^{(\,j)}(\nbeta_{i-1}-\nbeta_{i}) \alpha_{i}^{\,j-2} \right) \nonumber \\ & = \omwb{n_1, \dots ,n_{i-2}, 0 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, 0 , \nbeta_{i+1}, \dots ,\nbeta_\ell} \Big(\frac{\alpha_i^{-2} - \alpha_{i-1}^{-2} }{\alpha_{i-1}+\alpha_i } + \sum_{j = 0}^\infty h^{(\,j + 3)}(\nbeta_{i}-\nbeta_{i-1}) \sum_{a=0}^{\,j} (-1)^{\,j-a} \alpha_{i-1}^{a} \alpha_{i}^{\,j-a} \Big) \; . \label{eqn:non_binomial_terms} \nonumber \end{align} \tag{ E.10 }$$

The singular term $\newcommand{\al}{\alpha} \frac{\alpha_i^{-2} - \alpha_{i-1}^{-2} }{\alpha_{i-1}+\alpha_i } = \frac{1}{\alpha_{i-1} \alpha_i} (\frac{1}{\alpha_i} - \frac{1}{\alpha_{i-1}})$ will eventually cancel among different $B_{i, +}-B_{i, -}$. The remaining contribution may be rewritten into a form where we can easily read off the coefficient of a given monomial in the $\newcommand{\al}{\alpha} \alpha_i$,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle & \omwb{n_1, \dots ,n_{i-2}, 0 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, 0 , \nbeta_{i+1}, \dots ,\nbeta_\ell} \sum_{j=0}^\infty h^{(\,j + 3)}(\nbeta_{i}-\nbeta_{i-1}) \sum_{a=0}^{\,j} (-1)^{\,j-a} \alpha_{i-1}^{a} \alpha_{i}^{\,j-a} \nonumber \\ & \quad = \omwb{n_1, \dots ,n_{i-2}, 0 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, 0 , \nbeta_{i+1}, \dots ,\nbeta_\ell} \sum_{m,n = 0}^\infty h^{(m + n + 3)}(\nbeta_{i}-\nbeta_{i-1}) (-1)^{n} \alpha_{i-1}^{m} \alpha_{i}^{n} \; , \label{eqn:non_binomial_part} \nonumber \end{align} \tag{ E.11 }$$

which gives rise to the last line of equation (3.5).

For the contributions $C_{i, \pm}$ we have

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle C_{i,+} & = \sum_{j,k = 0}^\infty \sum_{p=0}^{k} \left(\begin{array}{@{}c@{}}{k} \nonumber \\{\,p}\end{array}\right) \alpha_{i-1}^{k+j-p-2} \alpha_i^{\,p} \omwb{n_1, \dots ,n_{i-2}, k+1 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, ~~~\nbeta_{i} ,~~~\, \nbeta_{i+1}, \dots ,\nbeta_\ell} h^{(\,j)}(\nbeta_{i} - \nbeta_{i-1}) \nonumber \\ & = \sum_{j,k = 0}^\infty \sum_{p=0}^{k-1} \left(\begin{array}{@{}c@{}}{k} \nonumber \\{\,p}\end{array}\right) \alpha_{i-1}^{k+j-p-1} \alpha_i^{\,p} \omwb{n_1, \dots ,n_{i-2}, k+1 ,n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, ~~~\nbeta_{i} ,~~~ \nbeta_{i+1}, \dots ,\nbeta_\ell} h^{(\,j+1)}(\nbeta_{i} - \nbeta_{i-1}) \nonumber \\ & \qquad + \sum_{k = 0}^\infty \sum_{p=0}^{k} \left(\begin{array}{@{}c@{}}{k+1} \nonumber \\{\,p}\end{array}\right) \alpha_{i-1}^{k-p-1} \alpha_i^{\,p} \omwb{n_1, \dots ,n_{i-2}, k+2 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2},~~~ \nbeta_{i},~~~ \nbeta_{i+1}, \dots ,\nbeta_\ell} h^{(0)}(\nbeta_{i} - \nbeta_{i-1}) \nonumber \\ & \qquad + \alpha_{i-1}^{-2} \sum_{a = 0}^\infty h^{(0)}(\nbeta_{i}-\nbeta_{i-1}) \alpha_i^a \omwb{n_1, \dots ,n_{i-2}, a+1 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2},~~~ \nbeta_{i},~~~ \nbeta_{i+1}, \dots ,\nbeta_\ell} \ . \label{eqn:binomial_part} \nonumber \end{align} \tag{ E.12 }$$

The sums proportional to $\newcommand{\al}{\alpha} \alpha_{i-1}^{-2}$ cancel among the corresponding contributions from C_i,+ and C_i−1,−. For the cases i  =  2 and i  =  r it is canceled by the corresponding sums in the last line of equation (E.4). We then find for the remaining part of C_i,+ 

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\nbeta}{b} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle & \sum_{j,k = 0}^\infty \sum_{p=0}^{k-1} \left(\begin{array}{@{}c@{}}{k} \nonumber \\{\,p}\end{array}\right) \alpha_{i-1}^{k+j-p-1} \alpha_i^{\,p} \omwb{n_1, \dots ,n_{i-2}, k+1 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, ~~~\nbeta_{i} ,~~~ \nbeta_{i+1}, \dots ,\nbeta_\ell} h^{(\,j+1)}(\nbeta_{i-1} - \nbeta_{i}) \nonumber \\ & \qquad + \sum_{k = 0}^\infty \sum_{p=0}^{k} \left(\begin{array}{@{}c@{}}{k+1} \nonumber \\{\,p}\end{array}\right) \alpha_{i-1}^{k-p-1} \alpha_i^{\,p} \omwb{n_1, \dots ,n_{i-2}, k+2 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2},~~~ \nbeta_{i} ,~~~ \nbeta_{i+1}, \dots ,\nbeta_\ell} h^{(0)}(\nbeta_{i-1} - \nbeta_{i}) \nonumber \\ & \quad = \sum_{m,n = 0}^\infty \alpha_{i-1}^{m-1} \alpha_i^n \sum_{j=0}^{m} \left(\begin{array}{@{}c@{}}{m+n-j} \nonumber \\{n}\end{array}\right) \omwb{n_1, \dots ,n_{i-2}, m+n+1-j , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, ~\quad ~~~\nbeta_{i} ,~~~~\quad \nbeta_{i+1}, \dots ,\nbeta_\ell} h^{(\,j+1)}(\nbeta_{i-1} - \nbeta_{i}) \nonumber \\ & \qquad + \sum_{m,n = 0}^\infty \alpha_{i-1}^{m-1} \alpha_i^n \left(\begin{array}{@{}c@{}}{m+n+1} \nonumber \\{n}\end{array}\right) \omwb{n_1, \dots ,n_{i-2}, m+n+2 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2},\quad ~~~ \nbeta_{i} ,~~~\quad \nbeta_{i+1}, \dots ,\nbeta_\ell} h^{(0)}(\nbeta_{i-1} - \nbeta_{i}) \nonumber \\ & \quad = \sum_{m,n = 0}^\infty \alpha_{i-1}^{m-1} \alpha_i^{n} \sum_{k=0}^{m+1} \left(\begin{array}{@{}c@{}}{n+k} \nonumber \\{k}\end{array}\right) \omwb{n_1, \dots ,n_{i-2}, n+k+1 , n_{i+1}, \dots ,n_\ell}{\nbeta_1, \dots ,\nbeta_{i-2}, ~~\quad \nbeta_{i} ,\quad ~~ \, \nbeta_{i+1}, \dots ,\nbeta_\ell} h^{(m-k+1)}(\nbeta_{i-1} - \nbeta_{i}) \; , \label{eqn:binomial_part_contd} \nonumber \end{align} \tag{ E.13 }$$

which, in combination with an analogous contribution from C_i,−, gives rise to the second and third line of equation (3.5).

The derivation of the differential equation given in appendix E was a priori only valid for generic twists. In this appendix, we provide an argument why it also holds for proper rational twists $\newcommand{\ZQ}{{\mathbb Q}} b_i \in \ZQ$. The only difference is that in equation (E.1) we now need to carefully keep track of the effect of deforming the domain of integration $[0, 1]$ infinitesimally to obtain $[0, 1]_{\varepsilon}$ (see figure 3). The key point is that the integral along $[0, 1]_{\varepsilon}$ is the same as along $[0, 1]$ up to terms which vanish in the limit, and the differential equation (E.1) goes through without change.

As in the case of generic twists, it will be convenient to define a generating function

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\TgenRe}[2]{{{T}_{\varepsilon}^{R}}\left[\begin{array}{cccc}#1\nonumber \\#2\end{array}\right]} \newcommand{\TgenR}[2]{{\tilde{T}_{\varepsilon}^{R}}\left[\begin{array}{cccc}#1\nonumber \\#2\end{array}\right]} \newcommand{\Tgenr}[2]{T^{R}\left[\begin{array}{cccc}#1\nonumber \\#2\end{array}\right]} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \Tgenr{\alpha_1,\dots,\alpha_\ell}{b_1,\dots, b_\ell} & = \lim_{\varepsilon \rightarrow 0} \TgenRe{\alpha_1 , \dots , \alpha_\ell}{b_1 , \dots , b_\ell} \; , \nonumber \end{align} \tag{ F.1 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\TgenRe}[2]{{{T}_{\varepsilon}^{R}}\left[\begin{array}{cccc}#1\nonumber \\#2\end{array}\right]} \newcommand{\TgenR}[2]{{\tilde{T}_{\varepsilon}^{R}}\left[\begin{array}{cccc}#1\nonumber \\#2\end{array}\right]} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\dd}{\mathrm{d}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\ga}{\gamma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \TgenRe{\alpha_1 , \dots , \alpha_\ell}{b_1 , \dots , b_\ell} & = \int_{0\leqslant t_i \leqslant t_{i+1} \leqslant 1} (\gamma_R^* \Omega(z_1 - b_1, \alpha_1,\tau) \dd z_1) \dots (\gamma_R^* \Omega(z_\ell - b_\ell, \alpha_\ell,\tau) \dd z_\ell) \; , \label{eqn:tgenr_def1} \nonumber \end{align} \tag{ F.2 }$$

where $\newcommand{\ga}{\gamma} \gamma_R = [0, 1]_\varepsilon$ and $\newcommand{\ga}{\gamma} \gamma_R^*$ denotes its pullback. Here and throughout this appendix, we will denote the pullback of the coordinate z_i along $\newcommand{\ga}{\gamma} \gamma_R$ by t_i. We note that in the case where all b_i are generic twists, we may pass to the limit $\varepsilon \to 0$ immediately and integrate along the line $\newcommand{\ga}{\gamma} \gamma_R(t) = t$, which leads to equation (3.1). Pulling back $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \newcommand{\dd}{\mathrm{d}} \Omega(z, \alpha, \tau) \dd z$ along $\newcommand{\ga}{\gamma} \gamma_R$, we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\ga}{\gamma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \gamma_R^* \Omega(z_i - b_i, \alpha_i,\tau) \dd z_i = \gamma_R^* \underbrace{{\rm e}^{-2 \pi {\rm i} r_i \alpha_i} F(z_i - b_i,\alpha_i,\tau) \dd z_i }_{= \ \tilde \Omega(z_i - b_i, \alpha_i, \tau) \dd z_i} + \mathcal{O}(\varepsilon) \; , \label{eqn:better_omega} \nonumber \end{align} \tag{ F.3 }$$

since ${\rm Im}(z_i)$ is of order ε on $[0, 1]_{\varepsilon}$.

The resulting form $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \newcommand{\dd}{\mathrm{d}} \tilde{\Omega}(z, \alpha, \tau) \dd z$ is now meromorphic; therefore the integral over $\newcommand{\al}{\alpha} \newcommand{\Om}{\Omega} \newcommand{\dd}{\mathrm{d}} \tilde{\Omega}(z, \alpha, \tau)\dd z$ along any path depends exclusively on its homotopy class. In particular, since the paths $[0, 1]_{\varepsilon}$ are all homotopic for sufficiently small ε, for every such ε we get²⁰

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\TgenR}[2]{{\tilde{T}_{\varepsilon}^{R}}\left[\begin{array}{cccc}#1\nonumber \\#2\end{array}\right]} \newcommand{\Tgenr}[2]{T^{R}\left[\begin{array}{cccc}#1\nonumber \\#2\end{array}\right]} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\dd}{\mathrm{d}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\ga}{\gamma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \Tgenr{\alpha_1 , \dots , \alpha_\ell}{b_1 , \dots , b_\ell} & = \lim_{\varepsilon \to 0} \TgenR{\alpha_1 , \dots , \alpha_\ell}{b_1 , \dots , b_\ell}=\TgenR{\alpha_1 , \dots , \alpha_\ell}{b_1 , \dots , b_\ell} \; , \nonumber \\ \TgenR{\alpha_1 , \dots , \alpha_\ell}{b_1 , \dots , b_\ell} & = \int_{0\leqslant t_i \leqslant t_{i+1} \leqslant 1} (\gamma_R^* \tilde \Omega(z_1 - b_1, \alpha_1,\tau) \dd z_1) \dots (\gamma_R^* \tilde \Omega(z_\ell - b_\ell, \alpha_\ell,\tau) \dd z_\ell) \; . \label{eqn:tgenr_def2} \nonumber \end{align} \tag{ F.4 }$$

Furthermore, we define the intermediate object

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\TgenTREz}[3]{{\tilde{T}_{\varepsilon}^{R}}\left[\begin{array}{c}#1\nonumber \\#2\end{array};#3\right]} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\dd}{\mathrm{d}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\ga}{\gamma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \TgenTREz{\alpha_1 , \dots , \alpha_i}{b_1 , \dots , b_i}{z_{i+1}} & = \int_{0\leqslant t_1 \leqslant \ldots \leqslant t_i \leqslant t_{i+1}} (\gamma_R^* \tilde \Omega(z_1 - b_1, \alpha_1,\tau) \dd z_1) \dots (\gamma_R^* \tilde \Omega(z_i - b_i, \alpha_i,\tau) \dd z_i) \nonumber \\ & = \int_0^{t_{i+1}} \gamma_R^* \left(\tilde \Omega(z_i - b_i, \alpha_i,\tau) \TgenTREz{\alpha_1 , \dots , \alpha_i}{b_1 , \dots , b_i}{z_i} \dd z_i \right) \label{eqn:tgenr_inter_merda} \nonumber \end{align} \tag{ F.5 }$$

with 0  <  t_i+1  <  1. It satisfies

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\TgenTREz}[3]{{\tilde{T}_{\varepsilon}^{R}}\left[\begin{array}{c}#1\nonumber \\#2\end{array};#3\right]} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle \partial_{z_{i+1}} \TgenTREz{\alpha_1 , \dots , \alpha_i}{b_1 , \dots , b_i}{z_{i+1}} = \tilde \Omega(z_{i+1} - b_i, \alpha_i,\tau) \TgenTREz{\alpha_1 , \dots , \alpha_{i-1}}{b_1 , \dots , b_{i-1}}{z_{i+1}} . \label{eqn:tgenr_inter_merda_2} \nonumber \end{align} \tag{ F.6 }$$

Using the above setup, we now show that all essential aspects of the computation (E.1) in appendix E remain unchanged. Firstly, we compute the τ-derivative of $\newcommand{\ga}{\gamma} \newcommand{\Om}{\Omega} \gamma_R^* \tilde \Omega$ and obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\ga}{\gamma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle 2 \pi {\rm i} \partial_\tau \gamma_R^* &\left(\tilde \Omega(z_i - b_i, \alpha_i,\tau) \dd z_i \right)\nonumber \\ &= 2 \pi {\rm i} \partial_\tau \left({\rm e}^{-2 \pi {\rm i} r_i \alpha_i } F(\gamma_R(t_i) - b_i, \alpha_i,\tau) \dd \gamma_R(t_i) \right) \nonumber \\ &= {\rm e}^{-2 \pi {\rm i} r_i \alpha_i } \left[ (- 2 \pi {\rm i} r_i \partial_{\gamma_R(t_i)} + \partial_{\gamma_R(t_i)} \partial_{\alpha_i}) F(\gamma_R(t_i) - b_i, \alpha_i,\tau) \right] \dd \gamma_R(t_i) \nonumber \\ &= \gamma_R^* \left(\partial_{z_i} \partial_{\alpha_i} \tilde \Omega(z_i - b_i, \alpha_i,\tau) \dd z_i \right) , \label{eqn:tau_diff_pb} \nonumber \end{align} \tag{ F.7 }$$

using the mixed heat equation (B.4) for $\newcommand{\al}{\alpha} F(z, \alpha, \tau)$ in the second step. In particular, $\newcommand{\ga}{\gamma} \newcommand{\Om}{\Omega} \gamma_R^*\tilde \Omega$ itself satisfies a mixed-heat type equation. Secondly, we may exchange the τ-derivative with the integration

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\TgenR}[2]{{\tilde{T}_{\varepsilon}^{R}}\left[\begin{array}{cccc}#1\nonumber \\#2\end{array}\right]} \newcommand{\TgenTREz}[3]{{\tilde{T}_{\varepsilon}^{R}}\left[\begin{array}{c}#1\nonumber \\#2\end{array};#3\right]} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\dd}{\mathrm{d}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\ga}{\gamma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle 2 \pi {\rm i} \partial_\tau \TgenR{\alpha_1 , \dots , \alpha_\ell}{b_1 , \dots , b_\ell} = \sum_{i=1}^\ell \ & \int_{0 \leqslant t_{j-1} \leqslant t_j \leqslant 1 } \ \prod^\ell_{j > i} (\gamma_R^* \tilde \Omega(z_j - b_j, \alpha_j,\tau) \dd z_j) \label{eqn:diff_eq_prt} \nonumber \\ & \times \int_{0}^{t_{i+1}} \gamma_R^* \left((\partial_{z_i} \partial_{\alpha_i} \tilde \Omega(z_i - b_i, \alpha_i,\tau)) \TgenTREz{\alpha_1 , \dots , \alpha_{i-1}}{b_1 , \dots , b_{i-1}}{z_i} \dd z_i \right) , \nonumber \end{align} \tag{ F.8 }$$

since the τ-derivative of the integrand is bounded on the domain of integration. Rewriting the ith integration using integration by parts yields

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\TgenTREz}[3]{{\tilde{T}_{\varepsilon}^{R}}\left[\begin{array}{c}#1\nonumber \\#2\end{array};#3\right]} \newcommand{\Tgen}[2]{T\left[\begin{array}{c}#1\nonumber \\#2\end{array}\right]} \newcommand{\dd}{\mathrm{d}} \newcommand{\Om}{\Omega} \newcommand{\la}{\lambda} \newcommand{\ga}{\gamma} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle & \int_{0}^{t_{i+1}} \gamma_R^* \left((\partial_{z_i} \partial_{\alpha_i} \tilde \Omega(z_i - b_i, \alpha_i,\tau)) \TgenTREz{\alpha_1 , \dots , \alpha_{i-1}}{b_1 , \dots , b_{i-1}}{z_i} \dd z_i \right) \nonumber \\ & = \gamma_R^* \left((\partial_{\alpha_i} \tilde \Omega(z_{i+1} - b_i, \alpha_i,\tau)) \TgenTREz{\alpha_1 , \dots , \alpha_{i-1}}{b_1 , \dots , b_{i-1}}{z_{i+1}} \right) \label{eqn:ith_int} \nonumber \\ & \qquad - \int_{0}^{t_{i+1}} \gamma_R^* \left((\partial_{\alpha_i} \tilde \Omega(z_i - b_i, \alpha_i,\tau)) \tilde \Omega(z_{i-1} - b_{i-1}, \alpha_{i-1},\tau) \TgenTREz{\alpha_1 , \dots , \alpha_{i-2}}{b_1 , \dots , b_{i-2}}{z_i} \dd z_i \right) , \nonumber \end{align} \tag{ F.9 }$$

as in the case of generic twists. Therefore, we may proceed further as in the computation of equation (E.1) and arrive at the same result simply by virtue of the replacements $\newcommand{\TL}{T} \TL \rightarrow \TL^R_\varepsilon$ and $\newcommand{\Om}{\Omega} \Omega \rightarrow \tilde \Omega$.

The purpose of this appendix is to gather constant terms and q-expansions of teMZVs as well as selected relations relevant to the one-loop open-string amplitude in section 4.

G.1. Constant terms for generic twists

We start by listing a few simple examples for constant terms of teMZVs with generic twists. These constant terms are obtained from equation (3.18) by comparing the coefficients of words in the non-commutative variables ${\rm ad}^n_{x_b}(y)$ on both sides, see equation (3.14) for the change of alphabet between the two sides. At length one, specializing equations (3.11) and (3.12) to $r=\frac{1}{2}$ yields

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \omwbc{n}{\tauh} = \left\{\begin{array}{@{}ll@{}} \frac{2^{n-1} -1}{2^{n-2}}\zeta_n &: \ n \ {{\rm even}} \nonumber \\ 0&: \ n\ {{\rm odd}} , \end{array}\right. \label{exlength1} \nonumber \end{align} \tag{ G.1 }$$

see equation (2.24) for the $\newcommand{\om}{\omega} \omega_0(\ldots)$ notation. This immediately implies that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\ZN}{{\mathbb N}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \omwbc{\ldots, 2k-1, \ldots}{\ldots, \tauh, \ldots} = 0 , \ \ \ \ \ \ k \in \ZN , \label{exlengthn} \nonumber \end{align} \tag{ G.2 }$$

regardless on the position of the combined letter $\newcommand{\be}{\beta} \newcommand{\e}{{\rm e}} \begin{array}{@{}c@{}} 2k-1 \\ \tau/2\end{array}$. Similarly, higher-length examples include

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \omwbc{n,0}{\tauh,0} &= \left\{\begin{array}{@{}ll@{}} \frac{2^{n-1} -1}{2^{n-1}}\zeta_n &: \ n \ {{\rm even}} \nonumber \\ 0&: \ n\ {{\rm odd}} \end{array}\right. , \ \ \ \ \ \ \omwbc{1,n}{0,\tauh} = \left\{\begin{array}{@{}ll@{}} (-{\rm i}\pi) \frac{2^{n-1} -1}{2^{n-1}}\zeta_n &: \ n \ {{\rm even}} \nonumber \\ 0&: \ n\ {{\rm odd}} \end{array}\right. \nonumber \\ \omwbc{2,0,0}{0,0,0} &= - \frac{\zeta_2}{3} , \ \ \ \ \ \ \omwbc{0,1,0,0}{0,0,0,0} = \frac{3 \zeta_3}{4\pi^2} , \ \ \ \ \ \ \ \ \ \ \ \omwbc{0,3,0,0}{0,0,0,0}=0 \label{exlength3} \nonumber \\ \omwbc{1,0,2}{0,0,\tauh} &= - \frac{{\rm i} \pi^3}{24} , \ \ \ \ \ \ \omwbc{0,2,2}{0,0,\tauh} = - \frac{\pi^4}{108} , \ \ \ \ \ \ \omwbc{0,1,0,2}{0,0,0,\tauh} = \frac{\zeta_3}{8} , \nonumber \end{align} \tag{ G.3 }$$

and we obtain the following examples with more general twists $\newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\La}{\Lambda} b \in (\Lambda_N+\Lambda_N\tau) \setminus \Lambda^\times_N$:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauv}{\frac{\tau}{4}} \newcommand{\taud}{\frac{\tau}{3}} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \label{eqn:real_twists_firstline_2nd} \omwbc{1}{\oneh + \taud} &= -\frac{{\rm i} \pi}{3}, \quad \;\omwbc{2, 1, 0, 1}{\tauh, \tauv, 0, 0} = - \frac{{\rm i} \pi}{48} \zeta_3 + \frac{5}{8} \zeta_4, \nonumber \end{align} \tag{ G.4 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauzf}{\frac{2\tau}{5}} \newcommand{\tauf}{\frac{\tau}{5}} \newcommand{\tauv}{\frac{\tau}{4}} \newcommand{\taud}{\frac{\tau}{3}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \omwbc{1,1}{\taud,\tauf} &= - \frac{3}{5} \zeta_2, \quad \,\, \omwbc{3,1,0,1}{\tauzf,0,0,\tauv} = - \frac{9}{125} \zeta_2 \zeta_3. \label{eqn:real_twists_2nd} \nonumber \end{align} \tag{ G.5 }$$

Given a twist $b=s+r\tau$ with $r\neq 0$, the constant term does not depend on s (see equations (3.11) and (3.12)).

Up to weight five and length three (respectively weight three and length four), we have checked the constant-term procedure for consistency with Fay relations among teMZVs which can be derived along the lines of [10]. As already noted above, the constant-term procedure discussed in section 3.2 covers all eMZVs occurring in the one-loop open-string amplitude at the orders considered in section 4. The examples presented here address all teMZVs relevant to the calculations in section 4.3.

G.2. Constant terms for proper rational twists

For the proper rational twist $b=\frac{1}{2}$ we arrive at the following examples of constant terms

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \label{eqn:real_twists_firstline} \omwbc{1}{\oneh} &= -{\rm i} \pi, \qquad &\omwbc{2 , 0 , 1}{\oneh , 0 , \oneh} = \frac{{\rm i} \pi^3}{24} - \zeta_2 \log(2), \nonumber \end{align} \tag{ G.6 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \omwbc{0 , 1}{0 , \oneh} &= - \frac{{\rm i} \pi}{2} + \log(2) , \;\;&\omwbc{1 , 0 , 0}{\oneh , 0 , 0} = - \frac{{\rm i} \pi}{8} - \frac{\log(2)}{2} , \label{eqn:real_twists} \nonumber \end{align} \tag{ G.7 }$$

while twists $\newcommand{\La}{\Lambda} b \in \Lambda^\times_3$ give rise to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\BZ}[2]{\zeta\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\onezd}{\frac{2}{3}} \newcommand{\oned}{\frac{1}{3}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \label{eqn:real_twists_firstline_2nda} \omwbc{1}{\oned} &= -{\rm i} \pi, \;\omwbc{1, 1}{\oned, \onezd} = {\rm i} \pi \left(\BZ{1}{{\rm e}^{4\pi {\rm i} /3}} - \BZ{1}{{\rm e}^{2 \pi {\rm i} /3}} \right) - 3 \zeta_2 , \nonumber \end{align} \tag{ G.8 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\BZ}[2]{\zeta\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\oned}{\frac{1}{3}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \omwbc{1,1,2}{0,0,\oned} &= \frac{5}{2} \zeta_4, \omwbc{1,1,0}{\oned,0,0} = -\frac{{\rm i} \pi}{2} \BZ{1}{{\rm e}^{2\pi {\rm i} /3}} + \frac{1}{2} \BZ{2}{{\rm e}^{2\pi {\rm i} /3}} - \BZ{1,1}{{\rm e}^{4\pi {\rm i} /3},{\rm e}^{2 \pi {\rm i} /3}} , \label{eqn:real_twists_2nda} \nonumber \end{align} \tag{ G.9 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\BZ}[2]{\zeta\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwbc}[2]{\omm_0\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\oned}{\frac{1}{3}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \omwbc{1, 1, 0, 1}{\oned, 0, 0, 0} &= \frac{{\rm i} \pi}{4} \zeta_2 + \zeta_2 \BZ{1}{{\rm e}^{2\pi {\rm i}/3}} + \frac{{\rm i} \pi}{4} \BZ{2}{{\rm e}^{2\pi {\rm i}/3}} + \frac{1}{4} \BZ{3}{{\rm e}^{2\pi {\rm i} /3}} \nonumber \\ & \quad - \frac{{\rm i} \pi}{2} \BZ{1,1}{{\rm e}^{4\pi {\rm i} /3},{\rm e}^{2\pi {\rm i} /3}} - \frac{1}{2} \BZ{1,2}{{\rm e}^{4\pi {\rm i} /3},{\rm e}^{2\pi {\rm i} /3}} - \frac{1}{2} \BZ{2,1}{{\rm e}^{4\pi {\rm i} /3},{\rm e}^{2\pi {\rm i} /3}} , \label{eqn:awesome_dude} \nonumber \end{align} \tag{ G.10 }$$

see equation (D.2) for the definition of cyclotomic MZVs $\newcommand{\be}{\beta} \newcommand{\ze}{\zeta} \newcommand{\z}{\zeta} \newcommand{\si}{\sigma} \newcommand{\s}{\sigma} \newcommand{\e}{{\rm e}} \newcommand{\BZ}[2]{\zeta\left(\begin{array}{c}#1\\#2\end{array}\right)} \BZ{n_1, n_2, \ldots, n_r}{\sigma_1, \sigma_2, \ldots, \sigma_r}$.

Again, consistency of the constant-term procedure with Fay relations among teMZVs has been checked up to weights five and three at lengths three and four, respectively.

G.3. teMZV relations for the string amplitude

The simplification of the string amplitude in section 4.3 requires several relations among teMZVs and eMZVs. The subsequent identities involving teMZVs can be proven by comparing both the constant terms encoded in equation (3.18) and the τ-derivatives equation (3.5) of both sides. The relations among eMZVs follow from a combination of Fay and shuffle identities [10] and are listed at https://tools.aei.mpg.de/emzv/. At the second order in $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap$, we make use of

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle 2\omwb{1 ,1 , 0}{0 ,0 , 0} - \omwb{1 , 0}{0 , 0}^2 &= \omwb{2 , 0, 0}{0 , 0, 0} + \frac{5 \zeta_2}{6}\label{weight2Bnoq} \nonumber \end{align} \tag{ G.11 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle 2\omwb{1 ,1 , 0}{\tauh,\tauh, 0} - \omwb{1 , 0}{\tauh, 0}^2 &= \omwb{2 , 0, 0}{0 , 0, 0} + \frac{\zeta_2}{3}\; , \label{weight2Anoq} \nonumber \end{align} \tag{ G.12 }$$

while the simplifications at the order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^3$ are based on the relations

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle & \omwb{1,0}{0,0}^3 - 3 \omwb{1,0}{0,0} \omwb{1 ,1,0}{0,0,0} + 3 \omwb{1,1,1,0}{0,0,0,0} \nonumber \\ & \ \ \ = \frac{1}{6} \, \omwb{0,3,0,0}{0,0,0,0} - \frac{1}{4} \, \zeta_3 - 4 \, \zeta_2\, \omwb{0,1,0,0}{0,0,0,0} \label{weight2Dnoq} \nonumber \end{align} \tag{ G.13 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle & \omwb{1,0}{\tauh,0}^3 -3 \omwb{1,0}{\tauh,0} \omwb{1 ,1,0}{\tauh,\tauh,0} +3 \omwb{1,1,1,0}{\tauh,\tauh,\tauh,0} \nonumber \\ & \ \ \ = \frac{1}{6} \, \omwb{0,3,0,0}{0,0,0,0} - \frac{1}{4} \, \zeta_3 +2 \, \zeta_2\, \omwb{0,1,0,0}{0,0,0,0} \; . \label{weight2Cnoq} \nonumber \end{align} \tag{ G.14 }$$

G.4. q-expansions for the string amplitude

The eMZVs seen in the final results equations (4.55) and (H.26) for the integrals in the string amplitude have the following q-expansions

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \omega(0,0,2) &=- \frac{\zeta_2 }{3}+ 2 \sum_{n,m=1}^{\infty} \frac{m }{n^2} q^{mn} \nonumber \\ \omega(0,1,0,0) &=\frac{3\zeta_3 }{4\pi^2} + \frac{3}{2\pi^2} \sum_{n,m=1}^{\infty} \frac{1 }{n^3} q^{mn} \label{qexps} \nonumber \\ \omega(0,3,0,0) &= -3 \sum_{n,m=1}^{\infty} \frac{m^2 }{n^3} q^{mn} \; . \nonumber \end{align} \tag{ G.15 }$$

These expressions follow from repeatedly integrating the Eisenstein series equation (3.6) in the τ-derivatives equation (3.5) of the eMZVs in question. Analogous q-expansions for teMZVs with twists $\newcommand{\tauh}{\frac{\tau}{2}} b_i \in \left\{0, \tauh\right\}$ can be determined based on

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\oneh}{\frac{1}{2}} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle f^{(k)}\left(\tauh\right) = \left\{\begin{array}{@{}ll@{}} \ \ \frac{2^{k-1}-1}{2^{k-2}} \zeta_k - \frac{2 (2 \pi {\rm i})^k }{(k-1)!} \sum\limits_{n,m=1}^\infty \left(n-\frac{1}{2}\right)^{k-1} q^{m(n-\oneh)} &: \ k {{\rm ~even~}} \nonumber \\ 0 &: \ k {{\rm ~odd~}} \end{array}\right. \; . \label{eqn:fn_th} \nonumber \end{align} \tag{ G.16 }$$

In this appendix, we investigate the low-energy expansion of the integral equation (4.6) associated with the color factor ${\rm Tr}(t^1 t^2 t^3) {\rm Tr}(t^4)$ in the one-loop four-point open-string amplitude. The representations equations (4.12) and (4.13) of the Green functions allow to cast the integral into the form

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\GL}{\Gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle I_{123|4}(q) = \int_{123}^{4} \prod_{1\leqslant i<j}^3 \sum_{n_{ij}=0}^{\infty} \frac{(s_{ij} \GLarg{1}{0 }{x_{ij}})^{n_{ij}} }{n_{ij}!} \prod_{j=1}^3 \sum_{n_{j4}=0}^{\infty} \frac{\left(s_{j4} \GLarg{1}{\tauh}{x_{j4} } \right)^{n_{j4} } }{n_{j4}!} \label{eqn:newnewnewrep} \nonumber \end{align} \tag{ H.1 }$$

analogous to equation (4.23), where $c(q)$ cancels by momentum conservation equation (4.5), and the integration measure is defined by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{\mathrm{d}} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\de}{\delta} \newcommand{\be}{\beta} \displaystyle \int_{123}^{4} &= \int_{0}^{1} \dd x_4 \int_{0}^{1} \dd x_3 \int_{0}^{x_3} \dd x_2 \int_{0}^{x_2} \dd x_1\, \delta(x_1) \label{eqn:int_bdy_31} \nonumber \\ &=\int_{0}^{1} \dd x_4 \, \delta(x_4)\, \bigg(\int_{0}^{1} \dd x_3 \int_{0}^{x_3} \dd x_2 \int_{0}^{x_2} \dd x_1 + {\rm cyc}(x_1,x_2,x_3) \bigg) \; . \nonumber \end{align} \tag{ H.2 }$$

One can check through the change of variables $x_i \rightarrow 1-x_i$ and symmetry properties of the Green function that the measure $\int_{132}^{4}$ with x₂ and x₃ interchanged yields the same result for the integral equation (H.1). Hence, one can equivalently employ the simpler measure equation (4.22) with $x_i \in [0, 1]$ tailored to the color structure ${\rm Tr}(t^1 t^2) {\rm Tr}(t^3 t^4)$ and rewrite

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\GL}{\Gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle I_{123|4}(q) = \frac{1}{2} \int_{12}^{34}\prod_{1\leq i<j}^3 \sum_{n_{ij}=0}^{\infty} \frac{ \left(s_{ij} \GLarg{1}{ 0 }{x_{ij}} \right)^{n_{ij}} }{n_{ij}!}\prod_{j=1}^3 \sum_{n_{j4}=0}^{\infty} \frac{ \left(s_{j4} \GLarg{1}{ \tauh}{x_{j4} } \right)^{n_{j4} } }{ n_{j4}!} \ .\label{eqn:evennewnewnewrep} \nonumber \end{align} \tag{ H.3 }$$

In the remainder of this appendix, we will follow the steps of section 4.3 to expand the integral I_123|4(q) to the order $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^3$, where the representation equation (H.3) is most convenient for practical purposes. While the elliptic iterated integrals in this expansion lead to teMZVs for each monomial in s_ij, the final results for the orders $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}$ turn out to comprise eMZVs only.

H.1. Structure of the leading orders ${ \newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}}$

We start by classifying the inequivalent integrals w.r.t. the cycle structure of ${\rm Tr}(t^1 t^2 t^3)\, {\rm Tr}(t^4)$ which occur at the orders $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}$ of equation (H.1): With the shorthands $P_{ij}=P(x_{ij})$ and

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\GL}{\Gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \hat Q_{ij}=Q(x_{ij}) - c(q) = \GLarg{1}{\tauh}{x_{ij} } \; , \label{eqn:hatQ} \nonumber \end{align} \tag{ H.4 }$$

we have two inequivalent cases at the first order,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\co}{~,\quad} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e_1^1= \int_{123}^{4} P_{12} \co e_2^1= \int_{123}^{4} \hat Q_{14} \, , \label{31npl9} \nonumber \end{align} \tag{ H.5 }$$

six cases at the second order,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\co}{~,\quad} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e_1^2 &= \frac{1}{2} \int_{123}^{4} P_{12}^2 \co e^2_3 = \int_{123}^{4} P_{12} P_{13} \co e^2_5 = \int_{123}^{4} P_{12} \hat Q_{34} \label{31npl10} \nonumber \\ e_2^2 &= \frac{1}{2} \int_{123}^{4}\hat Q_{14}^2 \co e^2_4 = \int_{123}^{4} \hat Q_{14} \hat Q_{24} \co e^2_6 = \int_{123}^{4} P_{12} \hat Q_{14} , \nonumber \end{align} \tag{ H.6 }$$

and fourteen cases at the third order:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\co}{~,\quad} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e_1^3 &= \frac{1}{6} \int_{123}^{4} P_{12}^3 \co \ \ \ \ \ \ \ \ \hspace{0.8mm} e_6^3= \frac{1}{2} \int_{123}^{4} \hat Q_{14}^2 \hat Q_{24} , \ \ \ \ \ \ \ \ \hspace{0.8mm} e_{11}^3= \int_{123}^{4} P_{12} P_{13} \hat Q_{14} \ \, \nonumber \\ e_2^3 &= \frac{1}{6} \int_{123}^{4} \hat Q_{14}^3 \co \ \ \ \ \ \ \ \ \hspace{0.3mm} e_7^3= \frac{1}{2} \int_{123}^{4} P_{12}^2 \hat Q_{14} , \ \ \ \ \ \ \ \ \hspace{0.8mm} e_{12}^3= \int_{123}^{4} P_{12} P_{13} \hat Q_{24} \ \, \nonumber \\ e_3^3 &= \frac{1}{2} \int_{123}^{4} P_{12}^2 P_{13} \co \ \ \ \ \hspace{0.3mm} e_8^3= \frac{1}{2} \int_{123}^{4} P_{12}^2 \hat Q_{34} , \ \ \ \ \ \ \ \ \hspace{0.8mm} e_{13}^3= \int_{123}^{4} P_{12} \hat Q_{14} \hat Q_{24} \ \label{31npl11} \nonumber \\ e_4^3 &= \int_{123}^{4} P_{12} P_{13} P_{23} \co \ \ \ e_9^3= \frac{1}{2} \int_{123}^{4} P_{12} \hat Q_{14}^2 , \ \ \ \ \ \ \ \ \hspace{0.8mm} e_{14}^3= \int_{123}^{4} P_{12} \hat Q_{14} \hat Q_{34} \ \nonumber \\ e_5^3 &= \int_{123}^{4} \hat Q_{14} \hat Q_{24} \hat Q_{34} \co e_{10}^3= \frac{1}{2} \int_{123}^{4} P_{12} \hat Q_{34}^2 \; . \nonumber \end{align} \tag{ H.7 }$$

H.2. teMZVs at orders ${ \newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}}$

As a next step, we evaluate the above $e^i_j$ in terms of teMZVs, using the equivalence of the measures $\int_{123}^4$ and $\frac{1}{2}\int^{34}_{12}$ noted in equations (H.1) and (H.3). By largely recycling the results of section 4.3, we obtain the following expressions at the first order,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e_{1}^1&= \frac{1}{2}d^1_1 =\frac{1}{2} \omwb{1,0}{0,0} , \ \ \ \ \ \ e_{2}^1 =\frac{1}{2}d^1_2 \, \big|_{c(q)\rightarrow 0}= \frac{1}{2} \omwb{1,0}{\tauh,0} \; , \label{31eqn:d12} \nonumber \end{align} \tag{ H.8 }$$

the following ones at the second order,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^2_1 &= \frac{1}{2} d^{2}_{1} \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1 , 1 , 0 }{0 , 0 , 0} , \ \ \ \ \ \ \ \ \ e^2_{3}= \frac{1}{2} d^{2}_{4} \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1 , 0 }{0 , 0}^2 \label{31eqn:d21} \nonumber \end{align} \tag{ H.9 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^2_2 &= \frac{1}{2} d^{2}_{2} \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1 , 1 , 0 }{\tauh, \tauh, 0} , \ \ \ \ \ \ \ \ e^2_4= \frac{1}{2} d^{2}_{5} \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1 , 0 }{\tauh, 0 }^2 \label{31eqn:d22} \nonumber \end{align} \tag{ H.10 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^2_5 &= \frac{1}{2} d^{2}_{3} \, \big|_{c(q)\rightarrow 0} =\frac{1}{2} \omwb{1 , 0 }{0, 0 }\omwb{1 , 0 }{\tauh, 0 } = e^2_6 \; , \label{31eqn:d25} \nonumber \end{align} \tag{ H.11 }$$

and the following ones at the third order:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^{3}_{1} & = \frac{1}{2} d^3_1= \frac{1}{2} \omwb{1 , 1 , 1 , 0 }{0 , 0 , 0, 0} \label{31eqn:d31} \nonumber \end{align} \tag{ H.12 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^{3}_{2} & = \frac{1}{2} d^3_2 \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1 , 1 , 1 , 0}{\tauh, \tauh, \tauh, 0} \label{31eqn:d32} \nonumber \end{align} \tag{ H.13 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^{3}_{3} &= \frac{1}{2} d^3_5 = \frac{1}{2} \omwb{1 , 1 , 0 }{0 , 0 , 0} \omwb{1 , 0 }{0 , 0} \label{31eqn:d33} \nonumber \end{align} \tag{ H.14 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle e^{3}_{4} &=\frac{1}{2} \omwb{1,0}{0,0}^3 - \zeta_2 \, \omwb{0 , 1 , 0 , 0 }{0 , 0 , 0 , 0}+ \frac{1}{6} \, \omwb{0 , 3 , 0 , 0 }{0 , 0 , 0 , 0} \label{31eqn:d34} \nonumber \end{align} \tag{ H.15 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^{3}_{5} &= \frac{1}{2} d^3_{12} \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1 , 0}{\tauh , 0}^3 \label{31eqn:d35} \nonumber \end{align} \tag{ H.16 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^{3}_{6} &= \frac{1}{2} d^3_7 \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1 , 1 , 0 }{\tauh, \tauh, 0} \omwb{1 , 0 }{\tauh, 0} \label{31eqn:d37} \nonumber \end{align} \tag{ H.17 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^3_{7} &= \frac{1}{2} d^3_3 \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1 , 1 , 0 }{0 , 0 , 0} \omwb{1 , 0}{\tauh , 0} = e^3_{8} \label{31eqn:d38} \nonumber \end{align} \tag{ H.18 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^3_{9} &= \frac{1}{2} d^3_4 \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1 , 1 , 0 }{\tauh, \tauh, 0} \omwb{1 , 0}{0 , 0} = e^3_{10} \label{31eqn:d39} \nonumber \end{align} \tag{ H.19 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^3_{11} &= \frac{1}{2} d^3_8 \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1 , 0}{0 , 0}^2 \omwb{1 , 0}{\tauh , 0} = e^3_{12} \label{31eqn:d311} \nonumber \end{align} \tag{ H.20 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^3_{13} &= \frac{1}{2} d^3_{11} \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1,0}{\tauh,0}^2 \omwb{1,0}{0,0} + \frac{1}{6} \, \omwb{0 , 3 , 0 , 0 }{0 , 0 , 0 , 0} \label{31eqn:d313} \nonumber \end{align} \tag{ H.21 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle e^3_{14} &= \frac{1}{2} d^3_9 \, \big|_{c(q)\rightarrow 0} = \frac{1}{2} \omwb{1,0}{\tauh,0}^2 \omwb{1,0}{0,0} \ . \label{31eqn:d314} \nonumber \end{align} \tag{ H.22 }$$

Expressions for $d_{11}^3$ (see equation (4.45)) and $e^3_4$ are derived in appendix I.

H.3. Assembling the orders ${ \newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}}$

Similar to equation (4.47), momentum conservation²¹ leaves the following contributions to the integral equation (H.1) at the orders $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}$,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ap}{\alpha'} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle I_{123|4}(q) = \frac{1}{2}\ &+\ (s_{12}^2 + s_{12}s_{23} + s_{23}^2) (2 e^2_1 + 2 e^2_2 - e^2_3 - e^2_4) \label{simp31int} \nonumber \\ & + \ s_{12}s_{23}(s_{12}+s_{23})\,(-3 e^3_1 - 3 e^3_2 + 3 e^3_3 - e^3_4 - e^3_5 + 3 e^3_6 - 3 e^3_{13} + 3 e^3_{14}) + \ {{\mathcal O}}(\ap^4) \; , \nonumber \end{align} \tag{ H.23 }$$

in agreement with the $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 1}$ results of [31]. The combinations of $e^i_j$ can be expressed in terms of eMZVs

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle I_{123|4}(q) \, \big|_{s_{12}^2 + s_{12}s_{23} + s_{23}^2} &= \omwb{1,1 ,0}{0,0,0} - \frac{1}{2} \omwb{1,0}{0,0}^2 + \omwb{1,1,0}{\tauh,\tauh ,0} - \frac{1}{2} \omwb{1,0}{\tauh,0}^2 \nonumber \\ &= \frac{7\zeta_2 }{12} + \omega(0,0,2) \label{eqn:stuffA} \nonumber \end{align} \tag{ H.24 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle I_{123|4}(q) \, \big|_{s_{12}s_{23}(s_{12}+s_{23})} &= \frac{3}{2} \omwb{1,0}{0,0}\omwb{1,1,0}{0,0,0} -\frac{3}{2} \omwb{1,1,1,0}{0,0,0,0} -\frac{1}{2} \omwb{1,0}{0,0}^3 \nonumber \\ &\ \ \ \ +\frac{3}{2} \omwb{1,0}{\tauh,0}\omwb{1,1,0}{\tauh,\tauh,0} -\frac{3}{2} \omwb{1,1,1,0}{\tauh,\tauh,\tauh,0} -\frac{1}{2} \omwb{1,0}{\tauh,0}^3 \nonumber \\ & \ \ \ \ + \zeta_2 \,\omega(0,1,0,0) -\frac{2}{3}\, \omega(0,3,0,0) \nonumber \\ &= 2\, \zeta_2 \,\omega(0,1,0,0) - \frac{5}{6}\, \omega(0,3,0,0) +\frac{\zeta_3}{4} \; , \label{eqn:stuffB} \nonumber \end{align} \tag{ H.25 }$$

where we have used the teMZV relations of appendix G.3.

H.4. Summary of the orders ${ \newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}}$

Once we insert the simplified expressions equations (H.24) and (H.25) for the combinations of $e^i_j$, the leading low-energy orders of the integral equation (H.23) boil down to the following eMZVs:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ap}{\alpha'} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle I_{123|4}(q) & = \frac{1}{2} +(s_{12}^2 + s_{12}s_{23} + s_{23}^2) \Big(\frac{7\zeta_2 }{12} + \omega(0,0,2) \Big) \nonumber \\ &\quad + s_{12}s_{23}(s_{12}+s_{23}) \Big(2\, \zeta_2 \,\omega(0,1,0,0) - \frac{5}{6}\, \omega(0,3,0,0) +\frac{\zeta_3}{4} \Big) + {{\mathcal O}}(\ap^4) \; . \label{31result} \nonumber \end{align} \tag{ H.26 }$$

From the constant terms of the eMZVs gathered in equation (G.3), our result equation (H.26) is consistent with the tree-level expression

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ap}{\alpha'} \newcommand{\Ga}{\Gamma} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \newcommand{\al}{\alpha} \displaystyle I_{123|4}(q) &= - \frac{1}{\pi^2} \bigg[ \frac{\Gamma(s_{12}) \Gamma(s_{23}) }{\Gamma(1+s_{12}+s_{23})} + {\rm cyc}(1,2,3) \bigg] + {{\mathcal O}}(q)\nonumber \\ &= \frac{1}{2} + \frac{1}{4}\zeta_2 (s_{12}^2 + s_{12}s_{23} + s_{23}^2) + \frac{1}{2} \zeta_3 s_{12}s_{23}(s_{12}+s_{23}) + {{\mathcal O}}(q,\ap^4) \; , \nonumber \end{align} \tag{ H.27 }$$

which is known to arise at the q⁰ order of the integral I_123|4(q) [32].

In this appendix, we will derive the results equations (4.45) and (H.15) for $d^3_{11}$ and $e^3_4$, respectively. These integrals are the most difficult cases at the orders $\newcommand{\al}{\alpha} \newcommand{\ap}{\alpha'} \ap^{\leqslant 3}$ because the Green functions form closed subcycles such as $P_{ij}P_{ik}P_{jk}$ and $P_{ij}Q_{ik}Q_{jk}$. Identities between elliptic iterated integrals will be seen to yield answers in terms of teMZVs, and the extensions of these manipulations to all weights and lengths [30] guarantee that each term in the low-energy expansion of the integrals equations (4.23) and (H.1) can be expressed in terms of teMZVs.

I.1. The ${d^3_{11}}$ integral from ${P_{ij}Q_{ik}Q_{jk}}$

The contributions to $d^3_{11}$ with at least one factor of $c(q)$ are equivalent to those in $d^3_9$, so it is sufficient to study

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\GL}{\Gamma} \newcommand{\dd}{\mathrm{d}} \newcommand{\Ga}{\Gamma} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \int_{12}^{34}P_{34}Q_{13}Q_{14} \, \big|_{c(q)\rightarrow 0} &= 2 \int^1_0 \dd x_4 \int^{x_4}_0 \dd x_3 \, \GLarg{1}{\tauh}{x_3} \GLarg{1}{\tauh}{x_4} \,\bigg(- \int^{x_4}_{x_3} \dd u\ f^{(1)}(u-{x_4}) \bigg) \nonumber \\ &= - 2 \int^1_0 \dd x_4\, \GLarg{1}{\tauh}{x_4} \int^{x_4}_0 \dd u \ f^{(1)}(u-{x_4}) \int^u_0 \dd x_3\,\GLarg{1}{\tauh}{x_3} \label{appG1} \nonumber \\ &= - 2 \int^1_0 \dd x_4 \, \GLarg{1}{\tauh}{x_4} \GLarg{1 &0 &1}{x_4 &0 &\tauh}{x_4}\; . \nonumber \end{align} \tag{ I.1 }$$

In the first step, we have exploited that the integration regions with $0\leqslant x_3 \leqslant x_4 \leqslant 1$ and $0\leqslant x_4 \leqslant x_3 \leqslant 1$ yield the same contributions by the symmetry of the integrand $P_{34}Q_{13}Q_{14}$ under exchange of x₃ and x₄. Moreover, we have used the integral representation $\newcommand{\dd}{\mathrm{d}} P_{34}=- \int^{x_4}_{x_3} \dd u\ f^{(1)}(u-{x_4})$ of the Green function, reparametrized the integration domain with $0 \leqslant x_3 \leqslant u \leqslant x_4$ and applied the definition equation (2.1) of elliptic iterated integrals.

The elliptic iterated integral $\newcommand{\be}{\beta} \newcommand{\Ga}{\Gamma} \newcommand{\GL}{\Gamma} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\\#2\end{array};#3\right)} \GLarg{1 &0 &1}{x_4 &0 &\tau/2 }{x_4}$ in the last line of equation (I.1) is not yet suitable for integration over x₄ in its present form due to the twofold appearance of the integration variable. As explained in [30], Fay relations among the weighting functions allow to derive a differential equation in x₄ whose integration yields the alternative representation

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\GL}{\Gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \GLarg{1 &0 &1}{x_4 &0 &\tauh}{x_4}&= \GLarg{0 &0 &2}{0 &0 &0 }{x_4} +\GLarg{0 &0 &2}{0 &0 &\tauh}{x_4} +\GLarg{0 &2 &0}{0 &\tauh&0}{x_4} \nonumber \\ & \ \ \ \ \ \ \ \ -\GLarg{0 &1 &1}{0 &\tauh&\tauh}{x_4} -\GLarg{0 &1 &1}{0 &\tauh&0 }{x_4} \ . \label{appG2} \nonumber \end{align} \tag{ I.2 }$$

After shuffle multiplication with $\newcommand{\be}{\beta} \newcommand{\Ga}{\Gamma} \newcommand{\GL}{\Gamma} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\\#2\end{array};#3\right)} \GLarg{1}{\tau/2 }{x_4}$, the integral over x₄ in equation (I.1) can be readily performed, e.g.

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLargv}[3]{\GL\left(\begin{array}{ccccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\GL}{\Gamma} \newcommand{\dd}{\mathrm{d}} \newcommand{\Ga}{\Gamma} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \int^1_0 \dd x_4 \,& \GLarg{1}{\tauh}{x_4} \GLarg{0 &1 &1}{0 &\tauh&0 }{x_4} \nonumber \\ &= \int^1_0 \dd x_4 \, \bigg[ \GLarg{1 &0 &1 &1}{\tauh&0 &\tauh&0 }{x_4} +2\GLarg{0 &1 &1 &1}{0 &\tauh&\tauh&0 }{x_4} +\GLarg{0 &1 &1 &1}{0 &\tauh&0 &\tauh}{x_4} \bigg] \nonumber \\ &= \GLargv{0 &1 &0 &1 &1}{0 &\tauh&0 &\tauh&0 }{1} +2\GLargv{0 &0 &1 &1 &1}{0 &0 &\tauh&\tauh&0 }{1} +\GLargv{0 &0 &1 &1 &1}{0 &0 &\tauh&0 &\tauh}{1} \nonumber \\ &=\omwb{1}{\tauh} \omwb{1, 1, 0, 0}{0, \tauh, 0, 0 } - \omwb{1, 1, 0, 0, 1}{0, \tauh, 0, 0, \tauh} \label{appG3} \nonumber \end{align} \tag{ I.3 }$$

for the rightmost term in equation (I.2). The shuffle operation in the last step of equation (I.3) together with $\newcommand{\be}{\beta} \newcommand{\om}{\omega} \newcommand{\omm}{\omega} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\\#2\end{array}\right)} \omwb{1}{\tau/2}=0$ reduces the number of terms and leads to the following end result

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \int_{12}^{34}P_{34}Q_{13}Q_{14} \, \big|_{c(q)\rightarrow 0} &= 2 \omwb{2, 0, 0, 0, 1}{0, 0, 0, 0, \tauh} + 2 \omwb{2, 0, 0, 0, 1}{\tauh, 0, 0, 0, \tauh}+ 2 \omwb{0, 2, 0, 0, 1}{0, \tauh, 0, 0, \tauh} \nonumber \\ & \ \ \ \ \ - 2 \omwb{1, 1, 0, 0, 1}{0, \tauh, 0, 0, \tauh} - 2 \omwb{1, 1, 0, 0, 1}{\tauh, \tauh, 0, 0, \tauh} \; . \label{appG4} \nonumber \end{align} \tag{ I.4 }$$

Finally, the expression for $d^3_{11}$ in equation (4.45) results from the eMZV relation

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle & 2 \omwb{2, 0, 0, 0, 1}{0, 0, 0, 0, \tauh} + 2 \omwb{2, 0, 0, 0, 1}{\tauh, 0, 0, 0, \tauh}+ 2 \omwb{0, 2, 0, 0, 1}{0, \tauh, 0, 0, \tauh} \label{appG4emzv} \nonumber \\ & \ \ \ \ \ \ \ \ \ \ - 2 \omwb{1, 1, 0, 0, 1}{0, \tauh, 0, 0, \tauh} - 2 \omwb{1, 1, 0, 0, 1}{\tauh, \tauh, 0, 0, \tauh} = \omwb{1,0}{0,0}\omwb{1,0}{\tauh,0}^2 + \frac{1}{3} \, \omwb{0 , 3 , 0 , 0 }{0 , 0 , 0 , 0} \; . \nonumber \end{align} \tag{ I.5 }$$

I.2. The ${e^3_{4}}$ integral from ${P_{ij}P_{ik}P_{jk}}$

A similar strategy applies to $e^3_4$ in equation (H.7),

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\GL}{\Gamma} \newcommand{\dd}{\mathrm{d}} \newcommand{\Ga}{\Gamma} \newcommand{\bi}{\bar\imath} \newcommand{\re}{{\rm Re}} \newcommand{\im}{{\rm Im}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle \int_{123}^{4}P_{12}P_{13}P_{23}&= \int^1_0 \dd x_3 \int^{x_3}_0 \dd x_2 \, \GLarg{1}{0 }{x_2} \GLarg{1}{0 }{x_3} \,\bigg(- \int^{x_3}_{x_2} \dd u\ f^{(1)}(u-{x_3}) \bigg) \nonumber \\ &= - \int^1_0 \dd x_3 \, \GLarg{1}{0 }{x_3} \GLarg{1 &0 &1}{x_3 &0 &0 }{x_3} , \label{appG6} \nonumber \end{align} \tag{ I.6 }$$

where the relevant identity among elliptic iterated integrals reads

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\nonumber \\#2\end{array};#3\right)} \newcommand{\GL}{\Gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \GLarg{1 &0 &1}{x_3 &0 &0 }{x_3}=& ~2\GLarg{0 &0 &2}{0 &0 &0 }{x_3} +\GLarg{0 &2 &0}{0 &0 &0}{x_3} \nonumber \\ & -2\GLarg{0 &1 &1}{0 &0 &0 }{x_3} +\zeta_2 \GLarg{0}{0}{x_3} \ . \label{appG8} \nonumber \end{align} \tag{ I.7 }$$

Note the extra term $\newcommand{\be}{\beta} \newcommand{\ze}{\zeta} \newcommand{\z}{\zeta} \newcommand{\Ga}{\Gamma} \newcommand{\GL}{\Gamma} \newcommand{\e}{{\rm e}} \newcommand{\GLarg}[3]{\GL\left(\begin{array}{cccc}#1\\#2\end{array};#3\right)} \zeta_2 \GLarg{0}{0}{x_3}$ in comparison to the analogous identity equation (I.2). Upon insertion into equation (I.6), we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle \int_{123}^{4}P_{12}P_{13}P_{23}= 2 \omm(2, 0, 0, 0, 1) + \omm(0, 2, 0, 0, 1) - 2 \omm(1, 1, 0, 0, 1) - \frac{1}{2} \zeta_2 \omm(1 , 0) \; , \label{appG9} \nonumber \end{align} \tag{ I.8 }$$

which translates into the expression equation (H.15) for $e^3_4$ by the eMZV relation

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\z}{\zeta} \newcommand{\ze}{\zeta} \newcommand{\be}{\beta} \displaystyle &2 \omm(2, 0, 0, 0, 1) +\omm(0, 2, 0, 0, 1) - 2 \omm(1, 1, 0, 0, 1) - \frac{1}{2} \zeta_2 \omm(1 , 0) \nonumber \\ & \ \ \ =\frac{1}{2} \omm(1,0)^3 - \zeta_2 \omm(0,1,0,0) + \frac{1}{6} \omm(0,3,0,0) \; . \label{fromdatabase} \nonumber \end{align} \tag{ I.9 }$$

For certain contributions to the integral equation (4.3) in the non-planar open-string amplitude, closed-form expressions in terms of $\newcommand{\te}{\textrm} \newcommand{\temzv}{{\rm teMZV}} \temzv$s can be given α to all orders in the $\newcommand{\al}{\alpha} \alpha'$-expansion. In the context and notation of section 4.3, we find

$$\begin{eqnarray}\newcommand{\myunderbrace}\newcommand{\dd}{\mathrm{d}} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} d_{n;1} &= \frac{1}{n!} \int_{12}^{34} P(x_{12})^n = \frac{ 1 }{ n! } \int_0^1 \dd x_2 \prod_{i=1}^n \int_0^{x_2} \dd y_i \; f^{(1)}(y_i) = \omega \mathop{\left(\begin{array}{c} {1, \dots , 1 ,0 \\ \underbrace{0,\dots, 0} ,0} \end{array}\right)}\limits_{n \mbox{ times}}\end{eqnarray} \tag{ J.1 }$$

$$\begin{eqnarray} &&\newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\GLarg}[2]{\begin{array}{c}#1\nonumber \\#2\end{array}} \newcommand{\G}{\Gamma} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle d_{n;2} &= \frac{1}{n!} \int_{12}^{34} Q(x_{13})^n = \frac{ 1 }{ n! } \int_{12}^{34} \left(\displaystyle\sum_{r=0}^n \left(\begin{array}{c}{n}\\{r}\end{array}\right) c(q)^{n-r} \left(\GLarg{1} {\tauh}; {x_3}\right)^r \right) \nonumber \\ & = \displaystyle\sum_{r=0}^n \frac{1}{(n-r)!} c(q)^{n-r} \omega\mathop{\left(\begin{array}{c}{1, \dots , 1 ,0 \\ \underbrace{\tauh,\dots , \tauh},0}\end{array}\right)}\limits_{r \mbox{ times}} \end{eqnarray} \tag{ J.2 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle d_{n,m;1} &= \frac{1}{n! m!} \int_{12}^{34} P(x_{12})^n P(x_{34})^m = d_{n;1} d_{m;1} \nonumber \\ &=\ \omega \mathop{\left(\begin{array}{c}{1, \dots , 1 ,0}\\{ \underbrace{0,\dots , 0} ,0}\end{array}\right)}\limits_{n \mbox{ times}} \qquad \omega \mathop{\left(\begin{array}{c}{1, \dots , 1 ,0}\\{ \underbrace{0,\dots , 0} ,0}\end{array}\right)}\limits_{m \mbox{ times}} \end{align} \tag{ J.3 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\omwb}[2]{\omm\left(\begin{array}{c}#1\nonumber \\#2\end{array}\right)} \newcommand{\tauh}{\frac{\tau}{2}} \newcommand{\omm}{\omega} \newcommand{\om}{\omega} \newcommand{\la}{\lambda} \newcommand{\s}{\sigma} \newcommand{\si}{\sigma} \newcommand{\be}{\beta} \displaystyle d_{n,m;2} &= \frac{1}{n! m!} \int_{12}^{34} P(x_{12})^n Q(x_{13})^m = d_{n;1} d_{m;2} \nonumber \\&= \omega \mathop{\left(\begin{array}{c}{1, \dots , 1 ,0}\\{ \underbrace{0,\dots , 0} ,0}\end{array}\right)}\limits_{n \mbox{ times}}\quad \; \; \sum_{r=0}^m \frac{1}{(m-r)!} c(q)^{m-r}\omega \mathop{\left(\begin{array}{c}{1, \dots , 1 ,0}\\{\underbrace{\tauh,\dots , \tauh},0}\end{array}\right)}\limits_{r \mbox{ times}}\end{align} \tag{ J.4 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d_{n,m;3} = \frac{1}{n! m!} \int_{12}^{34} Q(x_{13})^n Q(x_{14})^m = d_{n;2} d_{m;2} \label{eqn:dnm3} \nonumber \end{align} \tag{ J.5 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d_{n,m,p;1} = \frac{1}{n! m! p!} \int_{12}^{34} P(x_{12})^n P(x_{34})^m Q(x_{13})^{\,p} = d_{n;1} d_{m,p;2} = d_{n;1} d_{m;1} d_{p;2} \label{eqn:dnmp1} \nonumber \end{align} \tag{ J.6 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\lambda} \newcommand{\be}{\beta} \displaystyle d_{n,m,p;2} = \frac{1}{n! m! p!} \int_{12}^{34} P(x_{12})^n Q(x_{13})^m Q(x_{14})^{\,p} = d_{n;1} d_{m,p;3} = d_{n;1} d_{m;2} d_{p;2} \ . \label{eqn:dnmp2} \nonumber \end{align} \tag{ J.7 }$$