Towards closed strings as single-valued open strings at genus one

The main result of this work is the proposal

$$\begin{equation}{J}_{\overrightarrow{\eta }}^{\tau }=\mathrm{S}\mathrm{V}{B}_{\overrightarrow{\eta }}^{\tau }\end{equation} \tag{ 1.1 }$$

for a single-valued map SV at genus one which relates generating series ${B}_{\overrightarrow{\eta }}^{\tau }$ and ${J}_{\overrightarrow{\eta }}^{\tau }$ of open- and closed-string integrals, respectively, see (3.35). As summarized in figure 1, this induces an SV action on the eMZVs in the α'-expansion of the cylinder integrals ${B}_{\overrightarrow{\eta }}^{\tau }$ to be defined in (3.1). By comparing coefficients of dimensionless Mandelstam invariants α'k_i ⋅ k_j and formal expansion variables η_j , SV maps each eMZV generated by ${B}_{\overrightarrow{\eta }}^{\tau }$ to combinations of MGFs at the same order in the analogous expansion of the torus integrals ${J}_{\overrightarrow{\eta }}^{\tau }$ in (3.13). The integrands of ${B}_{\overrightarrow{\eta }}^{\tau }$ and ${J}_{\overrightarrow{\eta }}^{\tau }$ are assembled from combinations of doubly-periodic Kronecker–Eisenstein series ${\varphi }_{\overrightarrow{\eta }}^{\tau }$ in (2.13) known from [18, 59–61] and antielliptic functions $\bar{V(\dots \vert \tau )}$ that we introduce in (3.8) as tentative Betti–deRham duals of integration cycles on a cylinder boundary.

A key motivation and evidence for this construction stems from the degeneration limit τ → i∞ of the series ${B}_{\overrightarrow{\eta }}^{\tau }$ and ${J}_{\overrightarrow{\eta }}^{\tau }$. Within this limit, genus-zero integrals similar to those in open- and closed-string tree-level amplitudes are recovered, the latter being related by the single-valued map of MZVs [40–45]. The leading terms of the eMZVs in the τ → i∞ limit of ${B}_{\overrightarrow{\eta }}^{\tau }$ are certain Laurent polynomials in the modular parameter τ of the cylinder with MZVs in its coefficients. As visualized in the lower part of figure 1, the known single-valued map sv of MZVs [46, 48] is conjectured to yield the analogous Laurent polynomials in the degeneration limit τ → i∞ of the torus integrals ${J}_{\overrightarrow{\eta }}^{\tau }$. This is made precise in the conjecture (3.23)—a central prerequisite for (1.1)—which generalizes earlier observations in [51, 52] and has been proven at the leading orders in the formal expansion variable η_j at two points [16].

The earlier proposal for an elliptic single-valued map 'esv' in [51] concerns the full τ-dependence of certain generating series of eMZVs or the composing iterated Eisenstein integrals. This reference associates open-string prototypes (i.e. esv preimages) to the simplest closed-string integrals at genus one whose integrands are solely built from Green functions involving any number of punctures. On the one hand, the proposal for the single-valued map in [51] is contained in (1.1) upon symmetrizing over the integration cycles on its left-hand side and extracting the lowest order in η_j . On the other hand, the implementation of the single-valued map at the level of iterated Eisenstein integrals in the reference is very different from the proposal in the present work. In comparison to the proposal of [51], our SV action on iterated Eisenstein integrals in (4.30) does not necessarily generate real combinations and is therefore applicable to imaginary cusp forms and MGFs of different holomorphic and antiholomorphic modular weights. Moreover, in contrast to esv in [51], the SV map in (1.1) is observed to be compatible with shuffle multiplication in all known examples. Our SV map additionally introduces combinations of (conjecturally single-valued) MZVs and antiholomorphic terms, which are absent in [51].

At the time of writing, the antiholomorphic admixtures introduced by our SV map on iterated Eisenstein integrals at depth ⩾ 2 can only be fixed by indirect methods beyond the reach of open-string data. Instead, the explicit form of the SV action on iterated Eisenstein integrals has so far been extracted from the reality properties of closed-string generating series in [18] that extend the ${J}_{\overrightarrow{\eta }}^{\tau }$ series as described below. However, this limitation does not affect the formulation of our SV map at the level of the lattice-sum representation of MGFs: by virtue of the antielliptic functions $\bar{V(\dots \vert \tau )}$ in (3.8), the SV image of an arbitrary convergent eMZV given in (5.39) can be straightforwardly expressed in terms of lattice sums using the integration techniques of [8–10, 52] and, for certain weights, further simplified using the Mathematica package [62].

This work is organized as follows. We start by reviewing open- and closed-string integrals at genus zero and genus one as well as the basic definitions of single-valued MZVs, eMZVs and MGFs in section 2. Then, section 3 is dedicated to the modified generating series of open- and closed-string integrals as well as their relation through our proposed single-valued map at genus one. In particular, the central antielliptic integrands and the resulting proposal for an elliptic single-valued map can be found in sections 3.2 and 3.5, respectively. In section 4, we set the stage for generating explicit examples of single-valued eMZVs by introducing a new expansion method for open-string integrals over B-cycles and relating it to similar closed-string α'-expansions. This leads to the identifications of MGFs as single-valued eMZVs in section 5, where examples of the antielliptic integrands are related to earlier approaches to an elliptic single-valued map in the literature. The resulting lattice-sum representations of all single-valued convergent eMZVs are discussed in section 5.7. In the concluding section 6, we comment on the relation of string amplitudes to the generating series of this work and further directions.

We briefly review the basic disk (open-string) and sphere (closed-string) integrals for genus-zero world-sheets and how they are related by the genus-zero single-valued map.

2.1.1. Definitions of disk and sphere integrals

Massless tree-level n-point amplitudes of the open superstring [63] and the open bosonic string [64] can be expanded in a basis of iterated integrals [65]

$$\begin{equation}{Z}^{\text{tree}}(\gamma \vert \rho )=\underset{\mathfrak{D}(\gamma )}{\int }\frac{\left({\prod }_{j=1}^{n}\mathrm{d}{z}_{j}\right)}{\mathrm{v}\mathrm{o}\mathrm{l}\enspace {\mathrm{S}\mathrm{L}}_{2}(\mathbb{R})}\prod\limits _{1\leqslant i< j}^{n}\vert {z}_{ij}{\vert }^{-{s}_{ij}}\mathrm{P}\mathrm{T}(\rho (1,2,\dots ,n))\end{equation} \tag{ 2.1 }$$

over the boundary of a disk which we parametrize through the real line

$$\begin{equation}\mathfrak{D}(\gamma )=\left\{{z}_{j}\in \mathbb{R},-\infty < {z}_{\gamma (1)}< {z}_{\gamma (2)}< \cdots < {z}_{\gamma (n)}< \infty \right\}.\end{equation} \tag{ 2.2 }$$

The disk integrands involve dimensionless Mandelstam invariants

$$\begin{equation}{s}_{ij}=-\frac{{\alpha }^{\prime }}{2}{k}_{i}\cdot {k}_{j},\quad {k}_{j}^{2}=0\end{equation} \tag{ 2.3 }$$

and Parke–Taylor factors

$$\begin{equation}\mathrm{P}\mathrm{T}(\rho (1,2,\dots ,n))=\frac{1}{{z}_{\rho (1)\rho (2)}{z}_{\rho (2)\rho (3)}\dots {z}_{\rho (n)\rho (1)}},\quad {z}_{ij}={z}_{i}-{z}_{j}.\end{equation} \tag{ 2.4 }$$

The inverse $\mathrm{v}\mathrm{o}\mathrm{l}\enspace {\mathrm{S}\mathrm{L}}_{2}(\mathbb{R})$ in (2.1) instructs us to set any triplet of punctures to 0, 1, ∞, where the ${\mathrm{S}\mathrm{L}}_{2}(\mathbb{R})$ invariance of genus-zero integrands hinges on momentum conservation ${\sum }_{j=1}^{n}{k}_{j}=0$. Both the domains and the Parke–Taylor integrands are indexed via permutations γ, ρ ∈ S_n of the external legs 1, 2, ..., n. One can arrive at smaller bases of (n − 3)! cycles γ and Parke–Taylor orderings ρ via monodromy relations [66, 67] and integration by parts [63, 65], respectively.

Closed-string tree amplitudes in turn can be reduced to sphere integrals

$$\begin{align}\hfill {J}^{\text{tree}}(\gamma \vert \rho )& =\frac{1}{{\pi }^{n-3}}\underset{{\mathbb{C}}^{n-3}}{\int }\frac{\left({\prod }_{j=1}^{n}{\mathrm{d}}^{2}{z}_{j}\right)}{\mathrm{v}\mathrm{o}\mathrm{l}\enspace {\mathrm{S}\mathrm{L}}_{2}(\mathbb{C})}\hfill \\ \hfill & \quad \times \enspace \prod\limits _{1\leqslant i< j}^{n}\vert {z}_{ij}{\vert }^{-2{s}_{ij}}\bar{\mathrm{P}\mathrm{T}(\gamma (1,2,\dots ,n))}\mathrm{P}\mathrm{T}(\rho (1,2,\dots ,n))\hfill \end{align} \tag{ 2.5 }$$

involving ${\mathrm{d}}^{2}{z}_{j}=\frac{i}{2}\enspace \mathrm{d}{z}_{j}\wedge \mathrm{d}{\bar{z}}_{j}$ and permutations γ, ρ ∈ S_n of meromorphic and antimeromorphic Parke–Taylor factors subject to the same integration-by-parts relations as in the open-string case.

2.1.2. Single-valued map between disk and sphere integrals

The disk and sphere integrals (2.1) and (2.5) converge for a suitable range of the Re(s_ij ) and they admit a Laurent expansion in α', i.e. around the value s_ij = 0 of the dimensionless Mandelstam invariants (2.3). The coefficients in the α'-expansions of disk integrals Z^tree are MZVs [68, 69],

$$\begin{equation}{\zeta }_{{n}_{1},{n}_{2},\dots ,{n}_{r}}=\sum\limits _{0< {k}_{1}< {k}_{2}< \cdots < {k}_{r}}{k}_{1}^{-{n}_{1}}{k}_{2}^{-{n}_{2}}\dots {k}_{r}^{-{n}_{r}},\quad {n}_{r}\geqslant 2\end{equation} \tag{ 2.6 }$$

whose weight n₁ + n₂ +⋯+ n_r matches the order in α' beyond the low-energy limit (i.e. beyond the leading order in α'). The polynomial structure of the Z^tree in s_ij can for instance be generated from the Drinfeld associator [70] or Berends–Giele recursions [71], with explicit results available for download from [72, 73].

When applying the single-valued map [46, 48] of motivic [74] MZVs ⁸

$$\begin{equation}\mathrm{s}\mathrm{v}\enspace {\zeta }_{2k}=0,\qquad \mathrm{s}\mathrm{v}\enspace {\zeta }_{2k+1}=2{\zeta }_{2k+1},\qquad \mathrm{s}\mathrm{v}\enspace {\zeta }_{3,5}=-10{\zeta }_{3}{\zeta }_{5},\quad \mathrm{e}\mathrm{t}\mathrm{c}\end{equation} \tag{ 2.7 }$$

order by order in α', the disk and sphere integrals (2.1) and (2.5) are related by [40–45]

$$\begin{equation}{J}^{\text{tree}}(\gamma \vert \rho )=\mathrm{s}\mathrm{v}\enspace {Z}^{\text{tree}}(\gamma \vert \rho ).\end{equation} \tag{ 2.8 }$$

The first permutation γ in Z^tree and J^tree refers to a disk ordering (2.2) and an antimeromorphic Parke–Taylor factor (2.4), respectively, which are connected by a Betti–deRham duality [49, 57, 58]. The key result of this work is to identify similar pairs of cycles and antimeromorphic functions at genus one.

As a preparation for our proposal of a genus-one single-valued map, we now introduce the basic genus-one world-sheet integrals and the objects appearing in their α'-expansion.

2.2.1. Genus-one open-string A-cycle integrals

In the same way as disk integrals can be cast into a Parke–Taylor-type basis (2.1), the basis integrals for massless genus-one open-string amplitudes are claimed to be generated by [59, 60]

$$\begin{equation}{Z}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )=\underset{\mathfrak{A}(\gamma )}{\int }\left(\prod\limits _{j=2}^{n}\mathrm{d}{z}_{j}\right){\varphi }_{\overrightarrow{\eta }}^{\tau }(1,\rho (2,\dots ,n))\prod\limits _{1\leqslant i< j}^{n}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{A}}({z}_{ij},\tau )},\end{equation} \tag{ 2.9 }$$

where we have set z₁ = 0 by translation invariance. In this work we restrict to planar amplitudes with all state insertions on a single cylinder boundary (as opposed to non-planar amplitudes with punctures on both boundaries of the cylinder). We do not impose momentum conservation in a genus-one context and treat all the s_ij with 1 ⩽ i < j ⩽ n as independent. The ordering of the open-string punctures on a cylinder boundary is encoded in an integration domain on the A-cycle of a torus (see figure 2 for the standard parametrization) with $\tau \in i{\mathbb{R}}^{+}$ [75]

$$\begin{equation}\mathfrak{A}(\gamma )=\left\{{z}_{j}\in \mathbb{R},\enspace 0< {z}_{\gamma (2)}< {z}_{\gamma (3)}< \cdots < {z}_{\gamma (n)}< 1\right\},\end{equation} \tag{ 2.10 }$$

with similar integration domains [76] for the non-planar open-string integrals.

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The integrand of (2.9) features the open-string Green function on an A-cycle (which is chosen to enforce ${\mathcal{G}}_{\mathfrak{A}}(z,\tau )={\mathcal{G}}_{\mathfrak{A}}(-z,\tau )$ and ${\int }_{0}^{1}\mathrm{d}z\enspace {\mathcal{G}}_{\mathfrak{A}}(z,\tau )=0$ [51, 77])

$$\begin{equation}{\mathcal{G}}_{\mathfrak{A}}(z,\tau )=-\mathrm{log}\left(\frac{{\theta }_{1}(\vert z\vert ,\tau )}{\eta (\tau )}\right)+\frac{i\pi \tau }{6}+\frac{i\pi }{2},\quad z\in (-1,1)\end{equation} \tag{ 2.11 }$$

and the following combination of the doubly-periodic Kronecker–Eisenstein series [78]

$$\begin{equation}{\Omega}(z,\eta ,\tau )=\mathrm{exp}\left(2\pi i\eta \frac{\mathrm{I}\mathrm{m}\enspace z}{\mathrm{I}\mathrm{m}\enspace \tau }\right)\frac{{\theta }_{1}^{\prime }(0,\tau ){\theta }_{1}(z+\eta ,\tau )}{{\theta }_{1}(z,\tau ){\theta }_{1}(\eta ,\tau )}\end{equation} \tag{ 2.12 }$$

$$\begin{equation}{\varphi }_{\overrightarrow{\eta }}^{\tau }(1,2,\dots ,n)={\Omega}({z}_{12},{\eta }_{23\dots n},\tau ){\Omega}({z}_{23},{\eta }_{3\dots n},\tau )\cdots {\Omega}({z}_{n-1,n},{\eta }_{n},\tau )\end{equation} \tag{ 2.13 }$$

with η_ij...k = η_i + η_j +⋯+ η_k . ⁹ The permutation ρ ∈ S_n−1 in ${\varphi }_{\overrightarrow{\eta }}^{\tau }(1,\rho (2,\dots ,n))$ is taken to act on both the z_j and the formal expansion variables ${\eta }_{j}\in \mathbb{C}$ in (2.13). The conjectural basis (2.9) is a generating function of the world-sheet integrals over the Kronecker–Eisenstein coefficients f^(w)

$$\begin{equation}{\Omega}(z,\eta ,\tau )=\sum\limits _{w=0}^{\infty }{\eta }^{w-1}{f}^{(w)}(z,\tau )\end{equation} \tag{ 2.14 }$$

that occur in the integrands of genus-one open- and closed-string amplitudes [4, 52, 79], e.g.

$$\begin{equation}{f}^{(0)}(z,\tau )=1,\qquad {f}^{(1)}(z,\tau )={\partial }_{z}\enspace \mathrm{log}\enspace {\theta }_{1}(z,\tau )+2\pi i\frac{\mathrm{I}\mathrm{m}\enspace z}{\mathrm{I}\mathrm{m}\enspace \tau }.\end{equation} \tag{ 2.15 }$$

While the massless four-point genus-one amplitude of the open superstring [80] is proportional to the most singular ${\eta }_{j}^{-3}$-order of ${Z}_{\overrightarrow{\eta }}^{\tau }(\cdot \vert 1,2,3,4)$, the analogous amplitude of the open bosonic string additionally involves contributions of ${Z}_{\overrightarrow{\eta }}^{\tau }(\cdot \vert 1,2,3,4)$ (and its permutations in 2, 3, 4) at the orders of ${\eta }_{j}^{\pm 1}$ [76]. ¹⁰ The short-distance behavior ${f}^{(1)}(z,\tau )=\frac{1}{z}+\mathcal{O}(z)$ introduces kinematic poles into the α'-expansion of (2.9), and the remaining f^(w≠1)(z, τ) are regular for any $z\in \mathbb{C}$.

2.2.2. Genus-one closed-string integrals

In the same way as (2.9) is claimed to be a universal basis of genus-one open-string integrals, the integrals over the torus punctures for massless genus-one amplitudes in type II, heterotic and bosonic string theories should be generated by [61]

$$\begin{align}\hfill {Y}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )& ={(2i)}^{n-1}{\int }_{{\mathfrak{T}}^{n-1}}\left(\prod\limits _{j=2}^{n}{\mathrm{d}}^{2}{z}_{j}\right)\prod\limits _{1\leqslant i< j}^{n}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{T}}({z}_{ij},\tau )}\bar{{\varphi }_{\overrightarrow{\eta }}^{\tau }(1,\gamma (2,\dots ,n))}\hfill \\ \hfill & \quad \times \enspace {\varphi }_{(\tau -\bar{\tau })\overrightarrow{\eta }}^{\tau }(1,\rho (2,\dots ,n))\hfill \end{align} \tag{ 2.16 }$$

with z₁ = 0. The remaining z_j are integrated over the torus $\mathfrak{T}=\frac{\mathbb{C}}{\tau \mathbb{Z}+\mathbb{Z}}$ with modular parameter $\tau \in \mathbb{H}=\left\{\tau \in \mathbb{C},\mathrm{I}\mathrm{m}\enspace \tau > 0\right\}$. The closed-string Green function

$$\begin{equation}{\mathcal{G}}_{\mathfrak{T}}(z,\tau )=-\mathrm{log}{\left\vert \frac{{\theta }_{1}(z,\tau )}{\eta (\tau )}\right\vert }^{2}+\frac{2\pi {(\mathrm{I}\mathrm{m}\enspace z)}^{2}}{\mathrm{I}\mathrm{m}\enspace \tau }\end{equation} \tag{ 2.17 }$$

is chosen to be modular invariant and to obey ${\int }_{\mathfrak{T}}{\;\mathrm{d}}^{2}z\enspace {\mathcal{G}}_{\mathfrak{T}}(z,\tau )=0$, and its holomorphic derivatives parallel those of the open-string Green function ${\mathcal{G}}_{\mathfrak{A}}(z,\tau )$ in (2.11),

$$\begin{align}\hfill {\partial }_{z}{\mathcal{G}}_{\mathfrak{T}}(z,\tau )& =-{f}^{(1)}(z,\tau ),\hfill & \hfill 2\pi i{\partial }_{\tau }{\mathcal{G}}_{\mathfrak{T}}(u\tau +v,\tau )& =-{f}^{(2)}(u\tau +v,\tau )\quad (\text{fixed}\;u,v)\hfill \\ \hfill {\partial }_{v}{\mathcal{G}}_{\mathfrak{A}}(v,\tau )& =-{f}^{(1)}(v,\tau ),\hfill & \hfill 2\pi i{\partial }_{\tau }{\mathcal{G}}_{\mathfrak{A}}(v,\tau )& =-{f}^{(2)}(v,\tau )-2{\zeta }_{2},\hfill \end{align} \tag{ 2.18 }$$

where $u,v\in \mathbb{R}$ parametrize the covering space of the torus and the f^(w)(z, τ) with z = uτ + v are defined by (2.15). The second arguments $(\tau -\bar{\tau }){\eta }_{j}$ and ${\bar{\eta }}_{j}$ of the Kronecker–Eisenstein series and their complex conjugates in (2.16) have been chosen such that each order in the η_j - and α'-expansion gives rise to modular forms of purely antiholomorphic modular weight ¹¹ .

When assembling genus-one amplitudes of open and closed strings from the series ${Z}_{\overrightarrow{\eta }}^{\tau }$ and ${Y}_{\overrightarrow{\eta }}^{\tau }$, it remains to dress the component integrals in their η_j -expansions with kinematic factors that carry the dependence on the external polarizations. The latter are determined from the conformal-field-theory correlators of the vertex operators, see e.g. [81, 82], and are unaffected by our proposal for the single-valued map at genus one.

2.2.3. Differential equations in τ

Based on the differential equations (2.18) of the Green functions and integration by parts in the z_j , the open- and closed-string integrals (2.9) and (2.16) were shown in [60, 61] to obey the differential equations

$$\begin{align}\hfill 2\pi i{\partial }_{\tau }{Z}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )& =\sum\limits _{k=0}^{\infty }(1-k){G}_{k}(\tau )\sum\limits _{\alpha \in {S}_{n-1}}{r}_{\overrightarrow{\eta }}{{({{\epsilon}}_{k})}_{\rho }}^{\alpha }{Z}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \alpha )\hfill \\ \hfill 2\pi i{\partial }_{\tau }{Y}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )& =\sum\limits _{k=0}^{\infty }(1-k){(\tau -\bar{\tau })}^{k-2}{G}_{k}(\tau )\sum\limits _{\alpha \in {S}_{n-1}}{R}_{\overrightarrow{\eta }}{{({{\epsilon}}_{k})}_{\rho }}^{\alpha }{Y}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \alpha ),\hfill \end{align} \tag{ 2.19 }$$

respectively. The right-hand sides involve holomorphic Eisenstein series G₀ = −1 and

$$\begin{equation}{G}_{k}(\tau )=\sum\limits _{\genfrac{}{}{0pt}{}{m,n\in \mathbb{Z}}{(m,n)\ne (0,0)}}\frac{1}{{(m\tau +n)}^{k}},\quad k\geqslant 4\end{equation} \tag{ 2.20 }$$

as well as (n − 1)! × (n − 1)! matrices ${r}_{\overrightarrow{\eta }}({{\epsilon}}_{k}),{R}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ independent of τ that vanish for k = 2 and $k\in 2\mathbb{N}-1$. This means in particular that G₂(τ) does not appear in (2.19).

The two-point instances are

$$\begin{equation}\begin{aligned}\hfill {r}_{{\eta }_{2}}({{\epsilon}}_{0})& ={s}_{12}\left(\frac{1}{{\eta }_{2}^{2}}+2{\zeta }_{2}-\frac{1}{2}{\partial }_{{\eta }_{2}}^{2}\right)\hfill \\ \hfill {r}_{{\eta }_{2}}({{\epsilon}}_{k})& ={R}_{{\eta }_{2}}({{\epsilon}}_{k})={s}_{12}{\eta }_{2}^{k-2},\quad k\geqslant 4\hfill \\ \hfill {R}_{{\eta }_{2}}({{\epsilon}}_{0})& ={s}_{12}\left(\frac{1}{{\eta }_{2}^{2}}-\frac{1}{2}{\partial }_{{\eta }_{2}}^{2}\right)-2\pi i{\bar{\eta }}_{2}{\partial }_{{\eta }_{2}}.\hfill \end{aligned}\end{equation} \tag{ 2.21 }$$

The notation _k reflects the expectation that the ${r}_{\overrightarrow{\eta }}({{\epsilon}}_{k}),{R}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ are matrix representations of Tsunogai's derivation algebra [83] and obey relations such as (see [17, 84, 85] for similar relations at higher weight and depth)

$$\begin{equation}[{r}_{\overrightarrow{\eta }}({{\epsilon}}_{10}),{r}_{\overrightarrow{\eta }}({{\epsilon}}_{4})]-3[{r}_{\overrightarrow{\eta }}({{\epsilon}}_{8}),{r}_{\overrightarrow{\eta }}({{\epsilon}}_{6})]=0.\end{equation} \tag{ 2.22 }$$

The all-multiplicity formulae for these (n − 1)! × (n − 1)! representations in [60, 61] manifest that the ${r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ are linear in the s_ij , i.e. proportional to α', and their closed-string analogues ${R}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ additionally involve terms $\sim \hspace{-2.5pt}{\bar{\eta }}_{j}{\partial }_{{\eta }_{j}}$ independent of α' (with ${s}_{12\dots n}={\sum }_{1\leqslant i< j}^{n}{s}_{ij}$):

$$\begin{equation}{R}_{\overrightarrow{\eta }}({{\epsilon}}_{k})=\begin{cases}{r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})\hfill & :\enspace k\geqslant 4\hfill \\ {r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})-2{\zeta }_{2}{s}_{12\dots n}-2\pi i\sum\limits _{j=2}^{n}{\bar{\eta }}_{j}{\partial }_{{\eta }_{j}}\hfill & :\enspace k=0\hfill \end{cases}.\end{equation} \tag{ 2.23 }$$

2.2.4. Basic definitions of eMZVs and MGFs

We shall now review the definitions of the eMZVs and MGFs that occur as the expansion coefficients of the above genus-one integrals. The η_j - and α'-expansion of the open-string integrals ${Z}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$ in (2.9) gives rise to A-cycle eMZVs [4–6]

$$\begin{align}\hfill \omega ({n}_{1},{n}_{2},\dots ,{n}_{r}\vert \tau )& =\underset{0< {z}_{1}< {z}_{2}< \cdots < {z}_{r}< 1}{\int }\mathrm{d}{z}_{1}\enspace {f}^{({n}_{1})}({z}_{1},\tau )\enspace \mathrm{d}{z}_{2}\enspace {f}^{({n}_{2})}({z}_{2},\tau )\enspace \dots \mathrm{d}{z}_{r}\enspace {f}^{({n}_{r})}({z}_{r},\tau )\hfill \end{align} \tag{ 2.24 }$$

introduced by Enriquez [3] which are said to carry weight n₁ + n₂ +⋯+ n_r and length r. Endpoint divergences in case of n₁ = 1 or n_r = 1 are shuffle-regularized as in section 2.2.1 of [4]. The specific eMZVs at a given order of ${Z}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$ in s_ij and η_j can be obtained from the differential equations (2.19) along with the initial values ${Z}_{\overrightarrow{\eta }}^{\tau \to i\infty }(\gamma \vert \rho )$ in [60] or from matrix representations of the elliptic KZB associator [86, 87].

The closed-string integrals ${Y}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$ in (2.16) in turn introduce multiple sums over the momentum lattice of a torus [52, 61]

$$\begin{equation}{\Lambda}=\mathbb{Z}+\tau \mathbb{Z},\qquad {{\Lambda}}^{\prime }={\Lambda}{\backslash}\left\{0\right\}\end{equation} \tag{ 2.25 }$$

that are known as MGFs [10, 11]. With the removal of p = 0 from Λ, they can be thought of as infrared-regulated and discretized versions of Feynman integrals on a torus. The MGFs associated with Feynman graphs of dihedral topology are defined by ¹²

$$\begin{equation}\enspace \mathcal{C}\left[\begin{matrix}\hfill {a}_{1}\hfill & \hfill {a}_{2}\hfill & \hfill \dots \hfill & \hfill {a}_{r}\hfill \\ \hfill {b}_{1}\hfill & \hfill {b}_{2}\hfill & \hfill \dots \hfill & \hfill {b}_{r}\hfill \end{matrix}\right]=\sum\limits _{{p}_{1},{p}_{2},\dots ,{p}_{r}\in {{\Lambda}}^{\prime }}\frac{\delta ({p}_{1}+{p}_{2}+\cdots +{p}_{r})}{{p}_{1}^{{a}_{1}}{\bar{p}}_{1}^{{b}_{1}}{p}_{2}^{{a}_{2}}{\bar{p}}_{2}^{{b}_{2}}\dots {p}_{r}^{{a}_{r}}{\bar{p}}_{r}^{{b}_{r}}},\end{equation} \tag{ 2.26 }$$

and more general topologies are for instance discussed in [11, 62]. The simplest examples of dihedral MGFs (2.26) have two columns and are associated with one-loop graphs on the world-sheet

$$\begin{equation}\mathcal{C}\left[\begin{matrix}\hfill a\hfill & \hfill 0\hfill \\ \hfill b\hfill & \hfill 0\hfill \end{matrix}\right]=\sum\limits _{p\in {{\Lambda}}^{\prime }}\frac{1}{{p}^{a}{\bar{p}}^{b}},\end{equation} \tag{ 2.27 }$$

whereas $\mathcal{C}\left[\begin{matrix}{a}_{1}& {a}_{2}& \dots & {a}_{r}\\ {b}_{1}& {b}_{2}& \dots & {b}_{r}\end{matrix}\right]$ are referred to as (r − 1)-loop MGFs. As long as the entries obey a + b > 2, the lattice sums (2.27) are absolutely convergent and the one-loop MGFs are expressible in terms of non-holomorphic Eisenstein series E_k (τ) and their Cauchy–Riemann derivatives

$$\begin{align}\hfill {E}_{k}(\tau )={\left(\frac{\mathrm{I}\mathrm{m}\enspace \tau }{\pi }\right)}^{k}\mathcal{C}\left[\begin{matrix}\hfill k\hfill & \hfill 0\hfill \\ \hfill k\hfill & \hfill 0\hfill \end{matrix}\right],\quad {\nabla }^{m}{E}_{k}(\tau )& =\frac{{(\mathrm{I}\mathrm{m}\enspace \tau )}^{k+m}}{{\pi }^{k}}\frac{(k+m-1)!}{(k-1)!}\mathcal{C}\left[\begin{matrix}\hfill k+m\hfill & \hfill 0\hfill \\ \hfill k-m\hfill & \hfill 0\hfill \end{matrix}\right],\hfill \\ \hfill {\bar{\nabla }}^{m}{E}_{k}(\tau )& =\frac{{(\mathrm{I}\mathrm{m}\enspace \tau )}^{k+m}}{{\pi }^{k}}\frac{(k+m-1)!}{(k-1)!}\mathcal{C}\left[\begin{matrix}\hfill k-m\hfill & \hfill 0\hfill \\ \hfill k+m\hfill & \hfill 0\hfill \end{matrix}\right],\hfill \end{align} \tag{ 2.28 }$$

where ∇ = 2i(Im τ)²∂_τ and $\bar{\nabla }=-2i{(\mathrm{I}\mathrm{m}\enspace \tau )}^{2}{\partial }_{\bar{\tau }}$. As will be detailed below, both eMZVs (2.24) and MGFs such as (2.26) can be represented via iterated integrals of holomorphic Eisenstein series ${G}_{k}=\mathcal{C}\left[\begin{matrix}k& 0\\ 0& 0\end{matrix}\right]$ defined by (2.20). Both eMZVs [17] and MGFs [11, 62, 88–90] exhibit a multitude of relations over rational combinations of MZVs, all of which are automatically exposed in their iterated-Eisenstein-integral representation ¹³ . A computer implementation for the decomposition of a large number of eMZVs and MGFs into basis elements is available in [62, 92], respectively.

Instead of parametrizing the cylinder boundary through the A-cycle of a torus as in (2.9), one can perform a modular S transformation

$$\begin{align}\hfill {B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )& ={Z}_{\overrightarrow{\eta }}^{-1/\tau }(\gamma \vert \rho )\hfill \\ \hfill & =\underset{\mathfrak{B}(\gamma )}{\int }\left(\prod\limits _{j=2}^{n}\mathrm{d}{z}_{j}\right){\varphi }_{\tau \overrightarrow{\eta }}^{\tau }(1,\rho (2,\dots ,n))\prod\limits _{1\leqslant i< j}^{n}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{B}}({z}_{ij},\tau )}\hfill \end{align} \tag{ 3.1 }$$

to attain a parametrization through the B-cycle (recalling that z₁ = 0 and $\tau \in i{\mathbb{R}}^{+}$)

$$\begin{align}\hfill \mathfrak{B}(\gamma )& =\underset{j=1{}}{\overset{n}{\bigoplus}}{\mathfrak{B}}_{j}(\gamma )\hfill \\ \hfill {\mathfrak{B}}_{j}(\gamma )& =\left\{{z}_{i}=\tau {u}_{i},-\frac{1}{2}< {u}_{\gamma (j+1)}< {u}_{\gamma (j+2)}< \cdots < {u}_{\gamma (n)}< 0< {u}_{\gamma (2)}\right.\hfill \\ \hfill & \quad \left.< {u}_{\gamma (3)}< \cdots < {u}_{\gamma (j)}< \frac{1}{2}\right\},\hfill \end{align} \tag{ 3.2 }$$

where ${u}_{i}\in \mathbb{R}$, and the B-cycle Green function ${\mathcal{G}}_{\mathfrak{B}}(z,\tau )$ is constructed in two steps: first, we define ${\mathcal{G}}_{\mathfrak{B}}(z,\tau )$ for z on the line (0, τ) by [77]

$$\begin{equation}{\mathcal{G}}_{\mathfrak{B}}(z,\tau )={\mathcal{G}}_{\mathfrak{A}}\left(\frac{z}{\tau },-\frac{1}{\tau }\right)=-\frac{i\pi {z}^{2}}{\tau }-\mathrm{log}\left(\frac{{\theta }_{1}(z,\tau )}{\eta (\tau )}\right)-\frac{i\pi }{6\tau }+i\pi ,\quad z\in (0,\tau ).\end{equation} \tag{ 3.3 }$$

Then, we extend this to z ∈ (−τ, 0) by imposing ${\mathcal{G}}_{\mathfrak{B}}(z,\tau )={\mathcal{G}}_{\mathfrak{B}}(-z,\tau )$ for compatibility with (2.11) under modular S transformations, leading to the combined expression

$$\begin{equation}{\mathcal{G}}_{\mathfrak{B}}(u\tau ,\tau )=-i\pi {u}^{2}\tau -\mathrm{log}\left(\frac{{\theta }_{1}(\vert u\vert \tau ,\tau )}{\eta (\tau )}\right)-\frac{i\pi }{6\tau }+i\pi ,\quad u\in (-1,1).\end{equation} \tag{ 3.4 }$$

Instead of integrating over z_i = τu_i with u_i ∈ (0, 1), we have chosen the representative ${u}_{i}\in (-\frac{1}{2},\frac{1}{2})$ of the B-cycle in order to facilitate the comparison with genus-zero integration cycles as τ → i∞. Figure 3 illustrates the integration cycle (3.2) in both the z_j and ${\sigma }_{j}={\mathrm{e}}^{2\pi i{z}_{j}}$ variables (the latter becoming the coordinates on the sphere as τ → i∞), where ${z}_{j}\in i\mathbb{R}$ and ${\sigma }_{j}\in {\mathbb{R}}^{+}$ for purely imaginary choices of τ. Note that non-planar versions of the B-cycle integrals involve additional punctures at ${z}_{j}\in \frac{1}{2}+i\mathbb{R}$ or negative σ_j ∈ (−q^−1/2, −q^1/2).

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The modular transformation ${\Omega}(z,\eta ,-\frac{1}{\tau })=\tau {\Omega}(\tau z,\tau \eta ,\tau )$ of the doubly-periodic Kronecker–Eisenstein series (2.12) leads to the rescaling η_j → τη_j in the subscript of the ρ-dependent integrand ${\varphi }_{\tau \overrightarrow{\eta }}^{\tau }$ of (3.1).

The doubly-periodic integrands ${\varphi }_{\overrightarrow{\eta }}^{\tau }$ in (2.13) are non-holomorphic, so their complex conjugates in (2.16) obey

$$\begin{equation}{\partial }_{{z}_{j}}\bar{{\varphi }_{\overrightarrow{\eta }}^{\tau }(1,\gamma (2,\dots ,n))}=\frac{2\pi i{\bar{\eta }}_{j}}{\tau -\bar{\tau }}\bar{{\varphi }_{\overrightarrow{\eta }}^{\tau }(1,\gamma (2,\dots ,n))}\end{equation} \tag{ 3.5 }$$

which leads to the terms $\sim \hspace{-2.5pt}{\bar{\eta }}_{j}{\partial }_{{\eta }_{j}}$ in the closed-string derivations ${R}_{\overrightarrow{\eta }}({{\epsilon}}_{0})$ in (2.23). This introduces a tension between the open- and closed-string differential equation (2.19) such that the ${\varphi }_{\overrightarrow{\eta }}^{\tau }$ do not qualify as Betti–deRham duals of open-string integration cycles. In order to generalize the interplay of Parke–Taylor factors (2.4) with single-valued integration [48, 49] to genus one, the factor of $\bar{{\varphi }_{\overrightarrow{\eta }}^{\tau }(\dots )}$ in the ${Y}_{\overrightarrow{\eta }}^{\tau }$ integrals (2.16) needs to be replaced by an antimeromorphic function that is still well-defined on the torus, i.e. the complex conjugate of an elliptic function in all of z₁, z₂, ..., z_n .

Such elliptic functions of n punctures can be generated by cycles of Kronecker–Eisenstein series [79]

$$\begin{equation}{\Omega}({z}_{12},\xi ,\tau ){\Omega}({z}_{23},\xi ,\tau )\dots {\Omega}({z}_{n1},\xi ,\tau ){=:}{\xi }^{-n}\sum\limits _{w=0}^{\infty }{\xi }^{w}{V}_{w}(1,2,\dots ,n\vert \tau ),\end{equation} \tag{ 3.6 }$$

where V_w has holomorphic modular weight w. Even though the individual Kronecker–Eisenstein series Ω are not meromorphic in the z_j , the V_w are elliptic functions since the non-holomorphic phase factors in (2.12) cancel from the cyclic product in (3.6). The simplest examples are

$$\begin{align}\hfill {V}_{0}\left(1,2,\dots ,n\vert \tau \right)& =1\hfill \\ \hfill {V}_{1}\left(1,2,\dots ,n\vert \tau \right)& =\sum\limits _{j=1}^{n}{f}^{\left(1\right)}\left({z}_{j}-{z}_{j+1},\tau \right)\hfill \\ \hfill {V}_{2}\left(1,2,\dots ,n\vert \tau \right)& =\sum\limits _{j=1}^{n}{f}^{\left(2\right)}\left({z}_{j}-{z}_{j+1},\tau \right)+\sum\limits _{1\leqslant j< k}^{n}{f}^{\left(1\right)}\left({z}_{j}-{z}_{j+1},\tau \right)\hfill \\ \hfill & \quad \times \enspace {f}^{\left(1\right)}\left({z}_{k}-{z}_{k+1},\tau \right)\hfill \end{align} \tag{ 3.7 }$$

with z_n+1 = z₁ and Kronecker–Eisenstein coefficients f^(w) defined by (2.14), also see (5.2) for the analogous expressions at general w.

As will be detailed below, these elliptic functions degenerate to suitable combinations of Parke–Taylor factors when forming the linear combinations

$$\begin{equation}V(1,2,\dots ,n\vert \tau )=\sum\limits _{w=0}^{n-2}\frac{{V}_{w}(1,2,\dots ,n\vert \tau )}{{(2\pi i)}^{w}\enspace (n-w-1)!}\end{equation} \tag{ 3.8 }$$

such as (see sections 5.4 and 5.5 for detailed discussions of the three- and four-point examples)

$$\begin{align}\hfill V\left(1,2\vert \tau \right)& =1,\qquad V\left(1,2,3\vert \tau \right)=\frac{1}{2}+\frac{{V}_{1}\left(1,2,3\vert \tau \right)}{2\pi i}\hfill \\ \hfill V\left(1,2,3,4\vert \tau \right)& =\frac{1}{6}+\frac{1}{2}\frac{{V}_{1}\left(1,2,3,4\vert \tau \right)}{2\pi i}+\frac{{V}_{2}\left(1,2,3,4\vert \tau \right)}{{\left(2\pi i\right)}^{2}}\hfill \\ \hfill V\left(1,2,3,4,5\vert \tau \right)& =\frac{1}{24}+\frac{1}{6}\frac{{V}_{1}\left(1,2,3,4,5\vert \tau \right)}{2\pi i}+\frac{1}{2}\frac{{V}_{2}\left(1,2,3,4,5\vert \tau \right)}{{\left(2\pi i\right)}^{2}}\hfill \\ \hfill & \quad +\frac{{V}_{3}\left(1,2,3,4,5\vert \tau \right)}{{\left(2\pi i\right)}^{3}}.\hfill \end{align} \tag{ 3.9 }$$

To lend credence to this definition of the V-function, let us see how their properties parallel those of the genus-zero case: the Betti–deRham duality at genus zero relies on the simple-pole residues

$$\begin{equation}{\mathrm{R}\mathrm{e}\mathrm{s}}_{{z}_{j}={z}_{j\pm 1}}\mathrm{P}\mathrm{T}(1,2,\dots ,j,\dots ,n)=\pm \mathrm{P}\mathrm{T}(1,2,\dots ,j-1,j+1,\dots ,n)\end{equation} \tag{ 3.10 }$$

of the Parke–Taylor factors (2.4). These residues correspond to the situation when two neighboring points of the disk ordering (2.2) at z_j = z_j±1 come together, which is crucial for sphere integrals being single-valued disk integrals [43, 45].

Similarly, at genus one, the generating function (3.6) of the elliptic V_w functions exposes the recursive structure of their simple-pole residues

$$\begin{equation}{\mathrm{R}\mathrm{e}\mathrm{s}}_{{z}_{j}={z}_{j\pm 1}}{V}_{w}(1,2,\dots ,j,\dots ,n)=\pm {V}_{w-1}(1,2,\dots ,j-1,j+1,\dots ,n)\end{equation} \tag{ 3.11 }$$

and the absence of higher poles in z_j − z_j±1. Consequently, the pole structure of the elliptic combinations (3.8)

$$\begin{equation}{\mathrm{R}\mathrm{e}\mathrm{s}}_{{z}_{j}={z}_{j\pm 1}}V(1,2,\dots ,j,\dots ,n)=\pm \frac{1}{2\pi i}V(1,2,\dots ,j-1,j+1,\dots ,n)\end{equation} \tag{ 3.12 }$$

mirrors the boundaries of the open-string integration cycles as z_j = z_j±1, i.e. one recovers mutually consistent V-functions and cycles at multiplicity n − 1 in both cases ¹⁴ .

The absence of V_w with w ⩾ n − 1 in (3.8) can be understood from

• The vanishing of V_n−1(1, 2, ..., n|τ) since (3.6) would otherwise be an elliptic function of ξ with a simple pole at the origin [79]
• The breakdown of uniform transcendentality when expanding Koba–Nielsen integrals involving V_n (1, 2, ..., n|τ) [52] (which is in tension with the transcendentality properties of open-string integrals [60])
• The fact that V_w⩾n+1(1, 2, ..., n|τ) is expressible in terms of G_w−k V_k (1, 2, ..., n|τ) with k ⩽ n − 2 [79]

Similar to the closed-string integrals ${Y}_{\overrightarrow{\eta }}^{\tau }$, we define an (n − 1)! × (n − 1)! matrix of torus integrals

$$\begin{align}\hfill {J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )& ={(2i)}^{n-1}{\int }_{{\mathfrak{T}}^{n-1}}\left(\prod\limits _{j=2}^{n}{\mathrm{d}}^{2}{z}_{j}\right)\prod\limits _{1\leqslant i< j}^{n}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{T}}({z}_{ij},\tau )}\bar{V(1,\gamma (2,\dots ,n))}\hfill \\ \hfill & \quad \times \enspace {\varphi }_{(\tau -\bar{\tau })\overrightarrow{\eta }}^{\tau }(1,\rho (2,\dots ,n))\hfill \end{align} \tag{ 3.13 }$$

indexed by permutations γ, ρ ∈ S_n−1 of (3.8) and (2.13). Note that the cyclic symmetry

$$\begin{equation}{V}_{w}(2,\dots ,n,1\vert \tau )={V}_{w}(1,2,\dots ,n\vert \tau ),\qquad V(2,\dots ,n,1\vert \tau )=V(1,2,\dots ,n\vert \tau )\end{equation} \tag{ 3.14 }$$

exposed by the generating function (3.6) has been used to bring the integrand of (3.13) into the form of $\bar{V(1,\dots \vert \tau )}$.

The modular S transformation in (3.1) maps the A-cycle eMZVs (2.24) in the η_j - and α'-expansion of ${Z}_{\overrightarrow{\eta }}^{\tau }$ to B-cycle eMZVs [3] in the analogous expansion of ${B}_{\overrightarrow{\eta }}^{\tau }$. As detailed in [51, 77, 93], the asymptotic expansion of B-cycle eMZVs as τ → i∞ is governed by Laurent polynomials in $T=\pi \tau \in i{\mathbb{R}}^{+}$ whose coefficients are $\mathbb{Q}$-linear combinations of MZVs, for instance

$$\begin{equation}\begin{aligned}\hfill \omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)& =-\frac{{T}^{2}}{180}-\frac{{\zeta }_{2}}{2}+\frac{i{\zeta }_{3}}{T}+\frac{3{\zeta }_{4}}{2{T}^{2}}+\mathcal{O}\left({\mathrm{e}}^{2iT}\right)\hfill \\ \hfill \omega \left(0,0,1,0\left\vert \right.-\frac{1}{\tau }\right)& =\frac{iT}{120}-\frac{i{\zeta }_{2}}{4T}-\frac{3{\zeta }_{3}}{4{T}^{2}}+\frac{3i{\zeta }_{4}}{4{T}^{3}}+\mathcal{O}\left({\mathrm{e}}^{2iT}\right)\hfill \\ \hfill \omega \left(0,0,3,0\left\vert \right.-\frac{1}{\tau }\right)& =\frac{i{T}^{3}}{1260}-\frac{3i{\zeta }_{4}}{4T}-\frac{9{\zeta }_{5}}{2{T}^{2}}+\frac{15i{\zeta }_{6}}{2{T}^{3}}+\mathcal{O}\left({\mathrm{e}}^{2iT}\right).\hfill \end{aligned}\end{equation} \tag{ 3.15 }$$

The suppressed terms $\mathcal{O}({\mathrm{e}}^{2iT})$ are series in q = e^2πiτ = e^2iT with Laurent polynomials in T as their coefficients.

The MGFs (2.26) in the η_j - and α'-expansion of (2.16) admit similar expansions around the cusp, where the leading term is a Laurent polynomial in y = π Im τ instead of T. The coefficients in the Laurent polynomials of MGFs were shown to be $\mathbb{Q}$-linear combinations of MZVs ¹⁵ [15] and are conjectured to be single-valued MZVs [10, 12]. Simple examples of the asymptotics of MGFs include

$$\begin{equation}\begin{aligned}\hfill {E}_{2}\left(\tau \right)& =\frac{{y}^{2}}{45}+\frac{{\zeta }_{3}}{y}+\mathcal{O}\left({\mathrm{e}}^{-2y}\right)\hfill \\ \hfill \frac{\pi \bar{\nabla }{E}_{2}\left(\tau \right)}{{y}^{2}}& =\frac{2y}{45}-\frac{{\zeta }_{3}}{{y}^{2}}+\mathcal{O}\left({\mathrm{e}}^{-2y}\right)\hfill \\ \hfill {E}_{3}\left(\tau \right)& =\frac{2{y}^{3}}{945}+\frac{3{\zeta }_{5}}{4{y}^{2}}+\mathcal{O}\left({\mathrm{e}}^{-2y}\right),\hfill \end{aligned}\end{equation} \tag{ 3.16 }$$

see (2.28) for the lattice-sum representations of the non-holomorphic Eisenstein series.

In a variety of examples, the Laurent polynomials of MGFs and B-cycle eMZVs have been related by an extension of the single-valued map (2.7) to [16, 51, 52]

$$\begin{equation}\mathrm{s}\mathrm{v}\enspace T=2iy{\Leftrightarrow}\text{sv}\enspace \mathrm{log}(q)=\mathrm{log}\vert q{\vert }^{2}.\end{equation} \tag{ 3.17 }$$

By (3.15) and (3.16), for instance, the Laurent polynomials of $\omega (0,0,2\vert -\frac{1}{\tau })\to {E}_{2}(\tau )$ as well as $\omega (0,0,1,0\vert -\frac{1}{\tau })\to -\frac{3}{8}\frac{\pi \bar{\nabla }{E}_{2}(\tau )}{{y}^{2}}$ and $\omega (0,0,3,0\vert -\frac{1}{\tau })\to 3{E}_{3}(\tau )$ are related by (3.17).

The A-cycle eMZVs in ${Z}_{\overrightarrow{\eta }}^{\tau }$, by contrast, enjoy a Fourier expansion in q = e^2πiτ whose coefficients are $\mathbb{Q}[{(2\pi i)}^{-1}]$ combinations of MZVs [3, 17] and do not feature any analogues of the Laurent polynomials in the expansion of ${B}_{\overrightarrow{\eta }}^{\tau }$. This is yet another indication besides their differential equations that the B-cycle integrals (3.1) are a more suitable starting point for comparison with closed-string integrals than their A-cycle counterparts (2.9).

As a particular convenience of the elliptic combinations (3.8) in the integrands of ${J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$, their degeneration at the cusp gives rise to Parke–Taylor factors in n + 2 punctures (σ₁ = 1 by z₁ = 0)

$$\begin{equation}{\sigma }_{j}={\mathrm{e}}^{2\pi i{z}_{j}},\quad {\sigma }_{+}=0,\quad {\sigma }_{-}\to \infty .\end{equation} \tag{ 3.18 }$$

Since the non-holomorphic exponentials of ${\Omega}(z,\xi ,\tau )=\mathrm{exp}(2\pi i\xi \frac{\mathrm{I}\mathrm{m}\enspace z}{\mathrm{I}\mathrm{m}\enspace \tau })F(z,\xi ,\tau )$ cancel from the cyclic products in (3.6), one can determine the asymptotics of V(...|τ) as τ → i∞ by using the degeneration of the holomorphic Kronecker–Eisenstein series

$$\begin{equation}F({z}_{ij},\xi ,\tau )=\pi \enspace \mathrm{cot}(\pi \xi )+i\pi \frac{{\sigma }_{i}+{\sigma }_{j}}{{\sigma }_{i}-{\sigma }_{j}}+\mathcal{O}(q).\end{equation} \tag{ 3.19 }$$

The relative factors of the V_w in (3.8) have been engineered to obtain the following cyclic combinations of Parke–Taylor factors at the cusp,

$$\begin{align}\hfill \underset{\tau \to i\infty }{\mathrm{lim}}\enspace \frac{V(1,2,\dots ,n\vert \tau )}{{\sigma }_{1}{\sigma }_{2}\dots {\sigma }_{n}}& ={(-1)}^{n-1}\enspace \underset{{\sigma }_{-}\to \infty }{\mathrm{lim}}\vert {\sigma }_{-}{\vert }^{2}\left[\mathrm{P}\mathrm{T}(+,n,n-1,\dots ,2,1,-)\right.\hfill \\ \hfill & \quad \left.+\mathrm{c}\mathrm{y}\mathrm{c}(1,2,\dots ,n)\right],\hfill \end{align} \tag{ 3.20 }$$

which have featured in the context of one-loop gauge-theory amplitudes in ambitwistor string theories [94]. The denominators on the left-hand side of (3.20) arise from $\mathrm{d}{z}_{j}=\frac{\mathrm{d}{\sigma }_{j}}{2\pi i{\sigma }_{j}}$, and the factor of |σ₋|² on the right-hand side identifies functions on a degenerate torus with SL₂-fixed expressions at genus zero [65]. Given that Parke–Taylor factors are Betti–deRham dual to disk orderings $\mathfrak{D}(\dots )$ in (2.2), the τ → i∞ asymptotics of ${J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$ should yield the single-valued map of suitably chosen disk integrals. In fact, upon rewriting the B-cycle ordering in terms of the σ_j variables (3.18), each of the contributions ${\mathfrak{B}}_{j}(\gamma )$ in (3.2) and figure 3 degenerates to a single disk ordering

$$\begin{equation}{\mathfrak{B}}_{j}\left(\gamma \right)\enspace {\vert }_{\tau \to i\infty }={\left(-1\right)}^{n-1}\mathfrak{D}\left(+,\gamma \left(j,j-1,\dots ,3,2\right),1,\gamma \left(n,n-1,\dots ,j+1\right),-\right),\end{equation} \tag{ 3.21 }$$

such that the overall B-cycle ordering $\mathfrak{B}(\gamma )={\bigoplus}_{j=1}^{n}{\mathfrak{B}}_{j}(\gamma )$ at the cusp becomes the Betti–deRham dual to the cyclic combination of Parke–Taylor factors in (3.20),

$$\begin{equation}\mathfrak{B}\left(2,3,\dots ,n\right)\enspace {\vert }_{\tau \to i\infty }={\left(-1\right)}^{n-1}\underset{j=1{}}{\overset{n}{\bigoplus}}\mathfrak{D}\left(+,j,j-1,\dots ,3,2,1,n,n-1,\dots ,j+1,-\right).\end{equation} \tag{ 3.22 }$$

Hence, the tree-level result (2.8) provides evidence for our central conjecture

$$\begin{equation}{J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )\enspace {\vert }_{\text{LP}}=\mathrm{s}\mathrm{v}\enspace {B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )\enspace {\vert }_{\text{LP}},\end{equation} \tag{ 3.23 }$$

where the notation |_LP instructs to only keep the Laurent polynomials in τ and Im τ while discarding any contribution $\sim \hspace{-2.5pt}q,\bar{q}$. The conjectural part of (3.23) concerns the non-constant terms in the Laurent polynomials, i.e. corrections ∼(log q)^±1 to the expansion around the cusp q = 0, so it is not implied by the Betti–deRham duality of (3.2) and (3.20) which only holds at the cusp. That is why we support (3.23) by extensive tests at low orders in η_j , α' as detailed below, and by the fact that the asymptotic expansions of the Green functions are related by the single-valued map with sv log σ_ij = log|σ_ij |²,

$$\begin{align}\hfill {\mathfrak{G}}_{\mathfrak{B}}({z}_{ij},\tau )\enspace {\vert }_{\text{LP}}& =-\frac{iT}{6}-\frac{i{\zeta }_{2}}{T}+\frac{1}{2}(\mathrm{log}\enspace {\sigma }_{i}+\mathrm{log}\enspace {\sigma }_{j})-\mathrm{log}\vert {\sigma }_{ij}\vert \hfill \\ \hfill & \quad +\frac{i{(\mathrm{log}\enspace {\sigma }_{i}-\mathrm{log}\enspace {\sigma }_{j})}^{2}}{4T}\hfill \end{align} \tag{ 3.24 }$$

$$\begin{align}{\mathfrak{G}}_{\mathfrak{T}}({z}_{ij},\tau )\enspace {\vert }_{\text{LP}}& =\frac{y}{3}+\mathrm{log}\vert {\sigma }_{i}\vert +\mathrm{log}\vert {\sigma }_{j}\vert -2\enspace \mathrm{log}\vert {\sigma }_{ij}\vert +\frac{{\left(\mathrm{log}\vert {\sigma }_{i}\vert -\mathrm{log}\vert {\sigma }_{j}\vert \right)}^{2}}{2y}\\ & =\mathrm{s}\mathrm{v}\enspace {\mathfrak{G}}_{\mathfrak{B}}({z}_{ij},\tau )\enspace {\vert }_{\text{LP}}.\end{align} \tag{ 3.25 }$$

Note that the absolute value in (3.24) is due to the argument |u|τ of θ₁ in (3.4). For the two-point instances ${B}_{\overrightarrow{\eta }}^{\tau }$ and ${J}_{\overrightarrow{\eta }}^{\tau }$ of the open- and closed-string integrals (3.1) and (3.13), the Laurent polynomials in the asymptotics at the cusp can be determined [95] by mild generalizations of the techniques in [16, 55] (also see [56] for an alternative approach to the closed-string case):

$$\begin{align}\hfill {B}_{\eta }^{\tau }(2\vert 2)\enspace {\vert }_{\text{LP}}& =\mathrm{exp}\left(-\frac{i{s}_{12}T}{6}-\frac{i{s}_{12}{\zeta }_{2}}{T}\right)\left\{\left[i\mathrm{cot}({\eta }_{2}T)+1\right]\mathrm{exp}\left(\frac{i{s}_{12}}{4T}{\partial }_{{\eta }_{2}}^{2}\right)\frac{1}{{s}_{12}+2{\eta }_{2}}\right.\hfill \\ \hfill & \quad \left.\times \left[\frac{{\Gamma}(1+\frac{{s}_{12}}{2}+{\eta }_{2}){\Gamma}(1-{s}_{12})}{{\Gamma}(1-\frac{{s}_{12}}{2}+{\eta }_{2})}-{\mathrm{e}}^{iT(\frac{{s}_{12}}{2}+{\eta }_{2})}\right]+\left[i\mathrm{cot}({\eta }_{2}T)-1\right]\right.\hfill \\ \hfill & \quad \left.\times \mathrm{exp}\left(\frac{i{s}_{12}}{4T}{\partial }_{{\eta }_{2}}^{2}\right)\frac{1}{{s}_{12}-2{\eta }_{2}}\left[\frac{{\Gamma}(1+\frac{{s}_{12}}{2}-{\eta }_{2}){\Gamma}(1-{s}_{12})}{{\Gamma}(1-\frac{{s}_{12}}{2}-{\eta }_{2})}-{\mathrm{e}}^{iT(\frac{{s}_{12}}{2}-{\eta }_{2})}\right]\right.\hfill \\ \hfill & \quad \left.+\frac{1}{{s}_{12}}\enspace \mathrm{exp}\left(\frac{i{s}_{12}}{4T}{\partial }_{{\eta }_{2}}^{2}\right)\left[\frac{{\Gamma}(1+\frac{{s}_{12}}{2}-{\eta }_{2}){\Gamma}(1-{s}_{12})}{{\Gamma}(1-\frac{{s}_{12}}{2}-{\eta }_{2})}\right.\right.\hfill \\ \hfill & \left.\quad \left.-\frac{{\Gamma}(1+\frac{{s}_{12}}{2}+{\eta }_{2}){\Gamma}(1-{s}_{12})}{{\Gamma}(1-\frac{{s}_{12}}{2}+{\eta }_{2})}\right]\right\}\hfill \end{align} \tag{ 3.26 }$$

$$\begin{align}\hfill {J}_{\eta }^{\tau }(2\vert 2)\enspace {\vert }_{\text{LP}}& =\mathrm{exp}\left(\frac{{s}_{12}y}{3}\right)\left\{\left[i\mathrm{cot}(2i{\eta }_{2}y)+1\right]\mathrm{exp}\left(\frac{{s}_{12}}{8y}{\partial }_{{\eta }_{2}}^{2}\right)\frac{1}{{s}_{12}+2{\eta }_{2}}\right.\hfill \\ \hfill & \quad \times \left.\left[\frac{{\Gamma}(1+\frac{{s}_{12}}{2}+{\eta }_{2}){\Gamma}(1-{s}_{12}){\Gamma}(1+\frac{{s}_{12}}{2}-{\eta }_{2})}{{\Gamma}(1-\frac{{s}_{12}}{2}+{\eta }_{2}){\Gamma}(1+{s}_{12}){\Gamma}(1-\frac{{s}_{12}}{2}-{\eta }_{2})}-{\mathrm{e}}^{-y({s}_{12}+2{\eta }_{2})}\right]\right.\hfill \\ \hfill & \quad \left.+\left[i\mathrm{cot}(2i{\eta }_{2}y)-1\right]\mathrm{exp}\left(\frac{{s}_{12}}{8y}{\partial }_{{\eta }_{2}}^{2}\right)\frac{1}{{s}_{12}-2{\eta }_{2}}\right.\hfill \\ \hfill & \quad \times \left.\left[\frac{{\Gamma}(1+\frac{{s}_{12}}{2}+{\eta }_{2}){\Gamma}(1-{s}_{12}){\Gamma}(1+\frac{{s}_{12}}{2}-{\eta }_{2})}{{\Gamma}(1-\frac{{s}_{12}}{2}+{\eta }_{2}){\Gamma}(1+{s}_{12}){\Gamma}(1-\frac{{s}_{12}}{2}-{\eta }_{2})}-{\mathrm{e}}^{-y({s}_{12}-2{\eta }_{2})}\right]\right\}\hfill \end{align} \tag{ 3.27 }$$

These two-point expressions are easily seen to line up with the all-multiplicity claim (3.23) since

$$\begin{equation}\mathrm{s}\mathrm{v}\enspace \left[\frac{{\Gamma}(1-a){\Gamma}(1-b)}{{\Gamma}(1-a-b)}\right]=\frac{{\Gamma}(1-a){\Gamma}(1-b){\Gamma}(1+a+b)}{{\Gamma}(1+a){\Gamma}(1+b){\Gamma}(1-a-b)},\end{equation} \tag{ 3.28 }$$

and the last line of (3.26) therefore vanishes under sv. Moreover, we have checked the three-point Laurent polynomials to obey (3.23) to the orders in the s_ij - and η_j -expansions where MGFs such as (2.26) of total modular weight 10 occur ¹⁶ . Finally, we have checked (3.23) to hold at four points to the orders where MGFs of total modular weight 8 occur, at least for contributions from ${\varphi }_{\overrightarrow{\eta }}^{\tau }$ in (2.13) without any singular factors of f⁽¹⁾(z_ij , τ). ¹⁷ These checks are based on Enriquez' methods [3] (also see appendix B of [51]) to determine the Laurent polynomials of B-cycle eMZVs. The Laurent polynomials for all B-cycle eMZVs with (length + weight) ⩽16 obtained from an FORM implementation [96] of these methods are available for download [97].

While the two-point Laurent-polynomials generated by (3.26) and (3.27) only involve Riemann zeta values, higher-point examples also introduce irreducible MZVs of depth ⩾ 2. The appearance of ζ_3,5 in B-cycle Laurent polynomials is later on exemplified in (4.27) and (5.37). Moreover, the appearance of ζ_3,5,3 in open- and closed-string calculations at three points in agreement with (3.23) was observed in section 3.3.5 of [51], based on earlier closed-string computations [12].

The holomorphic derivatives of the ${B}_{\overrightarrow{\eta }}^{\tau }$ and ${J}_{\overrightarrow{\eta }}^{\tau }$-integrals (3.1) and (3.13) can be easily deduced from (2.19): in the open-string case, the modular S transformation relating ${B}_{\overrightarrow{\eta }}^{\tau }={Z}_{\overrightarrow{\eta }}^{-1/\tau }$ and the modular weight (k, 0) of the holomorphic Eisenstein series G_k give rise to

$$\begin{equation}\frac{\partial }{\partial \mathrm{log}\enspace q}{B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )=\frac{1}{{(2\pi i)}^{2}}\sum\limits _{k=0}^{\infty }(1-k){\tau }^{k-2}{G}_{k}(\tau )\sum\limits _{\alpha \in {S}_{n-1}}{r}_{\overrightarrow{\eta }}{{({{\epsilon}}_{k})}_{\rho }}^{\alpha }{B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \alpha ),\end{equation} \tag{ 3.29 }$$

see [60] for the n-point derivations ${r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ and (2.21) for their two-point examples. Since the single-valued map at genus zero acts on transcendental constants, we have passed to the differential operator $\frac{\partial }{\partial \mathrm{log}\enspace q}={(2\pi i)}^{-1}{\partial }_{\tau }$ in comparison to (2.19) and in preparation for the extended single-valued map to be introduced below around (3.32).

In the closed-string case, the $\bar{V(\dots \vert \tau )}$ in (3.13) are not affected by holomorphic derivatives, and one can import a simplified version of the differential equations in [61] where contributions $\sim {\bar{\hspace{-2.5pt}\eta }}_{j}{\partial }_{{\eta }_{j}}$ are absent,

$$\begin{align}\frac{\partial }{\partial \mathrm{log}\enspace q}{J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )& =\frac{1}{{(2\pi i)}^{2}}\sum\limits _{k=0}^{\infty }(1-k){(\tau -\bar{\tau })}^{k-2}{G}_{k}(\tau )\\ & \quad \times \enspace \sum\limits _{\alpha \in {S}_{n-1}}\mathrm{s}\mathrm{v}\enspace {r}_{\overrightarrow{\eta }}{{({{\epsilon}}_{k})}_{\rho }}^{\alpha }{J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \alpha ).\end{align} \tag{ 3.30 }$$

By the differential equation (2.18) of the Green functions, also the term ∼ζ₂ in ${r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})$ is absent which we have indicated through the sv notation,

$$\begin{equation}\mathrm{s}\mathrm{v}\enspace {r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})=\left\{\begin{array}{cccc}{r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})\enspace & :\enspace k\geqslant 4\\ {r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})-2{\zeta }_{2}{s}_{12\dots n}\enspace & :\enspace k=0\end{array}\right.,\end{equation} \tag{ 3.31 }$$

where ${r}_{\overrightarrow{\eta }}({{\epsilon}}_{2})=0$. The building blocks of the closed-string differential operator in (3.30) are related to those in the open-string analogue (3.29) through an extension SV of the single-valued map (recall that T = πτ and y = π Im τ)

$$\begin{align}\hfill \mathrm{S}\mathrm{V}\enspace \left[{(2\pi i\tau )}^{k-2}\frac{{G}_{k}(\tau )}{{(2\pi i)}^{k}}{r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})\right]& =\mathrm{S}\mathrm{V}\enspace \left[{(2iT)}^{k-2}\right]\mathrm{S}\mathrm{V}\enspace \left[\frac{{G}_{k}(\tau )}{{(2\pi i)}^{k}}\right]\mathrm{S}\mathrm{V}\enspace \left[{r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})\right]\hfill \\ \hfill & ={(-4y)}^{k-2}\frac{{G}_{k}(\tau )}{{(2\pi i)}^{k}}\enspace \mathrm{s}\mathrm{v}\enspace {r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})\hfill \\ \hfill & =\frac{1}{{(2\pi i)}^{2}}{(\tau -\bar{\tau })}^{k-2}{G}_{k}(\tau )\mathrm{s}\mathrm{v}\enspace {r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})\hfill \end{align} \tag{ 3.32 }$$

which is taken to preserve the properties of sv,

$$\begin{equation}\mathrm{S}\mathrm{V}\enspace {\zeta }_{{n}_{1},{n}_{2},\dots ,{n}_{r}}=\mathrm{s}\mathrm{v}\enspace {\zeta }_{{n}_{1},{n}_{2},\dots ,{n}_{r}},\quad \mathrm{S}\mathrm{V}\enspace T=2iy\end{equation} \tag{ 3.33 }$$

and to furthermore preserve (2πi)^−k G_k (τ) (the inverse powers of π ensuring rational coefficients in the q-expansion) and the η_j -variables, cf (3.32). In other words, the differential operator ${\mathcal{O}}_{\overrightarrow{\eta }}^{\tau }={(2\pi i)}^{-2}{\sum }_{k=0}^{\infty }(1-k){\tau }^{k-2}{G}_{k}(\tau ){r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ appearing in (3.29) and its closed-string analogue in (3.30) are related by

$$\begin{align}\hfill \frac{\partial {B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )}{\partial \mathrm{log}\enspace q}& =\sum\limits _{\alpha \in {S}_{n-1}}{\mathcal{O}}_{\overrightarrow{\eta }}^{\tau }(\rho \vert \alpha ){B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \alpha ){\leftrightarrow}\hfill \\ \hfill \frac{\partial {J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )}{\partial \mathrm{log}\enspace q}& =\sum\limits _{\alpha \in {S}_{n-1}}\left[\mathrm{S}\mathrm{V}\enspace {\mathcal{O}}_{\overrightarrow{\eta }}^{\tau }(\rho \vert \alpha )\right]{J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \alpha ).\hfill \end{align} \tag{ 3.34 }$$

From the above discussion, both the τ → i∞ asymptotics and the differential operators of the open- and closed-string integrals ${B}_{\overrightarrow{\eta }}^{\tau }$ and ${J}_{\overrightarrow{\eta }}^{\tau }$ are related by the SV map (3.33). Hence, we propose that the solutions of (3.34) yield an appropriate extension of the SV map

$$\begin{equation}{J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )=\mathrm{S}\mathrm{V}\enspace {B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho ).\end{equation} \tag{ 3.35 }$$

This proposal is key the result of this work, relating the open-string integrals ${B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$ in (3.1) with integration ordering γ to the closed-string integrals ${J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$, where the ordering γ governs the singularity structure of the antielliptic integrand $\bar{V}$ in (3.13). By construction, this SV map commutes with the holomorphic τ-derivative and, under the assumption (3.23), it is consistent at the level of the Laurent polynomials at the cusp. Compatibility of a single-valued map at genus one with τ → i∞ generalizes the fact that the single-valued map of multiple polylogarithms commutes with evaluation [46]. Moreover, by the evidence to be discussed in section 4.3, the SV map is expected to be compatible with the shuffle product. As we will see in the next section, the α'-expansion of (3.35) induces an elliptic single-valued map for the eMZVs generated by ${B}_{\overrightarrow{\eta }}^{\tau }$ which yields the MGFs generated by ${J}_{\overrightarrow{\eta }}^{\tau }$.

Let us consider the scope of our definition (3.35). Firstly, not all the holomorphic iterated Eisenstein integrals appear in the α'-expansion of ${B}_{\overrightarrow{\eta }}^{\tau }$. As was discussed in [17, 18] and will become clearer when we discuss the α'-expansion of the solution of (3.34), relations among the ${r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ such as (2.22) lead to dropouts of certain iterated Eisenstein integrals from eMZVs and ${Y}_{\overrightarrow{\eta }}^{\tau }$ and thereby from ${B}_{\overrightarrow{\eta }}^{\tau }$ and ${J}_{\overrightarrow{\eta }}^{\tau }$. Hence, (3.35) does not comprise the SV map for the combinations of iterated Eisenstein integrals affected by these dropouts, starting with double integrals involving G₄ and G₁₀.

By contrast, the SV map of arbitrary convergent eMZVs can be extracted from (3.35) at sufficiently high multiplicity: as will be detailed in section 5.7, for ω(n₁, ..., n_r |τ) in (2.24) with given entries n_j (where n₁, n_r ≠ 1), one can engineer a combination of genus-one open-string integrals, where the desired eMZV occurs at the zeroth order in s_ij . ¹⁸

Finally, one could wonder whether holomorphic cusp forms lead to ambiguities in the definition of the V(1, ..., n|τ) in (3.8): ¹⁹ Starting from n = 14 points, their defining properties including simple-pole residues, the modularity of their constituents and their behavior (3.20) at the cusp are unchanged when adding combinations of holomorphic cusp forms and lower-weight V_w (1, ..., n|τ). However, adding a cusp form without any z_j -dependent coefficient to V(1, ..., n|τ) leads to a contradiction with the requirement that the τ-independent η¹⁻ⁿ order of ${B}_{\overrightarrow{\eta }}^{\tau }$ is mapped to the same term ∼η¹⁻ⁿ in the corresponding ${J}_{\overrightarrow{\eta }}^{\tau }$ integral. Products of V_w (1, ..., n|τ) with cusp forms in turn would violate the pole structure (3.12) that reflects the boundary structure of the dual cycles. Hence, the above requirements do not leave any room to modify V(1, ..., n|τ) by holomorphic cusp forms.

Given that the antielliptic $\bar{{V}_{w}(\dots \vert \tau )}$-functions (3.6) carry modular weight (0, w), their combinations $\bar{V(\dots \vert \tau )}$ (3.8) mix different modular weights. Hence, the α'-expansion of the generating function (3.13) with $\bar{V(\dots \vert \tau )}$ in the integrand mixes modular forms of different weight, even at fixed order in η_j . One may wish to isolate the contributions at fixed modular weights and study

$$\begin{align}\hfill {J}_{w,\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )& =\frac{{(2i)}^{n-1}}{{(-2\pi i)}^{w}}{\int }_{{\mathfrak{T}}^{n-1}}\left(\prod\limits _{j=2}^{n}{\mathrm{d}}^{2}{z}_{j}\right)\prod\limits _{1\leqslant i< j}^{n}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{T}}({z}_{ij},\tau )}\hfill \\ \hfill & \quad \times \enspace \bar{{V}_{w}(1,\gamma (2,\dots ,n)\vert \tau )}{\varphi }_{(\tau -\bar{\tau })\overrightarrow{\eta }}^{\tau }(1,\rho (2,\dots ,n)),\hfill \end{align} \tag{ 3.36 }$$

with 0 ⩽ w ⩽ n − 2, where the terms at homogeneity degree m in the η_j are modular forms of weight (0, 1 − n − m + w). One can still identify combinations of integration cycles (3.2) to write (3.36) at fixed modular weight w and ordering γ as the single-valued version of known open-string integrals: each V_w (1, 2, ..., n|τ) with w ⩽ n − 2 is expressible via permutation sums

$$\begin{equation}{V}_{w}(1,2,\dots ,n\vert \tau )={(2\pi i)}^{w}\sum\limits _{\gamma \in {S}_{n-1}}{c}_{w,\gamma }V(1,\gamma (2,\dots ,n))\end{equation} \tag{ 3.37 }$$

with coefficients ${c}_{w,\gamma }\in \mathbb{Q}$, e.g.

$$\begin{equation}{V}_{0}(1,\dots ,n\vert \tau )=1=\sum\limits _{\gamma \in {S}_{n-1}}V(1,\gamma (2,\dots ,n))\end{equation} \tag{ 3.38 }$$

$$\begin{equation}{V}_{1}(1,2,3\vert \tau )=i\pi \left[V(1,2,3\vert \tau )-V(1,3,2\vert \tau )\right]\end{equation} \tag{ 3.39 }$$

$$\begin{equation}{V}_{1}(1,2,3,4\vert \tau )=2\pi i\left[V(1,2,3,4\vert \tau )-V(1,4,3,2\vert \tau )\right]\end{equation} \tag{ 3.40 }$$

$$\begin{align}\hfill {V}_{2}(1,2,3,4\vert \tau )& =\frac{{(2\pi i)}^{2}}{6}\left[2V(1,2,3,4\vert \tau )+2V(1,4,3,2\vert \tau )-V(1,2,4,3\vert \tau )\right.\hfill \\ \hfill & \left.\quad -V(1,3,4,2\vert \tau )-V(1,3,2,4\vert \tau )-V(1,4,2,3\vert \tau )\right].\hfill \end{align} \tag{ 3.41 }$$

These relations and coefficients c_w,γ can be traced back to the symmetries of the V_w -functions including the cyclicity (3.14), the reflection property

$$\begin{equation}{V}_{w}(1,2,\dots ,n\vert \tau )={(-1)}^{w}{V}_{w}(n,\dots ,2,1\vert \tau )\end{equation} \tag{ 3.42 }$$

and corollaries of the Fay identity [98] which have been discussed in [79, 99]. An independent method based on the degeneration (3.20) to determine the c_w,γ is described in appendix A. As a result, there are less than (n − 1)! independent permutations V_w (1, γ(2, ..., n)|τ) at fixed 0 ⩽ w ⩽ n − 2 and n ⩾ 3. Their counting is governed by the unsigned Stirling number S_{n−1,n−w−1} of the first kind (where S_a,b counts the number of permutations of a elements with b disjoint cycles) as exemplified in table 1.

In particular, permutations of V_w=n−2(1, ..., n|τ) are related by Kleiss–Kuijf relations [99, 100]

such as

$$\begin{equation}{V}_{1}(1,2,3\vert \tau )=-{V}_{1}(1,3,2\vert \tau ),\qquad {V}_{2}(1,2,3,4\vert \tau )+\mathrm{c}\mathrm{y}\mathrm{c}(2,3,4)=0,\end{equation} \tag{ 3.44 }$$

consistent with the counting S_n−1,1 = (n − 2)! of independent permutations.

Given the decomposition (3.37) of a given V_w function with rational coefficients c_w,γ , one can by (3.35) write each ${J}_{w,\overrightarrow{\eta }}^{\tau }$ integral (3.36) as a combination of single-valued B-cycle integrals

$$\begin{equation}{J}_{w,\overrightarrow{\eta }}^{\tau }(2,\dots ,n\vert \rho )=\mathrm{S}\mathrm{V}\sum\limits _{\gamma \in {S}_{n-1}}{c}_{w,\gamma }{B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho ).\end{equation} \tag{ 3.45 }$$

For instance, the equivalent

$$\begin{align}\hfill {J}_{0,{\eta }_{2},{\eta }_{3}}^{\tau }(2,3\vert \rho )& =\mathrm{S}\mathrm{V}\left[{B}_{{\eta }_{2},{\eta }_{3}}^{\tau }(2,3\vert \rho )+{B}_{{\eta }_{2},{\eta }_{3}}^{\tau }(3,2\vert \rho )\right]\hfill \\ \hfill {J}_{1,{\eta }_{2},{\eta }_{3}}^{\tau }(2,3\vert \rho )& =\frac{1}{2}\mathrm{S}\mathrm{V}\left[{B}_{{\eta }_{2},{\eta }_{3}}^{\tau }(2,3\vert \rho )-{B}_{{\eta }_{2},{\eta }_{3}}^{\tau }(3,2\vert \rho )\right]\hfill \end{align} \tag{ 3.46 }$$

of (3.35) together with (3.38) and (3.39) suggests to assign a formal 'dual modular weight' 0 and 1 to the symmetric and antisymmetric three-point cycles, respectively,

$$\begin{align}\hfill \mathfrak{B}(2,3)+\mathfrak{B}(3,2)& {\leftrightarrow}\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}\enspace \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\enspace \mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\enspace 0\hfill \\ \hfill \mathfrak{B}(2,3)-\mathfrak{B}(3,2)& {\leftrightarrow}\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}\enspace \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\enspace \mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\enspace 1.\hfill \end{align} \tag{ 3.47 }$$

Similarly, combining (3.35) with (3.38), (3.40) and (3.41) leads to the following dual modular weights (d.m.w.) for four-point cycles

$$\begin{align}\hfill \mathfrak{B}(2,3,4)+\mathfrak{B}(4,3,2)+\mathfrak{B}(2,4,3)+\mathfrak{B}(3,4,2)+\mathfrak{B}(3,2,4)+\mathfrak{B}(4,2,3)& {\leftrightarrow}\mathrm{d}.\mathrm{m}.\mathrm{w}.\enspace 0\hfill \\ \hfill \mathfrak{B}(2,3,4)-\mathfrak{B}(4,3,2)& {\leftrightarrow}\mathrm{d}.\mathrm{m}.\mathrm{w}.\enspace 1\hfill \\ \hfill 2\mathfrak{B}(2,3,4)+2\mathfrak{B}(4,3,2)-\mathfrak{B}(2,4,3)-\mathfrak{B}(3,4,2)-\mathfrak{B}(3,2,4)-\mathfrak{B}(4,2,3)& {\leftrightarrow}\mathrm{d}.\mathrm{m}.\mathrm{w}.\enspace 2,\hfill \end{align} \tag{ 3.48 }$$

see section 5.5 for a more detailed discussion of the weight-two case. Finally, the all-multiplicity formula (3.37) translates into

$$\begin{equation}\sum\limits _{\gamma \in {S}_{n-1}}{c}_{w,\gamma }\mathfrak{B}(\gamma (2,3,\dots ,n)){\leftrightarrow}\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}\enspace \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\enspace \mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\enspace w,\end{equation} \tag{ 3.49 }$$

see appendix A for the rational coefficients c_w,γ and table 1 for the counting of independent n-point cycles with dual modular weight w.

We shall now derive the structure of the α'-expansion of ${B}_{\overrightarrow{\eta }}^{\tau }$ and ${J}_{\overrightarrow{\eta }}^{\tau }$ by repeating the key steps of [18] in solving the differential equation (2.19) of ${Y}_{\overrightarrow{\eta }}^{\tau }$. The first step is to introduce redefined generating series ${\hat{B}}_{\overrightarrow{\eta }}^{\tau }$ and ${\hat{J}}_{\overrightarrow{\eta }}^{\tau }$ by

$$\begin{equation}{\hat{B}}_{\overrightarrow{\eta }}^{\tau }=\mathrm{exp}\left(-\frac{{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{2\pi i\tau }\right){B}_{\overrightarrow{\eta }}^{\tau },\qquad {\hat{J}}_{\overrightarrow{\eta }}^{\tau }=\mathrm{exp}\left(-\frac{\mathrm{s}\mathrm{v}\enspace {r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{2\pi i(\tau -\bar{\tau })}\right){J}_{\overrightarrow{\eta }}^{\tau }.\end{equation} \tag{ 4.2 }$$

After this redefinition, the k = 0 terms involving $\frac{{G}_{0}}{{\tau }^{2}}$ and $\frac{{G}_{0}}{{(\tau -\bar{\tau })}^{2}}$ with G₀ = −1 are absent from the analogues of the differential equations (3.29) and (3.30), see (4.4) and (4.5) below. Throughout this section, we suppress the permutations γ, ρ labeling ${B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$ and ${J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$, and all matrix representations ${r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ are understood to act matrix-multiplicatively on the ρ-entry.

Since the ${r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ are expected (as tested for a wide range of k and n) to inherit the ad-nilpotency relations of the derivation algebra,

$$\begin{equation}{\mathrm{a}\mathrm{d}}_{{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}^{k-1}{r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})=0,\quad k\geqslant 4\enspace \end{equation} \tag{ 4.3 }$$

the combinations $\mathrm{exp}(-\frac{{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{2\pi i\tau }){r}_{\overrightarrow{\eta }}({{\epsilon}}_{k\geqslant 4})\mathrm{exp}(\frac{{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{2\pi i\tau })$ in the differential equations of the redefined integrals (4.2) truncate to a finite number of terms and we obtain

$$\begin{equation}2\pi i{\partial }_{\tau }{\hat{B}}_{\overrightarrow{\eta }}^{\tau }=\sum\limits _{k=4}^{\infty }(1-k)\sum\limits _{j=0}^{k-2}\frac{1}{j!}{\left(\frac{-1}{2\pi i}\right)}^{j}{\tau }^{k-2-j}{G}_{k}(\tau ){r}_{\overrightarrow{\eta }}\left({\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{j}({{\epsilon}}_{k})\right){\hat{B}}_{\overrightarrow{\eta }}^{\tau }\end{equation} \tag{ 4.4 }$$

$$\begin{equation}2\pi i{\partial }_{\tau }{\hat{J}}_{\overrightarrow{\eta }}^{\tau }=\sum\limits _{k=4}^{\infty }(1-k)\sum\limits _{j=0}^{k-2}\frac{1}{j!}{\left(\frac{-1}{2\pi i}\right)}^{j}{(\tau -\bar{\tau })}^{k-2-j}{G}_{k}(\tau ){r}_{\overrightarrow{\eta }}\left({\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{j}({{\epsilon}}_{k})\right){\hat{J}}_{\overrightarrow{\eta }}^{\tau }.\end{equation} \tag{ 4.5 }$$

We have used that ${\mathrm{a}\mathrm{d}}_{{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}(\cdot )=[{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0}),\cdot ]={\mathrm{a}\mathrm{d}}_{\text{sv}{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}(\cdot )$ (since the term ∼ζ₂ in (3.31) suppressed by sv is commutative) and employ the shorthands

$$\begin{equation}{r}_{\overrightarrow{\eta }}\left({\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{j}({{\epsilon}}_{k})\right)={\mathrm{a}\mathrm{d}}_{{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}^{j}{r}_{\overrightarrow{\eta }}({{\epsilon}}_{k}),\qquad {r}_{\overrightarrow{\eta }}({{\epsilon}}_{{k}_{1}}{{\epsilon}}_{{k}_{2}})={r}_{\overrightarrow{\eta }}({{\epsilon}}_{{k}_{1}}){r}_{\overrightarrow{\eta }}({{\epsilon}}_{{k}_{2}}).\end{equation} \tag{ 4.6 }$$

In equations (4.4), (4.5) and below, we use the derivative with respect to τ instead of log q as compared to section 3.5 above.

By the differential equation

$$\begin{equation}2\pi i{\partial }_{\tau }\mathcal{E} \left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right]=-{(2\pi i)}^{2-{k}_{\ell }}{(2\pi i\tau )}^{{j}_{\ell }}{G}_{{k}_{\ell }}(\tau )\mathcal{E} \left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell -1}\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell -1}\end{matrix};\tau \right]\end{equation} \tag{ 4.7 }$$

of the iterated Eisenstein integrals (4.1), one can solve the differential equation (4.4) of the generating series through the path-ordered exponential

$$\begin{align}\hfill {\hat{B}}_{\overrightarrow{\eta }}^{\tau }& =\sum\limits _{\ell =0}^{\infty }\sum\limits _{\genfrac{}{}{0pt}{}{{k}_{1},{k}_{2},\dots ,{k}_{\ell }}{=4,6,8,\dots }}\sum\limits _{{j}_{1}=0}^{{k}_{1}-2}\sum\limits _{{j}_{2}=0}^{{k}_{2}-2}\dots \sum\limits _{{j}_{\ell }=0}^{{k}_{\ell }-2}\left(\prod\limits _{i=1}^{\ell }\frac{{(-1)}^{{j}_{i}}({k}_{i}-1)}{({k}_{i}-{j}_{i}-2)!}\right)\mathcal{E} \left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right]\hfill \\ \hfill & \quad \times {r}_{\overrightarrow{\eta }}\left({\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{{k}_{\ell }-{j}_{\ell }-2}({{\epsilon}}_{{k}_{\ell }})\dots {\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{{k}_{2}-{j}_{2}-2}({{\epsilon}}_{{k}_{2}}){\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{{k}_{1}-{j}_{1}-2}({{\epsilon}}_{{k}_{1}})\right){\hat{B}}_{\overrightarrow{\eta }}^{i\infty }\hfill \end{align} \tag{ 4.8 }$$

for some initial value ${\hat{B}}_{\overrightarrow{\eta }}^{i\infty }$ to be discussed below. By inverting the redefinition (4.2) and moving the exponential to act directly on the initial value, we obtain the open-string analogue

$$\begin{align}\hfill {B}_{\overrightarrow{\eta }}^{\tau }& =\sum\limits _{\ell =0}^{\infty }\sum\limits _{\genfrac{}{}{0pt}{}{{k}_{1},{k}_{2},\dots ,{k}_{\ell }}{=4,6,8,\dots }}\sum\limits _{{j}_{1}=0}^{{k}_{1}-2}\sum\limits _{{j}_{2}=0}^{{k}_{2}-2}\dots \sum\limits _{{j}_{\ell }=0}^{{k}_{\ell }-2}\left(\prod\limits _{i=1}^{\ell }\frac{{(-1)}^{{j}_{i}}({k}_{i}-1)}{({k}_{i}-{j}_{i}-2)!}\right)\beta \left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right]\hfill \\ \hfill & \quad \times {r}_{\overrightarrow{\eta }}\left({\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{{k}_{\ell }-{j}_{\ell }-2}({{\epsilon}}_{{k}_{\ell }})\dots {\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{{k}_{2}-{j}_{2}-2}({{\epsilon}}_{{k}_{2}}){\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{{k}_{1}-{j}_{1}-2}({{\epsilon}}_{{k}_{1}})\right)\mathrm{exp}\left(\frac{{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{2\pi i\tau }\right){\hat{B}}_{\overrightarrow{\eta }}^{i\infty }\hfill \end{align} \tag{ 4.9 }$$

of the key result for the α'-expansion of ${Y}_{\overrightarrow{\eta }}^{\tau }$ in (3.11) of [18]. In commuting $\mathrm{exp}(\frac{{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{2\pi i\tau })$ past the ${{\epsilon}}_{{k}_{j}}$, the iterated Eisenstein integrals are rearranged into the combinations

$$\begin{equation}\beta \left[\begin{matrix}{j}_{1}\\ {k}_{1}\end{matrix};\tau \right]=\sum\limits _{{p}_{1}=0}^{{k}_{1}-{j}_{1}-2}\left(\genfrac{}{}{0pt}{}{{k}_{1}-{j}_{1}-2}{{p}_{1}}\right){\left(\frac{i}{2T}\right)}^{{p}_{1}}\mathcal{E} \left[\begin{matrix}{j}_{1}+{p}_{1}\\ {k}_{1}\end{matrix};\tau \right]\end{equation} \tag{ 4.10 }$$

and more generally

$$\begin{align}\hfill \beta \left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right]& =\sum\limits _{{p}_{1}=0}^{{k}_{1}-{j}_{1}-2}\sum\limits _{{p}_{2}=0}^{{k}_{2}-{j}_{2}-2}\dots \sum\limits _{{p}_{\ell }=0}^{{k}_{\ell }-{j}_{\ell }-2}\left(\genfrac{}{}{0.0pt}{}{{k}_{1}-{j}_{1}-2}{{p}_{1}}\right)\hfill \\ \hfill & \quad \times \left(\genfrac{}{}{0.0pt}{}{{k}_{2}-{j}_{2}-2}{{p}_{2}}\right)\cdots \left(\genfrac{}{}{0.0pt}{}{{k}_{\ell }-{j}_{\ell }-2}{{p}_{\ell }}\right)\hfill \\ \hfill & \quad \times {\left(\frac{i}{2T}\right)}^{{p}_{1}+{p}_{2}+\cdots +{p}_{\ell }}\mathcal{E} \left[\begin{matrix}{j}_{1}+{p}_{1}& {j}_{2}+{p}_{2}& \dots & {j}_{\ell }+{p}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right].\hfill \end{align} \tag{ 4.11 }$$

Note that (4.9) is an alternative ²⁰ organization of open-string α'-expansions at genus one as compared to [59, 60]. Non-planar B-cycle integrals obey the same differential equation (3.29) as the planar ones and therefore have an α'-expansion of the same form (4.9), only their initial values ${\hat{B}}_{\overrightarrow{\eta }}^{i\infty }$ need to be adapted to the non-planar integration cycle.

The modified iterated Eisenstein integrals β[...] in (4.11) satisfy the differential equations

$$\begin{align}\hfill 2\pi i{\tau }^{2}{\partial }_{\tau }\beta \left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right]& =\sum\limits _{i=1}^{\ell }({k}_{i}-{j}_{i}-2)\beta \left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{i-1}& {j}_{i}+1& {j}_{i+1}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{i-1}& {k}_{i}& {k}_{i+1}& \dots & {k}_{\ell }\end{matrix};\tau \right]\hfill \\ \hfill & \quad -{\delta }_{{j}_{\ell },{k}_{\ell }-2}{\tau }^{{k}_{\ell }}{G}_{{k}_{\ell }}(\tau )\beta \left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell -1}\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell -1}\end{matrix};\tau \right],\hfill \end{align} \tag{ 4.12 }$$

which allows us to directly check that (4.9) obeys (3.29). The integrals β[...] inherit the property that they vanish for τ → i∞ from the $\mathcal{E}[\dots ]$. Note that the definition (4.11) is equivalent to integral representations such as

$$\begin{align}\hfill \beta \left[\begin{matrix}{j}_{1}\\ {k}_{1}\end{matrix};\tau \right]& =-\frac{{(2\pi i)}^{1+{j}_{1}-{k}_{1}}}{{\tau }^{{k}_{1}-{j}_{1}-2}}\underset{i\infty {}}{\overset{\tau }{\int }}\mathrm{d}{\tau }_{1}\enspace {G}_{{k}_{1}}({\tau }_{1}){(\tau -{\tau }_{1})}^{{k}_{1}-{j}_{1}-2}{\tau }_{1}^{{j}_{1}}\hfill \\ \hfill \beta \left[\begin{matrix}{j}_{1}& {j}_{2}\\ {k}_{1}& {k}_{2}\end{matrix};\tau \right]& =\frac{{(2\pi i)}^{2+{j}_{1}+{j}_{2}-{k}_{1}-{k}_{2}}}{{\tau }^{{k}_{1}+{k}_{2}-{j}_{1}-{j}_{2}-4}}\underset{i\infty {}}{\overset{\tau }{\int }}\mathrm{d}{\tau }_{2}\enspace {G}_{{k}_{2}}({\tau }_{2}){(\tau -{\tau }_{2})}^{{k}_{2}-{j}_{2}-2}{\tau }_{2}^{{j}_{2}}\hfill \\ \hfill & \quad \times \enspace \underset{i\infty {}}{\overset{{\tau }_{2}}{\int }}\mathrm{d}{\tau }_{1}\enspace {G}_{{k}_{1}}({\tau }_{1}){(\tau -{\tau }_{1})}^{{k}_{1}-{j}_{1}-2}{\tau }_{1}^{{j}_{1}}.\hfill \end{align} \tag{ 4.13 }$$

The definition (4.11) of the β[...] preserves the shuffle relations of the iterated Eisenstein integrals (4.1), for instance

$$\begin{align}\hfill \mathcal{E} \left[\begin{matrix}{j}_{1}\\ {k}_{1}\end{matrix};\tau \right]\mathcal{E} \left[\begin{matrix}{j}_{2}\\ {k}_{2}\end{matrix};\tau \right]& =\mathcal{E} \left[\begin{matrix}{j}_{1}\enspace {j}_{2}\\ {k}_{1}\enspace {k}_{2}\end{matrix};\tau \right]+\mathcal{E} \left[\begin{matrix}{j}_{2}\enspace {j}_{1}\\ {k}_{2}\enspace {k}_{1}\end{matrix};\tau \right]{\Rightarrow}\beta \left[\begin{matrix}{j}_{1}\\ {k}_{1}\end{matrix};\tau \right]\beta \left[\begin{matrix}{j}_{2}\\ {k}_{2}\end{matrix};\tau \right]\hfill \\ \hfill & =\beta \left[\begin{matrix}{j}_{1}\enspace {j}_{2}\\ {k}_{1}\enspace {k}_{2}\end{matrix};\tau \right]+\beta \left[\begin{matrix}{j}_{2}\enspace {j}_{1}\\ {k}_{2}\enspace {k}_{1}\end{matrix};\tau \right].\hfill \end{align} \tag{ 4.14 }$$

One can extend the above strategy to expand ${B}_{\overrightarrow{\eta }}^{\tau }$ via (4.4) to the ${J}_{\overrightarrow{\eta }}^{\tau }$ integrals. The idea is to solve their differential equation (4.5) order by order in α' via

$$\begin{equation}2\pi i{\partial }_{\tau }{\mathcal{E}}^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right]=-{(2\pi i)}^{2-{k}_{\ell }+{j}_{\ell }}{(\tau -\bar{\tau })}^{{j}_{\ell }}{G}_{{k}_{\ell }}(\tau ){\mathcal{E}}^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell -1}\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell -1}\end{matrix};\tau \right],\end{equation} \tag{ 4.15 }$$

using the combinations ${\mathcal{E}}^{\text{sv}}$ of holomorphic iterated Eisenstein integrals (4.1) and their complex conjugates introduced in [18]. Their depth ℓ = 1 instances are completely known from the reference

$$\begin{equation}{\mathcal{E}}^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}\\ {k}_{1}\end{matrix};\tau \right]=\sum\limits _{{r}_{1}=0}^{{j}_{1}}{(-2\pi i\bar{\tau })}^{{r}_{1}}\left(\genfrac{}{}{0.0pt}{}{{j}_{1}}{{r}_{1}}\right)\left(\mathcal{E} \left[\begin{matrix}{j}_{1}-{r}_{1}\\ {k}_{1}\end{matrix};\tau \right]+{(-1)}^{{j}_{1}-{r}_{1}}\bar{\mathcal{E} \left[\begin{matrix}{j}_{1}-{r}_{1}\\ {k}_{1}\end{matrix};\tau \right]}\right),\end{equation} \tag{ 4.16 }$$

and their generalizations to depth ℓ ⩾ 2 involve antiholomorphic integration constants $\bar{\alpha [\dots ]}$,

$$\begin{align}\hfill {\mathcal{E}}^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}& {j}_{2}\\ {k}_{1}& {k}_{2}\end{matrix};\tau \right]& =\sum\limits _{{r}_{1}=0}^{{j}_{1}}\sum\limits _{{r}_{2}=0}^{{j}_{2}}{(-2\pi i\bar{\tau })}^{{r}_{1}+{r}_{2}}\left(\genfrac{}{}{0.0pt}{}{{j}_{1}}{{r}_{1}}\right)\left(\genfrac{}{}{0.0pt}{}{{j}_{2}}{{r}_{2}}\right)\hfill \\ \hfill & \quad \times \left\{\mathcal{E} \left[\begin{matrix}{j}_{1}-{r}_{1}& {j}_{2}-{r}_{2}\\ {k}_{1}& {k}_{2}\end{matrix};\tau \right]+{(-1)}^{{j}_{1}-{r}_{1}}\bar{\mathcal{E} \left[\begin{matrix}{j}_{1}-{r}_{1}\\ {k}_{1}\end{matrix};\tau \right]}\mathcal{E} \left[\begin{matrix}{j}_{2}-{r}_{2}\\ {k}_{2}\end{matrix};\tau \right]\right.\hfill \\ \hfill & \quad \left.+{(-1)}^{{j}_{1}+{j}_{2}-{r}_{1}-{r}_{2}}\bar{\mathcal{E} \left[\begin{matrix}{j}_{2}-{r}_{2}& {j}_{1}-{r}_{1}\\ {k}_{2}& {k}_{1}\end{matrix};\tau \right]}\right\}+\bar{\alpha \left[\begin{matrix}{j}_{1}& {j}_{2}\\ {k}_{1}& {k}_{2}\end{matrix};\tau \right]}.\hfill \end{align} \tag{ 4.17 }$$

The integration constants $\bar{\alpha [\dots ]}$ are invariant under τ → τ + 1 since the ${\mathcal{E}}^{\text{sv}}[\dots ]$ and the contributions from the $\mathcal{E}[\dots ],\bar{\mathcal{E}[\dots ]}$ to (4.17) are. They are known on a case-by-case basis, for instance

$$\begin{equation}\begin{aligned}\hfill \alpha \left[\begin{matrix}1& 0\\ 4& 4\end{matrix}\right]& =\alpha \left[\begin{matrix}0& 1\\ 4& 4\end{matrix}\right]=0,\hfill \\ \hfill \alpha \left[\begin{matrix}2& 0\\ 4& 4\end{matrix}\right]& =\frac{2{\zeta }_{3}}{3}\left(\mathcal{E} \left[\begin{matrix}0\\ 4\end{matrix}\right]+\frac{i\pi \tau }{360}\right)=-\alpha \left[\begin{matrix}0& 2\\ 4& 4\end{matrix}\right],\hfill \\ \hfill \alpha \left[\begin{matrix}2& 1\\ 4& 4\end{matrix}\right]& =\frac{2{\zeta }_{3}}{3}\left(2\pi i\tau \mathcal{E} \left[\begin{matrix}0\\ 4\end{matrix}\right]-\mathcal{E} \left[\begin{matrix}1\\ 4\end{matrix}\right]-\frac{{\pi }^{2}{\tau }^{2}}{360}\right)=-\alpha \left[\begin{matrix}1& 2\\ 4& 4\end{matrix}\right],\hfill \end{aligned}\end{equation} \tag{ 4.18 }$$

and the complete list of $\bar{\alpha \left[\begin{matrix}{j}_{1}& {j}_{2}\\ {k}_{1}& {k}_{2}\end{matrix}\right]}$ at k₁ + k₂ ⩽ 12 can be found in an ancillary file on the journal website of this work. The integration constants at arbitrary depth can be determined from the reality properties of the ${Y}_{\overrightarrow{\eta }}^{\tau }$ integrals [18]. The method in the reference to fix the $\bar{\alpha [\dots ]}$ hinges on the fact that the coefficients in the η_j - and ${\bar{\eta }}_{j}$-expansion of ${Y}_{\overrightarrow{\eta }}^{\tau }$ are closed under complex conjugation. For the n-point ${J}_{\overrightarrow{\eta }}^{\tau }$-series in turn the antiholomorphic modular weights $\bar{w}$ of the integrands $\bar{V(\dots )}$ in (3.13) are bounded by $\bar{w}\leqslant n-2$, so the complex conjugates of higher orders in the η_j -expansion are not part of the series. Hence, in the present formulation, the expansion of the ${Y}_{\overrightarrow{\eta }}^{\tau }$ in [18] is a necessary input to obtain well-defined ${\mathcal{E}}^{\text{sv}}$. This expansion depends on the knowledge of the initial values of ${Y}_{\overrightarrow{\eta }}^{\tau }$ which is currently available from sphere integrals to arbitrary weight only for two points and is under investigation for higher multiplicity [95]. Still, the torus-integral- and lattice-sum representations of single-valued eMZVs in section 5.7 do not require any knowledge of ${Y}_{\overrightarrow{\eta }}^{i\infty }$ and $\bar{\alpha [\dots ]}$.

By repeating the steps toward (4.8) and (4.9), we arrive at the structure of the α'-expansion

$$\begin{align}\hfill {J}_{\overrightarrow{\eta }}^{\tau }& =\sum\limits _{\ell =0}^{\infty }\sum\limits _{\genfrac{}{}{0pt}{}{{k}_{1},{k}_{2},\dots ,{k}_{\ell }}{=4,6,8,\dots }}\sum\limits _{{j}_{1}=0}^{{k}_{1}-2}\sum\limits _{{j}_{2}=0}^{{k}_{2}-2}\dots \sum\limits _{{j}_{\ell }=0}^{{k}_{\ell }-2}\left(\prod\limits _{i=1}^{\ell }\frac{{(-1)}^{{j}_{i}}({k}_{i}-1)}{({k}_{i}-{j}_{i}-2)!}\right){\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right]\hfill \\ \hfill & \quad \times {r}_{\overrightarrow{\eta }}\left({\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{{k}_{\ell }-{j}_{\ell }-2}({{\epsilon}}_{{k}_{\ell }})\dots {\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{{k}_{2}-{j}_{2}-2}({{\epsilon}}_{{k}_{2}}){\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{{k}_{1}-{j}_{1}-2}({{\epsilon}}_{{k}_{1}})\right)\mathrm{exp}\left(-\frac{\mathrm{s}\mathrm{v}\enspace {r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{4y}\right){\hat{J}}_{\overrightarrow{\eta }}^{i\infty }\hfill \end{align} \tag{ 4.19 }$$

with an initial value ${\hat{J}}_{\overrightarrow{\eta }}^{i\infty }$ to be discussed below and the combinations analogous to (4.11) [18]

$$\begin{align}\hfill {\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right]& =\sum\limits _{{p}_{1}=0}^{{k}_{1}-{j}_{1}-2}\sum\limits _{{p}_{2}=0}^{{k}_{2}-{j}_{2}-2}\dots \sum\limits _{{p}_{\ell }=0}^{{k}_{\ell }-{j}_{\ell }-2}\left(\genfrac{}{}{0.0pt}{}{{k}_{1}-{j}_{1}-2}{{p}_{1}}\right)\hfill \\ \hfill & \quad \times \left(\genfrac{}{}{0.0pt}{}{{k}_{2}-{j}_{2}-2}{{p}_{2}}\right)\cdots \left(\genfrac{}{}{0.0pt}{}{{k}_{\ell }-{j}_{\ell }-2}{{p}_{\ell }}\right){\left(\frac{1}{4y}\right)}^{{p}_{1}+{p}_{2}+\cdots +{p}_{\ell }}\hfill \\ \hfill & \quad \times {\mathcal{E}}^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}+{p}_{1}& {j}_{2}+{p}_{2}& \dots & {j}_{\ell }+{p}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right].\hfill \end{align} \tag{ 4.20 }$$

The expansion (4.19) solves (3.30) since the β^sv inherit their differential equation from (4.15),

$$\begin{align}\hfill & 2\pi i{(\tau -\bar{\tau })}^{2}{\partial }_{\tau }{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell }\end{matrix};\tau \right]\hfill \\ \hfill & \qquad =\sum\limits _{i=1}^{\ell }({k}_{i}-{j}_{i}-2){\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{i-1}& {j}_{i}+1& {j}_{i+1}& \dots & {j}_{\ell }\\ {k}_{1}& {k}_{2}& \dots & {k}_{i-1}& {k}_{i}& {k}_{i+1}& \dots & {k}_{\ell }\end{matrix};\tau \right]\hfill \\ \hfill & \qquad \quad -{\delta }_{{j}_{\ell },{k}_{\ell }-2}{(\tau -\bar{\tau })}^{{k}_{\ell }}{G}_{{k}_{\ell }}(\tau ){\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}& {j}_{2}& \dots & {j}_{\ell -1}\\ {k}_{1}& {k}_{2}& \dots & {k}_{\ell -1}\end{matrix};\tau \right],\hfill \end{align} \tag{ 4.21 }$$

see (4.12) for the holomorphic counterpart for ∂_τ β[...]. Both the ${\mathcal{E}}^{\text{sv}}[\dots ]$ and the β^sv[...] are expected to preserve the shuffle multiplication of their holomorphic counterparts (4.1) and (4.11): the differential equations (4.15) and (4.20) recursively imply that shuffle relations among ${\mathcal{E}}^{\text{sv}}[\dots ]$ and the β^sv[...] can at most be violated by antiholomorphic functions such as the integration constants $\bar{\alpha [\dots ]}$ in (4.17). ²¹ All examples of $\alpha \left[\begin{matrix}{j}_{1}& {j}_{2}\\ {k}_{1}& {k}_{2}\end{matrix}\right]$ up to including k₁ + k₂ = 12 were checked to preserve the shuffle relations, and their explicit form can also be found in an ancillary file to the arXiv submission of this work. Note that these checks cover the more intricate cases with (k₁, k₂) = (4, 6) and (k₁, k₂) = (4, 8) where imaginary cusp forms occur among the MGFs [18, 62].

The ${\mathcal{E}}^{\text{sv}}$ and β^sv are expected to occur in Brown's generating series of single-valued iterated Eisenstein integrals [13, 14, 101]. The construction of non-holomorphic modular forms in the references—so-called equivariant iterated Eisenstein integrals—are obtained by augmenting their single-valued counterparts by combinations of MZVs and objects of lower depth. At depth one, the equivariant iterated Eisenstein integrals are non-holomorphic Eisenstein series along with their Cauchy–Riemann derivatives [13, 14, 101]. From their representation [18]

$$\begin{equation}\begin{aligned}\hfill {(\pi \nabla )}^{m}{E}_{k}& ={\left(-\frac{1}{4}\right)}^{m}\frac{(2k-1)!}{(k-1)!(k-1-m)!}\left\{-{\beta }^{\text{sv}}\left[\begin{matrix}\hfill k-1+m\hfill \\ \hfill 2k\hfill \end{matrix}\right]+\frac{2{\zeta }_{2k-1}}{(2k-1){(4y)}^{k-1-m}}\right\}\hfill \\ \hfill {E}_{k}& =\frac{(2k-1)!}{{[(k-1)!]}^{2}}\left\{-{\beta }^{\text{sv}}\left[\begin{matrix}\hfill k-1\hfill \\ \hfill 2k\hfill \end{matrix}\right]+\frac{2{\zeta }_{2k-1}}{(2k-1){(4y)}^{k-1}}\right\}\hfill \\ \hfill \frac{{(\pi \bar{\nabla })}^{m}{E}_{k}}{{y}^{2m}}& =\frac{{(-4)}^{m}(2k-1)!}{(k-1)!(k-1-m)!}\left\{-{\beta }^{\text{sv}}\left[\begin{matrix}\hfill k-1-m\hfill \\ \hfill 2k\hfill \end{matrix}\right]+\frac{2{\zeta }_{2k-1}}{(2k-1){(4y)}^{k-1+m}}\right\},\hfill \end{aligned}\end{equation} \tag{ 4.22 }$$

the β^sv are seen to take the role of the single-valued rather than equivariant iterated Eisenstein integrals at depth one. At higher depth, the precise relation of the β^sv to Brown's construction is an open question at the time of writing.

It remains to specify the initial values ${\hat{B}}_{\overrightarrow{\eta }}^{i\infty }$ and ${\hat{J}}_{\overrightarrow{\eta }}^{i\infty }$ in the α'-expansions (4.9) and (4.19). The Laurent-polynomial contributions from the asymptotics (3.24) and (3.25) of the Green functions are still functions of τ and need to be translated into series that solely depend on η_j and s_ij . Following the construction of a similar initial value for ${Y}_{\overrightarrow{\eta }}^{\tau }$ in section 3.4 of [18], we import the constant parts ∼τ⁰ and ∼(Im τ)⁰ of the respective Laurent polynomials

$$\begin{equation}{\hat{B}}_{\overrightarrow{\eta }}^{i\infty }=\mathrm{exp}\left(\frac{i{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{2T}\right){B}_{\overrightarrow{\eta }}^{\tau }\enspace {\left\vert \right.}_{\text{LP}}{\left.\right\vert }_{{\tau }^{0}}\end{equation} \tag{ 4.23 }$$

$$\begin{equation}{\hat{J}}_{\overrightarrow{\eta }}^{i\infty }=\mathrm{exp}\left(\frac{\mathrm{s}\mathrm{v}\enspace {r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{4y}\right){J}_{\overrightarrow{\eta }}^{\tau }{\vert }_{\text{LP}}\enspace {\vert }_{{(\mathrm{I}\mathrm{m}\enspace \tau )}^{0}}.\end{equation} \tag{ 4.24 }$$

In both cases, the exponentials ensure that the negative powers of T in ${B}_{\overrightarrow{\eta }}^{\tau }\enspace {\vert }_{\text{LP}}$ and y in ${J}_{\overrightarrow{\eta }}^{\tau }\enspace {\vert }_{\text{LP}}$ disappear order by order in α'. Hence, (4.23) and (4.24) pick up the lowest powers of T, y present in $\mathrm{exp}\left(\frac{i{r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{2T}\right){B}_{\overrightarrow{\eta }}^{\tau }\enspace {\vert }_{\text{LP}}$ and $\mathrm{exp}\left(\frac{\mathrm{s}\mathrm{v}\enspace {r}_{\overrightarrow{\eta }}({{\epsilon}}_{0})}{4y}\right){J}_{\overrightarrow{\eta }}^{\tau }\enspace {\vert }_{\text{LP}}$. The leading α'- and η₂-orders of the two-point initial values following from the expressions in (3.26) and (3.27) are

$$\begin{align}\hfill {\hat{B}}_{{\eta }_{2}}^{i\infty }& =\frac{1}{{\eta }_{2}}\left[1+\frac{1}{6}{s}_{12}^{2}{\zeta }_{2}+\frac{1}{12}{s}_{12}^{3}{\zeta }_{3}+\frac{131}{720}{s}_{12}^{4}{\zeta }_{4}+{s}_{12}^{5}\left(\frac{17}{360}{\zeta }_{2}{\zeta }_{3}+\frac{43{\zeta }_{5}}{720}\right)+\mathcal{O}({s}_{12}^{6})\right]\hfill \\ \hfill & \quad +{\eta }_{2}\left[-2{\zeta }_{2}-{s}_{12}{\zeta }_{3}-\frac{29}{12}{s}_{12}^{2}{\zeta }_{4}-{s}_{12}^{3}\left(\frac{1}{3}{\zeta }_{2}{\zeta }_{3}+\frac{5{\zeta }_{5}}{6}\right)\right.\hfill \\ \hfill & \quad \left.-{s}_{12}^{4}\left(\frac{1}{12}{\zeta }_{3}^{2}+\frac{87{\zeta }_{6}}{40}\right)+\mathcal{O}({s}_{12}^{5})\right]\hfill \\ \hfill & \quad +{\eta }_{2}^{3}\left[-2{\zeta }_{4}+{s}_{12}(2{\zeta }_{2}{\zeta }_{3}-{\zeta }_{5})+{s}_{12}^{2}\left(\frac{{\zeta }_{3}^{2}}{2}-\frac{33{\zeta }_{6}}{8}\right)\right.\hfill \\ \hfill & \quad \left.+{s}_{12}^{3}\left(\frac{9}{4}{\zeta }_{3}{\zeta }_{4}+\frac{3}{2}{\zeta }_{2}{\zeta }_{5}-\frac{7{\zeta }_{7}}{4}\right)+\mathcal{O}({s}_{12}^{4})\right]\hfill \\ \hfill & \quad +{\eta }_{2}^{5}\left[-2{\zeta }_{6}+{s}_{12}(2{\zeta }_{4}{\zeta }_{3}+2{\zeta }_{2}{\zeta }_{5}-{\zeta }_{7})+{s}_{12}^{2}\left(-{\zeta }_{2}{\zeta }_{3}^{2}+{\zeta }_{3}{\zeta }_{5}-\frac{43{\zeta }_{8}}{6}\right)+\mathcal{O}({s}_{12}^{3})\right]\hfill \\ \hfill & \quad +{\eta }_{2}^{7}\left[-2{\zeta }_{8}+{s}_{12}(2{\zeta }_{3}{\zeta }_{6}+2{\zeta }_{4}{\zeta }_{5}+2{\zeta }_{2}{\zeta }_{7}-{\zeta }_{9})+\mathcal{O}({s}_{12}^{2})\right]\hfill \\ \hfill & \quad +{\eta }_{2}^{9}\left[-2{\zeta }_{10}+\mathcal{O}({s}_{12})\right]+\mathcal{O}({\eta }_{2}^{11})\hfill \end{align} \tag{ 4.25 }$$

as well as

$$\begin{align}\hfill {\hat{J}}_{{\eta }_{2}}^{i\infty }& =\frac{1}{{\eta }_{2}}\left[1+\frac{1}{6}{s}_{12}^{3}{\zeta }_{3}+\frac{43}{360}{s}_{12}^{5}{\zeta }_{5}+\mathcal{O}\left({s}_{12}^{6}\right)\right]\hfill \\ \hfill & \quad +{\eta }_{2}\left[-2{s}_{12}{\zeta }_{3}-\frac{5}{3}{s}_{12}^{3}{\zeta }_{5}-\frac{1}{3}{s}_{12}^{4}{\zeta }_{3}^{2}+\mathcal{O}\left({s}_{12}^{5}\right)\right]\hfill \\ \hfill & \quad +{\eta }_{2}^{3}\left[-2{s}_{12}{\zeta }_{5}+2{s}_{12}^{2}{\zeta }_{3}^{2}-\frac{7}{2}{s}_{12}^{3}{\zeta }_{7}+\mathcal{O}\left({s}_{12}^{4}\right)\right]\hfill \\ \hfill & \quad +{\eta }_{2}^{5}\left[-2{s}_{12}{\zeta }_{7}+4{s}_{12}^{2}{\zeta }_{3}{\zeta }_{5}+\mathcal{O}\left({s}_{12}^{3}\right)\right]\hfill \\ \hfill & \quad +{\eta }_{2}^{7}\left[-2{s}_{12}{\zeta }_{9}+\mathcal{O}\left({s}_{12}^{2}\right)\right]+{\eta }_{2}^{9}\mathcal{O}\left({s}_{12}\right)+\mathcal{O}\left({\eta }_{2}^{11}\right).\hfill \end{align} \tag{ 4.26 }$$

Higher orders in s₁₂ and η₂ are readily available through the straightforward expansion of the exponentials and Γ-functions in (3.26) and (3.27). In particular, these two-point expressions imply that all the coefficients in the s₁₂- and η₂-expansions are combinations of Riemann zeta values for ${\hat{B}}_{{\eta }_{2}}^{i\infty }$ and odd Riemann zeta values for ${\hat{J}}_{{\eta }_{2}}^{i\infty }$.

Starting from n = 3 points, the initial values ${\hat{B}}_{\overrightarrow{\eta }}^{i\infty }$ will also feature irreducible MZVs of higher depth. Based on Enriquez' method to generate the Laurent polynomial of B-cycle eMZVs [3] (also see appendix B of [51]) we have determined the three-point initial values to certain orders, and the results are included in an ancillary file to the arXiv submission of this article. To the orders under consideration, we find the following coefficients of ζ_3,5

$$\begin{align}\hfill {\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert 2,3)\enspace {\vert }_{{\zeta }_{3,5}}& =\frac{1}{10}({\eta }_{23}-2{\eta }_{3})(2{\eta }_{23}-{\eta }_{3})({\eta }_{23}+{\eta }_{3})\hfill \\ \hfill & \quad \times \left[2{\eta }_{23}^{2}{s}_{13}-2{\eta }_{23}{\eta }_{3}{s}_{12}-4{\eta }_{23}{\eta }_{3}{s}_{13}+2{\eta }_{3}^{2}{s}_{13}\right.\hfill \\ \hfill & \quad \left.-2{\eta }_{23}{\eta }_{3}{s}_{23}+{\eta }_{23}{s}_{12}{s}_{13}+{\eta }_{3}{s}_{13}{s}_{23}+\mathcal{O}({s}_{ij}^{3})\right]\hfill \\ \hfill {\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert 3,2)\enspace {\vert }_{{\zeta }_{3,5}}& =-\frac{1}{10}({\eta }_{2}-2{\eta }_{23})(2{\eta }_{2}-{\eta }_{23})({\eta }_{2}+{\eta }_{23})\hfill \\ \hfill & \quad \times \left[2{\eta }_{2}^{2}{s}_{12}-4{\eta }_{2}{\eta }_{23}{s}_{12}+2{\eta }_{23}^{2}{s}_{12}-2{\eta }_{2}{\eta }_{23}{s}_{13}\right.\hfill \\ \hfill & \quad \left.-2{\eta }_{2}{\eta }_{23}{s}_{23}-{\eta }_{23}{s}_{12}{s}_{13}-{\eta }_{2}{s}_{12}{s}_{23}+\mathcal{O}({s}_{ij}^{3})\right],\hfill \end{align} \tag{ 4.27 }$$

which by the single-valued maps sv ζ_3,5 = −10ζ₃ ζ₅ and sv ζ₃ ζ₅ = 4ζ₃ ζ₅ enter the closed-string initial values via

$$\begin{align}\hfill {\hat{J}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert \rho (2,3))\enspace {\vert }_{{\zeta }_{3}{\zeta }_{5}}& =-10{\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert \rho (2,3))\enspace {\vert }_{{\zeta }_{3,5}}\hfill \\ \hfill & \quad +4{\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert \rho (2,3))\enspace {\vert }_{{\zeta }_{3}{\zeta }_{5}}.\hfill \end{align} \tag{ 4.28 }$$

The coefficients of ζ_3,5 and ζ₃ ζ₅ in (4.27) and (4.28) are extracted after reducing all MZVs at weight 8 to $\mathbb{Q}$-linear combinations of $\left\{{\zeta }_{8},{\zeta }_{2}{\zeta }_{3}^{2},{\zeta }_{3}{\zeta }_{5},{\zeta }_{3,5}\right\}$ [102]. Similarly, the MZV ζ_3,5,3 seen in Laurent polynomials of both B-cycle integrals [51] and modular graph functions [12] will occur in both ${\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }$ and ${\hat{J}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }$. The contributions to ${\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }$ involving MZVs of weight up to and including four can be found in appendix B.

Since the initial values are obtained from the Laurent polynomials and the exponents in (4.23) and (4.24) are related by the single-valued map, the conjecture (3.23) supported by tree-level results and extensive genus-one tests is equivalent to

$$\begin{equation}{\hat{J}}_{\overrightarrow{\eta }}^{i\infty }(\gamma \vert \rho )=\mathrm{s}\mathrm{v}\enspace {\hat{B}}_{\overrightarrow{\eta }}^{i\infty }(\gamma \vert \rho ),\end{equation} \tag{ 4.29 }$$

in agreement with (4.25) and (4.26).

The τ → i∞ asymptotics of n-point A-cycle integrals (2.9) has been expressed in terms of (n + 2)-point disk integrals (2.1) in suitable kinematic limits [60]. Similarly, the Laurent polynomials of n-point genus-one integrals ${B}_{\overrightarrow{\eta }}^{\tau },{J}_{\overrightarrow{\eta }}^{\tau }$ are determined by genus-zero integrals at multiplicity n + 2 and below, see (3.26) and (3.27) for the explicit two-point result. As will be further investigated in [95], the main challenge is to determine the admixture of lower-point genus-zero integrals that generalize the subtraction of ${\mathrm{e}}^{iT(\frac{{s}_{12}}{2}\pm {\eta }_{2})}$ and ${\mathrm{e}}^{-y({s}_{12}\pm 2{\eta }_{2})}$ from the Γ-functions in (3.26) and (3.27).

The proposed single-valued map (3.35) can now also be studied at the level of the α'-expansions. Using (4.29) and (3.31), we find that one obtains ${J}_{\overrightarrow{\eta }}^{\tau }$ as the single-valued version of ${B}_{\overrightarrow{\eta }}^{\tau }$ if the coefficients obey

$$\begin{equation}{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}{j}_{2}\dots {j}_{\ell }\\ {k}_{1}{k}_{2}\dots {k}_{\ell }\end{matrix};\tau \right]=\mathrm{S}\mathrm{V}\enspace \beta \left[\begin{matrix}{j}_{1}{j}_{2}\dots {j}_{\ell }\\ {k}_{1}{k}_{2}\dots {k}_{\ell }\end{matrix};\tau \right].\end{equation} \tag{ 4.30 }$$

This follows from the relation (4.29) among the initial values and the form of the ${r}_{\overrightarrow{\eta }}$ operators in the respective α'-expansions, recalling that ${r}_{\overrightarrow{\eta }}({\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{j}({{\epsilon}}_{k}))=\mathrm{s}\mathrm{v}\enspace {r}_{\overrightarrow{\eta }}({\mathrm{a}\mathrm{d}}_{{{\epsilon}}_{0}}^{j}({{\epsilon}}_{k}))$.

On the one hand, (4.30) fixes the single-valued map of the eMZVs in the expansion of ${B}_{\overrightarrow{\eta }}^{\tau }$ that enter through the iterated Eisenstein integrals β[...]. On the other hand, (4.30) only applies to the combinations β[...] and β^sv[...] that occur in the path-ordered exponentials (4.9) and (4.19). The SV map of individual $\beta \left[\begin{matrix}{j}_{1}& \dots & {j}_{\ell }\\ {k}_{1}& \dots & {k}_{\ell }\end{matrix};\tau \right]$ remains undetermined whenever the relations in the derivation algebra such as (2.22) lead to dropouts of certain β[...] and β^sv[...] from ${B}_{\overrightarrow{\eta }}^{\tau }$ and ${J}_{\overrightarrow{\eta }}^{\tau }$ (starting with cases at (k₁, k₂) = (10, 4) at depth ℓ = 2 and any instance where j_i > k_i − 2).

Since the factors of $\frac{i}{2T}$ and $\frac{1}{4y}$ in (4.11) and (4.20) are furthermore related by SV, (4.30) is equivalent to

$$\begin{equation}{\mathcal{E}}^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}{j}_{1}{j}_{2}\dots {j}_{\ell }\\ {k}_{1}{k}_{2}\dots {k}_{\ell }\end{matrix};\tau \right]=\mathrm{S}\mathrm{V}\enspace \mathcal{E} \left[\begin{matrix}{j}_{1}{j}_{2}\dots {j}_{\ell }\\ {k}_{1}{k}_{2}\dots {k}_{\ell }\end{matrix};\tau \right],\end{equation} \tag{ 4.31 }$$

again up to cases where the relations in the derivation algebra cause dropouts. For instance, (4.16) implies that the single-valued version of holomorphic Eisenstein integrals (4.1) at depth one is given by

$$\begin{equation}\mathrm{S}\mathrm{V}\enspace \mathcal{E} \left[\begin{matrix}j\\ k\end{matrix};\tau \right]=\sum\limits _{r=0}^{j}{(-2\pi i\bar{\tau })}^{r}\left(\genfrac{}{}{0.0pt}{}{j}{r}\right)\left(\mathcal{E} \left[\begin{matrix}j-r\\ k\end{matrix};\tau \right]+{(-1)}^{j-r}\bar{\mathcal{E} \left[\begin{matrix}j-r\\ k\end{matrix};\tau \right]}\right),\end{equation} \tag{ 4.32 }$$

where the contributions on the right-hand side can be recognized as

$$\begin{align}\hfill \sum\limits _{r=0}^{j}{(-2\pi i\bar{\tau })}^{r}\left(\genfrac{}{}{0.0pt}{}{j}{r}\right)\mathcal{E} \left[\begin{matrix}j-r\\ k\end{matrix};\tau \right]& ={(2\pi i)}^{1-k+j}{\int }_{\tau }^{i\infty }\mathrm{d}{\tau }_{1}\enspace {({\tau }_{1}-\bar{\tau })}^{j}{G}_{k}({\tau }_{1}),\hfill \\ \hfill {(-1)}^{j}\sum\limits _{r=0}^{j}{(2\pi i\bar{\tau })}^{r}\left(\genfrac{}{}{0.0pt}{}{j}{r}\right)\bar{\mathcal{E} \left[\begin{matrix}j-r\\ k\end{matrix};\tau \right]}& =-{(2\pi i)}^{1-k+j}{\int }_{\bar{\tau }}^{i\infty }\mathrm{d}{\bar{\tau }}_{1}\enspace {({\bar{\tau }}_{1}-\bar{\tau })}^{j}\bar{{G}_{k}({\tau }_{1})},\hfill \end{align} \tag{ 4.33 }$$

respectively. Note that (4.30) fixes the SV map of all the $\mathcal{E} \left[\begin{matrix}j\\ k\end{matrix};\tau \right]$ at depth one with k ⩾ 4 and 0 ⩽ j ⩽ k − 2 since the caveats related to relations in the derivation algebra beyond (4.3) only affect iterated Eisenstein integrals of depth ℓ ⩾ 2.

The iterated Eisenstein integrals defined in (4.1) may be reorganized in terms of [17]

$$\begin{equation}{\mathcal{E}}_{0}({k}_{1},{k}_{2},\dots ,{k}_{\ell };\tau )=2\pi i{\int }_{\tau }^{i\infty }\mathrm{d}{\tau }_{\ell }\enspace \frac{{G}_{{k}_{\ell }}^{0}({\tau }_{\ell })}{{(2\pi i)}^{{k}_{\ell }}}\enspace {\mathcal{E}}_{0}({k}_{1},{k}_{2},\dots ,{k}_{\ell -1};\tau )\end{equation} \tag{ 4.34 }$$

with ${k}_{i}\in 2{\mathbb{N}}_{0}$ and ${\mathcal{E}}_{0}(;\tau )=1$. By subtracting the zero mode of the holomorphic Eisenstein series

$$\begin{align}\hfill {G}_{k}^{0}(\tau )& ={G}_{k}(\tau )-2{\zeta }_{k}\quad \text{for}\enspace k > 0\enspace \text{even},\hfill \\ \hfill {G}_{0}^{0}(\tau )& ={G}_{0}(\tau )=-1,\hfill \end{align} \tag{ 4.35 }$$

the integrals (4.34) are made to converge if k₁ > 0, and all other cases are shuffle-regularized based on ${\mathcal{E}}_{0}(0;\tau )=2\pi i\tau$.

At depth one, they are related to the holomorphic iterated Eisenstein integrals via [51]

$$\begin{equation}\mathcal{E} \left[\begin{matrix}j\\ k\end{matrix};\tau \right]=j!\enspace {\mathcal{E}}_{0}({0}^{j},k\enspace ;\tau )+\frac{{B}_{k}{(2\pi i\tau )}^{j+1}}{k!(j+1)}\end{equation} \tag{ 4.36 }$$

with Bernoulli numbers B_k . From this, we see that the implicit action of SV on these functions at depth one is given by

$$\begin{equation}\mathrm{S}\mathrm{V}\enspace {\mathcal{E}}_{0}({0}^{j},k\enspace ;\tau )=\frac{1}{j!}{\mathcal{E}}^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}j\\ k\end{matrix};\tau \right]-\frac{{B}_{k}{(-4y)}^{j+1}}{k!(j+1)!}.\end{equation} \tag{ 4.37 }$$

By (4.32), (4.36) and the shuffle relation

$$\begin{equation}{\mathcal{E}}_{0}({0}^{j},k;\tau )=\sum\limits _{r=0}^{j}\frac{{(-1)}^{j-r}}{r!}{(2\pi i\tau )}^{r}{\mathcal{E}}_{0}(k,{0}^{j-r};\tau ),\end{equation} \tag{ 4.38 }$$

two equivalent formulations of (4.37) are

$$\begin{align}\hfill \mathrm{S}\mathrm{V}\enspace {\mathcal{E}}_{0}({0}^{j},k\enspace ;\tau )& ={\mathcal{E}}_{0}({0}^{j},k\enspace ;\tau )+\sum\limits _{r=1}^{j}\bar{{\mathcal{E}}_{0}({0}^{r}\enspace ;\tau )}{\mathcal{E}}_{0}({0}^{j-r},k\enspace ;\tau )+\bar{{\mathcal{E}}_{0}(k,{0}^{j}\enspace ;\tau )}\hfill \\ \hfill \mathrm{S}\mathrm{V}\enspace {\mathcal{E}}_{0}(k,{0}^{j}\enspace ;\tau )& ={\mathcal{E}}_{0}(k,{0}^{j}\enspace ;\tau )+\sum\limits _{r=1}^{j}\bar{{\mathcal{E}}_{0}({0}^{j-r},k\enspace ;\tau )}{\mathcal{E}}_{0}({0}^{r}\enspace ;\tau )+\bar{{\mathcal{E}}_{0}({0}^{j},k\enspace ;\tau )},\hfill \end{align} \tag{ 4.39 }$$

which match the expectation from [103].

At two points, the general definitions (3.1) and (3.13) only admit a single permutation of the integrands and cycle in ${B}_{{\eta }_{2}}^{\tau }={B}_{{\eta }_{2}}^{\tau }(2\vert 2)$ and ${J}_{{\eta }_{2}}^{\tau }={J}_{{\eta }_{2}}^{\tau }(2\vert 2)$,

$$\begin{align}\hfill {B}_{{\eta }_{2}}^{\tau }& ={\int }_{-\tau /2}^{\tau /2}\frac{\mathrm{d}{z}_{2}}{\tau {\eta }_{2}}\sum\limits _{a=0}^{\infty }{\tau }^{a}{\eta }_{2}^{a}{f }_{12}^{\enspace (a)}\enspace {\mathrm{e}}^{{s}_{12}{\mathcal{G}}_{\mathfrak{B}}({z}_{12},\tau )}\hfill \\ \hfill {J}_{{\eta }_{2}}^{\tau }& ={\int }_{\mathfrak{T}}\frac{{\mathrm{d}}^{2}{z}_{2}}{\mathrm{I}\mathrm{m}\enspace \tau }\enspace \frac{1}{{\eta }_{2}}\sum\limits _{a=0}^{\infty }{(\tau -\bar{\tau })}^{a}{\eta }_{2}^{a}{f }_{12}^{\enspace (a)}\enspace {\mathrm{e}}^{{s}_{12}{\mathcal{G}}_{\mathfrak{T}}({z}_{12},\tau )},\hfill \end{align} \tag{ 5.3 }$$

and we introduce the following notation for component integrals

$$\begin{align}\hfill {B}_{(a)}^{\tau }& ={B}_{{\eta }_{2}}^{\tau }\enspace {\left\vert \right.}_{{\eta }_{2}^{a-1}}={\tau }^{a}{\int }_{-\tau /2}^{\tau /2}\frac{\mathrm{d}{z}_{2}}{\tau }{f }_{12}^{\enspace (a)}\enspace {\mathrm{e}}^{{s}_{12}{\mathcal{G}}_{\mathfrak{B}}({z}_{12},\tau )}\hfill \\ \hfill {J}_{(a)}^{\tau }& ={J}_{{\eta }_{2}}^{\tau }\enspace {\left\vert \right.}_{{\eta }_{2}^{a-1}}={(\tau -\bar{\tau })}^{a}{\int }_{\mathfrak{T}}\frac{{\mathrm{d}}^{2}{z}_{2}}{\mathrm{I}\mathrm{m}\enspace \tau }\enspace {f }_{12}^{\enspace (a)}\enspace {\mathrm{e}}^{{s}_{12}{\mathcal{G}}_{\mathfrak{T}}({z}_{12},\tau )}.\hfill \end{align} \tag{ 5.4 }$$

Then, combining the initial values (4.25) and (4.26) with the α'-expansions (4.9) and (4.19) yields expressions like

$$\begin{align}\hfill {B}_{(0)}^{\tau }& =1+{s}_{12}^{2}\left(-3\beta \left[\begin{matrix}1\\ 4\end{matrix};\tau \right]+\frac{{\zeta }_{2}}{6}+\frac{i{\zeta }_{3}}{2T}+\frac{3{\zeta }_{4}}{4{T}^{2}}\right)\hfill \\ \hfill & \quad +{s}_{12}^{3}\left(-5\beta \left[\begin{matrix}2\\ 6\end{matrix};\tau \right]+12{\zeta }_{2}\beta \left[\begin{matrix}0\\ 4\end{matrix};\tau \right]+\frac{{\zeta }_{3}}{12}+\frac{19i{\zeta }_{4}}{24T}-\frac{{\zeta }_{5}}{4{T}^{2}}+\frac{{\zeta }_{2}{\zeta }_{3}}{{T}^{2}}-\frac{4i{\zeta }_{6}}{3{T}^{3}}\right)\hfill \\ \hfill & \quad +\mathcal{O}({s}_{12}^{4})\hfill \\ \hfill {B}_{(2)}^{\tau }& =-2{\zeta }_{2}+{s}_{12}\left(3\beta \left[\begin{matrix}2\\ 4\end{matrix};\tau \right]-{\zeta }_{3}+\frac{3i{\zeta }_{4}}{T}\right)\hfill \\ \hfill & \quad +{s}_{12}^{2}\left(10\beta \left[\begin{matrix}3\\ 6\end{matrix};\tau \right]-18{\zeta }_{2}\beta \left[\begin{matrix}1\\ 4\end{matrix};\tau \right]-\frac{29{\zeta }_{4}}{12}-\frac{i{\zeta }_{5}}{T}+\frac{3i{\zeta }_{2}{\zeta }_{3}}{T}+\frac{43{\zeta }_{6}}{8{T}^{2}}\right)\hfill \\ \hfill & \quad +\mathcal{O}({s}_{12}^{3})\hfill \\ \hfill {B}_{(4)}^{\tau }& =-2{\zeta }_{4}+{s}_{12}\left(5\beta \left[\begin{matrix}4\\ 6\end{matrix};\tau \right]-6{\zeta }_{2}\beta \left[\begin{matrix}2\\ 4\end{matrix};\tau \right]+2{\zeta }_{2}{\zeta }_{3}-{\zeta }_{5}-\frac{11i{\zeta }_{6}}{2T}\right)+\mathcal{O}({s}_{12}^{2})\hfill \end{align} \tag{ 5.5 }$$

as well as

$$\begin{equation}\begin{aligned}\hfill {J}_{(0)}^{\tau }& =1+{s}_{12}^{2}\left(-3{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}1\\ 4\end{matrix};\tau \right]+\frac{{\zeta }_{3}}{2y}\right)+{s}_{12}^{3}\left(-5{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}2\\ 6\end{matrix};\tau \right]+\frac{{\zeta }_{3}}{6}+\frac{{\zeta }_{5}}{8{y}^{2}}\right)+\mathcal{O}({s}_{12}^{4})\hfill \\ \hfill {J}_{(2)}^{\tau }& ={s}_{12}(3{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}2\\ 4\end{matrix};\tau \right]-2{\zeta }_{3})+{s}_{12}^{2}\left(10{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}3\\ 6\end{matrix};\tau \right]-\frac{{\zeta }_{5}}{y}\right)+\mathcal{O}({s}_{12}^{3})\hfill \\ \hfill {J}_{(4)}^{\tau }& ={s}_{12}(5{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}4\\ 6\end{matrix};\tau \right]-2{\zeta }_{5})+\mathcal{O}({s}_{12}^{2})\hfill \end{aligned}\end{equation} \tag{ 5.6 }$$

upon extracting suitable powers of η₂. The action of SV on the ζ_n , T, β[...] as in (3.33) and (4.30) relates ${J}_{(a)}^{\tau }=\mathrm{S}\mathrm{V}\enspace {B}_{(a)}^{\tau }$ as expected from (3.35). Examples of β[...] beyond depth one occur at the next orders in s_ij , e.g.

$$\begin{align}\hfill {B}_{(0)}^{\tau }\enspace {\left\vert \right.}_{{s}_{12}^{4}}& =-21\beta \left[\begin{matrix}3\\ 8\end{matrix};\tau \right]+9\beta \left[\begin{matrix}1& 1\\ 4& 4\end{matrix};\tau \right]-18\beta \left[\begin{matrix}2& 0\\ 4& 4\end{matrix};\tau \right]-\frac{{\zeta }_{2}}{2}\beta \left[\begin{matrix}1\\ 4\end{matrix};\tau \right]+40{\zeta }_{2}\beta \left[\begin{matrix}1\\ 6\end{matrix};\tau \right]\hfill \\ \hfill & \quad +6{\zeta }_{3}\beta \left[\begin{matrix}0\\ 4\end{matrix};\tau \right]-\frac{3i{\zeta }_{3}}{2T}\beta \left[\begin{matrix}1\\ 4\end{matrix};\tau \right]-\frac{18i{\zeta }_{4}}{T}\beta \left[\begin{matrix}0\\ 4\end{matrix};\tau \right]-\frac{9{\zeta }_{4}}{4{T}^{2}}\beta \left[\begin{matrix}1\\ 4\end{matrix};\tau \right]\hfill \\ \hfill & \quad +\frac{131{\zeta }_{4}}{720}+\frac{5i{\zeta }_{5}}{12T}+\frac{i{\zeta }_{2}{\zeta }_{3}}{12T}+\frac{23{\zeta }_{6}}{32{T}^{2}}+\frac{{\zeta }_{3}^{2}}{8{T}^{2}}-\frac{9i{\zeta }_{3}{\zeta }_{4}}{8{T}^{3}}\hfill \\ \hfill & \quad +\frac{i{\zeta }_{2}{\zeta }_{5}}{{T}^{3}}-\frac{3i{\zeta }_{7}}{8{T}^{3}}+\frac{85{\zeta }_{8}}{192{T}^{4}}\hfill \end{align} \tag{ 5.7 }$$

as well as

$$\begin{align}\hfill {J}_{(0)}^{\tau }\enspace {\left\vert \right.}_{{s}_{12}^{4}}& =-21{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}3\\ 8\end{matrix};\tau \right]+9{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}1& 1\\ 4& 4\end{matrix};\tau \right]-18{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}2& 0\\ 4& 4\end{matrix};\tau \right]+12{\zeta }_{3}{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}0\\ 4\end{matrix};\tau \right]\hfill \\ \hfill & \quad -\frac{3{\zeta }_{3}}{2y}{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}1\\ 4\end{matrix};\tau \right]+\frac{5{\zeta }_{5}}{12y}-\frac{{\zeta }_{3}^{2}}{8{y}^{2}}+\frac{3{\zeta }_{7}}{32{y}^{3}}\hfill \\ \hfill {J}_{(0)}^{\tau }\enspace {\left.\right\vert }_{{s}_{12}^{5}}& =-135{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}4\\ 10\end{matrix};\tau \right]-60{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}3& 0\\ 6& 4\end{matrix};\tau \right]+15{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}1& 2\\ 4& 6\end{matrix};\tau \right]+15{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}2& 1\\ 6& 4\end{matrix};\tau \right]\hfill \\ \hfill & \quad -60{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}2& 1\\ 4& 6\end{matrix};\tau \right]-\frac{1}{2}{\zeta }_{3}{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}1\\ 4\end{matrix};\tau \right]+\frac{6{\zeta }_{5}}{y}{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}0\\ 4\end{matrix};\tau \right]-\frac{3{\zeta }_{5}}{8{y}^{2}}{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}1\\ 4\end{matrix};\tau \right]\hfill \\ \hfill & \quad +40{\zeta }_{3}{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}1\\ 6\end{matrix};\tau \right]-\frac{5{\zeta }_{3}}{2y}{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}2\\ 6\end{matrix};\tau \right]+\frac{43{\zeta }_{5}}{360}+\frac{{\zeta }_{3}^{2}}{12y}\hfill \\ \hfill & \quad +\frac{7{\zeta }_{7}}{32{y}^{2}}-\frac{3{\zeta }_{3}{\zeta }_{5}}{16{y}^{3}}+\frac{15{\zeta }_{9}}{128{y}^{4}}.\hfill \end{align} \tag{ 5.8 }$$

The above α'-expansions at two points have been generated in earlier work in terms of eMZVs [51, 59] and MGFs [10, 11, 61], respectively. The results in the references include

$$\begin{align}\hfill {B}_{(0)}^{\tau }& =1+{s}_{12}^{2}\left(\frac{1}{2}\omega \left(0,0,2\vert -\frac{1}{\tau }\right)+\frac{5{\zeta }_{2}}{12}\right)\hfill \\ \hfill & \quad +{s}_{12}^{3}\left(\frac{1}{18}\omega \left(0,0,3,0\vert -\frac{1}{\tau }\right)-\frac{4}{3}{\zeta }_{2}\omega \left(0,0,1,0\vert -\frac{1}{\tau }\right)+\frac{{\zeta }_{3}}{12}\right)\hfill \\ \hfill & \quad +{s}_{12}^{4}\left(-\omega \left(0,0,0,2,2\vert -\frac{1}{\tau }\right)-\frac{5}{4}\omega \left(0,0,0,0,4\vert -\frac{1}{\tau }\right)+\frac{1}{8}\omega \left(0,0,4\vert -\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \left.\quad +\frac{5}{8}\omega {\left(0,0,2\vert -\frac{1}{\tau }\right)}^{2}+\frac{13}{24}{\zeta }_{2}\omega \left(0,0,2\vert -\frac{1}{\tau }\right)-2{\zeta }_{2}\omega \left(0,0,0,0,2\vert -\frac{1}{\tau }\right)+\frac{343{\zeta }_{4}}{576}\right)\hfill \\ \hfill & \quad +\mathcal{O}({s}_{12}^{5})\hfill \\ \hfill {J}_{(0)}^{\tau }& =1+\frac{1}{2}{s}_{12}^{2}{E}_{2}(\tau )+\frac{1}{6}{s}_{12}^{3}\left({E}_{3}(\tau )+{\zeta }_{3}\right)+{s}_{12}^{4}\left({E}_{2,2}(\tau )+\frac{3}{20}{E}_{4}(\tau )+\frac{1}{8}{E}_{2}^{2}(\tau )\right)\hfill \\ \hfill & \quad +\mathcal{O}({s}_{12}^{5}),\hfill \end{align} \tag{ 5.9 }$$

where for instance (by comparison with (5.5) and (5.6))

$$\begin{equation}\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)=-6\beta \left[\begin{matrix}1\\ 4\end{matrix};\tau \right]-\frac{{\zeta }_{2}}{2}+\frac{i{\zeta }_{3}}{T}+\frac{3{\zeta }_{4}}{2{T}^{2}},\qquad {E}_{2}(\tau )=-6{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}1\\ 4\end{matrix};\tau \right]+\frac{{\zeta }_{3}}{y},\end{equation} \tag{ 5.10 }$$

and the modular transformation may be evaluated to yield [51, 93]

$$\begin{equation}\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)=-\frac{{T}^{2}}{180}-\frac{{\zeta }_{2}}{6}+\frac{3{\zeta }_{4}}{2{T}^{2}}+\omega (0,0,2\vert \tau )+\frac{8i}{T}{\zeta }_{2}\omega (0,1,0,0\vert \tau ).\end{equation} \tag{ 5.11 }$$

For the ${s}_{12}^{2}$-order in (5.9), the component version ${J}_{(a)}^{\tau }=\mathrm{S}\mathrm{V}\enspace {B}_{(a)}^{\tau }$ of ${J}_{{\eta }_{2}}^{\tau }=\mathrm{S}\mathrm{V}\enspace {B}_{{\eta }_{2}}^{\tau }$ implies that

$$\begin{equation}\mathrm{S}\mathrm{V}\enspace \omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)={E}_{2}(\tau )\end{equation} \tag{ 5.12 }$$

and a similar analysis for higher orders in s₁₂ and at a ≠ 0 yields for instance

$$\begin{equation}\begin{aligned}\hfill \mathrm{S}\mathrm{V}\enspace \omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)& =2\pi \nabla {E}_{2}\left(\tau \right),\qquad \hfill & \hfill \mathrm{S}\mathrm{V}\enspace \omega \left(0,0,4\left\vert \right.-\frac{1}{\tau }\right)& =-\frac{4}{3}\pi \nabla {E}_{3}\left(\tau \right)\hfill \\ \hfill \mathrm{S}\mathrm{V}\enspace \omega \left(0,0,3,0\left\vert \right.-\frac{1}{\tau }\right)& =3{E}_{3}\left(\tau \right),\qquad \hfill & \hfill \mathrm{S}\mathrm{V}\enspace \omega \left(0,5\left\vert \right.-\frac{1}{\tau }\right)& =-\frac{4}{3}{\left(\pi \nabla \right)}^{2}{E}_{3}\left(\tau \right).\hfill \end{aligned}\end{equation} \tag{ 5.13 }$$

At depth two, relating ${B}_{(0)}^{\tau }{\vert }_{{s}_{12}^{4}}{\leftrightarrow}{J}_{(0)}^{\tau }{\vert }_{{s}_{12}^{4}}$ (see (5.7) and (5.8)) or ${B}_{(2)}^{\tau }{\vert }_{{s}_{12}^{3}}{\leftrightarrow}{J}_{(2)}^{\tau }{\vert }_{{s}_{12}^{3}}$ yields

$$\begin{align}\hfill {E}_{2,2}(\tau )& =\mathrm{S}\mathrm{V}\enspace \left(-\frac{7}{5}\omega \left(0,0,0,0,4\left\vert \right.-\frac{1}{\tau }\right)-\omega \left(0,0,0,2,2\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.+\frac{1}{2}\omega {\left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)}^{2}+\frac{3}{20}\omega \left(0,0,4\left\vert \right.-\frac{1}{\tau }\right)\right)\hfill \\ \hfill \pi \nabla {E}_{2,2}(\tau )& =\mathrm{S}\mathrm{V}\enspace \left(-\frac{1}{60}\omega \left(0,5\left\vert \right.-\frac{1}{\tau }\right)+\frac{3}{5}\omega \left(0,0,0,5\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.-\frac{1}{2}\omega \left(0,0,2,3\left\vert \right.-\frac{1}{\tau }\right)\right),\hfill \end{align} \tag{ 5.14 }$$

where the combinations

$$\begin{align}\hfill {E}_{2,2}& ={\left(\frac{\mathrm{I}\mathrm{m}\enspace \tau }{\pi }\right)}^{4}\left(\mathcal{C}\left[\begin{matrix}\hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \end{matrix}\right]-\frac{9}{10}\mathcal{C}\left[\begin{matrix}\hfill 4\hfill & \hfill 0\hfill \\ \hfill 4\hfill & \hfill 0\hfill \end{matrix}\right]\right)\hfill \\ \hfill & =-18{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}2& 0\\ 4& 4\end{matrix}\right]+12{\zeta }_{3}{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}0\\ 4\end{matrix}\right]+\frac{5{\zeta }_{5}}{12y}-\frac{{\zeta }_{3}^{2}}{4{y}^{2}}\hfill \\ \hfill \pi \nabla {E}_{2,2}& =\frac{{(\mathrm{I}\mathrm{m}\enspace \tau )}^{5}}{{\pi }^{3}}\left(\mathcal{C}\left[\begin{matrix}\hfill 3\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{matrix}\right]-\frac{8}{5}\mathcal{C}\left[\begin{matrix}\hfill 5\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{matrix}\right]\right)\hfill \\ \hfill & =9{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}2& 1\\ 4& 4\end{matrix}\right]-6{\zeta }_{3}{\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}1\\ 4\end{matrix}\right]-\frac{5{\zeta }_{5}}{12}+\frac{{\zeta }_{3}^{2}}{2y}\hfill \end{align} \tag{ 5.15 }$$

of MGFs (2.26) are engineered to avoid G₈ in the differential equations [51]. The systematics of depth-one relations between eMZVs and non-holomorphic Eisenstein series including higher-weight generalizations of (5.13) is detailed in appendix C.1 and leads to the closed-form results

$$\begin{align}\hfill & \mathrm{S}\mathrm{V}\enspace \sum\limits _{j=0}^{\ell -1}\frac{{B}_{j}}{j!}\omega \left({0}^{\ell -j},2k+\ell \left\vert \right.-\frac{1}{\tau }\right)\hfill \\ \hfill & \qquad ={(-1)}^{\ell }\frac{(k+\ell -1)!}{(2k+\ell -1)!}{(-4\pi \nabla )}^{k}{E}_{k+\ell },\qquad k\geqslant 0,\enspace \ell \geqslant 1,\enspace k+\ell \geqslant 2\hfill \\ \hfill & \mathrm{S}\mathrm{V}\enspace \sum\limits _{j=0}^{\ell -1}\frac{{B}_{j}}{j!}\omega \left({0}^{\ell -j},\ell -2k\left\vert \right.-\frac{1}{\tau }\right)\hfill \\ \hfill & \qquad ={(-1)}^{\ell +k}\frac{(\ell -k-1)!}{(\ell -1)!}\frac{{(\pi \bar{\nabla })}^{k}{E}_{\ell -k}}{{(2y)}^{2k}},\quad k\geqslant 0,\enspace \ell -2k\geqslant 1,\enspace \ell -k\geqslant 2\hfill \end{align} \tag{ 5.16 }$$

for combinations of eMZVs of different length that are weighted by Bernoulli numbers B_j [104]. Similarly, the analogue of (5.14) for the MGF ${E}_{2,3}={(\frac{\mathrm{I}\mathrm{m}\enspace \tau }{\pi })}^{5}\left(\mathcal{C}\left[\begin{matrix}3& 1& 1\\ 3& 1& 1\end{matrix}\right]-\frac{43}{35}\mathcal{C}\left[\begin{matrix}5& 0\\ 5& 0\end{matrix}\right]\right)$ and its holomorphic derivatives is spelled out in appendix C.2.

Note that the single-valued map of A-cycle eMZVs at argument τ rather than $-\frac{1}{\tau }$ generically leads to combinations of MGFs of different modular weights. For instance, changing the argument $-\frac{1}{\tau }$ to τ in (5.12) gives rise to

$$\begin{equation}\mathrm{S}\mathrm{V}\enspace \omega (0,0,2\vert \tau )=-\frac{{y}^{2}}{15}+{E}_{2}(\tau )+\frac{\pi \bar{\nabla }{E}_{2}(\tau )}{y}\end{equation} \tag{ 5.17 }$$

instead of a single modular invariant E₂(τ). This can be seen by expressing all of ω(0, 0, 2|τ), E₂(τ) and $\bar{\nabla }{E}_{2}(\tau )$ in terms of convergent iterated Eisenstein integrals (4.34) and applying their single-valued map (4.39). ²² Alternatively, (5.17) can be deduced by setting $\tau \to -\frac{1}{\tau }$ in (5.11) and exploiting the result $\mathrm{S}\mathrm{V}\enspace \omega (0,1,0,0\vert -\frac{1}{\tau })=\frac{3\pi \bar{\nabla }{E}_{2}(\tau )}{8{y}^{2}}$ that will be extracted from a four-point example in section 5.5. The much cleaner result (5.12) for $\mathrm{S}\mathrm{V}\enspace \omega (0,0,2\vert -\frac{1}{\tau })$ as compared to SV  ω(0, 0, 2|τ) is another manifestation of the fact that the differential equation (3.29) of B-cycle integrals are more closely related to the closed-string counterparts (3.30) than the A-cycle differential equations in (2.19).

At n ⩾ 3 points, most of the antielliptic functions in (3.9) introduce non-constant $\bar{{f}_{ij}^{(a)}}$ into the closed-string integrands, except for the simplest case V₀(1, 2, ..., n|τ) = 1 dual to a permutation sum over B-cycles, see (3.38),

$$\begin{align}\hfill {J}_{0,\overrightarrow{\eta }}^{\tau }(\ast \vert \rho )& ={\int }_{{\mathfrak{T}}^{n-1}}\left(\prod\limits _{j=2}^{n}\frac{{\mathrm{d}}^{2}{z}_{j}}{\mathrm{I}\mathrm{m}\enspace \tau }\right)\prod\limits _{1\leqslant i< j}^{n}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{T}}({z}_{ij},\tau )}\hfill \\ \hfill & \quad \times \enspace \rho \left\{\prod\limits _{k=2}^{n}\sum\limits _{{a}_{k}=0}^{\infty }{(\tau -\bar{\tau })}^{{a}_{k}}{({\eta }_{k,k+1\dots n})}^{{a}_{k}-1}{f}_{k-1,k}^{({a}_{k})}\right\}\hfill \\ \hfill & =\mathrm{S}\mathrm{V}\sum\limits _{\gamma \in {S}_{n-1}}{B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho ).\hfill \end{align} \tag{ 5.18 }$$

As indicated by the *-notation, the ${J}_{0,\overrightarrow{\eta }}^{\tau }(\ast \vert \rho )$ integral (3.36) on the left-hand side is independent of the ordering * since its integrand V₀ is. The symmetrized open-string integrals on the right-hand side were studied in [16, 51, 77] as the generating series of holomorphic graph functions,

$$\begin{align}\hfill \sum\limits _{\gamma \in {S}_{n-1}}{B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )& =\left(\underset{-\tau /2{}}{\overset{\tau /2}{\int }}\prod\limits _{j=2}^{n}\frac{\mathrm{d}{z}_{j}}{\tau }\right)\prod\limits _{1\leqslant i< j}^{n}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{B}}({z}_{ij},\tau )}\hfill \\ \hfill & \quad \times \enspace \rho \left\{\prod\limits _{k=2}^{n}\sum\limits _{{a}_{k}=0}^{\infty }{\tau }^{{a}_{k}}{({\eta }_{k,k+1\dots n})}^{{a}_{k}-1}{f}_{k-1,k}^{({a}_{k})}\right\},\hfill \end{align} \tag{ 5.19 }$$

where each puncture is integrated independently over the entire B-cycle. More specifically, the references considered the components ${f}_{k-1,k}^{({a}_{k})}\to {f}_{k-1,k}^{(0)}=1$ at the most singular order in the η_j ,

where the dependence on the permutation ρ drops out, and the integrands at fixed order in s_ij are polynomials in B-cycle Green functions. In passing to the second line, each monomial in ${\mathcal{G}}_{\mathfrak{B}}({z}_{ij},\tau )$ is mapped to a graph Γ that labels the B-cycle graph functions B[Γ], where a factor of ${\mathcal{G}}_{\mathfrak{B}}({z}_{ij},\tau )$ is represented by an edge connecting vertices z_i and z_j . One-particle reducible graphs Γ_1PR lead to vanishing B[Γ_1PR] since ${\int }_{-\tau /2}^{\tau /2}\mathrm{d}z\enspace {\mathcal{G}}_{\mathfrak{B}}(z,\tau )=0$, i.e. higher orders of (5.20) stem from all combinations of one-particle irreducible graphs with four and more edges in total. Any B[Γ] is expressible in terms of B-cycle eMZVs [51] since the α'-expansion of each component integral of the series ${B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$ is.

Similarly, modular graph functions D[Γ] (as opposed to MGFs) were defined [10, 11] by n-point torus integrals over monomials in ${\mathcal{G}}_{\mathfrak{T}}({z}_{ij},\tau )$, where each torus Green function is again visualized through an edge between vertices z_i and z_j . The D[Γ] associated with dihedral graphs Γ are proportional to the lattice sums (2.26) with a_j = b_j , and also more complicated graph topologies can be straightforwardly translated into lattice sums.

The generating series of n-point modular graph functions resides at the most singular order of (5.18) w.r.t. η_j where the insertions of ${f}_{ij}^{(a)}$ are absent,

and we have D[Γ_1PR] = 0 by ${\int }_{\mathfrak{T}}{\mathrm{d}}^{2}z\enspace {\mathcal{G}}_{\mathfrak{T}}(z,\tau )=0$. As a consequence of (5.18) at the most singular order in the η_j , modular graph functions are single-valued B-cycle graph functions,

$$\begin{equation}{M}_{n}^{\text{closed}}=\mathrm{S}\mathrm{V}\enspace {M}_{n}^{\text{open}}{\Leftrightarrow}\mathbf{D}[{\Gamma}]=\mathrm{S}\mathrm{V}\enspace \mathbf{B}[{\Gamma}],\end{equation} \tag{ 5.22 }$$

which ultimately follows from the 'Betti–deRham duality' between V₀ = 1 and the symmetrized cycles ${\sum }_{\gamma \in {S}_{n-1}}\mathfrak{B}(\gamma (2,\dots ,n))$.

The relations in (5.22) have firstly appeared in [51] with a proposal 'esv' for an elliptic single-valued map in the place of SV. The esv map of [51] has the same action (3.33) on MZVs and Laurent polynomials in τ as the SV map in this work. In particular, all pairs of B-cycle eMZVs and modular graph functions related via $\mathrm{e}\mathrm{s}\mathrm{v}\enspace \omega (\dots \vert -\frac{1}{\tau })\sim \mathbf{D}[\dots ]$ in the reference are also related via $\mathrm{S}\mathrm{V}\enspace \omega (\dots \vert -\frac{1}{\tau })\sim \mathbf{D}[\dots ]$ as a consequence of (5.22). For suitable representations of the q-series of eMZVs via ${\mathcal{E}}_{0}$ defined by (4.34), the Fourier expansions of all modular graph functions up to weight six could be reproduced from the replacement ${\mathcal{E}}_{0}\to 2\enspace \mathrm{R}\mathrm{e}({\mathcal{E}}_{0})$ prescribed by esv [51]. However, it was an open problem in the reference to reconcile esv with the shuffle property of iterated Eisenstein integrals. The SV action (4.30) in turn is expected to be compatible with the shuffle multiplication of the β[...] and β^sv[...] by the discussion in section 4.3 as detailed below (4.21).

Note that subleading orders in the η_j -expansion of (5.18) generate infinite families of additional relation between MGFs and single-valued eMZVs beyond (5.22). The comparison of open- and closed-string integrals with additional insertions of ${f }_{12}^{\enspace ({a}_{2})}{f }_{23}^{\enspace ({a}_{3})}\dots {f }_{n-1,n}^{\enspace ({a}_{n})}$ identifies MGFs of various modular weights as single-valued B-cycle eMZVs.

The simplest instance of ${J}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )=\mathrm{S}\mathrm{V}\enspace {B}_{\overrightarrow{\eta }}^{\tau }(\gamma \vert \rho )$ with non-constant antielliptic integrands $\bar{V(\dots \vert \tau )}$ occurs at three points. The single-valued map relates an antisymmetric integration cycle on the open-string side in (3.46) to the closed-string integral

$$\begin{align}\hfill {J}_{1,{\eta }_{2},{\eta }_{3}}^{\tau }\left(2,3\vert 2,3\right)& =-\frac{1}{2\pi i}{\int }_{{\mathfrak{T}}^{2}}\frac{{\mathrm{d}}^{2}{z}_{2}\enspace {\mathrm{d}}^{2}{z}_{3}}{{\left(\mathrm{I}\mathrm{m}\enspace \tau \right)}^{2}}\prod\limits _{1\leqslant i< j}^{3}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{T}}\left({z}_{ij},\tau \right)}\hfill \\ \hfill & \quad \times \sum\limits _{a,b=0}^{\infty }{\left(\tau -\bar{\tau }\right)}^{a+b}{\eta }_{23}^{a-1}{\eta }_{3}^{b-1}{f }_{12}^{\enspace \left(a\right)}{f }_{23}^{\left(b\right)}\left(\bar{{f }_{12}^{\enspace \left(1\right)}}+\bar{{f }_{23}^{\left(1\right)}}+\bar{{f }_{31}^{\left(1\right)}}\right).\hfill \end{align} \tag{ 5.23 }$$

Since contributions with even a + b integrate to zero, the simplest component integrals involve permutations of ${f }_{12}^{\enspace (1)}\bar{{f }_{12}^{\enspace (1)}}$ or ${f }_{23}^{\enspace (1)}\bar{{f }_{12}^{\enspace (1)}}$ at the orders of ${\eta }_{3}^{-1}$ or ${\eta }_{23}^{-1}$,

$$\begin{align}\hfill {J}_{1,{\eta }_{2},{\eta }_{3}}^{\tau }\left(2,3\vert 2,3\right)\enspace {\vert }_{{\eta }_{23}^{0}{\eta }_{3}^{-1}}& =-\frac{\mathrm{I}\mathrm{m}\enspace \tau }{\pi }{\int }_{{\mathfrak{T}}^{2}}\frac{{\mathrm{d}}^{2}{z}_{2}\enspace {\mathrm{d}}^{2}{z}_{3}}{{\left(\mathrm{I}\mathrm{m}\enspace \tau \right)}^{2}}\prod\limits _{1\leqslant i< j}^{3}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{T}}\left({z}_{ij},\tau \right)}\hfill \\ \hfill & \quad \times \enspace {f }_{12}^{\enspace \left(1\right)}\left(\bar{{f }_{12}^{\enspace \left(1\right)}}+\bar{{f }_{23}^{\left(1\right)}}+\bar{{f }_{31}^{\left(1\right)}}\right)\hfill \\ \hfill & =\frac{1}{{s}_{12}}+\frac{{s}_{123}^{2}}{2{s}_{12}}{E}_{2}+\frac{{s}_{123}^{3}}{6{s}_{12}}\left({E}_{3}+{\zeta }_{3}\right)+\frac{{s}_{13}{s}_{23}}{2}\left(3{E}_{3}+{\zeta }_{3}\right)\hfill \\ \hfill & \quad +\frac{{s}_{123}^{4}}{{s}_{12}}\left({E}_{2,2}+\frac{3}{20}{E}_{4}+\frac{1}{8}{E}_{2}^{2}\right)\hfill \\ \hfill & \quad +\enspace {s}_{13}{s}_{23}{s}_{123}\left(\frac{9}{2}{E}_{2,2}+\frac{21}{20}{E}_{4}\right)+\mathcal{O}\left({s}_{ij}^{4}\right),\hfill \end{align} \tag{ 5.24 }$$

which furnish the simplest examples of kinematic poles $\sim {s}_{ij}^{-1}$ in a ${J}_{\overrightarrow{\eta }}^{\tau }$-series. The corresponding antisymmetrized B-cycle integral features the same types of kinematic poles in component integrals involving ${f}_{ij}^{(1)}$, e.g. ²³

$$\begin{align}\hfill & \frac{1}{2}\left[{B}_{{\eta }_{2},{\eta }_{3}}^{\tau }(2,3\vert 2,3)-{B}_{{\eta }_{2},{\eta }_{3}}^{\tau }(3,2\vert 2,3)\right]\enspace {\vert }_{{\eta }_{23}^{0}{\eta }_{3}^{-1}}\hfill \\ \hfill & \qquad =\frac{1}{\tau }{\int }_{-\frac{\tau }{2}< {z}_{2}< {z}_{3}< \frac{\tau }{2}}\mathrm{d}{z}_{2}\enspace \mathrm{d}{z}_{3}\enspace {f }_{12}^{\enspace (1)}\prod\limits _{1\leqslant i< j}^{3}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{B}}({z}_{ij},\tau )}\hfill \\ \hfill & \qquad =\frac{1}{{s}_{12}}+\frac{{s}_{123}^{2}}{{s}_{12}}\left(\frac{1}{2}\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)+\frac{5{\zeta }_{2}}{12}\right)\hfill \\ \hfill & \qquad \quad +\frac{{s}_{123}^{3}}{{s}_{12}}\left(\frac{1}{18}\omega \left(0,0,3,0\left\vert \right.-\frac{1}{\tau }\right)-\frac{4}{3}{\zeta }_{2}\omega \left(0,0,1,0\left\vert \right.-\frac{1}{\tau }\right)+\frac{{\zeta }_{3}}{12}\right)\hfill \\ \hfill & \qquad \quad +{s}_{13}{s}_{23}\left(\frac{1}{2}\omega \left(0,0,3,0\left\vert \right.-\frac{1}{\tau }\right)+\frac{{\zeta }_{3}}{4}\right)\hfill \\ \hfill & \qquad \quad +\frac{{s}_{123}^{4}}{{s}_{12}}\left(-\omega \left(0,0,0,2,2\left\vert \right.-\frac{1}{\tau }\right)-\frac{5}{4}\omega \left(0,0,0,0,4\left\vert \right.-\frac{1}{\tau }\right)+\frac{1}{8}\omega \left(0,0,4\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \left.\quad \qquad +\frac{5}{8}\omega {\left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)}^{2}+\frac{13}{24}{\zeta }_{2}\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)-2{\zeta }_{2}\omega \left(0,0,0,0,2\left\vert \right.-\frac{1}{\tau }\right)+\frac{343{\zeta }_{4}}{576}\right)\hfill \\ \hfill & \quad \qquad -{s}_{13}{s}_{23}{s}_{123}\left(\frac{9}{2}\omega \left(0,0,0,2,2\left\vert \right.-\frac{1}{\tau }\right)+\frac{21}{4}\omega \left(0,0,0,0,4\left\vert \right.-\frac{1}{\tau }\right)-\frac{1}{2}\omega \left(0,0,4\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \left.\quad \qquad -\frac{9}{4}\omega {\left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)}^{2}+\frac{1}{2}{\zeta }_{2}\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)-3{\zeta }_{2}\omega \left(0,0,0,0,2\left\vert \right.-\frac{1}{\tau }\right)+\frac{11{\zeta }_{4}}{40}\right)\hfill \\ \hfill & \qquad \quad +\mathcal{O}({s}_{ij}^{4}).\hfill \end{align} \tag{ 5.25 }$$

We have used that, by the antisymmetry ${f }_{12}^{\enspace (1)}=-{f }_{21}^{(1)}$ of the integrand, the contribution from the ordering $-\frac{\tau }{2}< {z}_{3}< {z}_{2}< \frac{\tau }{2}$ is minus that of the ordering $-\frac{\tau }{2}< {z}_{2}< {z}_{3}< \frac{\tau }{2}$. Comparison of (5.25) with (5.24) confirms the relation (3.46) under the SV map at the respective orders in s_ij and η_j . Up to the restriction of the Koba–Nielsen factor to three instead of five punctures, (5.24) and (5.25) are the type of integrals over ${f}_{ij}^{(1)}\bar{{f}_{pq}^{(1)}}$ seen in genus-one five-point amplitudes of type II superstrings [105, 106].

The esv map [51] has also been applied to the four-gluon amplitude of the heterotic string [52], where the torus integral ²⁴

$$\begin{align}\hfill {J}_{\text{het}}^{\tau }& =\frac{1}{{(2\pi i)}^{2}}{\int }_{{\mathfrak{T}}^{3}}\left(\prod\limits _{j=2}^{4}\frac{{\mathrm{d}}^{2}{z}_{j}}{\mathrm{I}\mathrm{m}\enspace \tau }\right)\bar{{V}_{2}(1,2,3,4\vert \tau )}\prod\limits _{1\leqslant i< j}^{4}{\mathrm{e}}^{{s}_{ij}{\mathcal{G}}_{\mathfrak{T}}({z}_{ij},\tau )}\hfill \\ \hfill & ={J}_{2,{\eta }_{2},{\eta }_{3},{\eta }_{4}}^{\tau }(2,3,4\vert 2,3,4)\enspace {\vert }_{{\eta }_{234}^{-1}{\eta }_{34}^{-1}{\eta }_{4}^{-1}}\hfill \end{align} \tag{ 5.26 }$$

was related to the open-string integration cycle dual to (3.39). More specifically, the MGFs in [52]

$$\begin{align}\hfill {J}_{\text{het}}^{\tau }\enspace {\vert }_{{k}_{j}^{2}=0}& =-\frac{3{s}_{13}\pi \bar{\nabla }{E}_{2}}{4{y}^{2}}-({s}_{13}^{2}+2{s}_{12}{s}_{23})\frac{\pi \bar{\nabla }{E}_{3}}{6{y}^{2}}+{s}_{13}({s}_{12}{s}_{23}-{s}_{13}^{2})\hfill \\ \hfill & \quad \times \enspace \left(\frac{\pi \bar{\nabla }{E}_{4}}{5{y}^{2}}+\frac{3{E}_{2}\bar{\nabla }{E}_{2}}{2{y}^{2}}+\frac{3\bar{\nabla }{E}_{2,2}}{{y}^{2}}\right)+\mathcal{O}({s}_{ij}^{4})\hfill \end{align} \tag{ 5.27 }$$

were proposed to be the single-valued versions of the eMZVs in the α'-expansion of

$$\begin{align}\hfill {B}_{\text{het}}^{\tau }& =\frac{1}{6}\left[2{B}_{234}^{\tau }+2{B}_{432}^{\tau }-{B}_{243}^{\tau }-{B}_{342}^{\tau }-{B}_{324}^{\tau }-{B}_{423}^{\tau }\right]\hfill \\ \hfill {B}_{ijk}^{\tau }& ={B}_{{\eta }_{2},{\eta }_{3},{\eta }_{4}}^{\tau }(i,j,k\vert 2,3,4)\enspace {\vert }_{{\eta }_{234}^{-1}{\eta }_{34}^{-1}{\eta }_{4}^{-1}},\hfill \end{align} \tag{ 5.28 }$$

namely [4]

$$\begin{align}\hfill {B}_{\text{het}}^{\tau }\enspace {\vert }_{{k}_{j}^{2}=0}& =-2{s}_{13}\omega \left(0,1,0,0\left\vert \right.-\frac{1}{\tau }\right)-\frac{2}{3}\left({s}_{13}^{2}+2{s}_{12}{s}_{23}\right)\hfill \\ \hfill & \quad \times \enspace \left[\omega \left(0,1,0,1,0\left\vert \right.-\frac{1}{\tau }\right)+\omega \left(0,1,1,0,0\left\vert \right.-\frac{1}{\tau }\right)\right]+\frac{4}{3}{s}_{13}\left({s}_{13}^{2}-{s}_{12}{s}_{23}\right)\hfill \\ \hfill & \quad \times \enspace \left[\omega \left(0,0,1,0,0,2\left\vert \right.-\frac{1}{\tau }\right)+\omega \left(0,0,0,1,0,2\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \left.\quad -\omega \left(0,1,0,1,1,0\left\vert \right.-\frac{1}{\tau }\right)-{\zeta }_{2}\omega \left(0,1,0,0\left\vert \right.-\frac{1}{\tau }\right)\right]+\mathcal{O}\left({s}_{ij}^{4}\right).\hfill \end{align} \tag{ 5.29 }$$

As indicated by ${\vert }_{{k}_{j}^{2}=0}$, the α'-expansions (5.27) and (5.29) have been obtained in the limit of four-point on-shell kinematics with two independent Mandelstam invariants instead of six. However, the relation (3.35) between n-point closed-string and single-valued open-string integrals is conjectured to be valid for the $\frac{1}{2}n(n-1)$ independent Mandelstam variables {s_ij , 1 ⩽ i < j ⩽ n} with s_ij = s_ji . At four points, the corollary

$$\begin{align}\hfill {J}_{2,{\eta }_{2},{\eta }_{3},{\eta }_{4}}^{\tau }(2,3,4\vert \rho )& =\frac{1}{6}\mathrm{S}\mathrm{V}\enspace \left[2{B}_{{\eta }_{2},{\eta }_{3},{\eta }_{4}}^{\tau }(2,3,4\vert \rho )+2{B}_{{\eta }_{2},{\eta }_{3},{\eta }_{4}}^{\tau }(4,3,2\vert \rho )\right.\hfill \\ \hfill & \quad \left.-{B}_{{\eta }_{2},{\eta }_{3},{\eta }_{4}}^{\tau }(2,4,3\vert \rho )-{B}_{{\eta }_{2},{\eta }_{3},{\eta }_{4}}^{\tau }(3,4,2\vert \rho )\right.\hfill \\ \hfill & \quad \left.-{B}_{{\eta }_{2},{\eta }_{3},{\eta }_{4}}^{\tau }(3,2,4\vert \rho )-{B}_{{\eta }_{2},{\eta }_{3},{\eta }_{4}}^{\tau }(4,2,3\vert \rho )\right]\hfill \end{align} \tag{ 5.30 }$$

of the relation (3.41) between V₂(1, 2, 3, 4|τ) and permutations of the V(1, 2, 3, 4|τ) functions is claimed to hold for all of {s₁₂, s₁₃, s₂₃, s₁₄, s₂₄, s₃₄} independent. The coefficient of ${\eta }_{234}^{-1}{\eta }_{34}^{-1}{\eta }_{4}^{-1}$ in (5.30) with ρ = 2, 3, 4 then implies

$$\begin{equation}{J}_{\text{het}}^{\tau }=\mathrm{S}\mathrm{V}\enspace {B}_{\text{het}}^{\tau }\end{equation} \tag{ 5.31 }$$

and explains the relations between the α'-expansions (5.27) and (5.29) observed in [52] in the on-shell limit ${k}_{j}^{2}=0$. In particular, the prescription (4.30) for the single-valued map of the iterated-Eisenstein-integral representation of ${B}_{\text{het}}^{\tau }$ produces the complete $q,\bar{q}$-expansion of the MGFs in (5.27), whereas certain antiholomorphic contributions could not be reproduced by esv in [52].

By applying (5.31) at the level of the α'-expansions (5.27) and (5.29), one can infer

$$\begin{align}\hfill \frac{\pi \bar{\nabla }{E}_{2}}{{y}^{2}}& =\frac{8}{3}\enspace \mathrm{S}\mathrm{V}\enspace \omega \left(0,1,0,0\left\vert \right.-\frac{1}{\tau }\right)\hfill \\ \hfill \frac{\pi \bar{\nabla }{E}_{3}}{{y}^{2}}& =4\enspace \mathrm{S}\mathrm{V}\enspace \left[\omega \left(0,1,0,1,0\left\vert \right.-\frac{1}{\tau }\right)+\omega \left(0,1,1,0,0\left\vert \right.-\frac{1}{\tau }\right)\right]\hfill \\ \hfill \frac{\pi \bar{\nabla }{E}_{2,2}}{{y}^{2}}& =\mathrm{S}\mathrm{V}\enspace \left[\frac{8}{5}\omega \left(0,0,0,0,0,3\left\vert \right.-\frac{1}{\tau }\right)+\frac{2}{5}\omega \left(0,0,0,3\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.-\frac{11}{75}\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)-8\omega \left(0,0,0,0,1,2\left\vert \right.-\frac{1}{\tau }\right)+\frac{{\zeta }_{3}}{3}\right].\hfill \end{align} \tag{ 5.32 }$$

Moreover, higher orders in the η_j -expansion of (5.30) yield infinite families of relations between the α'-expansions of open- and closed-string integrals over additional factors ${f }_{1i}^{(a)}{f}_{ij}^{(b)}{f}_{jk}^{(c)}$.

We shall finally exemplify the appearance of cuspidal MGFs from single-valued open-string integrals whose Laurent polynomial at the order of ${q}^{0}{\bar{q}}^{0}$ vanishes. A systematic study of imaginary cusp forms among the two-loop MGFs can be found in [54], also see [62] for examples of real cusp forms. The simplest imaginary cusp forms occur among the lattice sums (2.26) at modular weights (5, 5) whose basis can be chosen ²⁵ to include [18]

$$\begin{align}\hfill {B}_{2,3}& ={\left(\frac{\mathrm{I}\mathrm{m}\enspace \tau }{\pi }\right)}^{5}\left(\mathcal{C}\left[\begin{matrix}\hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill \end{matrix}\right]-\bar{\mathcal{C}\left[\begin{matrix}\hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill \end{matrix}\right]}\enspace \right)+\frac{\left(\nabla {E}_{2}\right)\bar{\nabla }{E}_{3}-\left(\bar{\nabla }{E}_{2}\right)\nabla {E}_{3}}{6{\left(\mathrm{I}\mathrm{m}\enspace \tau \right)}^{2}}\hfill \\ \hfill {B}_{2,3}^{\prime }& ={B}_{2,3}+\frac{1}{2}{\left(\frac{\mathrm{I}\mathrm{m}\enspace \tau }{\pi }\right)}^{5}\left(\mathcal{C}\left[\begin{matrix}\hfill 0\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 0\hfill & \hfill 2\hfill \end{matrix}\right]-\bar{\mathcal{C}\left[\begin{matrix}\hfill 0\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 0\hfill & \hfill 2\hfill \end{matrix}\right]}\enspace \right)-\frac{21}{4}{E}_{2,3}-\frac{1}{2}{\zeta }_{3}{E}_{2}\hfill \end{align} \tag{ 5.33 }$$

The β^sv-representations involve double-integrals over G₄ G₆ [18],

$$\begin{align}\hfill {B}_{2,3}& =450{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill 6\hfill \end{matrix}\right]-450{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 3\hfill & \hfill 0\hfill \\ \hfill 6\hfill & \hfill 4\hfill \end{matrix}\right]+270{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 6\hfill & \hfill 4\hfill \end{matrix}\right]-270{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 4\hfill & \hfill 6\hfill \end{matrix}\right]\hfill \\ \hfill & \quad -3{\zeta }_{3}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 1\hfill \\ \hfill 4\hfill \end{matrix}\right]-300{\zeta }_{3}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 1\hfill \\ \hfill 6\hfill \end{matrix}\right]+\frac{45{\zeta }_{3}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 2\hfill \\ \hfill 6\hfill \end{matrix}\right]}{y}+\frac{45{\zeta }_{5}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 0\hfill \\ \hfill 4\hfill \end{matrix}\right]}{y}\hfill \\ \hfill & \quad -\frac{27{\zeta }_{5}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 1\hfill \\ \hfill 4\hfill \end{matrix}\right]}{4{y}^{2}}-\frac{13{\zeta }_{5}}{120},\hfill \\ \hfill {B}_{2,3}^{\prime }& =1260{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill 6\hfill \end{matrix}\right]-840{\zeta }_{3}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 1\hfill \\ \hfill 6\hfill \end{matrix}\right]+\frac{7{\zeta }_{5}}{240}-\frac{{\zeta }_{3}^{2}}{2y}-\frac{147{\zeta }_{7}}{64{y}^{2}}+\frac{21{\zeta }_{3}{\zeta }_{5}}{8{y}^{3}},\hfill \end{align} \tag{ 5.34 }$$

and the associated integration constants $\bar{\alpha [\dots ]}$ can be found in the reference and in an ancillary file within the arXiv submission of this article. Both B_2,3, B_2,3' and their Cauchy–Riemann derivatives drop out from ${J}_{{\eta }_{2}}^{\tau }$ and ${Y}_{{\eta }_{2}}^{\tau }$ at two points. At three points, one can identify their derivatives as single-valued eMZVs,

$$\begin{align}\hfill \pi \nabla {B}_{2,3}& =\mathrm{S}\mathrm{V}\enspace \left[-\frac{1}{2}\omega \left(0,0,2,2,2\left\vert \right.-\frac{1}{\tau }\right)-2\omega \left(0,0,0,1,5\left\vert \right.-\frac{1}{\tau }\right)-\frac{3}{8}\omega \left(0,0,0,2,4\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.+\frac{{\zeta }_{3}}{8}\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)+\frac{17}{48}\omega {\left(0,3\left\vert \right.-\frac{1}{\tau }\right)}^{2}+\frac{7}{8}\omega \left(0,0,4\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.-\frac{7}{2}\omega \left(0,0,0,3\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)+\frac{137}{16}\omega \left(0,0,0,0,6\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.-\frac{15}{32}\omega \left(0,0,6\left\vert \right.-\frac{1}{\tau }\right)\right]\hfill \\ \hfill \pi \nabla {B}_{2,3}^{\prime }& =\mathrm{S}\mathrm{V}\enspace \left[-\frac{1}{2}\omega \left(0,0,2,2,2\left\vert \right.-\frac{1}{\tau }\right)-11\omega \left(0,0,0,1,5\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.+\frac{295}{8}\omega \left(0,0,0,0,6\left\vert \right.-\frac{1}{\tau }\right)-\frac{25}{16}\omega \left(0,0,6\left\vert \right.-\frac{1}{\tau }\right)-11\omega \left(0,0,0,3\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.\times \omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)+\frac{11}{12}\omega {\left(0,3\left\vert \right.-\frac{1}{\tau }\right)}^{2}-\frac{1}{4}\omega \left(0,0,4\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)\right]\hfill \\ \hfill {\left(\pi \nabla \right)}^{2}{B}_{2,3}^{\prime }& =\mathrm{S}\mathrm{V}\enspace \left[-\frac{189}{80}\omega \left(0,0,2,5\left\vert \right.-\frac{1}{\tau }\right)+\frac{63}{160}\omega \left(0,0,4,3\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.+\frac{603}{40}\omega \left(0,0,0,7\left\vert \right.-\frac{1}{\tau }\right)-\frac{699}{320}\omega \left(0,7\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.-\frac{1323}{160}\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,4\left\vert \right.-\frac{1}{\tau }\right)\right]\hfill \\ \hfill {\left(\pi \nabla \right)}^{3}{B}_{2,3}^{\prime }& =\mathrm{S}\mathrm{V}\enspace \left[-\frac{63}{40}\omega \left(0,3,5\left\vert \right.-\frac{1}{\tau }\right)+\frac{567}{64}\omega \left(0,0,8\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.-\frac{63}{8}\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,5\left\vert \right.-\frac{1}{\tau }\right)\right]\hfill \end{align} \tag{ 5.35 }$$

by inspecting the contributions of ${f }_{12}^{\enspace (3)}{f }_{23}^{\enspace (3)}$ or ${f }_{12}^{\enspace (4)}$ to ${J}_{0,{\eta }_{2},{\eta }_{3}}^{\tau }$ and ${f }_{12}^{\enspace (3)}$ to ${J}_{1,{\eta }_{2},{\eta }_{3}}^{\tau }$. The appearance of the undifferentiated B_2,3 and B_2,3' is relegated to the ${J}_{\overrightarrow{\eta }}^{\tau }$-series at four points (or the ${Y}_{\overrightarrow{\eta }}^{\tau }$-series at three points [18]), and comparison with the B-cycle integrals yields

$$\begin{align}\hfill {B}_{2,3}& =\mathrm{S}\mathrm{V}\enspace \left[\frac{143}{20}\omega \left(0,0,0,0,0,5\left\vert \right.-\frac{1}{\tau }\right)-\frac{11}{2}\omega \left(0,0,0,0,1,4\left\vert \right.-\frac{1}{\tau }\right)+2\omega \left(0,0,0,0,2,3\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.-2\omega \left(0,0,0,1,2,2\left\vert \right.-\frac{1}{\tau }\right)-\frac{91}{40}\omega \left(0,0,0,5\left\vert \right.-\frac{1}{\tau }\right)-\frac{449}{7200}\omega \left(0,5\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.-\frac{5}{12}\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)+\frac{5}{2}\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,0,3\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.+\frac{23}{12}\omega \left(0,0,2,3\left\vert \right.-\frac{1}{\tau }\right)-\frac{15}{2}\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,0,0,2\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.+\frac{{\zeta }_{3}}{4}\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)-\frac{43{\zeta }_{5}}{480}\right]\hfill \\ \hfill {B}_{2,3}^{\prime }& =\mathrm{S}\mathrm{V}\enspace \left[\frac{463}{10}\omega \left(0,0,0,0,0,5\left\vert \right.-\frac{1}{\tau }\right)-22\omega \left(0,0,0,0,1,4\left\vert \right.-\frac{1}{\tau }\right)+5\omega \left(0,0,0,0,2,3\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.-2\omega \left(0,0,0,1,2,2\left\vert \right.-\frac{1}{\tau }\right)-\frac{121}{20}\omega \left(0,0,0,5\left\vert \right.-\frac{1}{\tau }\right)-\frac{1069}{3600}\omega \left(0,5\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.-\frac{1}{6}\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)+\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,0,3\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \quad \left.+\frac{25}{6}\omega \left(0,0,2,3\left\vert \right.-\frac{1}{\tau }\right)-24\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,0,0,2\left\vert \right.-\frac{1}{\tau }\right)-\frac{11{\zeta }_{5}}{96}\right].\hfill \end{align} \tag{ 5.36 }$$

The open-string counterparts of B_2,3, B_2,3' and their Cauchy–Riemann derivatives involve the simplest combinations of B-cycle eMZVs with an irreducible ζ_3,5 in their Laurent polynomials: the methods of [3] (also see appendix B of [51]) yield the following examples of τ → i∞ degenerations in (5.35) and (5.36),

$$\begin{align}\hfill {B}_{2,3}\enspace {\vert }_{\text{LP}}& =\mathrm{S}\mathrm{V}\enspace \left[\frac{i{T}^{3}{\zeta }_{2}}{5040}+\frac{29iT{\zeta }_{4}}{630}+\frac{{\zeta }_{3}{\zeta }_{2}}{8}-\frac{8153i{\zeta }_{6}}{2880T}-\frac{9{\zeta }_{5}{\zeta }_{2}}{8{T}^{2}}-\frac{73{\zeta }_{3}{\zeta }_{4}}{16{T}^{2}}+\frac{1837i{\zeta }_{8}}{240{T}^{3}}\right.\hfill \\ \hfill & \left.\quad +\frac{3i}{10{T}^{3}}(2{\zeta }_{3,5}+5{\zeta }_{3}{\zeta }_{5})-\frac{39{\zeta }_{5}{\zeta }_{4}}{8{T}^{4}}+\frac{45{\zeta }_{3}{\zeta }_{6}}{8{T}^{4}}+\frac{33i{\zeta }_{10}}{20{T}^{5}}\right]\hfill \\ \hfill & =0\hfill \\ \hfill {(\pi \nabla )}^{3}{B}_{2,3}^{\prime }\enspace {\vert }_{\text{LP}}& =\mathrm{S}\mathrm{V}\enspace \left[\frac{{T}^{8}}{17\enspace 280}+\frac{i{T}^{5}{\zeta }_{3}}{120}+\frac{{T}^{4}{\zeta }_{4}}{40}-\frac{7{T}^{2}{\zeta }_{6}}{80}-\frac{8211{\zeta }_{8}}{640}\right.\hfill \\ \hfill & \left.\quad +\frac{63{\zeta }_{3,5}}{40}+\frac{189i{\zeta }_{5}{\zeta }_{4}}{8T}+\frac{63i{\zeta }_{3}{\zeta }_{6}}{4T}+\frac{14\enspace 553{\zeta }_{10}}{160{T}^{2}}\right]\hfill \\ \hfill & =\frac{2{y}^{8}}{135}-\frac{8{y}^{5}{\zeta }_{3}}{15}-\frac{63{\zeta }_{3}{\zeta }_{5}}{4}.\hfill \end{align} \tag{ 5.37 }$$

One can see from the order of T⁻³ or y⁻³ that the cuspidal nature of B_2,3 hinges on the depth-two result sv ζ_3,5 = −10ζ₃ ζ₅. The non-vanishing Laurent polynomial of (π∇)³ B_2,3' is due to the real MGFs $-\frac{21}{4}{E}_{2,3}-\frac{1}{2}{\zeta }_{3}{E}_{2}$ in (5.33).

Note that the simplest instances of ζ_3,7 and ζ_3,5,3 arise in the Laurent polynomials of B-cycle eMZVs with MGFs (2.26) of weights ${\sum }_{j=1}^{r}({a}_{j}+{b}_{j})=12$ and 14 in their SV mage. The appearance of ζ_3,5,3 in modular graph functions and eMZVs can be found in [12, 51], respectively. While ζ_3,7 drops out from MGFs under the single-valued map, it enters for instance the T⁰-order of the Laurent polynomial of $\omega (0,3,7\vert -\frac{1}{\tau })$ whose SV image contributes to the quantity (π∇)⁴ B'_2,4 in section 9.2 of [62]

$$\begin{align}\hfill \omega \left(0,3,7\left\vert \right.-\frac{1}{\tau }\right)\enspace {\left\vert \right.}_{\text{LP}}& =-\frac{{T}^{10}}{1261\enspace 260}+\frac{2i{T}^{5}{\zeta }_{5}}{315}+\frac{2{T}^{4}{\zeta }_{6}}{63}+\frac{7i{T}^{3}{\zeta }_{7}}{45}+\frac{7{T}^{2}{\zeta }_{8}}{6}\hfill \\ \hfill & \quad -{\zeta }_{3,7}-14{\zeta }_{3}{\zeta }_{7}-6{\zeta }_{5}^{2}+\frac{27{\zeta }_{10}}{2}+\frac{84i{\zeta }_{11}}{T}+\frac{30i{\zeta }_{5}{\zeta }_{6}}{T}\hfill \\ \hfill & \quad +\frac{84i{\zeta }_{3}{\zeta }_{8}}{T}+\frac{1353{\zeta }_{12}}{2{T}^{2}}.\hfill \end{align} \tag{ 5.38 }$$

One can eventually find all $\mathbb{Q}$-independent MZVs ²⁶ in the Laurent polynomials of B-cycle eMZVs. This follows from both the degeneration limits of the elliptic KZB associator [3] and from the fact that any MZV is expressible via $\mathbb{Q}[2\pi i]$-linear combinations of multiple modular values [107].

Note that the Laurent polynomials of all B-cycle eMZVs with length + weight ⩽ 16 obtained from an FORM implementation [96] of the methods of [3, 51] are available for download from [97].

While the above combinations of single-valued eMZVs were tailored to obtaining a single MGF in the bases of [18, 51], we shall now give a closed formula for the single-valued map of individual eMZVs. The integrands of convergent A-cycle eMZVs (2.24) with length r and n₁, n_r ≠ 1 arise at the ${s}_{ij}^{0}$-order of the series ${Z}_{\overrightarrow{\eta }}^{\tau }$ at r + 1 points. After modular S transformation, one can obtain any convergent $\omega ({n}_{1},\dots ,{n}_{r}\vert -\frac{1}{\tau })$ by isolating suitable η_j -orders in the s_ij → 0 limit of ${\sum }_{\rho \in {S}_{r}}{B}_{\overrightarrow{\eta }}^{\tau }(1,2,\dots ,r+1\vert 1,\rho (2,3,\dots ,r+1))$, where the permutation sum over the orderings of the integrands (2.13) yields the integrands ${f }_{21}^{({n}_{1})}{f }_{31}^{({n}_{2})}\dots {f}_{r+1,1}^{({n}_{r})}$ in the definition (2.24) of eMZVs. Hence, our proposal (3.35) implies that $\mathrm{S}\mathrm{V}\enspace \omega ({n}_{1},\dots ,{n}_{r}\vert -\frac{1}{\tau })$ occurs at the corresponding orders of s_ij and η_j in the series ${J}_{\overrightarrow{\eta }}^{\tau }$, so their definition (3.13) leads to (n₁, n_r ≠ 1 and z₁ = 0)

$$\begin{align}\hfill \mathrm{S}\mathrm{V}\enspace \omega \left({n}_{1},{n}_{2},\dots ,{n}_{r}\left\vert \right.-\frac{1}{\tau }\right)& ={(\tau -\bar{\tau })}^{{n}_{1}+\cdots +{n}_{r}}{\int }_{{\mathfrak{T}}^{r}}\left(\prod\limits _{j=2}^{r+1}\frac{{\mathrm{d}}^{2}{z}_{j}}{\mathrm{I}\mathrm{m}\enspace \tau }\right)\hfill \\ \hfill & \quad \times \enspace \bar{V(1,2,\dots ,r+1\vert \tau )}{f }_{21}^{({n}_{1})}{f }_{31}^{({n}_{2})}\dots {f}_{r+1,1}^{({n}_{r})}.\hfill \end{align} \tag{ 5.39 }$$

By the techniques in [52], the integral on the right-hand side can be straightforwardly performed in terms of lattice sums over p = mτ + n ∈ Λ' in (2.25): after expressing the $\bar{{V}_{w}}$-functions in terms of $\bar{{f}_{jk}^{(w)}}$ via (5.2) and inserting the double Fourier expansions

$$\begin{equation}{f}_{jk}^{(w)}={(-1)}^{w-1}\sum\limits _{p\in {{\Lambda}}^{\prime }}\frac{{e}^{2\pi i\langle p,{z}_{jk}\rangle }}{{p}^{w}},\quad {z}_{jk}={u}_{jk}\tau +{v}_{jk},\quad \langle p,{z}_{jk}\rangle =m{v}_{jk}-n{u}_{jk}\end{equation} \tag{ 5.40 }$$

the integrals ${\int }_{\mathfrak{T}}\frac{{\mathrm{d}}^{2}{z}_{j}}{\mathrm{I}\mathrm{m}\enspace \tau }={\int }_{0}^{1}\mathrm{d}{u}_{j}{\int }_{0}^{1}\mathrm{d}{v}_{j}$ lead to momentum-conserving delta functions as seen in the dihedral MGFs (2.26). When visualizing each factor of ${f}_{jk}^{(w)}$ and $\bar{{f}_{jk}^{(w)}}$ in the integrand of (5.39) through an edge between vertices j and k, contributions from one-particle reducible graphs integrate to zero. There are at most r − 1 factors of $\bar{{f}_{jk}^{(w)}}$ from the $\bar{{V}_{w\leqslant r-1}(1,2,\dots ,r+1)}$ in (5.39), and the admissible pairs (j, k) are visualized via dashed lines in figure 4.

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One can anticipate from the example (reproducing (5.12))

$$\begin{align}\hfill \mathrm{S}\mathrm{V}\enspace \omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)& ={(\tau -\bar{\tau })}^{2}{\int }_{{\mathfrak{T}}^{3}}\left(\prod\limits _{j=2}^{4}\frac{{\mathrm{d}}^{2}{z}_{j}}{\mathrm{I}\mathrm{m}\enspace \tau }\right)\left\{\frac{1}{6}-\frac{\bar{{V}_{1}(1,2,3,4\vert \tau )}}{4\pi i}+\frac{\bar{{V}_{2}(1,2,3,4\vert \tau )}}{{(2\pi i)}^{2}}\right\}{f }_{41}^{(2)}\hfill \\ \hfill & ={\left(\frac{\mathrm{I}\mathrm{m}\enspace \tau }{\pi }\right)}^{2}{\int }_{{\mathfrak{T}}^{3}}\left(\prod\limits _{j=2}^{4}\frac{{\mathrm{d}}^{2}{z}_{j}}{\mathrm{I}\mathrm{m}\enspace \tau }\right){f }_{41}^{(2)}{f }_{41}^{(2)}={\left(\frac{\mathrm{I}\mathrm{m}\enspace \tau }{\pi }\right)}^{2}\sum\limits _{p\in {{\Lambda}}^{\prime }}\frac{1}{\vert p{\vert }^{4}}={E}_{2}(\tau )\hfill \end{align} \tag{ 5.41 }$$

that only small subset of the terms in $\bar{V(1,2,\dots ,r+1\vert \tau )}$ contribute to generic single-valued eMZVs—the right-hand side of (5.41) entirely stems from $\bar{{V}_{2}(1,2,3,4\vert \tau )}\to \bar{{f }_{41}^{(2)}}$. Apart from the restriction to one-particle irreducible graphs, only those $\bar{{V}_{w}(1,2,\dots ,r+1\vert \tau )}$ with parity ${(-1)}^{w}={(-1)}^{{n}_{1}+{n}_{2}+\cdots +{n}_{r}}$ contribute since lattice sums with odd overall modular weight vanish.

Note that the torus integrals in the expression (5.39) for single-valued eMZVs converge whenever the eMZVs themselves do: the convergence criterion n₁, n_r ≠ 1 rules out any double pole |z_jk |⁻² in the integrand (and kinematic poles ${s}_{jk}^{-1}$ in the Koba–Nielsen integral) since the only overlap between the solid and dashed lines in figure 4 occurs via ${f }_{12}^{\enspace ({n}_{1})}\bar{{f }_{12}^{\enspace (w)}}$ and ${f }_{1,r+1}^{({n}_{r})}\bar{{f }_{1,r+1}^{(w)}}$.

Based on the conjectural relation (3.23) between the Laurent polynomials, one can use (5.39) to infer the asymptotics of the MGFs on the right-hand side by importing the Laurent polynomials of the B-cycle eMZVs from [97] and applying the single-valued map. Moreover, any relation among eMZVs induces a relation among the MGFs through the lattice-sum representation of their single-valued images. Hence, the database of MGF relations [62] can be complemented by applying (5.39) to the eMZV relations on the website [92]. The lattice sums contributing to $\mathrm{S}\mathrm{V}\enspace \omega ({n}_{1},\dots ,{n}_{r}\vert -\frac{1}{\tau })$ have (anti-)holomorphic modular weights $(w,\bar{w})$ subject to w = n₁ +⋯+ n_r and $\bar{w}\leqslant r-1$. Accordingly, the Laurent polynomials of B-cycle eMZVs of length + weight ⩽ 16 [97] give access to those of various combinations of MGFs with $w+\bar{w}\leqslant 15$.

In the remainder of this section, we will study the single-valued eMZVs in (5.39) at fixed length r and comment on consistency checks with the properties of their SV preimages [17]. The condition n₁, n_r ≠ 1 for convergence is taken to hold throughout. At length r = 1, for instance, $\omega (2k\vert -\frac{1}{\tau })=-2{\zeta }_{2k}$ and $\omega (2k-1\vert -\frac{1}{\tau })=0$ with k ⩾ 1 are annihilated by SV, in lines with the vanishing of the torus integral over a single ${f }_{12}^{\enspace (w)}$ at w ≠ 0.

5.7.1. Length-two examples ω(n₁, n₂)

Single-valued eMZVs of length r = 2 take the form (5.39)

$$\begin{align}\hfill \mathrm{S}\mathrm{V}\enspace \omega \left({n}_{1},{n}_{2}\left\vert \right.-\frac{1}{\tau }\right)& ={(\tau -\bar{\tau })}^{{n}_{1}+{n}_{2}}{\int }_{{\mathfrak{T}}^{2}}\frac{{\mathrm{d}}^{2}{z}_{2}}{\mathrm{I}\mathrm{m}\enspace \tau }\frac{{\mathrm{d}}^{2}{z}_{3}}{\mathrm{I}\mathrm{m}\enspace \tau }{f }_{21}^{({n}_{1})}{f }_{31}^{({n}_{2})}\hfill \\ \hfill & \quad \times \begin{cases}\frac{1}{2}\quad \hfill & :\enspace {n}_{1}+{n}_{2}\enspace \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\hfill \\ -\frac{\bar{{V}_{1}(1,2,3\vert \tau )}}{2\pi i}\quad \hfill & :\enspace {n}_{1}+{n}_{2}\enspace \mathrm{o}\mathrm{d}\mathrm{d}\hfill \end{cases},\hfill \end{align} \tag{ 5.42 }$$

where the distinction between even and odd weight n₁ + n₂ stems from the vanishing of lattice sums with odd modular weight. At even n₁ + n₂ > 0, the integrand of (5.42) is proportional to ${f }_{21}^{({n}_{1})}{f }_{31}^{({n}_{2})}$ which corresponds to a one-particle reducible graph with

$$\begin{equation}\mathrm{S}\mathrm{V}\enspace \omega \left({n}_{1},{n}_{2}\left\vert \right.-\frac{1}{\tau }\right)\enspace {\left\vert \right.}_{{n}_{1}+{n}_{2} > 0\enspace \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}=0,\end{equation} \tag{ 5.43 }$$

in agreement with $\omega (2{k}_{1},2{k}_{2}\vert -\frac{1}{\tau })=2{\zeta }_{2{k}_{1}}{\zeta }_{2{k}_{2}}$ and $\omega (2{k}_{1}-1,2{k}_{2}-1\vert -\frac{1}{\tau })=0$ [17]. For odd weight n₁ + n₂, we keep n₂ ≠ 0 and distinguish the two cases n₁ = 0 and n₁ ≠ 0, where the only contributions of $\bar{{V}_{1}(1,2,3\vert \tau )}$ to the integral (5.42) stem from $\bar{{f }_{31}^{(1)}}$ and $\bar{{f }_{23}^{\enspace (1)}}$, respectively:

$$\begin{align}\hfill \mathrm{S}\mathrm{V}\enspace \omega \left(0,{n}_{2}\left\vert \right.-\frac{1}{\tau }\right)\enspace {\left\vert \right.}_{{n}_{2}\enspace \mathrm{o}\mathrm{d}\mathrm{d}}& =-\frac{{(\tau -\bar{\tau })}^{{n}_{2}}}{2\pi i}{\int }_{\mathfrak{T}}\frac{{\mathrm{d}}^{2}{z}_{3}}{\mathrm{I}\mathrm{m}\enspace \tau }{f }_{31}^{({n}_{2})}\bar{{f }_{31}^{(1)}}=\frac{i}{2\pi }{(\tau -\bar{\tau })}^{{n}_{2}}\mathcal{C}\left[\begin{matrix}\hfill {n}_{2}\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]\hfill \\ \hfill \mathrm{S}\mathrm{V}\enspace \omega \left({n}_{1},{n}_{2}\left\vert \right.-\frac{1}{\tau }\right)\enspace {\left\vert \right.}_{{\hspace{2pt}n}_{1}+{n}_{2}\enspace \mathrm{o}\mathrm{d}\mathrm{d}}^{{\hspace{2pt}n}_{1},{n}_{2}\enspace \ne \enspace 0}& =-\frac{{(\tau -\bar{\tau })}^{{n}_{1}+{n}_{2}}}{2\pi i}{\int }_{\mathfrak{T}}\frac{{\mathrm{d}}^{2}{z}_{2}}{\mathrm{I}\mathrm{m}\enspace \tau }\frac{{\mathrm{d}}^{2}{z}_{3}}{\mathrm{I}\mathrm{m}\enspace \tau }{f }_{21}^{({n}_{1})}{f }_{31}^{({n}_{2})}\bar{{f }_{23}^{\enspace (1)}}\hfill \\ \hfill & ={(-1)}^{{n}_{1}}\frac{i}{2\pi }{(\tau -\bar{\tau })}^{{n}_{1}+{n}_{2}}\mathcal{C}\left[\begin{matrix}\hfill {n}_{1}+{n}_{2}\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]\hfill \end{align} \tag{ 5.44 }$$

The resulting relation $\mathrm{S}\mathrm{V}\enspace \omega ({n}_{1},{n}_{2}\vert -\frac{1}{\tau })={(-1)}^{{n}_{1}}\mathrm{S}\mathrm{V}\enspace \omega (0,{n}_{1}+{n}_{2}\vert -\frac{1}{\tau })$ is consistent with (2.33) of [17] after discarding any SV ζ_2k with k ⩾ 1 from the equation of the reference. In combination with the vanishing of $\mathrm{S}\mathrm{V}\enspace \omega ({n}_{1},{n}_{2})\enspace {\vert }_{{n}_{1}+{n}_{2}\enspace \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}$, we conclude that all single-valued eMZVs of length two do not exceed one-loop MGFs, in lines with ω(0, 2k + 1) being iterated Eisenstein integrals of depth one.

5.7.2. Length-three examples ω(n₁, n₂, n₃)

Single-valued eMZVs of length three can be written as

$$\begin{align}\hfill \mathrm{S}\mathrm{V}\enspace \omega \left({n}_{1},{n}_{2},{n}_{3}\left\vert \right.-\frac{1}{\tau }\right)& ={(\tau -\bar{\tau })}^{{n}_{1}+{n}_{2}+{n}_{3}}{\int }_{{\mathfrak{T}}^{3}}\left(\prod\limits _{j=2}^{4}\frac{{\mathrm{d}}^{2}{z}_{j}}{\mathrm{I}\mathrm{m}\enspace \tau }\right){f }_{21}^{({n}_{1})}{f }_{31}^{({n}_{2})}{f }_{41}^{({n}_{3})}\hfill \\ \hfill & \quad \times \begin{cases}\frac{1}{6}+\frac{\bar{{V}_{2}(1,2,3,4\vert \tau )}}{{(2\pi i)}^{2}}\quad \hfill & :\enspace {n}_{1}+{n}_{2}+{n}_{3}\enspace \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\hfill \\ -\frac{\bar{{V}_{1}(1,2,3,4\vert \tau )}}{4\pi i}\quad \hfill & :\enspace {n}_{1}+{n}_{2}+{n}_{3}\enspace \mathrm{o}\mathrm{d}\mathrm{d}\hfill \end{cases}\hfill \end{align} \tag{ 5.45 }$$

after discarding lattice sums of odd modular weight. For $\mathrm{S}\mathrm{V}\enspace \omega (0,0,2k+2\vert -\frac{1}{\tau })$, this results in the one-loop MGFs in the second line of (C.2). Starting from weight 8, the bases of ω(n₁, n₂, n₃) require representatives with two non-zero entries such as ω(0, 3, 5), ω(0, 3, 7) [17], and their single-valued versions correspond to MGFs of depth two as exemplified in section 5.6. For any combination of three non-vanishing entries, the torus integral in (5.45) can be expressed in terms of two-loop MGFs $\left[\begin{matrix}{a}_{1}& {a}_{2}& {a}_{3}\\ {b}_{1}& {b}_{2}& {b}_{3}\end{matrix}\right]$ in (2.26) and products of one-loop MGFs,

$$\begin{align}\hfill & \mathrm{S}\mathrm{V}\enspace \omega \left({n}_{1},{n}_{2},{n}_{3}\left\vert \right.-\frac{1}{\tau }\right)\enspace {\left\vert \right.\enspace }_{{n}_{1}+{n}_{2}+{n}_{3}\enspace \mathrm{o}\mathrm{d}\mathrm{d}}^{{n}_{1},{n}_{2},{n}_{3}\ne 0}=0\hfill \\ \hfill & \mathrm{S}\mathrm{V}\enspace \omega \left({n}_{1},{n}_{2},{n}_{3}\left\vert \right.-\frac{1}{\tau }\right)\enspace {\left\vert \right.\enspace }_{{n}_{1}+{n}_{2}+{n}_{3}\enspace \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}^{{n}_{1},{n}_{2},{n}_{3}\ne 0}\hfill \\ \hfill & \qquad =\frac{{(\tau -\bar{\tau })}^{{n}_{1}+{n}_{2}+{n}_{3}}}{{(2\pi i)}^{2}}\left\{{(-1)}^{{n}_{3}}\hspace{2pt}\mathcal{C}\left[\begin{matrix}\hfill {n}_{1}\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]\mathcal{C}\left[\begin{matrix}\hfill {n}_{2}+{n}_{3}\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]\right.\hfill \\ \hfill & \qquad \quad \left.+{(-1)}^{{n}_{1}}\hspace{2pt}\mathcal{C}\left[\begin{matrix}\hfill {n}_{3}\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]\mathcal{C}\left[\begin{matrix}\hfill {n}_{1}+{n}_{2}\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]+\mathcal{C}\left[\begin{matrix}\hfill {n}_{1}\hfill & \hfill {n}_{2}\hfill & \hfill {n}_{3}\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \end{matrix}\right]\right\},\hfill \end{align} \tag{ 5.46 }$$

leading to iterated Eisenstein integrals of depth two.

5.7.3. Length four and beyond

Starting from single-valued eMZVs of length four

$$\begin{align}\hfill & \mathrm{S}\mathrm{V}\enspace \omega \left({n}_{1},{n}_{2},{n}_{3},{n}_{4}\left\vert \right.-\frac{1}{\tau }\right)\hfill \\ \hfill & \qquad ={(\tau -\bar{\tau })}^{{n}_{1}+{n}_{2}+{n}_{3}+{n}_{4}}{\int }_{{\mathfrak{T}}^{4}}\left(\prod\limits _{j=2}^{5}\frac{{\mathrm{d}}^{2}{z}_{j}}{\mathrm{I}\mathrm{m}\enspace \tau }\right){f }_{21}^{({n}_{1})}{f }_{31}^{({n}_{2})}{f }_{41}^{({n}_{3})}{f }_{51}^{({n}_{4})}\hfill \\ \hfill & \qquad \quad \times \begin{cases}\frac{1}{24}+\frac{\bar{{V}_{2}(1,2,3,4,5\vert \tau )}}{2{(2\pi i)}^{2}}\quad \hfill & :\enspace {n}_{1}+{n}_{2}+{n}_{3}+{n}_{4}\enspace \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\hfill \\ -\frac{\bar{{V}_{1}(1,2,3,4,5\vert \tau )}}{12\pi i}-\frac{\bar{{V}_{3}(1,2,3,4,5\vert \tau )}}{{(2\pi i)}^{3}}\quad \hfill & :\enspace {n}_{1}+{n}_{2}+{n}_{3}+{n}_{4}\enspace \mathrm{o}\mathrm{d}\mathrm{d}\hfill \end{cases},\hfill \end{align} \tag{ 5.47 }$$

MGFs of different modular weights may mix through the contributions of $\bar{{V}_{1}}$ and $\bar{{V}_{3}}$ to

$$\begin{equation}\mathrm{S}\mathrm{V}\enspace \omega \left(0,0,0,{n}_{4}\left\vert \right.-\frac{1}{\tau }\right)\enspace {\left\vert \right.}_{{n}_{4}\enspace \mathrm{o}\mathrm{d}\mathrm{d}}=-\frac{{(\tau -\bar{\tau })}^{n}}{12\pi i}\mathcal{C}\left[\begin{matrix}\hfill {n}_{4}\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]-\frac{{(\tau -\bar{\tau })}^{n}}{{(2\pi i)}^{3}}\mathcal{C}\left[\begin{matrix}\hfill {n}_{4}\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{matrix}\right]\end{equation} \tag{ 5.48 }$$

The first term drops out when adding $-\frac{1}{6}\mathrm{S}\mathrm{V}\enspace \omega (0,{n}_{4}\vert -\frac{1}{\tau })\enspace {\vert }_{{n}_{4}\hspace{2pt}\mathrm{o}\mathrm{d}\mathrm{d}}$ in (5.44) and explains the combinations of eMZVs of different length in the third line of (C.2). Similarly, the more general combinations (5.16) of single-valued eMZVs of different length that isolate one-loop MGFs can be understood from the combinations of $\bar{{V}_{w}}$ that contribute to higher-point $\bar{V}$-functions (3.8).

With multiple non-zero entries in $\mathrm{S}\mathrm{V}\enspace \omega ({n}_{1},{n}_{2},{n}_{3},{n}_{4}\vert -\frac{1}{\tau })$ of odd weight, the lattice sums from integrating $\bar{{V}_{3}}$ correspond to MGFs of trihedral topology. Similarly, single-valued eMZVs at length five introduce four-point MGFs of kite topology introduced in section 4.3 of [62]. We hope that their identification with single-valued eMZVs will facilitate the study of MGFs beyond the dihedral topology and result in efficient methods to determine their Laurent polynomials and relations at arbitrary weight.

The results of this work concern infinite families of configuration-space integrals at genus one, and their application to genus-one string amplitudes requires the following leftover steps:

For both open and closed strings, it remains to integrate over the modular parameter τ of the respective surface. In the closed-string case, τ-integrals over MGFs are typically performed on the basis of their Laplace equations [9, 108–112] and Poincaré-series representations [54, 113–118]. The τ-integration of open-string integrals has for instance been discussed in [119–122], and a general method applicable to arbitrary depth may be based on the representation of eMZVs in terms of iterated Eisenstein integrals (including their 'over-integrated' instances with kernels τ^j G_k at j > k − 2 [118]) and properties of multiple modular values [101]. It would be particularly interesting to relate closed-string and single-valued open-string integrals at genus one after integration over τ.

For open strings, the ${Z}_{\overrightarrow{\eta }}^{\tau }$- or ${B}_{\overrightarrow{\eta }}^{\tau }$-series are claimed to exhaust all the configuration-space integrands built from f^(k)(z_i − z_j , τ) that are inequivalent under Fay identities and integration by parts. Similarly, the ${Y}_{\overrightarrow{\eta }}^{\tau }$-series built from double copies of the open-string integrands is expected to contain all torus integrals of this type. Hence, by the arguments of [4, 52, 79], ${B}_{\overrightarrow{\eta }}^{\tau }$ and ${Y}_{\overrightarrow{\eta }}^{\tau }$ should ²⁷ capture the conformal-field-theory correlators in the integrands of n-point genus-one amplitudes of massless states (and possibly also of massive states) in bosonic, heterotic and type-II string theories. In all cases, the component integrals in the η_j -expansions of the ${Z}_{\overrightarrow{\eta }}^{\tau }$-, ${B}_{\overrightarrow{\eta }}^{\tau }$ or ${Y}_{\overrightarrow{\eta }}^{\tau }$ series need to be dressed with kinematic factors that are determined by the correlators and carry the polarization dependence of the respective string amplitude.

The integrands of ${J}_{\overrightarrow{\eta }}^{\tau }$ only involve antielliptic combinations of $\bar{{f}^{(k)}({z}_{i}-{z}_{j},\tau )}$ and omit infinite classes of component integrals of ${Y}_{\overrightarrow{\eta }}^{\tau }$. For a given genus-one closed-string amplitude, it is therefore not a priori clear if its correlator is generated by the integrand of ${J}_{\overrightarrow{\eta }}^{\tau }$. Still, the correlators for the four- and five-point type-II amplitudes can be recovered from the subsectors ${J}_{w,\overrightarrow{\eta }}^{\tau }$ of the ${J}_{\overrightarrow{\eta }}^{\tau }$-series at fixed modular weights: the four-point correlator of [80] resides at the η⁻³ order of ${J}_{0,{\eta }_{2},{\eta }_{3},{\eta }_{4}}^{\tau }$, and the five-point correlators of [105, 106] can be assembled from the most singular η-orders of ${J}_{w,{\eta }_{2},{\eta }_{3},{\eta }_{4},{\eta }_{5}}^{\tau }$ at w = 0, 1. Similarly, the four- and five-point amplitudes of gluons and gravitons in heterotic string theories can in principle be extracted from the same ${J}_{w,\overrightarrow{\eta }}^{\tau }$ which also appear in type II, where higher orders in η_j are needed to capture the bosonic sectors. It would be interesting to see if this pattern persists at higher points in supersymmetric amplitudes, and whether the ${J}_{\overrightarrow{\eta }}^{\tau }$ are sufficient to generate bosonic-string amplitudes at low multiplicity.

This work spawns a variety of further directions and open questions of relevance to both physicists and mathematicians:

The single-valued image of eMZVs is proposed to contain combinations of holomorphic iterated Eisenstein integrals and their complex conjugates denoted by β^sv and constructed from the α'-expansion of closed-string integrals in [18]. It would be important to work out their detailed relation to Brown's earlier construction of single-valued iterated Eisenstein integrals [13, 14]. In particular, it remains to relate the MZVs in the antiholomorphic contributions to β^sv (fixed from reality properties of Koba–Nielsen integrals in [18]) to the combinations of multiple modular values entering Brown's construction. This will hopefully bypass the need to use these reality properties as independent input for the construction of β^sv as done so far.

Several aspects of our construction are based on conjectures with strong support from a variety of non-trivial examples. As pointed out in the relevant passages in earlier parts of this work, it would be desirable to find mathematically rigorous proofs that

• Any Koba–Nielsen integral at genus one involving products and derivatives of Kronecker–Eisenstein coefficients f^(k)(z, τ) can be expanded in the coefficients of the series ${Z}_{\overrightarrow{\eta }}^{\tau }$ and ${Y}_{\overrightarrow{\eta }}^{\tau }$
• The matrices ${r}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ and ${R}_{\overrightarrow{\eta }}({{\epsilon}}_{k})$ in open- and closed-string differential equation (2.19) preserve the commutation relations of Tsunogai's derivations _k
• The single-valued images β^sv of iterated Eisenstein integrals satisfy shuffle relations, i.e. that the antiholomorphic integration constants α do not introduce any obstructions
• The coefficients of the s_ij - and η_j -expansion of the initial values ${\hat{J}}_{\overrightarrow{\eta }}^{i\infty }$ and ${\hat{Y}}_{\overrightarrow{\eta }}^{i\infty }$ are single-valued multiple zeta values

The proposal of the present work concerns single-valued integration [48, 49] in the modular parameter τ. An alternative approach is to recover MGFs from single-valued functions of torus punctures [10, 15]. In this context, it would be rewarding to find an explicit realization of single-valued integration in z for elliptic polylogarithms and their complex conjugates, for instance by building upon the ideas of [15] and the depth-one results in [123].

At genus zero, the identification of sphere integrals as single-valued disk integrals is equivalent to the Kawai–Lewellen–Tye (KLT) relations between closed-string and squares of open-string tree-level amplitudes [124]. Accordingly, one could wonder if the combinations of holomorphic and antiholomorphic iterated Eisenstein integrals in the β^sv or MGFs can arise from products of open-string type generating functions and their complex conjugates. If such a genus-one echo of KLT relations exists, then one can expect a close connection to the monodromy relations among open-string integrals [122, 125–127] and in particular their study in the light of twisted deRham theory [128]. And it could open up a new perspective on the quest for loop-level KLT relations to revisit the generating functions of closed-string integrals in the framework of chiral splitting [129, 130], by performing the α'-expansion at the level of the loop integrand.

A particularly burning question concerns a higher-genus realization of single-valued integration and the associated relations between open- and closed-string amplitudes. A promising first step could be to identify suitable holomorphic open-string analogues of the MGFs [131, 132] and modular graph tensors [133] at higher genus. More generally, the simplified correlators of maximally supersymmetric genus-two amplitudes at four points [134, 135] and five points [136, 137] provide valuable showcases of Koba–Nielsen integrals relevant to open- and closed-string scattering. Furthermore, the construction of the generating series in this work was inspired by extended families of genus-one Koba–Nielsen integrals that arise from heterotic or bosonic strings [52]. Hence, the genus-two correlators of the heterotic string and the combinations of theta functions studied in [138, 139] could give important clues on higher-genus versions of the elliptic functions and generating series in this work.

All data that support the findings of this study are included within the article (and any supplementary files).

The terms of even orders in η_j in the three-point initial values are given by

$$\begin{align}\hfill {\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert 2,3)\enspace {\vert }_{\text{even}}& =\frac{1}{{\eta }_{23}{\eta }_{3}}\left(\frac{1}{2}+\frac{{\zeta }_{2}}{12}({s}_{12}^{2}+{s}_{13}^{2}+{s}_{23}^{2})+\frac{{\zeta }_{3}}{24}({s}_{12}^{3}+{s}_{13}^{3}+{s}_{23}^{3})\right.\hfill \\ \hfill & \left.\quad +{\zeta }_{4}\left[\frac{131}{1440}({s}_{12}^{4}+{s}_{13}^{4}+{s}_{23}^{4})+\frac{5}{144}({s}_{12}^{2}{s}_{13}^{2}+{s}_{12}^{2}{s}_{23}^{2}+{s}_{23}^{2}{s}_{13}^{2})\right.\right.\hfill \\ \hfill & \quad \left.\left.+\frac{1}{18}{s}_{12}{s}_{13}{s}_{23}{s}_{123}\right]+\dots \right)\hfill \\ \hfill & \quad +\frac{{\eta }_{23}}{{\eta }_{3}}\left(-{\zeta }_{2}-\frac{{\zeta }_{3}}{2}{s}_{12}-{\zeta }_{4}\left[\frac{29{s}_{12}^{2}}{24}+\frac{5{s}_{13}^{2}}{12}+\frac{{s}_{13}{s}_{23}}{3}+\frac{5{s}_{23}^{2}}{12}\right]+\dots \right)\hfill \\ \hfill & \quad +\frac{{\eta }_{3}}{{\eta }_{23}}\left(-{\zeta }_{2}-\frac{{\zeta }_{3}}{2}{s}_{23}-{\zeta }_{4}\left[\frac{29{s}_{23}^{2}}{24}+\frac{5{s}_{12}^{2}}{12}+\frac{{s}_{12}{s}_{13}}{3}+\frac{5{s}_{13}^{2}}{12}\right]+\dots \right)\hfill \\ \hfill & \quad +\left(\frac{3{\zeta }_{2}{s}_{13}}{{s}_{123}}+\frac{{\zeta }_{3}}{2}{s}_{13}+\frac{15{\zeta }_{4}{s}_{13}}{4{s}_{123}}({s}_{12}^{2}+{s}_{12}{s}_{23}+{s}_{23}^{2})\right.\hfill \\ \hfill & \quad \left.-\frac{41}{24}{\zeta }_{4}({s}_{12}+{s}_{23}){s}_{13}+\frac{49{\zeta }_{4}{s}_{13}^{2}}{24}+\dots \right)\hfill \\ \hfill & \quad +{\eta }_{23}{\eta }_{3}(5{\zeta }_{4})-\frac{{\eta }_{23}^{3}}{{\eta }_{3}}{\zeta }_{4}-\frac{{\eta }_{3}^{3}}{{\eta }_{23}}{\zeta }_{4}+\cdots \hfill \end{align} \tag{ B.1 }$$

and

$$\begin{equation}{\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert 3,2)\enspace {\vert }_{\text{even}}={\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert 2,3)\enspace {\vert }_{\text{even}}\enspace {\vert }_{{s}_{12}{\leftrightarrow}{s}_{13}}^{{\eta }_{2}{\leftrightarrow}{\eta }_{3}}\end{equation} \tag{ B.2 }$$

with MZVs of weight ⩾ 5 in the ellipsis.

The terms of odd orders in η_j in the three-point initial values are given by

$$\begin{align}\hfill {\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert 2,3)\enspace {\vert }_{\text{odd}}& =\frac{1}{{\eta }_{3}}\left(\frac{1}{{s}_{12}}+\frac{{\zeta }_{2}{s}_{123}^{2}}{6{s}_{12}}+{\zeta }_{3}\left[\frac{{s}_{13}{s}_{23}}{4}+\frac{{s}_{123}^{3}}{12{s}_{12}}\right]\right.\hfill \\ \hfill & \quad \left.+{\zeta }_{4}\left[\frac{131{s}_{123}^{4}}{720{s}_{12}}-\frac{{s}_{23}{s}_{123}{s}_{13}}{20}\right]+\dots \right)\hfill \\ \hfill & \quad +\frac{1}{{\eta }_{23}}\left(\frac{1}{{s}_{23}}+\frac{{\zeta }_{2}{s}_{123}^{2}}{6{s}_{23}}+{\zeta }_{3}\left[\frac{{s}_{12}{s}_{13}}{4}+\frac{{s}_{123}^{3}}{12{s}_{23}}\right]\right.\hfill \\ \hfill & \quad \left.+{\zeta }_{4}\left[\frac{131{s}_{123}^{4}}{720{s}_{23}}-\frac{{s}_{12}{s}_{123}{s}_{13}}{20}\right]+\dots \right)\hfill \\ \hfill & \quad +{\eta }_{3}\left(-\frac{2{\zeta }_{2}}{{s}_{12}}-\frac{{\zeta }_{3}{s}_{123}}{{s}_{12}}+{\zeta }_{4}\left[-\frac{29{s}_{123}^{2}}{12{s}_{12}}+\frac{{s}_{13}}{4}\right]+\dots \right)\hfill \\ \hfill & \quad +{\eta }_{23}\left(-\frac{2{\zeta }_{2}}{{s}_{23}}-\frac{{\zeta }_{3}{s}_{123}}{{s}_{23}}+{\zeta }_{4}\left[-\frac{29{s}_{123}^{2}}{12{s}_{23}}+\frac{{s}_{13}}{4}\right]+\dots \right)\hfill \\ \hfill & \quad +\frac{{\eta }_{23}^{2}}{{\eta }_{3}}\left(-{\zeta }_{3}+{\zeta }_{4}\left[\frac{2}{3}{s}_{123}+\frac{{s}_{12}}{4}\right]+\dots \right)\hfill \\ \hfill & \quad +\frac{{\eta }_{3}^{2}}{{\eta }_{23}}\left(-{\zeta }_{3}+{\zeta }_{4}\left[\frac{2}{3}{s}_{123}+\frac{{s}_{23}}{4}\right]+\dots \right)\hfill \\ \hfill & \quad +{\eta }_{23}^{3}\left(-\frac{2{\zeta }_{4}}{{s}_{23}}+\dots \right)+{\eta }_{3}^{3}\left(-\frac{2{\zeta }_{4}}{{s}_{12}}+\dots \right)+\dots \hfill \end{align} \tag{ B.3 }$$

and

$$\begin{equation}{\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert 3,2)\enspace {\vert }_{\text{odd}}=-{\hat{B}}_{{\eta }_{2},{\eta }_{3}}^{i\infty }(2,3\vert 2,3)\enspace {\vert }_{\text{odd}}\enspace {\vert }_{{s}_{12}{\leftrightarrow}{s}_{13}}^{{\eta }_{2}{\leftrightarrow}{\eta }_{3}},\end{equation} \tag{ B.4 }$$

again with MZVs of weight ⩾ 5 in the ellipsis.

The simplest examples (5.12) and (5.13) of single-valued eMZVs extracted from the two-point integrals (5.4) are special cases of the SV map (4.30) on holomorphic iterated Eisenstein integrals. For their depth-one combinations $\beta \left[\begin{matrix}j\\ k\end{matrix}\right]$ in (4.10), the SV image ${\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}j\\ k\end{matrix}\right]$ yields the following Cauchy–Riemann derivatives ∇ = 2i(Im τ)²∂_τ of non-holomorphic Eisenstein series [18]

$$\begin{align}\hfill {\beta }^{\text{sv}}\left[\begin{matrix}\hfill k-1\hfill \\ \hfill 2k\hfill \end{matrix}\right]& =-\frac{{[(k-1)!]}^{2}}{(2k-1)!}{E}_{k}+\frac{2{\zeta }_{2k-1}}{(2k-1){(4y)}^{k-1}}\hfill \\ \hfill {\beta }^{\text{sv}}\left[\begin{matrix}\hfill k-1+m\hfill \\ \hfill 2k\hfill \end{matrix}\right]& =-\frac{{(-4)}^{m}(k-1)!\enspace (k-1-m)!\enspace {(\pi \nabla )}^{m}{E}_{k}}{(2k-1)!}+\frac{2{\zeta }_{2k-1}}{(2k-1){(4y)}^{k-1-m}}\hfill \\ \hfill {\beta }^{\text{sv}}\left[\begin{matrix}\hfill k-1-m\hfill \\ \hfill 2k\hfill \end{matrix}\right]& =-\frac{(k-1)!\enspace (k-1-m)!\enspace {(\pi \bar{\nabla })}^{m}{E}_{k}}{{(-4)}^{m}(2k-1)!{y}^{2m}}+\frac{2{\zeta }_{2k-1}}{(2k-1){(4y)}^{k-1+m}},\hfill \end{align} \tag{ C.1 }$$

also see (4.22). While the objects on the right-hand side are expressible in terms of the lattice sums $\left[\begin{matrix}a& 0\\ b& 0\end{matrix}\right]$ in (2.27) via (2.28), the $\beta \left[\begin{matrix}j\\ k\end{matrix}\right]$ are simple combinations of B-cycle eMZVs $\omega ({0}^{p},k\vert -\frac{1}{\tau })$, where 0^p stands for a sequence 0, 0, ..., 0 of p successive zeros. On these grounds, ${\beta }^{\mathrm{s}\mathrm{v}}\left[\begin{matrix}j\\ k\end{matrix}\right]=\mathrm{S}\mathrm{V}\enspace \beta \left[\begin{matrix}j\\ k\end{matrix}\right]$ translates into simple relations such as

$$\begin{equation}\begin{gathered}\mathrm{S}\mathrm{V}\enspace \omega \left(0,2k+1\left\vert \right.-\frac{1}{\tau }\right)=-\frac{{(\tau -\bar{\tau })}^{2k+1}}{2\pi i}\mathcal{C}\left[\begin{matrix}\hfill 2k+1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right],\quad k\geqslant 1\\ \mathrm{S}\mathrm{V}\enspace \omega \left(0,0,2k+2\left\vert \right.-\frac{1}{\tau }\right)=\frac{{(\tau -\bar{\tau })}^{2k+2}}{{(2\pi i)}^{2}}\mathcal{C}\left[\begin{matrix}\hfill 2k+2\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill \end{matrix}\right],\quad k\geqslant 0\\ \mathrm{S}\mathrm{V}\enspace \left(\omega \left(0,0,0,2k+3\left\vert \right.-\frac{1}{\tau }\right)-\frac{1}{6}\omega \left(0,2k+3\left\vert \right.-\frac{1}{\tau }\right)\right)=-\frac{{(\tau -\bar{\tau })}^{2k+3}}{{(2\pi i)}^{3}}\mathcal{C}\left[\begin{matrix}\hfill 2k+3\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{matrix}\right],\\ \quad k\geqslant -1\\ \mathrm{S}\mathrm{V}\enspace \left(\omega \left(0,0,0,0,2k+4\left\vert \right.-\frac{1}{\tau }\right)-\frac{1}{6}\omega \left(0,0,2k+4\left\vert \right.-\frac{1}{\tau }\right)\right)=\frac{{(\tau -\bar{\tau })}^{2k+4}}{{(2\pi i)}^{4}}\mathcal{C}\left[\begin{matrix}\hfill 2k+4\hfill & \hfill 0\hfill \\ \hfill 4\hfill & \hfill 0\hfill \end{matrix}\right],\\ \quad k\geqslant -1\end{gathered}\end{equation} \tag{ C.2 }$$

as well as

$$\begin{align}\hfill & \mathrm{S}\mathrm{V}\enspace \left(\omega \left(0,0,0,0,0,2k+5\left\vert \right.-\frac{1}{\tau }\right)-\frac{1}{6}\omega \left(0,0,0,2k+5\left\vert \right.-\frac{1}{\tau }\right)+\frac{7}{360}\omega \left(0,2k+5\left\vert \right.-\frac{1}{\tau }\right)\right)\hfill \\ \hfill & \qquad =-\frac{{(\tau -\bar{\tau })}^{2k+5}}{{(2\pi i)}^{5}}\mathcal{C}\left[\begin{matrix}\hfill 2k+5\hfill & \hfill 0\hfill \\ \hfill 5\hfill & \hfill 0\hfill \end{matrix}\right],\quad k\geqslant -2.\hfill \end{align} \tag{ C.3 }$$

The relative factors of $-\frac{1}{6}$ and $\frac{7}{360}$ among the eMZVs of different lengths are engineered to streamline the iterated-Eisenstein-integral representation [17] and generalize as follows [104]

$$\begin{equation}\mathrm{S}\mathrm{V}\enspace \sum\limits _{j=0}^{\ell -1}\frac{{B}_{j}}{j!}\omega \left({0}^{\ell -j},2k+\ell \left\vert \right.-\frac{1}{\tau }\right)={(-1)}^{\ell }\frac{{(\tau -\bar{\tau })}^{2k+\ell }}{{(2\pi i)}^{\ell }}\mathcal{C}\left[\begin{matrix}\hfill 2k+\ell \hfill & \hfill 0\hfill \\ \hfill \ell \hfill & \hfill 0\hfill \end{matrix}\right],\quad k\geqslant 1-\lceil \frac{\ell }{2}\rceil ,\enspace \ell \geqslant 1.\end{equation} \tag{ C.4 }$$

In obtaining (C.2) and (C.3) from (C.4), we have used SV ω(m) = 0  ∀  m ⩾ 1 and the following simplifications of the only eMZV ω(0^ℓ−1, 2k + ℓ$\vert \tau$ℓ) whose length and weight adds up to an even number [17],

$$\begin{align}\hfill \omega (0,0,2k+1\vert \tau )& =\frac{1}{2}\omega (0,2k+1)\hfill \\ \hfill \omega (0,0,0,2k\vert \tau )& =\frac{1}{2}\omega (0,0,2k)-\frac{1}{24}\omega (2k)\hfill \\ \hfill \omega (0,0,0,0,2k+1\vert \tau )& =\frac{1}{2}\omega (0,0,0,2k+1)-\frac{1}{24}\omega (0,2k+1).\hfill \end{align} \tag{ C.5 }$$

Based on the dictionary (2.28) between lattice sums $\left[\begin{matrix}a& 0\\ b& 0\end{matrix}\right]$ and non-holomorphic Eisenstein series, one can reformulate (C.4) as

$$\begin{equation}\mathrm{S}\mathrm{V}\enspace \sum\limits _{j=0}^{\ell -1}\frac{{B}_{j}}{j!}\omega \left({0}^{\ell -j},2k+\ell \left\vert \right.-\frac{1}{\tau }\right)={(-1)}^{\ell }\frac{(k+\ell -1)!}{(2k+\ell -1)!}{(-4\pi \nabla )}^{k}{E}_{k+\ell },\quad k\geqslant 0,\enspace \ell \geqslant 1,\end{equation} \tag{ C.6 }$$

where k = 0 needs to be excluded if ℓ = 1, for instance

$$\begin{align}\hfill \mathrm{S}\mathrm{V}\enspace \omega \left(0,2k+1\left\vert \right.-\frac{1}{\tau }\right)& =-\frac{k!}{(2k)!}{(-4\pi \nabla )}^{k}{E}_{k+1},\hfill & \hfill k\geqslant 1\\ \hfill \mathrm{S}\mathrm{V}\enspace \omega \left(0,0,2k+2\left\vert \right.-\frac{1}{\tau }\right)& =\frac{(k+1)!}{(2k+1)!}{(-4\pi \nabla )}^{k}{E}_{k+2},\hfill & \hfill k\geqslant 0.\end{align} \tag{ C.7 }$$

Moreover, by extending (C.4) to k → −k and applying the complex conjugate of (2.28), we also obtain antiholomorphic Cauchy–Riemann derivatives as single-valued eMZVs (with ℓ − 2k > 0),

$$\begin{align}\hfill \mathrm{S}\mathrm{V}\enspace \sum\limits _{j=0}^{\ell -1}\frac{{B}_{j}}{j!}\omega \left({0}^{\ell -j},\ell -2k\left\vert \right.-\frac{1}{\tau }\right)& ={(-1)}^{\ell }\frac{{(\tau -\bar{\tau })}^{\ell -2k}}{{(2\pi i)}^{\ell }}\mathcal{C}\left[\begin{matrix}\hfill \ell -2k\hfill & \hfill 0\hfill \\ \hfill \ell \hfill & \hfill 0\hfill \end{matrix}\right]\hfill \\ \hfill & ={(-1)}^{\ell +k}\frac{(\ell -k-1)!}{(\ell -1)!}\frac{{(\pi \bar{\nabla })}^{k}{E}_{\ell -k}}{{(2y)}^{2k}}.\hfill \end{align} \tag{ C.8 }$$

The simplest examples include

$$\begin{equation}\begin{aligned}\hfill \mathrm{S}\mathrm{V}\enspace \left(\omega \left(0,0,0,1\left\vert \right.-\frac{1}{\tau }\right)-\frac{1}{6}\omega \left(0,1\left\vert \right.-\frac{1}{\tau }\right)\right)& =\frac{\pi \bar{\nabla }{E}_{2}}{8{y}^{2}}\hfill \\ \hfill \mathrm{S}\mathrm{V}\enspace \left(\omega \left(0,0,0,0,2\left\vert \right.-\frac{1}{\tau }\right)-\frac{1}{6}\omega \left(0,0,2\left\vert \right.-\frac{1}{\tau }\right)\right)& =-\frac{\pi \bar{\nabla }{E}_{3}}{12{y}^{2}}\hfill \\ \hfill \mathrm{S}\mathrm{V}\enspace \left(\omega \left(0,0,0,0,0,1\left\vert \right.-\frac{1}{\tau }\right)-\frac{1}{6}\omega \left(0,0,0,1\left\vert \right.-\frac{1}{\tau }\right)+\frac{7}{360}\omega \left(0,1\left\vert \right.-\frac{1}{\tau }\right)\right)& =-\frac{{(\pi \bar{\nabla })}^{2}{E}_{3}}{192{y}^{4}}\hfill \\ \hfill \mathrm{S}\mathrm{V}\enspace \left(\omega \left(0,0,0,0,0,3\left\vert \right.-\frac{1}{\tau }\right)-\frac{1}{6}\omega \left(0,0,0,3\left\vert \right.-\frac{1}{\tau }\right)+\frac{7}{360}\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)\right)& =\frac{\pi \bar{\nabla }{E}_{4}}{16{y}^{2}},\hfill \end{aligned}\end{equation} \tag{ C.9 }$$

and the first two lines are equivalent to those in (5.32).

By inspecting the ${s}_{ij}^{4}$ order of the two-point integrals ${B}_{(0)}^{\tau },{J}_{(0)}^{\tau }$ and the ${s}_{ij}^{3}$ order of ${B}_{(2)}^{\tau },{J}_{(2)}^{\tau }$, we have obtained the representations (5.14) of E_2,2 and π∇E_2,2 as single-valued eMZVs. One can extract similar representations for E_2,3, π∇E_2,3 and (π∇)² E_2,3 from the ${s}_{ij}^{5}$ order of ${B}_{(0)}^{\tau },{J}_{(0)}^{\tau }$, the ${s}_{ij}^{4}$ order of ${B}_{(2)}^{\tau },{J}_{(2)}^{\tau }$ and the ${s}_{ij}^{3}$ order of ${B}_{(4)}^{\tau },{J}_{(4)}^{\tau }$, respectively:

$$\begin{align}\hfill {E}_{2,3}& =\mathrm{S}\mathrm{V}\enspace \left(-\frac{167}{35}\omega \left(0,0,0,0,0,5\left\vert \right.-\frac{1}{\tau }\right)+2\omega \left(0,0,0,0,1,4\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \left.\quad +\frac{97}{210}\omega \left(0,0,0,5\left\vert \right.-\frac{1}{\tau }\right)-\frac{1}{3}\omega \left(0,0,2,3\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \left.\quad +2\omega \left(0,0,0,0,2\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)+\frac{7}{200}\omega \left(0,5\left\vert \right.-\frac{1}{\tau }\right)\right)\hfill \\ \hfill \pi \nabla {E}_{2,3}& =\mathrm{S}\mathrm{V}\enspace \left(-\frac{1}{12}\omega {\left(0,3\left\vert \right.-\frac{1}{\tau }\right)}^{2}+\frac{13}{168}\omega \left(0,0,6\left\vert \right.-\frac{1}{\tau }\right)+\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,0,3\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \left.\quad -\frac{41}{28}\omega \left(0,0,0,0,6\left\vert \right.-\frac{1}{\tau }\right)+\frac{1}{2}\omega \left(0,0,0,2,4\left\vert \right.-\frac{1}{\tau }\right)\right)\hfill \\ \hfill {(\pi \nabla )}^{2}{E}_{2,3}& =\mathrm{S}\mathrm{V}\enspace \left(\frac{25}{336}\omega \left(0,7\left\vert \right.-\frac{1}{\tau }\right)+\frac{5}{8}\omega \left(0,3\left\vert \right.-\frac{1}{\tau }\right)\omega \left(0,0,4\left\vert \right.-\frac{1}{\tau }\right)-\frac{23}{28}\omega \left(0,0,0,7\left\vert \right.-\frac{1}{\tau }\right)\right.\hfill \\ \hfill & \left.\quad +\frac{1}{4}\omega \left(0,0,2,5\left\vert \right.-\frac{1}{\tau }\right)+\frac{1}{8}\omega \left(0,0,4,3\left\vert \right.-\frac{1}{\tau }\right)\right)\hfill \end{align} \tag{ C.10 }$$

The corresponding lattice-sum representations [51, 62] and β^sv representations [18] are given by

$$\begin{align}\hfill {E}_{2,3}& ={\left(\frac{\mathrm{I}\mathrm{m}\enspace \tau }{\pi }\right)}^{5}\left(\mathcal{C}\left[\begin{matrix}\hfill 3\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill 1\hfill \end{matrix}\right]-\frac{43}{35}\mathcal{C}\left[\begin{matrix}\hfill 5\hfill & \hfill 0\hfill \\ \hfill 5\hfill & \hfill 0\hfill \end{matrix}\right]\right)\hfill \\ \hfill & =-120{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill 6\hfill \end{matrix}\right]-120{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 3\hfill & \hfill 0\hfill \\ \hfill 6\hfill & \hfill 4\hfill \end{matrix}\right]+\frac{12{\zeta }_{5}}{y}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 0\hfill \\ \hfill 4\hfill \end{matrix}\right]+80{\zeta }_{3}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 1\hfill \\ \hfill 6\hfill \end{matrix}\right]\hfill \\ \hfill & \quad -\frac{{\zeta }_{5}}{36}+\frac{7{\zeta }_{7}}{16{y}^{2}}-\frac{{\zeta }_{3}{\zeta }_{5}}{2{y}^{3}}\hfill \\ \hfill \pi \nabla {E}_{2,3}& =\frac{{(\mathrm{I}\mathrm{m}\enspace \tau )}^{6}}{{\pi }^{4}}\left(3\mathcal{C}\left[\begin{matrix}\hfill 1\hfill & \hfill 1\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \end{matrix}\right]+2\mathcal{C}\left[\begin{matrix}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill \end{matrix}\right]-\frac{43}{7}\mathcal{C}\left[\begin{matrix}\hfill 6\hfill & \hfill 0\hfill \\ \hfill 4\hfill & \hfill 0\hfill \end{matrix}\right]\right)\hfill \\ \hfill & =90{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 2\hfill & \hfill 2\hfill \\ \hfill 4\hfill & \hfill 6\hfill \end{matrix}\right]+60{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 3\hfill & \hfill 1\hfill \\ \hfill 6\hfill & \hfill 4\hfill \end{matrix}\right]+30{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 4\hfill & \hfill 0\hfill \\ \hfill 6\hfill & \hfill 4\hfill \end{matrix}\right]\hfill \\ \hfill & \quad -60{\zeta }_{3}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 2\hfill \\ \hfill 6\hfill \end{matrix}\right]-12{\zeta }_{5}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 0\hfill \\ \hfill 4\hfill \end{matrix}\right]-\frac{6{\zeta }_{5}}{y}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 1\hfill \\ \hfill 4\hfill \end{matrix}\right]-\frac{7{\zeta }_{7}}{8y}+\frac{3{\zeta }_{3}{\zeta }_{5}}{2{y}^{2}}\hfill \\ \hfill {(\pi \nabla )}^{2}{E}_{2,3}& =\frac{{(\mathrm{I}\mathrm{m}\enspace \tau )}^{7}}{{\pi }^{3}}\left(4\mathcal{C}\left[\begin{matrix}\hfill 0\hfill & \hfill 2\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \end{matrix}\right]-4\mathcal{C}\left[\begin{matrix}\hfill 3\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]\mathcal{C}\left[\begin{matrix}\hfill 4\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill \end{matrix}\right]-\frac{62}{7}\mathcal{C}\left[\begin{matrix}\hfill 7\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill 0\hfill \end{matrix}\right]\right)\hfill \\ \hfill & =-45{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 6\hfill \end{matrix}\right]-15{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 3\hfill & \hfill 2\hfill \\ \hfill 6\hfill & \hfill 4\hfill \end{matrix}\right]-30{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 4\hfill & \hfill 1\hfill \\ \hfill 6\hfill & \hfill 4\hfill \end{matrix}\right]\hfill \\ \hfill & \quad +30{\zeta }_{3}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 3\hfill \\ \hfill 6\hfill \end{matrix}\right]+12{\zeta }_{5}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 1\hfill \\ \hfill 4\hfill \end{matrix}\right]+\frac{3{\zeta }_{5}}{2y}{\beta }^{\text{sv}}\left[\begin{matrix}\hfill 2\hfill \\ \hfill 4\hfill \end{matrix}\right]+\frac{7{\zeta }_{7}}{8}-\frac{3{\zeta }_{3}{\zeta }_{5}}{y}.\hfill \end{align} \tag{ C.11 }$$