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# Non-holomorphic modular forms from zeta generators

Daniele Dorigoni, Mehregan Doroudiani, Joshua Drewitt, Martijn Hidding, Axel Kleinschmidt, Oliver Schlotterer, Leila Schneps, Bram Verbeek

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## ► To cite this version:

Daniele Dorigoni, Mehregan Doroudiani, Joshua Drewitt, Martijn Hidding, Axel Kleinschmidt, et al. "Non-holomorphic modular forms from zeta generators." Journal of High Energy Physics, 2024, 2024 (10), pp.53. 10.1007/jhep10(2024)053. hal-05388828

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https://hal.science/hal-05388828v1

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Non-holomorphic modular forms from zeta generators

Daniele Dorigoni<sup>id,a</sup> Mehregan Doroudiani<sup>id,b</sup> Joshua Drewitt<sup>id,c</sup> Martijn Hidding<sup>id,d,e</sup>

Axel Kleinschmidt<sup>id,f</sup> Oliver Schlotterer<sup>id,g</sup> Leila Schneps<sup>id,h</sup> and Bram Verbeek<sup>id,i</sup>

**ABSTRACT:** We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL(2, Z) known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown’s framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes.

**KEYWORDS:** Differential and Algebraic Geometry, Superstrings and Heterotic Strings

**ARXIV EPRINT:** 2403.14816

<footnote><sup>a</sup>Centre for Particle Theory & Department of Mathematical Sciences, Durham University,
Lower Mountjoy, Stockton Road, Durham DH1 3LE, U.K.</footnote>

<footnote><sup>b</sup>Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut),
Am Mühlenberg 1, 14476 Potsdam, Germany</footnote>

<footnote><sup>c</sup>School of Mathematics, University of Bristol,
Queens Road, Bristol, BS8 1QU, U.K.</footnote>

<footnote><sup>d</sup>Department of Physics and Astronomy, Uppsala University,
Lägerhyddsvägen 1, 752 37 Uppsala, Sweden</footnote>

<footnote><sup>e</sup>Institute for Theoretical Physics, ETH Zurich,
Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland</footnote>

<footnote><sup>f</sup>International Solvay Institutes,
ULB-Campus Plaine CP231, Boulevard du Triomphe, 1050 Brussels, Belgium</footnote>

<footnote><sup>g</sup>CNRS, Sorbonne Université, Campus Pierre et Marie Curie,
4 place Jussieu, 75005 Paris, France
E-mail: daniele.dorigoni@durham.ac.uk, mehregan.doroudiani@aei.mpg.de,
josh.drewitt@bristol.ac.uk, mhidding@phys.ethz.ch,
axel.kleinschmidt@aei.mpg.de, oliver.schlotterer@physics.uu.se,
Leila.Schneps@imj-prg.fr, bram.verbeek@physics.uu.se</footnote>
# Contents

<table><tbody><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>1.1</td><td>Summary of results</td><td>3</td></tr><tr><td>1.2</td><td>Outline</td><td>12</td></tr><tr><td>2</td><td>Review and basics</td><td>13</td></tr><tr><td>2.1</td><td>Iterated integrals of holomorphic modular forms</td><td>13</td></tr><tr><td>2.2</td><td>Modular graph forms and beyond</td><td>19</td></tr><tr><td>2.3</td><td>sl<sub>2</sub> representations and Tsunogai's derivations</td><td>25</td></tr><tr><td>2.4</td><td>Zeta generators associated with primitive zeta values</td><td>32</td></tr><tr><td>2.5</td><td>Generating series</td><td>34</td></tr><tr><td>3</td><td>Generating series of equivariant iterated Eisenstein integrals</td><td>37</td></tr><tr><td>3.1</td><td>The holomorphic frame</td><td>37</td></tr><tr><td>3.2</td><td>Modular graph forms from zeta generators</td><td>46</td></tr><tr><td>3.3</td><td>Reinstating iterated integrals of holomorphic cusp forms</td><td>55</td></tr><tr><td>3.4</td><td>Reformulation in terms of Laplace equations</td><td>64</td></tr><tr><td>4</td><td>Modular properties</td><td>72</td></tr><tr><td>4.1</td><td>T-equivariance</td><td>73</td></tr><tr><td>4.2</td><td>S-equivariance</td><td>75</td></tr><tr><td>5</td><td>Reformulation in terms of commutative X<sub>i</sub>, Y<sub>i</sub></td><td>79</td></tr><tr><td>5.1</td><td>equivariant iterated integrals at modular depths one and two</td><td>79</td></tr><tr><td>5.2</td><td>Modular completion at modular depth three</td><td>84</td></tr><tr><td>5.3</td><td>equivariant integrals at arbitrary modular depth</td><td>89</td></tr><tr><td>6</td><td>Conclusion and outlook</td><td>94</td></tr><tr><td>A</td><td>Multiple zeta values and f-alphabet</td><td>96</td></tr><tr><td>A.1</td><td>The f-alphabet</td><td>97</td></tr><tr><td>A.2</td><td>Single-valued MZVs</td><td>99</td></tr><tr><td>B</td><td>Changing from holomorphic to modular frame</td><td>101</td></tr><tr><td>C</td><td>Comparing M<sub>k<sub>1</sub>,k<sub>2</sub>,k<sub>3</sub></sub> with combinations of E and c<sup>sv</sup></td><td>102</td></tr><tr><td>C.1</td><td>(Anti-)meromorphic building blocks</td><td>103</td></tr><tr><td>C.2</td><td>Assembling E<sup>eqv</sup></td><td>104</td></tr><tr><td>C.3</td><td>Comparing M<sub>k</sub> and M<sub>k<sub>1</sub>,k<sub>2</sub></sub> with E<sup>eqv</sup></td><td>105</td></tr><tr><td>C.4</td><td>Comparing M<sub>k<sub>1</sub>,k<sub>2</sub>,k<sub>3</sub></sub> with E<sup>eqv</sup></td><td>107</td></tr><tr><td>D</td><td>Proof of Lemma 2</td><td>108</td></tr><tr><td>E</td><td>Zeta generators at modular depth three</td><td>111</td></tr><tr><td>E.1</td><td>[ẑ<sub>w</sub>, e<sub>k</sub>] beyond modular depth two</td><td>111</td></tr><tr><td>E.2</td><td>Expansion of zeta generators</td><td>112</td></tr></tbody></table>
# 1 Introduction

The low-energy expansion of closed-string genus-one amplitudes introduces non-holomorphic modular forms for SL(2, $\mathbb{Z}$) [1–4] known as *modular graph forms* (MGFs) [5, 6]. In the physics literature, the study of MGFs informed S-duality properties of low-energy interactions of type IIB superstrings [1–3, 7, 8] and manifested striking connections between open- and closed-string amplitudes [4, 5, 9, 10].

Due to the intriguing behaviour of MGFs both in their expansion at the cusp of the moduli space [2, 3, 5] and in their differential structures [3, 6], these objects have attracted considerable interest in the mathematics literature [11–16]. In particular, the mathematical properties of MGFs stimulated Brown’s construction of *equivariant iterated Eisenstein integrals* [11, 13, 14] whose explicit form and valuable implications for MGFs will be the main subject of this work.

The connection between MGFs and equivariant iterated Eisenstein integrals is indirect. Brown has shown [13, 14] how to define non-holomorphic modular forms as combinations of non-modular iterated integrals of holomorphic Eisenstein series and their complex conjugates. The main challenge in constructing non-holomorphic modular forms from iterated Eisenstein integrals is to spell out the required admixtures of *multiple zeta values* (MZVs) known from the expansion of MGFs around the cusp $\tau \to i\infty$ [5, 12, 15, 17–19]. For those iterated Eisenstein integrals compatible with the differential structure of MGFs, the existence of modular completions via *single-valued MZVs*<fnref n="1" /> was proven in [13, 14]. In [28], involving many of the present authors, this prescription was aligned with iterated-integral representations of MGFs that are based on certain generating-series methods [29, 30] originating from string theory.

In this work, we shall provide a unified explicit description of the occurrence of MZVs in the modular completions of iterated Eisenstein integrals to their equivariant versions and their connection to MGFs. One of the key themes of our analysis will be that for equivariant iterated Eisenstein integrals the appearance of products and higher-depth instances of MZVs can be reconstructed from that of Riemann zeta values by means of certain algebraic structures. This phenomenon is familiar from the low-energy expansion of string amplitudes at genus zero [22] which relies on a particular description of (motivic) MZVs [31–33] reviewed below. In the present genus-one context, the information about the Riemann zeta values in equivariant iterated Eisenstein integrals is encoded via certain Lie-algebra generators acting on the generating series. These so-called *zeta generators* are determined from genus-zero structures in a companion paper [34].

MGFs are related only to specific linear combinations of equivariant iterated Eisenstein integrals, namely those that follow from the so-called Tsunogai relations [35, 36], see [14, 28, 30, 37]. In fact, MGFs are expressible in terms of iterated integrals involving only Eisenstein series (and their complex conjugates) and in addition the periods arising in this construction are conjecturally restricted solely to single-valued MZVs.

By contrast, the modular completion of iterated Eisenstein integrals to *generic* equivariant iterated Eisenstein integrals (i.e. including cases beyond MGFs) necessitates iterated integrals of holomorphic cusp forms [11, 13]. Referring to the number of iterated integrals as *modular*

<footnote n="1">¹Single-valued MZVs arise as special values of single-valued multiple polylogarithms where all branch-cuts have been removed, see [20, 21] for the introduction of single-valued MZVs and [22–27] for their significance in relating closed-string tree amplitudes to those of open strings.</footnote>
depth, it was shown in [11, 13, 28] that (critical and non-critical) L-values associated with holomorphic cusp forms appear at modular depth two. In the present work, we shall extend this analysis to modular depth three and also encounter “hybrid” double integrals mixing holomorphic Eisenstein series with cusp forms as part of the modular completion.

As another feature that sets in at modular depth three, we find new periods beyond MZVs and L-values that are closely related to these hybrid double integrals and their associated so-called multiple modular values [11, 38].<fnref n="2" />

The construction of equivariant iterated Eisenstein integrals and the results of this work are most conveniently presented at the level of generating series. The absence of holomorphic cusp forms in MGFs [6, 29, 30] is implemented through the relations among the non-commutative bookkeeping variables $\epsilon_k$ in the generating series of equivariant iterated Eisenstein integrals introduced in [14].<fnref n="3" /> The underlying derivations $\epsilon_k$ with even $k \ge 0$ have a long history, encode deep connections between (motivic) fundamental groups, modular forms and mixed elliptic motives [35, 36, 39–48] and satisfy the Tsunogai relations mentioned above. In our discussion of generic equivariant iterated Eisenstein integrals beyond MGFs, we shall pass to a variant of the $\epsilon_k$ that we call $e_k$ and that obey no relations other than $(\mathrm{ad}_{e_0})^j e_k = 0$ if $j \ge k-1$.

The non-commutative bookkeeping variables $e_k$ without Tsunogai relations will be seen to govern the modular completions via holomorphic cusp forms. Specialising $e_k \to \epsilon_k$ within the generating series of equivariant iterated Eisenstein integrals then reduces it to the generating series of MGFs where all combinations of multiple modular values conspire to MZVs.

The three main results of the present work can be briefly summarised as:

* making all sources of MZVs in Brown's construction of equivariant iterated Eisenstein integrals fully explicit;

* inferring the appearance of arbitrary MZVs in equivariant iterated Eisenstein integrals from that of simple Riemann zeta values;

* exemplifying the modular completions of iterated Eisenstein integrals to equivariant modular-depth-three combinations featuring double integrals involving holomorphic cusp forms and new periods beyond MZVs or L-values.

## 1.1 Summary of results

The goal of this work is to construct non-holomorphic modular forms from iterated integrals of holomorphic modular forms for SL(2, $\mathbb{Z}$), their complex conjugates and real constants such as MZVs. More specifically, the integration kernels are given by the one-forms $\tau^j G_k(\tau) d\tau$ and $\tau^j \Delta_k(\tau) d\tau$ with holomorphic Eisenstein series,

$$ G_k(\tau) := \sum_{(m,n) \in \mathbb{Z}^2 \setminus \{(0,0)\}} \frac{1}{(m\tau + n)^k}, \quad k \ge 4, \qquad (1.1) $$

<footnote n="2">²The equivariant iterated Eisenstein integrals of modular depth three involving hybrid double integrals discussed in this work suggest that our results on MZVs apply also beyond MGFs.</footnote>

<footnote n="3">³The generating series of MGFs furnished by closed-string genus-one integrals [29, 30] conjecturally realise matrix representations of the non-commutative $\epsilon_k$ such that holomorphic cusp forms cancel automatically.</footnote>
and Hecke-normalised holomorphic cusp forms $\Delta_k(\tau)$ of modular weight $k \ge 12$.<fnref n="4" /> The integer exponents of $\tau^j$ in the integration kernels are taken to range over $0 \le j \le k-2$ to ensure that the one-forms are closed under modular SL(2, $\mathbb{Z}$) transformations. The latter act as usual on the variable $\tau$ on the upper half-plane by

$$ \gamma \cdot \tau := \frac{a\tau + b}{c\tau + d} \quad \text{for} \quad \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}(2, \mathbb{Z}) \qquad (1.2) $$

and with holomorphic weight $k$ on holomorphic modular forms, i.e.,

$$ G_k(\gamma \cdot \tau) = (c\tau+d)^k G_k(\tau), \quad \Delta_k(\gamma \cdot \tau) = (c\tau+d)^k \Delta_k(\tau). \qquad (1.3) $$

However, it is easy to see that already the simplest integrals $\int_{\tau}^{i\infty} d\tau_1 \tau_1^{j_1} G_{k_1}(\tau_1)$ transform inhomogeneously under SL(2, $\mathbb{Z}$). After a modular transformation $\tau \to \gamma \cdot \tau$, the change of integration variable $\tau_1 \to \gamma \cdot \tau_1$ needed to transform the lower integration limit from $\gamma \cdot \tau$ back to $\tau$, does not preserve the cusp $i\infty$ in the upper integration limit, i.e. in general $\gamma^{-1} \cdot i\infty \neq i\infty$. More specifically, the generators $T : \tau \mapsto \tau+1$ and $S : \tau \mapsto -1/\tau$ of SL(2, $\mathbb{Z}$) map iterated Eisenstein integrals

$$ \int_{\tau}^{i\infty} \tau_{\ell}^{j\ell} G_{k_{\ell}}(\tau_{\ell}) d\tau_{\ell} \int_{\tau_{\ell}}^{i\infty} \dots \int_{\tau_3}^{i\infty} \tau_2^{j_2} G_{k_2}(\tau_2) d\tau_2 \int_{\tau_2}^{i\infty} \tau_1^{j_1} G_{k_1}(\tau_1) d\tau_1, \qquad (1.4) $$

to combinations of similar iterated integrals over the same path $\int_{\tau}^{i\infty}$ and inhomogeneous terms over paths $\int_{i\infty-1}^{i\infty}$ in case of $T$ and $\int_{0}^{i\infty}$ in case of $S$, for instance

$$ S : \int_{\tau}^{i\infty} \tau_1^j G_k(\tau_1) d\tau_1 \to (-1)^j \left\{ \int_{\tau}^{i\infty} \tau_1^{k-j-2} G_k(\tau_1) d\tau_1 - \int_{0}^{i\infty} \tau_1^{k-j-2} G_k(\tau_1) d\tau_1 \right\}. \qquad (1.5) $$

The same kinds of inhomogeneous terms arise for modular transformations of the iterated integrals (1.4) with cusp forms $\Delta_{k_i}$ in the place of some of the $G_{k_i}$. For all of these integrals, the endpoint divergence at $i\infty$ is regularised through the tangential-base-point method [11] which in the simplest cases amounts to $\int_{\tau}^{i\infty} \tau_1^j d\tau_1 = -\frac{1}{j+1} \tau^{j+1}$ for $j \ge 0$.

The inhomogeneous terms in the modular transformation of the holomorphic integrals (1.4) pose a major challenge for combining them to modular forms. Adding the complex conjugates suffices to cancel the inhomogeneous terms over $\int_{i\infty-1}^{i\infty}$ from the modular $T$-transformation. However, the period integrals over $\int_{0}^{i\infty}$ from the modular $S$-transformation such as the last term in (1.5) turn out to be harder to cancel by further additions. These period integrals again take the form of (1.4) with $\tau \to 0$ (also in case of $G_{k_i} \to \Delta_{k_i}$) and are known as multiple modular values [11].

The existence of non-holomorphic modular forms of this type was proven in Brown's work on equivariant iterated Eisenstein integrals [11, 13, 14] that augment (1.4) by complex conjugates of the same class of integrals and suitably chosen real coefficients — MZVs, L-values of holomorphic cusp forms and more general periods.

<footnote n="4">⁴Here and later in this work, we abuse notation in that Hecke-normalised cusp forms $\Delta_k$ are not uniquely specified by their weight $k$. We write $\Delta_k$ for an arbitrary weight-$k$ representative to avoid cluttering notation, and in all cases appearing in the examples of this paper, the space of cusp forms is one-dimensional.</footnote>
### 1.1.1 Main results on equivariant iterated Eisenstein integrals

The MGFs in the low-energy expansion of genus-one closed-string amplitudes only involve the equivariant versions of a subset of the general iterated Eisenstein integrals presented in (1.4). Brown's work [11, 13, 14] organises the holomorphic integrals into generating series which are most compactly represented as path-ordered exponentials

$$ \mathbb{I}_+(\epsilon_k; \tau) = \mathrm{P}\text{-exp} \left( \int_{\tau}^{i\infty} \sum_{k=4}^{\infty} \sum_{j=0}^{k-2} \frac{(k-1)}{j!} (2\pi i)^{1+j-k} \tau_1^j G_k(\tau_1) d\tau_1 \epsilon_k^{(j)} \right), \quad (1.6) $$

$$ \tilde{\mathbb{I}}_-(\epsilon_k; \tau) = \tilde{\mathrm{P}}\text{-exp} \left( - \int_{\bar{\tau}}^{-i\infty} \sum_{k=4}^{\infty} \sum_{j=0}^{k-2} \frac{(k-1)}{j!} (2\pi i)^{1+j-k} \bar{\tau}_1^j \overline{G_k(\tau_1)} d\bar{\tau}_1 \epsilon_k^{(j)} \right), $$

without any holomorphic cusp forms $\Delta_k$, see the discussion around (3.6) for our ordering conventions and the meaning of the tilde for the second series. The non-commutative variables $\epsilon_k^{(j)} = (\mathrm{ad}_{\epsilon_0})^j \epsilon_k$ with $k \ge 4$ and $0 \le j \le k-2$, and where $\mathrm{ad}_{\epsilon_0}(X) := [\epsilon_0, X]$, in (1.6) are built from the Tsunogai derivations $\epsilon_k$ mentioned above. They obey a wealth of relations such as

$$ [\epsilon_4, \epsilon_{10}] - 3[\epsilon_6, \epsilon_8] = 0, \quad (1.7) $$

along with corollaries from acting on such identities with $\mathrm{ad}_{\epsilon_0}$ (noting that $\epsilon_k^{(k-1)} = 0$) or more general $\mathrm{ad}_{\epsilon_\ell}$. Given that the iterated Eisenstein integral (1.4) enters the generating series (1.6) with coefficient $\epsilon_{k_1}^{(j_1)} \epsilon_{k_2}^{(j_2)} \dots \epsilon_{k_\ell}^{(j_\ell)}$, the relation (1.7) eliminates from $\mathbb{I}_+(\epsilon_k; \tau)$ one linear combination of the double Eisenstein integrals $\int_{\tau}^{i\infty} G_{k_2}(\tau_2) d\tau_2 \int_{\tau_2}^{i\infty} G_{k_1}(\tau_1) d\tau_1$ at $k_1+k_2=14$. These kinds of dropouts are essential to project the multiple modular values in the modular $S$-transformations of (1.6) to the world of MZVs, e.g. eliminating two more involved periods $\Lambda(\Delta_{12}, 12)$ and $c(\Delta_{12}, 12)$ [38] present in the individual $\int_0^{i\infty} \tau_2^{j_2} G_{k_2}(\tau_2) d\tau_2 \int_{\tau_2}^{i\infty} \tau_1^{j_1} G_{k_1}(\tau_1) d\tau_1$ at $k_1+k_2=14$.

Relations such as (1.7) among Tsunogai's derivations still leave $\mathbb{Q}[2\pi i]$ combinations of MZVs as inhomogeneous terms in the modular transformations of the generating series (1.6). As a first step, one can combine holomorphic iterated Eisenstein integrals and their complex conjugates through the composition $\tilde{\mathbb{I}}_-(\epsilon_k; \tau)\mathbb{I}_+(\epsilon_k; \tau)$ which transforms homogeneously under $T$. However, the cancellation of multiple modular values under modular $S$-transformation requires admixtures of single-valued MZVs. This is achieved in Brown's equivariant generating series [14]

$$ \mathbb{I}^{\mathrm{eqv}}(\epsilon_k; \tau) = \hat{\psi}^{\mathrm{sv}}(\tilde{\mathbb{I}}_-(\epsilon_k; \tau)) \mathbb{C}^{\mathrm{sv}}(\epsilon_k) \mathbb{I}_+(\epsilon_k; \tau), \quad (1.8) $$

by adding two ingredients:

(i) a constant series $\mathbb{C}^{\mathrm{sv}}(\epsilon_k)$ in $\epsilon_{k_i}^{(j_i)}$ with $\mathbb{Q}$-linear combinations of single-valued MZVs as coefficients, and,

(ii) a change of alphabet $\hat{\psi}^{\mathrm{sv}}: \epsilon_k \to \epsilon_k + \dots$ replacing each derivation by similar infinite series in single-valued MZVs and $\epsilon_{k_i}^{(j_i)}$.

While the explicit form of $\hat{\psi}^{\mathrm{sv}}$ was proposed in [28] for MGFs, a main result of the present work is a similar all-order description of the series $\mathbb{C}^{\mathrm{sv}}(\epsilon_k)$ in (1.8).
The key object to make both $\hat{\psi}^{\text{sv}}$ and $\mathbb{C}^{\text{sv}}(\epsilon_k)$ explicit is the following group-like series in single-valued MZVs,

$$ \begin{aligned} \mathbb{M}^{\text{sv}}(z_i) &:= \sum_{\substack{\ell=0 \\ i_1, i_2, \dots, i_\ell \in 2\mathbb{N}+1}}^{\infty} \sum z_{i_1} z_{i_2} \dots z_{i_\ell} \rho^{-1}(\text{sv}(f_{i_1} f_{i_2} \dots f_{i_\ell})) \\ &= 1 + 2 \sum_{i_1 \in 2\mathbb{N}+1} z_{i_1} \zeta_{i_1} + 2 \sum_{i_1, i_2 \in 2\mathbb{N}+1} z_{i_1} z_{i_2} \zeta_{i_1} \zeta_{i_2} + \sum_{i_1, i_2, i_3 \in 2\mathbb{N}+1} z_{i_1} z_{i_2} z_{i_3} \rho^{-1}(\text{sv}(f_{i_1} f_{i_2} f_{i_3})) + \dots, \end{aligned} \quad (1.9) $$

where the isomorphism $\rho$, which maps (motivic) MZVs into the $f$-alphabet [32, 33], and the single-valued map sv, which acts on $f_{i_1} f_{i_2} \dots f_{i_\ell}$ [20, 21], are reviewed in appendix A. The realisation of $\hat{\psi}^{\text{sv}}$ in [28] involves derivations $\{z_w, w \in 2\mathbb{N}+1\}$ associated with odd zeta values similar to Tsunogai's $\epsilon_k$:

$$ \begin{aligned} \hat{\psi}^{\text{sv}}(\epsilon_k^{(j)}) &= \mathbb{M}^{\text{sv}}(z_i)^{-1} \epsilon_k^{(j)} \mathbb{M}^{\text{sv}}(z_i) \\ &= \sum_{\substack{\ell=0 \\ i_1, i_2, \dots, i_\ell \in 2\mathbb{N}+1}}^{\infty} \sum [[\dots [[\epsilon_k^{(j)}, z_{i_1}], z_{i_2}], \dots], z_{i_\ell}] \rho^{-1}(\text{sv}(f_{i_1} f_{i_2} \dots f_{i_\ell})). \end{aligned} \quad (1.10) $$

Given that all brackets $[z_w, \epsilon_k^{(j)}]$ are expressible via nested commutators of $\epsilon_{k_i}^{(j_i)}$ [36, 48] (see [34] and appendix E.1 for recent higher-order computations), each term in the series expansion of $\hat{\psi}^{\text{sv}}(\epsilon_k^{(j)})$ and therefore $\hat{\psi}^{\text{sv}}(\tilde{\mathbb{I}}_-(\epsilon_k; \tau))$ boils down to words in $\epsilon_k^{(j)}$.

In order to ensure that the series (1.9) is well-defined to all orders, canonical choices of the $f$-alphabet and dual canonical choices of the $z_w$ (as well as similar generators $\sigma_w, \hat{z}_w$ and $\hat{\sigma}_w$ below) are described in [34]. The choices in the reference for instance eliminate the ambiguity of redefining $z_{11}$ by rational multiples of $[z_3, [z_3, z_5]]$.

As a key result of the present work, the leftover ingredient $\mathbb{C}^{\text{sv}}(\epsilon_k)$ in (1.8) is identified as a product of group-like series (1.9) in single-valued MZVs,

$$ \mathbb{C}^{\text{sv}}(\epsilon_k) = \mathbb{M}^{\text{sv}}(z_i)^{-1} \mathbb{M}^{\text{sv}}(\sigma_i). \quad (1.11) $$

Besides the derivations $z_w$ in the expression (1.10) for $\hat{\psi}^{\text{sv}}$, our new representation (1.11) of $\mathbb{C}^{\text{sv}}(\epsilon_k)$ involves zeta generators $\{\sigma_w, w \in 2\mathbb{N}+1\}$. As summarised in section 2.4 and detailed in [34], the zeta generators $\sigma_w$ in (1.11) combine infinite series in nested brackets of $\epsilon_k^{(j)}$ with the above $z_w$

$$ \sigma_w = z_w - \frac{1}{(w-1)!} \epsilon_{w+1}^{(w-1)} + \text{series in nested brackets of } \epsilon_{k_i}^{(j_i)}. \quad (1.12) $$

The infinite tower of these nested brackets is computable to any desired order from the methods of [34] and follows from the interplay of configuration-space integrals at genus zero and genus one. Each $\sigma_w$ already involves infinitely many brackets of two $\epsilon_{k_i}^{(j_i)}$ which can be extracted from the closed-form expression (2.83). In principle, the expansion (1.12) of zeta generators leaves ambiguities to absorb certain types of nested $\epsilon_{k_i}^{(j_i)}$-brackets into redefinitions of $z_w$. A canonical choice for $z_w$ that fixes these ambiguities is presented in section 3.2.3 and at the end of section 3.3.4.

Thanks to the particular combination of $\mathbb{M}^{\text{sv}}$ in (1.11), all the $z_w$ which by themselves go beyond the algebra of $\epsilon_k^{(j)}$ conspire to nested brackets $[[\dots [[\epsilon_k^{(j)}, z_{i_1}], z_{i_2}], \dots], z_{i_\ell}]$. Since
the latter boil down to Lie-algebra valued words in $\epsilon_k^{(j)}$, the same is true for the series $\mathbb{C}^{\text{sv}}(\epsilon_k)$ in (1.11) as expected [14].

After combining (1.8) with (1.10) and (1.11), our end result for the generating series of equivariant iterated Eisenstein integrals can be compactly written as

$$ \mathbb{I}^{\text{eqv}}(\epsilon_k; \tau) = \mathbb{M}^{\text{sv}}(z_i)^{-1} \tilde{\mathbb{I}}_{-}(\epsilon_k; \tau) \mathbb{M}^{\text{sv}}(\sigma_i) \mathbb{I}_{+}(\epsilon_k; \tau), \quad (1.13) $$

making the sources of single-valued MZVs in Brown's construction [14] fully explicit. By the above expressions for the series on the right-hand side, the coefficients $\mathcal{E}^{\text{eqv}}[\begin{matrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{matrix}; \tau]$ of $\epsilon_k^{(j_1)} \dots \epsilon_k^{(j_\ell)}$ in (1.13) are combinations of holomorphic iterated Eisenstein integrals (1.4) and their complex conjugates with $\mathbb{Q}$-linear combinations of single-valued MZVs as coefficients.

The series $\mathbb{M}^{\text{sv}}(z_i)^{-1}$ and $\mathbb{M}^{\text{sv}}(\sigma_i)$ in single-valued MZVs ensure that the $\mathcal{E}^{\text{eqv}}[\begin{matrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{matrix}; \tau]$ transform homogeneously under the generators $T$ and $S$ of the modular group, as seen in (3.27). More precisely, these modular transformations match those of the (product of) integration kernels $\tau_i^{j_i} G_{k_i} d\tau_i$ paired with words in $\epsilon_{k_i}^{(j_i)}$ via (1.6). That is why the $\mathcal{E}^{\text{eqv}}[\begin{matrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{matrix}; \tau]$ are referred to as equivariant iterated Eisenstein integrals. The sum $k_1 + \dots + k_\ell$ will be called their degree.

However, by the wealth of relations among the $\epsilon_{k_i}^{(j_i)}$ starting with (1.7), not all the $\mathcal{E}^{\text{eqv}}[\begin{matrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{matrix}; \tau]$ with $k_i \in 2\mathbb{N}+2$ and $0 \le j_i \le k_i-2$ appear independently in (1.13). Instead, the specific relation (1.7) implies that only three linear combinations of the four independent $\mathcal{E}^{\text{eqv}}[\begin{matrix} 0 & 0 \\ 4 & 10 \end{matrix}; \tau]$, $\mathcal{E}^{\text{eqv}}[\begin{matrix} 0 & 0 \\ 10 & 4 \end{matrix}; \tau]$, $\mathcal{E}^{\text{eqv}}[\begin{matrix} 0 & 0 \\ 6 & 8 \end{matrix}; \tau]$, $\mathcal{E}^{\text{eqv}}[\begin{matrix} 0 & 0 \\ 8 & 6 \end{matrix}; \tau]$ are accessible from (1.13), and similar dropouts occur at all higher degrees $k_1 + \dots + k_\ell \ge 16$. A separate line of results of this paper, to be summarised in section 1.1.3, is to fix this shortcoming of the $\epsilon_k^{(j)}$-valued series (1.13) and explicitly determine the individual $\mathcal{E}^{\text{eqv}}[\begin{matrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{matrix}; \tau]$ in a wide range of the $k_i$.

### 1.1.2 Connections with MGFs and genus zero

The modular properties of equivariant iterated Eisenstein integrals do not yet line up with the transformation law of modular forms, e.g. $\mathcal{E}^{\text{eqv}}[\begin{matrix} j \\ k \end{matrix}; \tau]$ is not $T$ invariant for $j > 0$ (see (3.27)). Still, one can systematically generate linear combinations

$$ \beta^{\text{eqv}}[\begin{matrix} j & j & \dots & j \\ \dots & k & \dots & k \end{matrix}] = \sum_{p=0}^{k-2-j} \sum_{\ell=0}^{j+p} \binom{k-j-2}{p} \binom{j+p}{\ell} \frac{(-2\pi i \bar{\tau})^{\ell}}{(4\pi \text{Im } \tau)^p} \mathcal{E}^{\text{eqv}}[\begin{matrix} \dots & j-\ell+p & \dots \\ \dots & k & \dots \end{matrix}], \quad (1.14) $$

(with a similar double sum over $p_i$ and $\ell_i$ for each column $\begin{matrix} j_i \\ k_i \end{matrix}$)<fnref n="5" /> which furnish modular forms of weight $(0, \sum_{i=1}^{\ell}(k_i - j_i - 2))$,

$$ \beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{matrix}; \frac{a\tau+b}{c\tau+d}] = \left( \prod_{i=1}^{\ell} (c\bar{\tau}+d)^{k_i-2j_i-2} \right) \beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{matrix}; \tau]. \quad (1.15) $$

$$ \beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}] = \sum_{p_1=0}^{k_1-2-j_1} \sum_{\ell_1=0}^{j_1+p_1} \sum_{p_2=0}^{k_2-2-j_2} \sum_{\ell_2=0}^{j_2+p_2} \binom{k_1-j_1-2}{p_1} \binom{k_2-j_2-2}{p_2} \binom{j_1+p_1}{\ell_1} \binom{j_2+p_2}{\ell_2} \\ \times \frac{(-2\pi i \bar{\tau})^{\ell_1+\ell_2}}{(4\pi \text{Im } \tau)^{p_1+p_2}} \mathcal{E}^{\text{eqv}}[\begin{matrix} j_1-\ell_1+p_1 & j_2-\ell_2+p_2 \\ k_1 & k_2 \end{matrix}]. $$

<footnote n="5">⁵For the case of two columns (1.14) reads explicitly</footnote>
An implementation of the linear combinations (1.14) at the level of generating series via simple SL$_2(\mathbb{C})$ transformations is described in section 3.1. Alternatively, the linear combinations (1.14) arise from the organisation of equivariant iterated Eisenstein integrals via polynomials in commutative bookkeeping variables $X_i, Y_i$ by rewriting powers $X^a Y^b$ with $a, b \in \mathbb{N}_0$ in terms of $(X-\tau Y)$ and $(X-\bar{\tau}Y)$ [13], see section 5.

As detailed in [28], MGFs are linear combinations of the non-holomorphic modular forms $\beta^{\text{eqv}}$ in (1.14), conjecturally with $\mathbb{Q}$-linear combinations of single-valued MZVs as coefficients. This follows from the differential equations of the generating series of closed-string genus-one integrals studied in [29, 30] which contain all MGFs in their low-energy expansion and involve conjectural matrix representations of the above derivations $\epsilon_k$.

The series $M^{\text{sv}}(z_i)$ and $M^{\text{sv}}(\sigma_i)$ defined by (1.9) entering the generating series (1.13) introduce single-valued MZVs into $\mathcal{E}^{\text{eqv}}$ and thereby into $\beta^{\text{eqv}}$. This offers a systematic way of generating the (conjecturally single-valued) MZVs in the expansion of MGFs at the cusp $\tau \to i\infty$ [5, 12]$^6$ which provided the key motivation for their further study in the mathematics literature [13–16]. As a particular virtue of the group-like series $M^{\text{sv}}$, the coefficients $z_w, \sigma_w$ of odd Riemann zeta values $\zeta_w$ determine those of all other MZVs composed of two or more letters in the $f$-alphabet via compositions of zeta generators. This interlocks products $2\zeta_{2m+1}\zeta_{2n+1} = \rho^{-1}(\text{sv}(f_{2m+1}f_{2n+1}))$ or (conjecturally) indecomposable higher-depth MZVs (starting with $\zeta_{3,3,5}^{\text{sv}}$ involving up to three letters $f_i$) to the appearance of $\zeta_w$ in equivariant iterated Eisenstein integrals.

The pattern in the expansion (1.9) of $M^{\text{sv}}$ where MZVs $\rho^{-1}(f_{i_1}f_{i_2}\dots)$ are accompanied by compositions $M_{i_1}M_{i_2}\dots$ of certain operators $M_i$ was observed earlier on in the low-energy expansion of string tree-level amplitudes [22, 49]. In these references, for the case of the configuration-space integrals at genus zero, the $M_{i\in 2\mathbb{N}+1}$ are matrices whose products determine the coefficients of all MZVs at arbitrary depth from those of Riemann zeta values. Hence, our result (1.13) that imports all MZVs in equivariant iterated Eisenstein integrals from series $M^{\text{sv}}$ generalises these elegant structures of genus-zero integrals to genus one.

Another echo of genus-zero structures in the genus-one results of this work can be found in the generating series of single-valued iterated Eisenstein integrals [14]

$$ \mathbb{I}^{\text{sv}}(\epsilon_k; \tau) = \mathbb{M}^{\text{sv}}(\sigma_i)^{-1} \tilde{\mathbb{I}}_{-}(\epsilon_k; \tau) \mathbb{M}^{\text{sv}}(\sigma_i) \mathbb{I}_{+}(\epsilon_k; \tau), \quad (1.16) $$

with $\mathbb{M}^{\text{sv}}(\sigma_i)^{-1}$ instead of $\mathbb{M}^{\text{sv}}(z_i)^{-1}$ as the leftmost factor on the right-hand side. In contrast to their equivariant counterparts in (1.13), the coefficients of $\epsilon_{k_1}^{(j_1)}\epsilon_{k_2}^{(j_2)}\dots\epsilon_{k_\ell}^{(j_\ell)}$ in $\mathbb{I}^{\text{sv}}(\epsilon_k; \tau)$ transform inhomogeneously under both $S$ and $T$. At the same time, the single-valued iterated Eisenstein integrals in (1.16) are canonically defined and do not depend on specific choices in the definition of $\mathbb{I}^{\text{eqv}}(\epsilon_k; \tau)$ (see [14] and section 3.2.3 for the ambiguities in the definition of $\mathbb{I}^{\text{eqv}}(\epsilon_k; \tau)$ and [34] for a preferable way of fixing them). Moreover, the conjugation of the antiholomorphic generating series in (1.16) by $\mathbb{M}^{\text{sv}}(\sigma_i)$ is in one-to-one correspondence to a similar conjugation formula at genus zero [50]

$$ \mathbb{G}_{\{0,1\}}^{\text{sv}}(x_i; z) = \mathbb{M}^{\text{sv}}(M_i)^{-1} \tilde{\mathbb{G}}_{\{0,1\}}(x_i; z) \mathbb{M}^{\text{sv}}(M_i) \mathbb{G}_{\{0,1\}}(x_i; z), \quad (1.17) $$

<footnote>$^6$See for instance [2, 3] for earlier results on the expansion of MGFs around the cusp and [15, 18, 30] for closed-form results on the MGFs in two-point genus-one integrals.</footnote>
for the generating series $\mathbb{G}_{\{0,1\}}^{\text{sv}}(x_i; z)$ of single-valued polylogarithms in one variable [51]. The series $\mathbb{G}_{\{0,1\}}(x_i; z)$ and $\tilde{\mathbb{G}}_{\{0,1\}}(x_i; z)$ on the right-hand side are path-ordered exponentials of the Knizhnik-Zamolodchikov connection $(\frac{x_0}{z} + \frac{x_1}{z-1})dz$ similar to (1.6). The operators $M_i$ in the series $\mathbb{M}^{\text{sv}}$ of (1.17) now refer to zeta generators at genus zero<fnref n="7" /> which normalise the braid operators $x_0$ and $x_1$ [52, 53]. Similar to the reasoning below (1.10), $\mathbb{M}^{\text{sv}}(M_i)^{-1}\tilde{\mathbb{G}}_{\{0,1\}}(x_i; z)\mathbb{M}^{\text{sv}}(M_i)$ is expressible in terms of $x_0, x_1$ after iteratively inserting commutators such as $[x_0, M_w] = 0$ or $[x_1, M_3] = [[x_0, x_1], x_0+x_1], x_1]$, also see the discussion of section 3.2.4.

Hence, our explicit description of the series of single-valued MZVs in the equivariant generating series (1.13) manifests an important analogy between genus zero and genus one: the construction of single-valued iterated Eisenstein integrals (1.16) at genus one closely follows that of single-valued genus-zero polylogarithms (1.17) in one variable. In both cases, the zeta generators in the conjugations of the antiholomorphic series need to be adapted to the variables $x_i$ or $\epsilon_k^{(j)}$ relevant to genus zero or one, respectively.

### 1.1.3 Main results beyond MGFs

Another body of results in this work concerns the generalisations of MGFs to non-holomorphic modular forms involving iterated integrals of (anti-)holomorphic cusp forms. Our starting point is to uplift the Tsunogai derivations $\epsilon_k^{(j)}$ in the path-ordered exponential (1.6) to a free algebra of symbols $e_k^{(j)} = (\text{ad}_{e_0})^j e_k$ with $k \ge 4$ and $0 \le j \le k-2$,

$$ \mathbb{I}_+(e_k; \tau) = \text{P-exp} \left( \int_{\tau}^{i\infty} \sum_{k=4}^{\infty} \sum_{j=0}^{k-2} \frac{(k-1)}{j!} (2\pi i)^{1+j} \tau_1^j \left[ \frac{\mathbb{G}_k(\tau_1)}{(2\pi i)^k} e_k^{(j)} + \sum_{\Delta \in S_k} \Delta(\tau_1) e_{\Delta^+}^{(j)} \right] d\tau_1 \right), \quad (1.18) $$

$$ \tilde{\mathbb{I}}_-(e_k; \tau) = \tilde{\text{P-exp}} \left( - \int_{\bar{\tau}}^{-i\infty} \sum_{k=4}^{\infty} \sum_{j=0}^{k-2} \frac{(k-1)}{j!} (2\pi i)^{1+j} \bar{\tau}_1^j \left[ \frac{\overline{\mathbb{G}_k(\tau_1)}}{(2\pi i)^k} e_k^{(j)} + \sum_{\Delta \in S_k} \overline{\Delta(\tau_1)} e_{\Delta^-}^{(j)} \right] d\bar{\tau}_1 \right). $$

The admixtures of holomorphic cusp forms are needed to find an equivariant completion of the additional iterated Eisenstein integrals in $\mathbb{I}_+(e_k; \tau)$ that are absent from $\mathbb{I}_+(e_k; \tau)$ due to Tsunogai's relations. The vanishing combinations of $\epsilon_k^{(j)}$ such as (1.7) translate into non-zero combinations of the free-algebra generators such as $P_{14}^2 \sim [e_4, e_{10}] - 3[e_6, e_8]$. More general combinations $P_w^d$ of this type described in section 2.3.2 appear as coefficients of various periods beyond MZVs in the modular $S$ transformations of the iterated Eisenstein integrals in $\mathbb{I}_+(e_k; \tau)$. The role of the cusp forms in (1.18) is to compensate many of these new periods which necessitates relations between the letters $e_{\Delta^\pm}$ and the combinations $P_w^d$ of $e_k^{(j)}$, for instance

$$ e_{\Delta_{12}^{\pm}} = \mp \frac{\Lambda(\Delta_{12}, 12)}{\Lambda(\Delta_{12}, 10)} P_{14}^2 + \frac{\Lambda(\Delta_{12}, 13)}{\Lambda(\Delta_{12}, 11)} P_{16}^3 + \dots, \quad (1.19) $$

with an infinite tower of $P_w^d$ and their brackets with $e_k^{(j)}$ in the ellipsis. As detailed in section 3.3.2, terms of higher degree in the expansion of $e_{\Delta^\pm}$ involve new periods, beyond ratios of L-values $\Lambda(\Delta_{12}, t)$ defined by (2.19). Most importantly, the goal of constructing an

<footnote n="7">⁷In slight abuse of notation, we employ the same symbol $M_i$ for the zeta generators in (1.17) and their matrix representations relevant to the aforementioned string tree-level amplitudes.</footnote>
equivariant series from $\mathbb{I}_+(e_k; \tau)$ and $\tilde{\mathbb{I}}_-(e_k; \tau)$ in (1.18) completely reduces the letters $e_{\Delta\pm}$ associated with holomorphic cusp forms to infinite series in the Eisenstein letters $e_k^{(j)}$.

The most ambitious claim of this work concerns the equivariant completion of the series (1.18) of iterated integrals of (anti-)holomorphic modular forms. The key idea is to uplift each ingredient in the series (1.8) of Eisenstein integrals with generators $\epsilon_k^{(j)}$ (subject to relations such as (1.7)) to the free algebra of $e_k^{(j)}$ via<fnref n="8" />

$$ \mathbb{I}^{\text{eqv}}(e_k; \tau) = \mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_{\bar{\omega}})^{-1} \tilde{\mathbb{I}}_-(e_k; \tau) \mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}}) \mathbb{I}_+(e_k; \tau). \quad (1.20) $$

Apart from the augmentation (1.18) of the (anti-)holomorphic generating series $\mathbb{I}_{\pm}$ by cusp forms, also the series $\mathbb{M}^{\text{sv}}$ in (1.9) combining single-valued MZVs with zeta generators requires an uplift in passing from $\epsilon_k^{(j)}$ to $e_k^{(j)}$. The series

$$ \mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}}) = \mathbb{M}^{\text{sv}}(\hat{\sigma}_i) + \dots, \quad \mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_{\bar{\omega}}) = \mathbb{M}^{\text{sv}}(\hat{z}_i) + \dots \quad (1.21) $$

in (1.20) first of all contain the single-valued MZVs of (1.9), where the associated zeta generators $\sigma_i \to \hat{\sigma}_i$ and $z_i \to \hat{z}_i$ are adapted to $\epsilon_k^{(j)} \to e_k^{(j)}$ as described in section 2.4.2. The extra terms in the ellipsis of (1.21) comprise new periods beyond MZVs that are collectively denoted by $\bar{\omega}$, see section 3.3.3 for the simplest examples.

Similar to the letters $e_{\Delta\pm}$ multiplying cusp forms in (1.18), the generators $\hat{\sigma}_{\bar{\omega}} = \hat{z}_{\bar{\omega}} + \dots$ associated with the new periods (see [38] for the association of generators with general primitive periods) are needed to ensure that the right-hand side of (1.20) is expressible via words in $e_k^{(i)}$. We expect that both the terms $\hat{\sigma}_{\bar{\omega}} - \hat{z}_{\bar{\omega}}$, usually referred to as the *geometric part* of $\hat{\sigma}_{\bar{\omega}}$, and the brackets $[\hat{z}_{\bar{\omega}}, e_k^{(j)}]$ boil down to nested brackets in $e_k^{(j)}$. Moreover, both of $\hat{\sigma}_{\bar{\omega}} - \hat{z}_{\bar{\omega}}$ and $[\hat{z}_{\bar{\omega}}, e_k^{(j)}]$ have to vanish in the specialisation $e_k^{(j)} \to \epsilon_k^{(j)}$ in order to recover the generating series (1.13) relevant to MGFs when adapting (1.20) to Tsunogai's derivation algebra. Hence, similar to the expansion of $e_{\Delta\pm}$ exemplified in (1.19), both of $\hat{\sigma}_{\bar{\omega}} - \hat{z}_{\bar{\omega}}$ and $[\hat{z}_{\bar{\omega}}, e_k^{(j)}]$ are naturally expressed in terms of the combinations $P_w^d$ that vanish upon $e_k^{(j)} \to \epsilon_k^{(j)}$, see for instance (3.76) and (3.77).

In practice, the expansions of $e_{\Delta\pm}$ as well as $\mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}})$ and $[\hat{z}_{\bar{\omega}}, e_k^{(j)}]$ in terms of $e_k^{(i)}$ is determined by imposing modularity of the series (1.20) at each modular depth and degree: following the discussion below (1.13), the coefficients $\mathcal{E}^{\text{eqv}}[j_1 \dots j_\ell; \tau]$ of $e_{k_1}^{(j_1)} \dots e_{k_\ell}^{(j_\ell)}$ in (1.20) are now individually accessible by the absence of bracket relations among the $e_k^{(j)}$. By adding suitable brackets of $e_k^{(j)}$ to the expressions for $e_{\Delta\pm}$, $\mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}})$ and $[\hat{z}_{\bar{\omega}}, e_k^{(j)}]$, these coefficients $\mathcal{E}^{\text{eqv}}$ are made to transform equivariantly under SL(2, $\mathbb{Z}$). Equivalently, their linear combinations $\beta^{\text{eqv}}$ defined by (1.14) are imposed to be modular forms.

The conditions for the modular properties (1.15) of $\beta^{\text{eqv}}[j_1 \dots j_\ell; \tau]$ are investigated separately for each choice $(k_1, \dots, k_\ell)$. At modular depth $\ell = 2$, the known results for $\beta^{\text{eqv}}[j_1 \ j_2 \ \dots \ j_\ell; \tau]$ [28] are translated into contributions $[e_{k_1}^{(j_1)}, e_{k_2}^{(j_2)}]$ to $e_{\Delta\pm}$ with ratios of L-values $\Lambda(\Delta, t)$ as coefficients. At modular depth $\ell = 3$, modularity of $\beta^{\text{eqv}}[j_1 \ j_2 \ j_3 \ \dots \ j_\ell; \tau]$ up to and including degree $k_1+k_2+k_3 = 20$ is used to infer contributions $[e_{k_1}^{(j_1)}, [e_{k_2}^{(j_2)}, e_{k_3}^{(j_3)}]]$ to both $e_{\Delta\pm}$

<footnote n="8">⁸The first part of this expression also realises an uplift of $\hat{\psi}^{\text{sv}}$ to generating series in the $e_k^{(j)}$; equation (1.8) only applies to expressions series in $\epsilon_k^{(j)}$.</footnote>
and $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\varpi)$, identifying new periods beyond MZVs and L-values in both cases. The contributions $\hat{z}_\varpi$ are usually referred to as the arithmetic part of the new generators $\hat{\sigma}_\varpi$ and do not yet enter the $\beta^{\text{eqv}}\begin{bmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{bmatrix}; \tau$ at any degree $k_1+k_2+k_3$. Instead, their leading contributions are easily anticipated from T-invariance at modular depth $\ell = 4$, see sections 3.3.4 and 4.2.

By assembling the modular forms $\beta^{\text{eqv}}$ from the coefficients in the generating series (1.20), we obtain explicit examples of the fact that the known modular completions $\beta^{\text{eqv}}\begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix}; \tau$ of double Eisenstein integrals [28] solely involve:<fnref n="9" />

* rational combinations of odd zeta values (possibly multiplied by Eisenstein integrals of modular depth one) and products $\zeta_{2a+1}\zeta_{2b+1}$;

* cusp form integrals over $\tau_1^j \Delta_k(\tau_1)$ or $\bar{\tau}_1^j \bar{\Delta}_k(\tau_1)$ along with rational multiples of $\frac{\Lambda(\Delta_k, t_1)}{\Lambda(\Delta_k, t_2)}$, where the L-values in the numerator are non-critical with $t_1 \in \mathbb{N}$ and $t_1 \ge k$ and those in the denominator are critical with $t_2 \in \{k-1, k-2\}$.<fnref n="10" />

More importantly, we pinpoint a variety of new ingredients in the modular completions $\beta^{\text{eqv}}\begin{bmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{bmatrix}; \tau$ of triple Eisenstein integrals:

* indecomposable single-valued MZVs $\rho^{-1}(\text{sv}(f_{2a+1}f_{2b+1}f_{2c+1}))$ beyond depth one in the constant terms (i.e. without any accompanying $\tau_i$ integrals) at degree $k_1+k_2+k_3 \ge 14$;

* new periods $\varpi$ associated with the new generators $\hat{\sigma}_\varpi$ discussed below (1.20) in the constant terms at degree $k_1+k_2+k_3 \ge 18$;

* double integrals involving an Eisenstein series and a holomorphic cusp form with $\mathcal{Q}[\frac{\Lambda(\Delta_k, t_1)}{\Lambda(\Delta_k, t_2)}]$ coefficients at degree $k_1+k_2+k_3 \ge 18$ (where again $\mathbb{N} \ni t_1 \ge k$ and $t_2 \in \{k-1, k-2\}$);

* additional new periods along with the integrals of a single cusp form at $k_1+k_2+k_3 \ge 18$.

Up to and including the coefficients of $e_{k_1}^{(j_1)} e_{k_2}^{(j_2)} e_{k_3}^{(j_3)}$ at degree $k_1+k_2+k_3=20$, our results for the generating series (1.20) of equivariant iterated Eisenstein integrals are confirmed both analytically and numerically. At higher degree or modular depth, equivariance is contingent on the relation (4.44) interlocking multiple modular values, the $\epsilon_k^{(j)} \to e_k^{(j)}$ uplift of zeta generators as well as the extension of $M^{\text{sv}}$ by new periods and generators $\varpi$ and $\hat{\sigma}_\varpi$. In section 5.3, we prove the existence of modular completions of iterated Eisenstein integrals over $\tau_i^{j_i} G_{k_i} d\tau_i$ of arbitrary degree and modular depth.

Nevertheless, it will be important to determine the explicit form of $\beta^{\text{eqv}}$ of modular depth $\ge 4$ in future work in order to gather information on the structure of the letters $e_{\Delta\pm}$ and the series $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\varpi)$: First, it will be interesting to understand the systematics of the new periods encountered at different modular depths and degrees. Second, by analogy with the group-like series (1.9) of MZVs and zeta generators, one may expect contributions of the form $\hat{\sigma}_i \hat{\sigma}_\varpi$ or $\hat{\sigma}_\varpi \hat{\sigma}_i$ to $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\varpi)$. It will be interesting to investigate these terms $\hat{\sigma}_i \hat{\sigma}_\varpi$ or $\hat{\sigma}_\varpi \hat{\sigma}_i$ from the modularity conditions for $\beta^{\text{eqv}}\begin{bmatrix} j_1 & j_2 & j_3 & j_4 \\ k_1 & k_2 & k_3 & k_4 \end{bmatrix}; \tau$ and to relate their coefficients

<footnote n="9">⁹In the following we do not list separately the fact that the coefficients contain rational numbers and factors of $2\pi i\bar{\tau}$ and $\pi \operatorname{Im}\tau$, see for example (1.14).</footnote>

<footnote n="10">¹⁰The restriction to these integer values in the critical strip for $t_2$ is possible due to Manin's theorem [54].</footnote>
to $f$-alphabet descriptions of $\zeta_i$ and $\bar{\omega}$. Third, it remains to prove our conjecture that the MZV sector of $\mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}})$ in (1.21) follows the group-like series (1.9) which is so far checked up to including modular depth three and degree 20.

## 1.2 Outline

The remainder of this work will elaborate on the ideas and results summarised in the previous section, supported by a multitude of examples. In section 2, we start by reviewing the key ingredients of our later construction of non-holomorphic modular forms and by fixing conventions on iterated integrals as well as the non-commuting variables in their generating series. Our new results on generating series of equivariant iterated Eisenstein integrals are presented in section 3, with a discussion of $\text{SL}_2$ frames for the non-commutative variables in section 3.1, an all-order account on the series in MGFs and Tsunogai derivations $\epsilon_k^{(j)}$ in section 3.2 and their extension to include holomorphic cusp in section 3.3 (focusing on modular depth three). In the same way as equivariant iterated Eisenstein integrals of modular depth two [11, 13] were made explicit [28] on the basis of Laplace eigenfunctions $F_{m,k}^{\pm(s)}$ [37], their counterparts $F_{m,k,\ell}^{\pm(s)}$ at modular depth three are briefly discussed in section 3.4. Section 4 provides a conditional proof for the modular properties of the equivariant generating series (1.20) where the key assumption is inductively proven in section 5. A variety of appendices contain background information on MZVs, explicit results on zeta generators and proofs of important lemmas.

The arXiv submission of this work is accompanied by several ancillary files which include:

• a Mathematica implementation (with examples) of the $sl_2$ projectors $t_{p,q}^d$ and $s_{p,q}^d$, introduced in section 2.3.1;

• all Pollack combinations $P_d^w$ of degree $w \le 20$ and $d \le 5$, discussed in section 2.3.2;

• uplifted zeta generators $\hat{\sigma}_w$ up to degree 20 and arithmetic relations up modular depth $\le 3$ and degree $\le 21$, as discussed in section 2.4.2;

• expressions for all $\beta^{\text{eqv}}$ at modular depth $\le 3$ and degree $\le 20$, and relevant new periods $\bar{\omega}$, $\Lambda$ introduced in section 3.3 typically to 300 digits of precision;

• the coefficients $c^{\text{sv}}$ of the generating series (3.15) in single-valued MZVs and new periods for modular depth $\le 3$ and degree $\le 20$;

• all modular invariant solutions $F_{m,k,\ell}^{\pm(s)}$ with degree $2m + 2k + 2\ell \le 20$ of inhomogeneous Laplace eigenvalue equations expressed in terms of $\beta^{\text{eqv}}$, as discussed in section 3.4;

• explicit isomorphism between single-valued MZVs and the $f$-alphabet following the method of [34] (also see appendix A) as well as the dictionary between MZVs and their single-valued versions — both up to weight 17.
## 2 Review and basics

In this section, we review the basic constructions appearing in this paper. A key ingredient are iterated integrals of holomorphic or anti-holomorphic modular forms of SL(2, $\mathbb{Z}$) which we introduce in section 2.1. A recap of modular graph forms and more general non-holomorphic modular forms can be found in section 2.2. The Tsunogai derivation algebra is briefly reviewed in section 2.3 while in section 2.4 we present derivations related to zeta values or zeta generators in short. Both types of derivations enter the generating series of real-analytic modular forms that we review in section 2.5.

### 2.1 Iterated integrals of holomorphic modular forms

Iterated integrals of modular forms have become a driving force in numerous recent developments in particle physics [55, 56] and string perturbation theory [57–59]. Our conventions here follow most closely the ones used in [28, 30].

#### 2.1.1 Definitions

We define holomorphic Eisenstein series $G_k$ for $k \ge 4$ an even integer by

$$G_k(\tau) := \sum_{(m,n) \in \mathbb{Z}^2 \setminus \{(0,0)\}} \frac{1}{(m\tau+n)^k} = 2\zeta_k + \frac{2(2\pi i)^k}{(k-1)!} \sum_{n=1}^{\infty} \sigma_{k-1}(n)q^n \quad (2.1)$$

with $\tau$ on the upper half plane $\mathbb{H} := \{\tau \in \mathbb{C} | \mathrm{Im}(\tau) > 0\}$ and $q := e^{2\pi i \tau}$. The Fourier coefficients in (2.1) are divisor sums $\sigma_{k-1}(n) := \sum_{d|n} d^{k-1}$. A relation between the zeta values $2\zeta_k$ and the Bernoulli numbers $B_k$ for even $k$ that we will make frequent use of is

$$2\zeta_k = -\mathrm{BF}_k(2\pi i)^k, \quad \mathrm{BF}_k := \frac{B_k}{k!}, \qquad (2.2)$$

introducing a factorially rescaled version $\mathrm{BF}_k$ of Bernoulli numbers $B_k$, which will appear quite often in the rest of the paper.

As already stated in (1.3), under a modular transformation $\gamma \in \mathrm{SL}(2, \mathbb{Z})$ we have

$$G_k(\gamma \cdot \tau) = (c\tau+d)^k G_k(\tau), \qquad (2.3)$$

which means that $G_k$ is a modular form of (holomorphic/anti-holomorphic) weight $(k, 0)$. More generally, a non-holomorphic modular form of weight $(w, \bar{w})$ will transform similarly with factors $(c\tau+d)^w (c\bar{\tau}+d)^{\bar{w}}$ and we call $w$ and $\bar{w}$ its holomorphic and anti-holomorphic modular weights, respectively. We recall that the modular group $\mathrm{SL}(2, \mathbb{Z})$ is generated by the two transformations $T : \tau \mapsto \tau+1$ and $S : \tau \mapsto -1/\tau$.

We will be interested in iterated integrals of holomorphic modular forms, starting from the integration kernels

$$\nu\begin{bmatrix} j \\ k \\ \tau \end{bmatrix} := (2\pi i)^{1+j-k} \tau^j G_k(\tau) \, d\tau, \qquad \overline{\nu\begin{bmatrix} j \\ k \\ \tau \end{bmatrix}} := (-1)^{j+1} (2\pi i)^{1+j-k} \bar{\tau}^j \overline{G_k(\tau)} \, d\bar{\tau}, \quad (2.4)$$

for which it is very easy using (2.3) to compute the two $\mathrm{SL}(2, \mathbb{Z})$ transformations

$$\nu\begin{bmatrix} j \\ k \\ \tau+1 \end{bmatrix} = \sum_{p=0}^{j} \binom{j}{p} (2\pi i)^{j-p} \nu\begin{bmatrix} p \\ k \\ \tau \end{bmatrix}, \qquad (2.5)$$

$$\nu\begin{bmatrix} j \\ k \\ -\frac{1}{\tau} \end{bmatrix} = (-1)^j (2\pi i)^{2+2j-k} \nu\begin{bmatrix} k-2-j \\ k \\ \tau \end{bmatrix}. \qquad (2.6)$$
From these kernels we construct homotopy invariant iterated integrals, given by the recursive definition

$$ 
\begin{aligned}
\mathcal{E}^{[\begin{smallmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{smallmatrix}; \tau]} &:= \int_{\tau}^{i\infty} \nu^{[\begin{smallmatrix} j_\ell \\ k_\ell \end{smallmatrix}; \tau_\ell]} \cdots \int_{\tau_3}^{i\infty} \nu^{[\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}; \tau_2]} \int_{\tau_2}^{i\infty} \nu^{[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}; \tau_1]} \\
&= (2\pi i)^{1+j_\ell-k_\ell} \int_{\tau}^{i\infty} \tau_\ell^{j_\ell} G_{k_\ell}(\tau_\ell) \mathcal{E}^{[\begin{smallmatrix} j_1 & \dots & j_{\ell-1} \\ k_1 & \dots & k_{\ell-1} \end{smallmatrix}; \tau_\ell]} \, d\tau_\ell, \quad (2.7)
\end{aligned}
$$

with the convention that $\mathcal{E}[\emptyset; \tau] = 1$. Here, we restrict the integers $j_i$ to the range $0 \le j_i \le k_i - 2$ for all of $i = 1, 2, \dots, \ell$. Note that in our convention the right-most columns correspond to the outer-most integrations of the iterated integral.

In the rest of this paper we will introduce different flavours of iterated integrals. When referring to objects such as $\mathcal{E}^{[\begin{smallmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{smallmatrix}; \tau]}$, we will consistently use the following terminology:

* The number $\ell$ of iterated integrals, i.e. the number of different integration kernels present, will be referred to as the *modular depth* of the iterated integral;

* The grading introduced by the sum $\sum_{i=1}^{\ell} k_i$ of the modular weights of the modular forms being integrated will be referred to as the *degree* of the iterated integral.

The simplest examples at modular depth one and two are

$$ 
\begin{aligned}
\mathcal{E}^{[\begin{smallmatrix} j_k \\ k \end{smallmatrix}; \tau]} &= \int_{\tau}^{i\infty} \nu^{[\begin{smallmatrix} j_k \\ k \end{smallmatrix}; \tau_1]} = (2\pi i)^{1+j-k} \int_{\tau}^{i\infty} \tau_1^j G_k(\tau_1) \, d\tau_1, \quad (2.8) \\
\mathcal{E}^{[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}; \tau]} &= \int_{\tau}^{i\infty} \nu^{[\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}; \tau_2]} \int_{\tau_2}^{i\infty} \nu^{[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}; \tau_1]} \\
&= (2\pi i)^{2+j_1+j_2-k_1-k_2} \int_{\tau}^{i\infty} \tau_2^{j_2} G_{k_2}(\tau_2) \, d\tau_2 \int_{\tau_2}^{i\infty} \tau_1^{j_1} G_{k_1}(\tau_1) \, d\tau_1.
\end{aligned}
$$

We note that the integrals as written are not well-defined since the zero mode $2\zeta_k$ of $G_k$ introduces endpoint divergences at the cusp $\tau_i \to i\infty$. These divergences can be dealt with by using the method of tangential-base-point regularisation [11] that we will employ throughout. Its simplest incarnation $\int_{\tau}^{i\infty} \tau_1^j d\tau_1 = -\frac{\tau^{j+1}}{j+1}$ propagates to higher modular depth $\ell$ by imposing the regularisation to preserve the shuffle relations of (2.7).

The vector space of holomorphic modular forms of weight $k$ can be decomposed into the one-dimensional space spanned by the holomorphic Eisenstein series $G_k$, and the vector space $S_k$ of holomorphic cusp forms. The first holomorphic cusp form arises for $S_{12}$ and is given by the Ramanujan cusp form $\Delta_{12}(\tau)$. Even though $S_{k=24}$ and $S_{k \ge 28}$ have dimensions $\ge 2$, we shall employ the abusive placeholder notation $\Delta_k$ for an unspecified cusp form of weight $k$ to avoid cluttering. Moreover, the examples in this paper only involve modular weight $k \le 20$ where no ambiguity arises.

Definitions similar to (2.7) apply when the holomorphic Eisenstein series $G_k(\tau)$ is replaced by a holomorphic cusp form $(2\pi i)^k \Delta_k(\tau)$ of weight $k$. Throughout this paper, we assume $\Delta_k \in S_k$ to be a normalised Hecke eigenform, so that its Fourier expansion starts as $\Delta_k = q + O(q^2)$, and the factor $(2\pi i)^k$ is included everywhere for uniformity with the
Fourier coefficients $\in \mathbb{Q}\pi^k$ of $G_k$. Our notation

$$ \nu[\Delta_k; \tau] := (2\pi i)^{j+1} \tau^j \Delta_k(\tau) \, d\tau, \qquad (2.9) $$

$$ \mathcal{E}[\Delta_k; \tau] := \int_{\tau}^{i\infty} \nu[\Delta_k; \tau_1] = (2\pi i)^{1+j} \int_{\tau}^{i\infty} \tau_1^j \Delta_k(\tau_1) \, d\tau_1, $$

closely follows (2.4) and (2.7), replacing the letter $k$ in the lower line by $\Delta_k$ for every instance of a cusp form instead of a holomorphic Eisenstein series. The range of the upper index is still $0 \le j \le k-2$. At higher modular depth, for instance, $\mathcal{E}[\Delta_k; \tau]$ denotes a double integral as in (2.8) over $\nu[\Delta_k; \tau_1]$ and $\nu[\Delta_k; \tau_2]$.

### 2.1.2 Modular properties

The iterated integral (2.7) does not have clean modular properties under the action of SL(2, $\mathbb{Z}$). Under the $T$-transformation we get for instance at modular depth one

$$ \begin{aligned} \mathcal{E}[\Delta_k; \tau+1] &= \int_{\tau+1}^{i\infty} \nu[\Delta_k; \tau_1] = \int_{\tau}^{i\infty-1} \nu[\Delta_k; \tau_1+1] \qquad (2.10) \\ &= \int_{\tau}^{i\infty} \nu[\Delta_k; \tau_1+1] + \int_{i\infty}^{i\infty-1} \nu[\Delta_k; \tau_1+1] \\ &= \sum_{p=0}^{j} (2\pi i)^{j-p} \binom{j}{p} \mathcal{E}[\Delta_k; \tau] + (2\pi i)^{1+j-k} \sum_{p=0}^{j} (-1)^{p+1} \binom{j}{p} \frac{2\zeta_k}{p+1}, \end{aligned} $$

where we used the kernels' transformation property (2.5), while the $T$-period integral is regularised using $\int_{i\infty}^{i\infty-1} \tau_1^j G_k(\tau_1) d\tau_1 = 2\zeta_k(-1)^{j+1}/(j+1)$ in agreement with the tangential-base-point scheme. It only has contributions from the zero mode of $G_k$. For holomorphic cusp forms the corresponding $T$-period integral vanishes and $\mathcal{E}[\Delta_k; \tau+1]$ is a linear combination of $\mathcal{E}[\Delta_k; \tau]$ with $0 \le p \le j$.

For higher modular depth, the general behaviour of iterated integrals under compositions of paths (in this case $(\tau, i\infty-1)$ versus $(\tau, i\infty)$ followed by $(i\infty, i\infty-1)$) leads to

$$ \begin{aligned} \mathcal{E}[\Delta_k; \tau+1] &= \sum_{r=0}^{\ell} \int_{\tau}^{i\infty} \nu[\Delta_k; \tau_r+1] \int_{\tau_r}^{i\infty} \dots \int_{\tau_2}^{i\infty} \nu[\Delta_k; \tau_1+1] \qquad (2.11) \\ & \times \int_{i\infty}^{i\infty-1} \nu[\Delta_k; \tau_\ell+1] \int_{\tau_\ell}^{i\infty-1} \dots \int_{\tau_{r+2}}^{i\infty-1} \nu[\Delta_k; \tau_{r+1}+1], \end{aligned} $$

where the entries of the forms $\nu[\Delta_k; \tau_i+1]$ on the right-hand side deconcatenate the word on the left-hand side in all ordering-preserving ways. The iterated integral over a path $(\tau, i\infty)$ in the first line of (2.11) yields combinations of $\mathcal{E}[\Delta_k; \tau]$ with $0 \le p_i \le j_i$ and $\mathbb{Q}[2\pi i]$ coefficients similar to the first term in (2.10). The $T$-period integral over a path $(i\infty, i\infty-1)$ in the second line of (2.11) evaluates to a rational multiple of $(2\pi i)^{\ell-r+j_{r+1}+\dots+j_\ell}$, generalising the second term in (2.10), and vanishes in passing to cusp forms $k_i \to \Delta_{k_i}$ for one or more of $i = r+1, \dots, \ell$.
Under the S-modular transformation, the analogue of the result (2.10) at modular depth one is

$$ 
\begin{aligned}
\mathcal{E}\begin{bmatrix} j \\ k \\ -\frac{1}{\tau} \end{bmatrix} &= \int_{-1/\tau}^{i\infty} \nu\begin{bmatrix} j_1 \\ k_1 \\ \tau_1 \end{bmatrix} = \int_{\tau}^{0} \nu\begin{bmatrix} j_1 \\ k_1 \\ -\frac{1}{\tau_1} \end{bmatrix} \qquad (2.12) \\
&= (-1)^j (2\pi i)^{2+2j-k} \int_{\tau}^{i\infty} \nu\begin{bmatrix} k-2-j \\ k \\ \tau \end{bmatrix} - (-1)^j (2\pi i)^{2+2j-k} \int_{0}^{i\infty} \nu\begin{bmatrix} k-2-j \\ k \\ \tau \end{bmatrix} \\
&= (-1)^j (2\pi i)^{2+2j-k} \mathcal{E}\begin{bmatrix} k-2-j \\ k \\ \tau \end{bmatrix} - (-1)^j (2\pi i)^{1+j-k} \mathbf{m}\begin{bmatrix} j \\ k \\ \end{bmatrix},
\end{aligned}
$$

where we used the kernel transformation property (2.6) and have defined the regularised S-period integral<fnref n="11" />

$$ \mathbf{m}\begin{bmatrix} j \\ k \end{bmatrix} := \int_{0}^{i\infty} \tau_1^j G_k(\tau_1) \, d\tau_1 = \begin{cases} \frac{-2\pi i \zeta_{k-1}}{k-1} & : j=0, \\ \frac{2(-1)^{j+1} j! (2\pi i)^{k-1-j}}{(k-1)!} \zeta_{j+1} \zeta_{j+2-k} : 0 < j \le k-2, \end{cases} \qquad (2.13) $$

that in this case has contributions from the non-zero modes of the modular form. The quantity $\mathbf{m}\begin{bmatrix} j \\ k \end{bmatrix}$ introduced here is a known period integral [60] but can be also thought of as the simplest instance of a *multiple modular value* (MMV) [11]. For the analogous iterated integrals over a path $(0, i\infty)$ at higher modular depth, numbers more complicated than (multiple) zeta values arise and their appearances is a central theme of our investigation: Given the definition

$$ \mathbf{m}\begin{bmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{bmatrix} = \int_0^{i\infty} \tau_\ell^{j_\ell} G_{k_\ell}(\tau_\ell) \, d\tau_\ell \int_{\tau_\ell}^{i\infty} \dots \int_{\tau_3}^{i\infty} \tau_2^{j_2} G_{k_2}(\tau_2) \, d\tau_2 \int_{\tau_2}^{i\infty} \tau_1^{j_1} G_{k_1}(\tau_1) \, d\tau_1, \quad (2.14) $$

of higher modular depth MMVs, the modular S-transformation of generic iterated Eisenstein integrals (2.7) can be derived from the composition-of-paths formula similar to (2.11) (here applied to $(\tau, 0)$ versus $(\tau, i\infty)$ followed by $(i\infty, 0)$),

$$ 
\begin{aligned}
\mathcal{E}\begin{bmatrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{bmatrix; -\frac{1}{\tau}] = \sum_{r=0}^{\ell} (2\pi i)^{r+j_1+\dots+j_r-k_1-\dots-k_r} \mathbf{m}\begin{bmatrix} j_1 & \dots & j_r \\ k_1 & \dots & k_r \end{bmatrix} \qquad (2.15) \\
\times (2\pi i)^{2(\ell-r)+2j_r+1+\dots+2j_\ell-k_r+1-\dots-k_\ell} (-1)^{j_r+1+\dots+j_\ell} \mathcal{E}\begin{bmatrix} k_{r+1}-j_{r+1}-2 & \dots & k_\ell-j_\ell-2 \\ k_{r+1} & \dots & k_\ell \end{bmatrix}; \tau]
\end{aligned}
$$

for instance

$$ 
\begin{aligned}
\mathcal{E}\begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix} - \frac{1}{\tau} &= (2\pi i)^{4+2j_1+2j_2-k_1-k_2} (-1)^{j_1+j_2} \mathcal{E}\begin{bmatrix} k_1-j_1-2 & k_2-j_2-2 \\ k_1 & k_2 \end{bmatrix}; \tau \qquad (2.16) \\
&\quad + (2\pi i)^{1+j_1-k_1} \mathbf{m}\begin{bmatrix} j_1 \\ k_1 \end{bmatrix} (2\pi i)^{2+2j_2-k_2} (-1)^{j_2} \mathcal{E}\begin{bmatrix} k_2-j_2-2 \\ k_2 \end{bmatrix}; \tau \\
&\quad + (2\pi i)^{2+j_1+j_2-k_1-k_2} \mathbf{m}\begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix}.
\end{aligned}
$$

Iterated integrals involving holomorphic cusp forms $\Delta_k \in S_k$ as in (2.9) give rise to generalisations of MMVs in their modular S-transformations. Our conventions

$$ 
\begin{aligned}
\mathbf{m}\begin{bmatrix} j_1 \\ \Delta_{k_1} \end{bmatrix} &= (2\pi i)^{k_1} \int_{0}^{i\infty} \tau_1^{j_1} \Delta_{k_1}(\tau_1) \, d\tau_1, \qquad (2.17) \\
\mathbf{m}\begin{bmatrix} j_1 & j_2 \\ k_1 & \Delta_{k_2} \end{bmatrix} &= (2\pi i)^{k_2} \int_{0}^{i\infty} \tau_2^{j_2} \Delta_{k_2}(\tau_2) \, d\tau_2 \int_{\tau_2}^{i\infty} \tau_1^{j_1} G_{k_1}(\tau_1) \, d\tau_1,
\end{aligned}
$$

<footnote n="11">¹¹Since Riemann zeta values $\zeta_n$ at negative integers $n$ are rational, vanish for negative even integers and are rational multiples of powers of $\pi$ for positive even integers, the expression for $\mathbf{m}\begin{bmatrix} j \\ k \end{bmatrix}$ in (2.13) evaluates to rational multiples of $\pi^k$ if $1 \le j \le k-1$ and involves the odd zeta value $\zeta_{k-1}$ at $j \in \{0, k-2\}$, e.g. $\mathbf{m}\begin{bmatrix} k-2 \\ k \end{bmatrix} = \frac{2\pi i \zeta_{k-1}}{k-1}$. The transcendental weight of $\mathbf{m}\begin{bmatrix} j \\ k \end{bmatrix}$ is $k$ for all values of $j$.</footnote>
ensure that (2.15) readily generalises to situations with some of the labels $k_i$ replaced by $\Delta_{k_i}$.
In both cases, the substitution rule for the integration kernels reads $G_k \to (2\pi i)^k \Delta_k$, leading
to Fourier coefficients in $\mathbb{K}\pi^k$ with $\mathbb{K}$ the number-field extension of $\mathbb{Q}$ defined by the Fourier
coefficients $\{a_k(n), n \in \mathbb{N}\}$ of the Hecke normalised cusp form $\Delta_k(\tau) = \sum_{n=1}^{\infty} a_k(n)q^n \in S_k$.<fnref n="12" />

Note that at modular depth one, the MMVs associated with a cusp form $\Delta_k$ simply correspond to critical completed L-values, i.e.

$$ \mathfrak{m}[\Delta_k^j] = (2\pi)^k i^{k+j+1} \Lambda(\Delta_k, j+1), \qquad (2.18) $$

where given a holomorphic cusp form $\Delta_k(\tau) = \sum_{n=1}^{\infty} a_k(n)q^n$, we defined its completed L-function via meromorphic analytic continuation in the variable $t$ of the Dirichlet series

$$ \Lambda(\Delta_k, t) := \frac{\Gamma(t)}{(2\pi)^t} \sum_{n=1}^{\infty} \frac{a_k(n)}{n^t}. \qquad (2.19) $$

Our aim is to build combinations of iterated integrals that have good modular properties. This will be achieved by considering very specific combinations of the iterated integrals above and their complex conjugates.

### 2.1.3 Alternative basis of integration kernels

The construction of modular forms from iterated Eisenstein integrals is facilitated by reorganising the integration kernels $\sim \tau_1^j G_k(\tau_1)$ in (2.4) to [28]

$$ \omega_+[\begin{smallmatrix} j \\ k \end{smallmatrix}; \tau, \tau_1] := \left(\frac{\tau - \tau_1}{4y}\right)^{k-2-j} (\tau - \tau_1)^j G_k(\tau_1) \frac{d\tau_1}{2\pi i}, \qquad (2.20) $$

$$ \omega_-[\begin{smallmatrix} j \\ k \end{smallmatrix}; \tau, \tau_1] := -\left(\frac{\tau - \bar{\tau}_1}{4y}\right)^{k-2-j} (\bar{\tau} - \bar{\tau}_1)^j \overline{G_k(\tau_1)} \frac{d\bar{\tau}_1}{2\pi i}, $$

where we restrict the integer $j$ to the range $0 \le j \le k-2$ and introduced the shorthand

$$ y := \pi \operatorname{Im} \tau = \frac{\pi}{2i} (\tau - \bar{\tau}). \qquad (2.21) $$

By their holomorphic (antiholomorphic) dependence on the integration variable $\tau_1$, the integration kernels $\omega_+$ and $\omega_-$ in (2.20) give rise to homotopy invariant iterated integrals [28]

$$ \beta_+[\begin{smallmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{smallmatrix}; \tau] := \int_{\tau}^{i\infty} \omega_+[\begin{smallmatrix} j_\ell \\ k_\ell \end{smallmatrix}; \tau, \tau_\ell] \dots \int_{\tau_3}^{i\infty} \omega_+[\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}; \tau, \tau_2] \int_{\tau_2}^{i\infty} \omega_+[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}; \tau, \tau_1], \qquad (2.22) $$

$$ \beta_-[\begin{smallmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{smallmatrix}; \tau] := \int_{\bar{\tau}}^{-i\infty} \omega_-[\begin{smallmatrix} j_\ell \\ k_\ell \end{smallmatrix}; \tau, \tau_\ell] \dots \int_{\bar{\tau}_3}^{-i\infty} \omega_-[\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}; \tau, \tau_2] \int_{\bar{\tau}_2}^{-i\infty} \omega_-[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}; \tau, \tau_1], $$

despite their non-holomorphic dependence on $\tau$. By isolating the powers of the integration variables $\tau_i$ in (2.20) via binomial expansion, one can straightforwardly relate the iterated integrals $\beta_\pm$ to the earlier iterated Eisenstein integrals $\mathcal{E}$ in (2.7) and their complex conjugates, with simple rational functions of $\tau$ and $\bar{\tau}$ as coefficients.

<footnote n="12">¹²The relation between $\mathbb{K}$ and the period polynomial of the cusp form is discussed in [54]. Note that the number field $\mathbb{K}$ differs from $\mathbb{Q}$ when $\dim S_k \neq 1$. For example at $k = 24$ we have $\dim S_{24} = 2$ and the associated number-field extension of $\mathbb{Q}$ is given by $\mathbb{K} = \mathbb{Q}(\sqrt{144169})$.</footnote>
In the previous basis of integration kernels ~ $\tau_1^j G_k(\tau_1)$, modular transformations mixed different values of $j$, see (2.5) and (2.6). The kernels in (2.20) by contrast are engineered to transform as modular forms of purely antiholomorphic modular weight $k-2-2j$,

$$ \omega_{\pm}[\begin{matrix} j \\ k \end{matrix}; \tau+1, \tau_1+1] = \omega_{\pm}[\begin{matrix} j \\ k \end{matrix}; \tau, \tau_1], \quad \omega_{\pm}[\begin{matrix} j \\ k \end{matrix}; -\frac{1}{\tau}, -\frac{1}{\tau_1}] = \bar{\tau}^{k-2-2j} \omega_{\pm}[\begin{matrix} j \\ k \end{matrix}; \tau, \tau_1]. \quad (2.23) $$

As a consequence, the terms of highest modular depth in the modular transformation of the iterated integrals (2.22) are those of modular forms, with lower-modular-depth corrections from T- and S-period integrals over $\int_{i\infty}^{i\infty-1}$ and $\int_{i\infty}^{0}$. At modular depth one, for instance, the integrals $\beta_{\pm}$ on the right-hand side of

$$ \beta_{\pm}[\begin{matrix} j \\ k \end{matrix}; \tau+1] = \beta_{\pm}[\begin{matrix} j \\ k \end{matrix}; \tau] \mp \frac{2\zeta_k}{2\pi i} \sum_{p_1=0}^{k-2-j} \sum_{p_2=0}^{j} \binom{k-2-j}{p_1} \binom{j}{p_2} \frac{\tau^{k-2-j-p_1}\bar{\tau}^{j-p_2}}{(p_1+p_2+1)(4y)^{k-2-j}}, \quad (2.24) $$

$$ \beta_{\pm}[\begin{matrix} j \\ k \end{matrix}; -\frac{1}{\tau}] = \bar{\tau}^{k-2-2j} \beta_{\pm}[\begin{matrix} j \\ k \end{matrix}; \tau] \\ - \frac{(\tau\bar{\tau})^{k-2-j}}{2\pi i (4y)^{k-2-j}} \sum_{p_1=0}^{k-2-j} \sum_{p_2=0}^{j} \binom{k-2-j}{p_1} \binom{j}{p_2} \frac{(\mp 1)^{p_1+p_2}}{\tau^{p_1}\bar{\tau}^{p_2}} \mathfrak{m}[\begin{matrix} p_1+p_2 \\ k \end{matrix}], $$

no longer share the mixing of different $j$ as seen in (2.12) and (2.10). The T-periods in (2.24) are easily seen to cancel from the sum $\beta_+[\begin{matrix} j \\ k \end{matrix}; \tau] + \beta_-[\begin{matrix} j \\ k \end{matrix}; \tau]$, see section 4 for a discussion of higher modular depth analogues in terms of generating series. The S-periods in (2.24) are combinations of the MMVs in (2.13) where $\mathfrak{m}[\begin{matrix} j \\ k \end{matrix}]$ at odd values of $j$ cancel from $\beta_+[\begin{matrix} j \\ k \end{matrix}; \tau] + \beta_-[\begin{matrix} j \\ k \end{matrix}; \tau]$. As a consequence of (2.23), the leading-modular-depth contributions to $\beta_{\pm}[\begin{matrix} j_1 \dots j_{\ell} \\ k_1 \dots k_{\ell} \end{matrix}; \frac{a\tau+b}{c\tau+d}]$ line up with modular forms of weight $(0, \sum_{i=1}^{\ell}(k_i-2-2j_i))$.

In close analogy with (2.9), we also introduce a variant of the forms $\omega_{\pm}$ and iterated integrals $\beta_{\pm}$ with holomorphic cusp forms $(2\pi i)^k \Delta_k$ in place of the Eisenstein series $G_k$, for instance

$$ \omega_+[\begin{matrix} j \\ \Delta_k \end{matrix}; \tau, \tau_1] := \left(\frac{\tau-\tau_1}{4y}\right)^{k-2-j} (\bar{\tau}-\tau_1)^j (2\pi i)^k \Delta_k(\tau_1) \frac{d\tau_1}{2\pi i}, \quad (2.25) $$

$$ \omega_-[\begin{matrix} j \\ \Delta_k \end{matrix}; \tau, \tau_1] := -\left(\frac{\tau-\bar{\tau}_1}{4y}\right)^{k-2-j} (\bar{\tau}-\bar{\tau}_1)^j (2\pi i)^k \overline{\Delta_k(\tau_1)} \frac{d\bar{\tau}_1}{2\pi i}, $$

as well as $\beta_+[\begin{matrix} j \\ \Delta_k \end{matrix}; \tau] = \int_{\tau}^{i\infty} \omega_+[\begin{matrix} j \\ \Delta_k \end{matrix}; \tau, \tau_1]$ and $\beta_-[\begin{matrix} j \\ \Delta_k \end{matrix}; \tau] = \int_{\bar{\tau}}^{-i\infty} \omega_-[\begin{matrix} j \\ \Delta_k \end{matrix}; \tau, \tau_1]$ with obvious generalisations to higher modular depth and arbitrary combinations of $G_k$ and $\Delta_k$ kernels.

### 2.1.4 Differential properties

From their definition, the iterated integrals (2.7) and their cusp-form generalisations satisfy simple differential equations with respect to $\tau$-derivatives:

$$ \begin{align} \partial_{\tau} \mathcal{E} \begin{bmatrix} j_1 & \dots & j_{\ell-1} & j_{\ell} \\ k_1 & \dots & k_{\ell-1} & k_{\ell} \end{bmatrix; \tau] &= -(2\pi i)^{1+j_{\ell}-k_{\ell}} \tau^{j_{\ell}} G_{k_{\ell}}(\tau) \mathcal{E} \begin{bmatrix} j_1 & \dots & j_{\ell-1} \\ k_1 & \dots & k_{\ell-1} \end{bmatrix; \tau], \nonumber \\ \partial_{\tau} \mathcal{E} \begin{bmatrix} j_1 & \dots & j_{\ell-1} & j_{\ell} \\ k_1 & \dots & k_{\ell-1} & \Delta_{k_{\ell}} \end{bmatrix; \tau] &= -(2\pi i)^{1+j_{\ell}} \tau^{j_{\ell}} \Delta_{k_{\ell}}(\tau) \mathcal{E} \begin{bmatrix} j_1 & \dots & j_{\ell-1} \\ k_1 & \dots & k_{\ell-1} \end{bmatrix; \tau], \nonumber \\ \partial_{\bar{\tau}} \mathcal{E} \begin{bmatrix} j_1 & \dots & j_{\ell-1} & j_{\ell} \\ k_1 & \dots & k_{\ell-1} & k_{\ell} \end{bmatrix; \tau] &= 0. \tag{2.26} \end{align} $$
The differential equations of the iterated integrals $\beta_{\pm}$ in (2.22) are modified by the $\tau$-dependence of the modular kernels $\omega_{\pm}$ in (2.20)

$$2\pi i(\tau - \bar{\tau})^2 \partial_{\tau} \beta_+ \begin{bmatrix} j_1 & j_2 & \cdots & j_{\ell} \\ k_1 & k_2 & \cdots & k_{\ell} \end{bmatrix}; \tau \bigg|_{\tau=\bar{\tau}} = \sum_{i=1}^{\ell} (k_i - j_i - 2) \beta_+ \begin{bmatrix} j_1 & \cdots & j_{i+1} & \cdots & j_{\ell} \\ k_1 & \cdots & k_i & \cdots & k_{\ell} \end{bmatrix}; \tau \quad (2.27) \\ - \delta_{j_{\ell}, k_{\ell}-2} (\tau - \bar{\tau})^{k_{\ell}} G_{k_{\ell}}(\tau) \beta_+ \begin{bmatrix} j_1 & j_2 & \cdots & j_{\ell-1} \\ k_1 & k_2 & \cdots & k_{\ell-1} \end{bmatrix}; \tau \bigg|_{\tau=\bar{\tau}}$$

$$2\pi i(\tau - \bar{\tau})^2 \partial_{\tau} \beta_{-} \begin{bmatrix} j_1 & j_2 & \cdots & j_{\ell} \\ k_1 & k_2 & \cdots & k_{\ell} \end{bmatrix}; \tau \bigg|_{\tau=\bar{\tau}} = \sum_{i=1}^{\ell} (k_i - j_i - 2) \beta_{-} \begin{bmatrix} j_1 & \cdots & j_{i+1} & \cdots & j_{\ell} \\ k_1 & \cdots & k_i & \cdots & k_{\ell} \end{bmatrix}; \tau \bigg|_{\tau=\bar{\tau}}$$

where the analogous differential equations for $\begin{bmatrix} j_1 & j_2 & \cdots & j_{\ell} \\ k_1 & k_2 & \cdots & k_{\ell} \end{bmatrix} \to \begin{bmatrix} j_1 & j_2 & \cdots & j_{\ell} \\ \Delta_{k_\ell} & \Delta_{k_\ell} & \cdots & \Delta_{k_\ell} \end{bmatrix}$ are again obtained by replacing $G_{k_\ell} \to (2\pi i)^{k_\ell} \Delta_{k_\ell}$ on the right-hand side. The derivatives with respect to $\bar{\tau}$ are given by

$$2\pi i(\tau - \bar{\tau})^2 \partial_{\tau} \beta_+ \begin{bmatrix} j_1 & j_2 & \cdots & j_{\ell} \\ k_1 & k_2 & \cdots & k_{\ell} \end{bmatrix}; \tau \bigg|_{\tau=\bar{\tau}} = -(4y)(k-2-2j)\beta_+ \begin{bmatrix} j_1 & j_2 & \cdots & j_{\ell} \\ k_1 & k_2 & \cdots & k_{\ell} \end{bmatrix}; \tau \quad (2.28) \\ - (4y)^2 \sum_{i=1}^{\ell} j_i \beta_+ \begin{bmatrix} j_1 & \cdots & j_{i-1} & \cdots & j_{\ell} \\ k_1 & \cdots & k_i & \cdots & k_{\ell} \end{bmatrix}; \tau \bigg|_{\tau=\bar{\tau}}$$

$$2\pi i(\tau - \bar{\tau})^2 \partial_{\tau} \beta_{-} \begin{bmatrix} j_1 & j_2 & \cdots & j_{\ell} \\ k_1 & k_2 & \cdots & k_{\ell} \end{bmatrix}; \tau \bigg|_{\tau=\bar{\tau}} = -(4y)(k-2-2j)\beta_{-} \begin{bmatrix} j_1 & j_2 & \cdots & j_{\ell} \\ k_1 & k_2 & \cdots & k_{\ell} \end{bmatrix}; \tau \quad (2.29) \\ - (4y)^2 \sum_{i=1}^{\ell} j_i \beta_{-} \begin{bmatrix} j_1 & \cdots & j_{i-1} & \cdots & j_{\ell} \\ k_1 & k_2 & \cdots & k_{\ell} \end{bmatrix}; \tau \bigg|_{\tau=\bar{\tau}} \\ - \delta_{j_{\ell}, 0} \frac{(4y)^2}{(2\pi i)^{k_{\ell}}} \overline{G_{k_{\ell}}(\tau)} \beta_{-} \begin{bmatrix} j_1 & j_2 & \cdots & j_{\ell-1} \\ k_1 & k_2 & \cdots & k_{\ell-1} \end{bmatrix}; \tau \bigg|_{\tau=\bar{\tau}}$$

The first terms on the right-hand sides are due to the non-vanishing anti-holomorphic modular weight of $\beta_{\pm}$ and can be accounted for by promoting $\partial_{\tau}$ to a Maaß derivative [61].

We will later on construct real-analytic combinations of $\mathcal{E}[\cdots]$ and $\overline{\mathcal{E}[\cdots]}$ as well as $\beta_+[\cdots]$ and $\beta_-[\cdots]$ with good modular properties that preserve the differential equations in $\partial_{\tau}$ at the cost of more complicated $\bar{\tau}$ derivatives.

## 2.2 Modular graph forms and beyond

Modular graph forms (MGFs) [5, 6] are non-holomorphic modular forms that arise from the low-energy expansion of configuration-space integrals in one-loop closed-string amplitudes, see [8, 57–59] for overview references. More precisely, MGFs are defined through a graphical organisation of the underlying conformal-field-theory correlators, assigning an edge for each (generalised) scalar propagator connecting pairs of punctures on the genus-one world-sheet.

Momenta on the compact world-sheet torus with complex modulus $\tau \in \mathbb{H}$ are quantised as $p = m\tau + n$ with $m, n \in \mathbb{Z}$. The Feynman integrals corresponding to a given graph are then discretised to sums over such momenta, where the conformal-field-theory building blocks of MGFs exclude the value $p = 0$ as an automatic infrared regulator. For this reason, a rather generic expression for MGFs associated with a dihedral graph $\Gamma$ with $r$ edges is given (when it converges) by

$$C_{\Gamma}(\tau) = N_{\Gamma} \sum_{p_1 \neq 0} \cdots \sum_{p_r \neq 0} \frac{\delta_{\Gamma}(\{p_i\})}{p_1^{a_1} \bar{p}_1^{b_1} \cdots p_r^{a_r} \bar{p}_r^{b_r}}, \quad (2.29)$$
<figure><img src="image_4.png" /><figcaption>Figure 1. Some modular graphs with associated MGFs indicated. The one-loop graphs in panels (a) and (b) have k links each while the two-loop graph in panel (c) has four links.</figcaption></figure>

see [6, 62] for discussions of trihedral MGFs and [57, 63, 64] for more general graph topologies. We have allowed for different exponents $a_i, b_i \in \mathbb{Z}$ of the momenta $p_i = m_i\tau + n_i$ and their complex conjugates $\bar{p}_i = m_i\bar{\tau} + n_i$. The numerator $\delta_{\Gamma}(\{p_i\})$ gathers the momentum-conserving Kronecker deltas of the graph arising at each vertex and thereby encodes its adjacency relations. The convention-dependent normalisation factor $N_{\Gamma}$ usually comprises integer powers of $\pi$ and $\mathrm{Im}\,\tau$ where the choices in numerous physics references assign modular weight $(0, \sum_{i=1}^r (b_i - a_i))$ to the MGF in (2.29). In graphical representations as in figure 1, the exponents of the momentum $p_i$ translate into labels $(a_i, b_i)$ of the associated edge.

On the one hand, lattice-sum representations as in (2.29) expose that MGFs are modular forms of SL(2, $\mathbb{Z}$). On the other hand, the wealth of algebraic and differential relations among MGFs [3, 6, 62, 65–67] are typically not evident from lattice sums.

### 2.2.1 Illustrative examples

The holomorphic Eisenstein series $G_k(\tau)$ introduced in (2.1) is related to a modular graph form associated with the one-loop graph in the left panel (a) of figure 1 via

$$G_{2k}(\tau) = \sum_{p_1 \neq 0} \cdots \sum_{p_k \neq 0} \frac{\delta(p_1-p_2) \cdots \delta(p_{k-1}-p_k) \delta(p_k-p_1)}{p_1^2 \cdots p_k^2} = \sum_{p \neq 0} \frac{1}{p^{2k}}. \quad (2.30)$$

(Here, the factor $N_{\Gamma}$ was chosen to match the definition of the holomorphic Eisenstein series (2.1).) Similarly, we can view the non-holomorphic Eisenstein series as a modular graph form associated with the same one-loop graph but different edge labels (1, 1),

$$E_k(\tau) = \frac{(\mathrm{Im}\,\tau)^k}{\pi^k} \sum_{p_1 \neq 0} \cdots \sum_{p_k \neq 0} \frac{\delta(p_1-p_2) \cdots \delta(p_{k-1}-p_k) \delta(p_k-p_1)}{|p_1|^2 \cdots |p_k|^2} = \frac{(\mathrm{Im}\,\tau)^k}{\pi^k} \sum_{p \neq 0} \frac{1}{|p|^{2k}}, \quad (2.31)$$

see the middle panel (b) of figure 1. As a final example, we consider the two-loop graph in the right panel (c) of figure 1, where simplifications of the momentum-conserving delta functions in the associated modular graph form yield the modular invariant

$$C_{2,1,1}(\tau) = \frac{(\mathrm{Im}\,\tau)^4}{\pi^4} \sum_{\substack{p_1, p_2 \neq 0 \\ p_1+p_2 \neq 0}} \frac{1}{|p_1|^2 |p_2|^2 |p_1+p_2|^4}. \quad (2.32)$$

The study of closely related two-loop MGFs denoted in general by $C_{a,b,c}(\tau)$ with more general lattice summands $|p_1|^{2b}|p_2|^{2c}|p_1+p_2|^{2a}$ considerably advanced the low-energy expansion of the one-loop four-graviton amplitude of type-II superstrings [1–3, 7], also see [68–70] for the analogous expansions at five points.
### 2.2.2 Differential equations

MGFs satisfy a rich network of algebraic relations that make it hard to find a canonical basis in terms of graphs and lattice sums.<fnref n="13" /> In order to understand the space of MGFs, it is instructive to study their differential equations with respect to $\tau$ and $\bar{\tau}$. There exists standard differential operators, called Maaß operators, that act on the non-holomorphic modular forms of weight $(A, B)$ [61]. Since the definition includes an explicit dependence on A and B, we will here use adapted versions [6]

$$ \nabla := 2i(\mathrm{Im} \, \tau)^2 \partial_{\tau}, \qquad \bar{\nabla} := -2i(\mathrm{Im} \, \tau)^2 \partial_{\bar{\tau}} \qquad (2.33) $$

that act on forms of weight $(0, w)$ and $(w, 0)$, respectively. In particular, these differential operators shift the modular weights according to $\nabla : (0, w) \to (0, w-2)$ or $\bar{\nabla} : (w, 0) \to (w-2, 0)$, which preserves the vanishing of the holomorphic or anti-holomorphic weight, respectively. In this way, $\nabla$ and $\bar{\nabla}$ can be applied several times.

As an example, we can act with $\nabla$ on the non-holomorphic Eisenstein series $E_k$ in (2.31) that is modular invariant and thus has weight (0, 0). One can check that

$$ (\pi \nabla)^k E_k = \frac{(2k-1)!}{(k-1)!} (\mathrm{Im} \, \tau)^{2k} G_{2k}, \qquad (2.34) $$

where both sides have modular weight $(0, -2k)$ since $\mathrm{Im} \, \tau$ has weight $(-1, -1)$ and $G_{2k}$ has weight $(2k, 0)$. On MGFs associated with higher-loop graphs, repeated action of $\nabla$ brings out holomorphic Eisenstein series multiplying simpler MGFs as for instance seen in

$$ (\pi \nabla)^3 (C_{2,1,1} - \frac{9}{10} E_4) = -6 (\mathrm{Im} \, \tau)^4 G_4 \pi \nabla E_2. \qquad (2.35) $$

Differential equations of the type (2.34) and (2.35) are at the heart of the sieve algorithm for analysing MGFs [6] and can be extended to generating functions of closed-string integrals to compactly address infinite families of MGFs through their low-energy expansion [29].

### 2.2.3 Iterated-integral representations

One consequence suggested by (2.34) is that the non-holomorphic Eisenstein series $E_k$ should be expressible as a $\tau$-integral over $G_{2k}$. Based on the integrals introduced in (2.7) and their differential equations (2.26), one can indeed verify that for instance [30, Eq. (4.25)]

$$ \pi \mathrm{Im} \, \tau E_2(\tau) = 12\pi^2 \tau \bar{\tau} \mathrm{Re} \, \mathcal{E}[\frac{0}{4}; \tau] - 6\pi(\tau+\bar{\tau}) \mathrm{Im} \, \mathcal{E}[\frac{1}{4}; \tau] - 3 \mathrm{Re} \, \mathcal{E}[\frac{2}{4}; \tau] + \zeta_3, \qquad (2.36) $$

in line with earlier representations of $E_k$ as iterated Eisenstein integrals [5, 9, 13, 72]. However, (2.36) and similar iterated-integral representations of more general MGFs substantially simplify once we change our basis of integration kernels towards (2.20). The alternative organisation (2.22) of iterated Eisenstein integrals brings the expression (2.36) for the non-holomorphic Eisenstein series $E_2$ into the more illuminating form

$$ \begin{aligned} E_2(\tau) &= -6(\beta_+[\frac{1}{4};\tau] + \beta_-[\frac{1}{4};\tau]) + \frac{\zeta_3}{y} \qquad (2.37) \\ &= -\frac{3}{2\pi y} \mathrm{Im} \int_{\tau}^{i\infty} d\tau_1 (\tau-\tau_1)(\bar{\tau}-\tau_1)G_4(\tau_1) + \frac{\zeta_3}{y}. \end{aligned} $$

<footnote n="13">¹³See [71] for bases of MGFs with exponents $\sum_i (a_i + b_i) \le 12$ and a Mathematica package with a variety of MGF manipulations and integration routines for one-loop closed-string amplitudes.</footnote>
For the iterated-integral representation of the two-loop MGF (2.32), passing from the basis of $\mathcal{E}$ and $\bar{\mathcal{E}}$ to the $\beta_{\pm}$ considerably streamlines the expressions and results in

$$C_{2,1,1}(\tau) = -18(\beta_+[\frac{2}{4}0;\tau] + \beta_+[0;\tau]\beta_-[\frac{2}{4};\tau] + \beta_-[0\frac{2}{4};\tau]) \quad (2.38) \\ -126(\beta_+[\frac{3}{8};\tau] + \beta_-[\frac{3}{8};\tau]) + 12\zeta_3(\beta_+[0;\tau] + \frac{\beta_-[\frac{2}{4};\tau]}{(4y)^2}) \\ + \frac{\pi^2\zeta_3\tau\bar{\tau}}{60y} + \frac{5\zeta_5}{12y} - \frac{\zeta_3^2}{4y^2} + \frac{9\zeta_7}{16y^3},$$

identifying this two-loop MGF as an iterated Eisenstein integral at modular depth two.

As these two examples illustrate, MGFs are closely related to iterated Eisenstein integrals and most conveniently written in their basis of $\beta_{\pm}$ instead of the integrals $\mathcal{E}$.

At this point, we caution the reader that the terms of modular depth zero are not the same as the Laurent polynomial (LP) of a modular graph form. The latter is given by the purely $y$-dependent terms in the Fourier zero mode of an MGF and for the case of $C_{2,1,1}$ is [3]

$$C_{2,1,1}|_{\text{LP}} = \frac{2}{14175}y^4 + \frac{\zeta_3}{45}y + \frac{5\zeta_5}{12y} - \frac{\zeta_3^2}{4y^2} + \frac{9\zeta_7}{16y^3}. \quad (2.39)$$

The terms of modular depth zero in the last line of (2.38) contain only some of the terms of the Laurent polynomial since the integrals $\beta_{\pm}$ have non-trivial contributions at the cusp. In particular, the first term $\sim \tau\bar{\tau}\zeta_3/y$ in the last line of (2.38) conspires with the leading terms $\beta_+[0] = \frac{i\pi^3\tau^3}{4320y^2} + O(q)$ and $\beta_-[0] = -\frac{i\pi^3\bar{\tau}^3}{270} + O(\bar{q})$ at the cusp to build up the T-invariant contribution $\zeta_3 y$ to the Laurent polynomial in (2.39). In general, the notion of modular depth introduced below (2.7) does not give the Laurent polynomial when projecting to modular depth zero.

### 2.2.4 Modular combinations of iterated integrals

By the modular T- and S-transformations of $\beta_{\pm}$ as in (2.24), iterated-integral representations of MGFs no longer manifest that they transform as modular forms. For instance, modular invariance of (2.36), (2.37) and (2.38) is tied to the interplay of iterated Eisenstein integrals of different modular depths with multiple zeta values (MZVs) reviewed in appendix A.

Non-holomorphic Eisenstein series and their derivatives under the Maaß operators in (2.33) realise the real-analytic modular completions of modular-depth-one integrals [13, 30]

$$E_k = -\frac{(2k-1)!}{(k-1)!^2} \beta^{\text{eqv}} \left[ \begin{matrix} k-1 \\ 2k \end{matrix} \right], \quad (2.40)$$

$$\sum_{p \neq 0} \frac{1}{p^a \bar{p}^b} = - \frac{\pi^b (2i)^{b-a} (a+b-1)!}{(\text{Im} \tau)^a (a-1)!(b-1)!} \beta^{\text{eqv}} \left[ \begin{matrix} a-1 \\ a+b \end{matrix} \right],$$

with $a, b \in \mathbb{N}$ and where the T- and S-cocycles in (2.24) cancel between the different contributions to

$$\beta^{\text{eqv}} \left[ \begin{matrix} j \\ k \end{matrix} \right] := \beta_+ \left[ \begin{matrix} j \\ k \end{matrix} \right] + \beta_- \left[ \begin{matrix} j \\ k \end{matrix} \right] - \frac{2\zeta_{k-1}}{(k-1)(4y)^{k-2-j}}. \quad (2.41)$$
Similarly, (2.38) identifies a modular-depth-two example of modular iterated integrals

$$C_{2,1,1} = -18\beta^{\text{eqv}}[\begin{matrix} 2 & 0 \\ 4 & 4 \end{matrix}] - 126\beta^{\text{eqv}}[\begin{matrix} 3 \\ 8 \end{matrix}], \quad (2.42)$$

with more intricate cancellations between the T- and S-cocycles of

$$\begin{aligned} \beta^{\text{eqv}}[\begin{matrix} 2 & 0 \\ 4 & 4 \end{matrix}] = & \beta_{+}[\begin{matrix} 2 & 0 \\ 4 & 4 \end{matrix}] + \beta_{-}[\begin{matrix} 2 \\ 4 \end{matrix}] + \beta_{-}[\begin{matrix} 0 & 2 \\ 4 & 4 \end{matrix}] \\ & - \frac{2\zeta_3}{3} \left( \beta_{+}[\begin{matrix} 0 \\ 4 \end{matrix}] + \frac{\beta_{-}[\begin{matrix} 2 \\ 4 \end{matrix}]}{(4y)^2} \right) - \frac{\pi^2 \tau \bar{\tau} \zeta_3}{1080y} - \frac{5\zeta_5}{216y} + \frac{\zeta_3^2}{72y^2}. \end{aligned} \quad (2.43)$$

The main goal of this work is to explicitly construct the modular completions of arbitrary iterated Eisenstein integrals $\beta_+$,

$$\beta^{\text{eqv}}\left[ \begin{matrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{matrix} \right] = \sum_{r=0}^{\ell} \beta_+ \left[ \begin{matrix} j_{r+1} & j_{r+2} & \dots & j_\ell \\ k_{r+1} & k_{r+2} & \dots & k_\ell \end{matrix} \right] \beta_- \left[ \begin{matrix} j_r & j_{r-1} & \dots & j_1 \\ k_r & k_{r-1} & \dots & k_1 \end{matrix} \right] + \dots, \quad (2.44)$$

where the ellipsis comprises iterated integrals of lower modular depth $\le \ell-1$ and rational functions of $\tau, \bar{\tau}$ with constant coefficients including MZVs such as to attain the modular equivariant transformation

$$\beta^{\text{eqv}}\left[ \begin{matrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{matrix} ; \frac{a\tau+b}{c\tau+d} \right] = \left( \prod_{i=1}^{\ell} (c\bar{\tau}+d)^{k_i-2j_i-2} \right) \beta^{\text{eqv}}\left[ \begin{matrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{matrix} ; \tau \right]. \quad (2.45)$$

While the modular-depth $\ell$ terms in (2.44) are by themselves T invariant, the modular-depth two example (2.43) illustrates that the cancellation of the T-cocycle of $\beta^{\text{eqv}}$ generically relies on the interplay of different modular depths. Modular S transformation of $\beta^{\text{eqv}}$ in turn is tied to even more intricate conspiracies between iterated integrals and MMVs (2.14) of different modular depths. The superscript of $\beta^{\text{eqv}}$ alludes to Brown's equivariant iterated Eisenstein integrals [14] where several modular iterated integrals (2.44) are combined with bookkeeping variable to be reviewed below that also transform under SL(2, $\mathbb{Z}$). Following Brown's approach, we will construct the modular completion in (2.44) via generating series and pinpoint the structure and explicit form of the series in MZVs of [14].

In slight abuse of terminology, we will sometimes attribute modular depth $\ell$ to the modular forms $\beta^{\text{eqv}}\left[ \begin{matrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{matrix} \right]$ even though some of the contributing products of holomorphic and antiholomorphic iterated Eisenstein integrals have combined modular depth $< \ell$, see for instance (2.41) and (2.43). By the linear-independence result for holomorphic iterated Eisenstein integrals in [73], there are no linear relations among $\beta^{\text{eqv}}$ with different entries $j_i, k_i$, though the generating-series construction of [14] and this work imply standard (non-linear) shuffle relations.

Note that the approach of [30] to the low-energy expansion of closed-string genus-one integrals organises MGFs in terms of real-analytic and T-invariant combinations $\beta^{\text{sv}}\left[ \begin{matrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{matrix} \right]$ of $\beta_\pm$ that share the leading-modular-depth terms on the right-hand side of (2.44). However, the $\beta^{\text{sv}}$ are lacking most of the S-invariant completion by terms of lower modular depth in the ellipsis of (2.44), e.g. the zeta values in (2.41) at modular depth one are absent from $\beta^{\text{sv}}\left[ \begin{matrix} j \\ k \end{matrix} \right]$. The systematics of MZV contributions to $\beta^{\text{sv}}$ at modular depth two and three was described in [28], and we will review the translation to $\beta^{\text{eqv}}$ in (2.88) below.
### 2.2.5 Modular iterated integrals beyond MGFs

All MGFs can be expressed by definition in terms of (manifestly modular) lattice sums over torus momenta $p_i$, while from their differential equations they can also be written in terms of iterated Eisenstein integrals. More precisely, the differential equations of MGFs [6] and their generating series [29] imply that the $\beta_\pm$ in their integral representations only involve Eisenstein series $G_k$ as their integration kernels and no holomorphic cusp forms. However, some of the modular completions of the iterated Eisenstein integrals in (2.44) necessitate cuspidal integration kernels [11, 13], say integrals $\beta_\pm$ of modular depth $\ell-1$ (see the discussion below (2.25)) contributing to $\beta^{\text{eqv}}$ of modular depth terms are solely built from Eisenstein kernels, modular integrals $\beta^{\text{eqv}}$ with cusp-form admixtures cannot be realised by MGFs.

Nevertheless, modular integrals $\beta^{\text{eqv}}$ at modular depth two including their cusp-form contributions have been constructed in [37, 74] using methods that prominently feature in the MGF literature, namely Laplace equations and Poincaré series. Two-loop MGFs $C_{a,b,c}$ including the example (2.32) obey inhomogeneous Laplace eigenvalue equations such as [3]

$$ (\Delta - 2)C_{2,1,1} = 9E_4 - E_2^2, \qquad (2.46) $$

with $\Delta := 4(\text{Im } \tau)^2 \partial_\tau \partial_{\bar{\tau}}$ the SL$_2$ invariant Laplace-Beltrami operator. This inspired the assembly of all the $\beta^{\text{eqv}}$ at modular depth two from solutions $F_{m,k}^{\pm(s)}$ to inhomogeneous Laplace equations with source terms built from $E_m, E_k$ and eigenvalues $s(s-1)$ with $s, m, k \in \mathbb{N}$ and $|k-m|+1 \le s \le k+m-1$. In case of (2.46), we can eliminate the source term $E_4$ by redefining $F_{2,2}^{+(2)} = -C_{2,1,1} + \frac{9}{10}E_4$ such that $(\Delta-2)F_{2,2}^{+(2)} = E_2^2$. The functions $F_{m,k}^{\pm(s)}$ and their generalisation to modular depth three are discussed in more detail in section 3.4.

Integration kernels involving holomorphic cusp forms arise for eigenvalues $s(s-1) \ge 30$, for example in the solution to the modular Laplace equation [37]

$$ (\Delta - 6.5)F_{4,4}^{+(6)} = E_4 E_4. \qquad (2.47) $$

The iterated-integral representation of $F_{4,4}^{+(6)}$ and its Maaß derivatives involves modular-depth-one integrals $\beta_\pm$ of the Hecke normalised Ramanujan discriminant cusp form $\Delta_{12}$ and their complex conjugates for various values of $j$. More generally, the cuspidal contributions to the modular invariant $F_{m,k}^{\pm(s)}$ are packaged into Laplace eigenfunctions [37]

$$ \begin{aligned} H_{\Delta_{2s}}^{\pm}(\tau) &= -\frac{(i\pi)^{2s-1}}{(s-1)!y^{s-1}} \int_{\tau}^{i\infty} d\tau_1 (\tau-\tau_1)^{s-1}(\tau_1-\tau)^{s-1}\Delta_{2s}(\tau_1) \pm \text{c.c.} \\ &= -\frac{1}{2(s-1)!} \left( \beta_+ \left[ \Delta_{2s}; \tau \right] \pm \beta_- \left[ \Delta_{2s}; \tau \right] \right), \end{aligned} \qquad (2.48) $$

with $(\Delta - s(s-1))H_{\Delta_{2s}}^{\pm} = 0$. The associated $\beta_\pm$ at larger and smaller values of $j$ correspond to the $(j+1-s)^{\text{th}} \tau$-derivative and $(s-j-1)^{\text{th}} \bar{\tau}$-derivative of $H_{\Delta_{2s}}^{\pm}$, respectively:

$$ \begin{aligned} \beta_+ \left[ \Delta_{2s}; \tau \right] \pm \beta_- \left[ \Delta_{2s}; \tau \right] &= -2(-4)^m (s-1-m)!(\pi\nabla)^m H_{\Delta_{2s}}^{\pm}(\tau), \\ \beta_+ \left[ \Delta_{2s}; \tau \right] \pm \beta_- \left[ \Delta_{2s}; \tau \right] &= \frac{-2(s-1-m)!(\pi\bar{\nabla})^m H_{\Delta_{2s}}^{\pm}(\tau)}{(-4)^m y^{2m}}. \end{aligned} \qquad (2.49) $$
Interestingly, the coefficients of these functions in $F_{m,k}^{\pm(s)}$ involve non-critical values of the completed L-function $\Lambda(\Delta_{2s}, t)$ in (2.19) associated with the cusp form $\Delta_{2s}$, where ‘non-critical’ refers to an evaluation at $t \ge 2s$. For instance, for the function $F_{4,4}^{+(6)}$ one has

$$ 
\begin{aligned}
F_{4,4}^{+(6)} &= 2450(\beta_+[\frac{4}{8}\frac{2}{8}]+\beta_+[\frac{2}{8}]\beta_-[\frac{4}{8}]+\beta_-[\frac{2}{8}\frac{4}{8}]) + 784(\beta_+[\frac{5}{8}\frac{1}{8}]+\beta_+[\frac{1}{8}]\beta_-[\frac{5}{8}]+\beta_-[\frac{1}{8}\frac{5}{8}]) \\
&\quad + \frac{98}{3}(\beta_+[\frac{6}{8}\frac{0}{8}]+\beta_+[\frac{0}{8}]\beta_-[\frac{6}{8}]+\beta_-[\frac{0}{8}\frac{6}{8}]) + \dots + \frac{7\Lambda(\Delta_{12}, 13)}{10365\Lambda(\Delta_{12}, 11)} H_{\Delta_{12}}^+ \\
&= 2450\beta^{\text{eqv}}[\frac{4}{8}\frac{2}{8}] + 784\beta^{\text{eqv}}[\frac{5}{8}\frac{1}{8}] + \frac{98}{3}\beta^{\text{eqv}}[\frac{6}{8}\frac{0}{8}],
\end{aligned}
\quad (2.50) $$

where the ellipsis denotes iterated Eisenstein integrals of lower modular depth and coefficients built from $\tau, \bar{\tau}$ and odd zeta values [37, 74]. The analogous iterated-integral representations of general $F_{m,k}^{\pm(s)}$ inform the contributions of $H_{\Delta_{2s}}^{\pm}$ or equivalently $\beta_{\pm}[\Delta_{2s}^{j}]$ to $\beta^{\text{eqv}}[\frac{j_1}{2m}\frac{j_2}{2k}]$, see section 3.4 of [28]$^{14}$ for further details. An earlier discussion of the modular completion of double Eisenstein integrals via modular-depth-one integrals of holomorphic cusp forms can be found in Brown’s work [11, 13].

The critical and non-critical L-values $\Lambda(\Delta_{12}, t)$ in (2.50) can be inferred from the $S$-cocycles arising in the iterated-integral representation of $F_{4,4}^{+(6)}$. These L-values exemplify that the MMVs governing the construction of modular iterated integrals (2.44) are in general periods outside the realm of (single-valued) MZVs but are essential ingredients of the full theory of iterated integrals of holomorphic modular forms [11, 13, 38]. Due to the connection of the cusp forms and their periods to Tsunogai’s derivation algebra to be reviewed in section 2.3 below, the appearance of new periods in equivariant iterated Eisenstein integrals can be anticipated from the commutator relations among these derivations. These structures are most conveniently incorporated by unifying the iterated integrals of Eisenstein series and cusp forms in generating series which will be a central theme of this paper.

## 2.3 sl₂ representations and Tsunogai’s derivations

In preparation for the generating series of iterated integrals in section 2.5 below, we shall now introduce their non-commutative expansion variables. The constructions are based on certain non-commuting letters $e_k$ for even $k \ge 4$ as well as an $sl_2$ algebra generated by

$$ 
\begin{aligned}
e_0 \text{ (raising operator),} \\
e_0^{\vee} \text{ (lowering operator),} \\
h = [e_0, e_0^{\vee}] \text{ (Cartan operator).}
\end{aligned}
\quad (2.51) $$

We can use this $sl_2$ algebra to define irreducible $sl_2$ representations of dimension $k-1$ by letting for $0 \le j \le k-2$

$$ e_k^{(j)} := \mathrm{ad}_{e_0}^j e_k = \underbrace{[e_0, [e_0, \dots [e_0, e_k]\dots]]}_{j} \quad \text{with nilpotency} \quad e_k^{(k-1)} = 0. \quad (2.52) $$

<footnote>$^{14}$The combinations of $\beta_{\pm}[\Delta_{2s}^j]$ in (2.48) and (2.49) are denoted by $\beta^{\text{sv}}[\Delta_{2s}^j] = \beta_+[\Delta_{2s}^j] \pm \beta_-[\Delta_{2s}^j]$ in [28]. In order to avoid unwieldy notation at higher modular depth (e.g. double integrals of a cusp form and an Eisenstein series) we do not use or extend the $\beta^{\text{sv}}$-notation for cuspidal contributions in this work.</footnote>
The sl₂ algebra acts on these elements by

$$ \mathrm{ad}_{e_0} e_k^{(j)} = e_k^{(j+1)}, \qquad (2.53) $$

$$ \mathrm{ad}_{e_0^\vee} e_k^{(j)} = j(k-1-j)e_k^{(j-1)}, $$

$$ \mathrm{ad}_{h} e_k^{(j)} = (2j+2-k)e_k^{(j)}, $$

which imply the following commutator relations for the diagonalised generator h

$$ [h, e_0] = 2e_0, \quad [h, e_0^\vee] = -2e_0^\vee. \qquad (2.54) $$

Given these relations, it is rather straightforward, and shortly extremely useful, to compute $(\mathrm{ad}_{e_0^\vee})^\ell$ on a generic element:

**Lemma 1** If we iterate $\ell$-times the action of $(\mathrm{ad}_{e_0^\vee})$ on $e_k^{(j)}$ we obtain

$$ (\mathrm{ad}_{e_0^\vee})^\ell e_k^{(j)} = \frac{j!(k+\ell-2-j)!}{(j-\ell)!(k-2-j)!} e_k^{(j-\ell)}, \qquad (2.55) $$

which can be easily proven by induction on $\ell$ starting from (2.53) and this is a standard result in sl₂ representation theory.

Given the sl₂ action (2.52) and (2.53) on the $(k-1)$-dimensional sl₂-module spanned by $e_k^{(j)}$ with $0 \le j \le k-2$, we will refer to $e_k^{(0)} = e_k$ as the lowest-weight vector of the module, and to $e_k^{(k-2)}$ as its highest-weight vector. Note that the highest-/lowest-weight vectors have highest/lowest eigenvalues with respect to the action of $\mathrm{ad}_h$, namely $k-2$ and $-(k-2)$.

We also define the Casimir operator

$$ \Omega := e_0 e_0^\vee + \frac{1}{4} h(h-2) \qquad (2.56) $$

that commutes with the whole sl₂ algebra and is related to the modular Laplace operator in functional realisation. On the representation generated from the lowest-weight vector $e_k$ the eigenvalue is

$$ \Omega e_k^{(j)} = \frac{k}{2} \left( \frac{k}{2} - 1 \right) e_k^{(j)}, \quad \text{for all } 0 \le j \le k-2. \qquad (2.57) $$

It is also useful to define the standard sl₂ (Weyl) reflection w that acts as follows on the members of any given $(k-1)$-dimensional module:

$$ w := e_0^\vee e^{-e_0} e_0^\vee, \quad w e_k^{(j)} w^{-1} = (-1)^j \frac{j!}{(k-2-j)!} e_k^{(k-2-j)}. \qquad (2.58) $$

This expression for $w e_k^{(j)} w^{-1}$ can be explicitly checked for example by using matrix representatives or by expanding the exponentials and using the relations (2.53).

### 2.3.1 Tensor products and highest-/lowest-weight vectors

We shall next review a few standard facts about tensor products of finite-dimensional sl₂ multiplets. Let $e_k^{(j)}$ denote the basis elements of a $(k-1)$-dimensional multiplet (with even $k$) for $0 \le j \le k-2$ and $V(e_k)$ the associated irreducible representation. Then we have the well-known decomposition of the tensor product of two representations

$$ V(e_p) \otimes V(e_q) = \bigoplus_{\substack{r=p-q+2 \\ r \in 2\mathbb{Z}}}^{p+q-2} V(e_r), \qquad (2.59) $$
where each irreducible representation $V(e_r)$ on the right-hand side occurs with multiplicity one.

Many of our expressions will be written in terms of generating series with coefficients made out of words of the form $e_{k_1}^{(j_1)} \cdots e_{k_\ell}^{(j_\ell)}$, where the order of the non-commutative letters $e_{k_i}^{(j_i)}$ is important. For both of $e_{k_1}^{(j_1)} e_{k_2}^{(j_2)}$ and $e_{k_2}^{(j_2)} e_{k_1}^{(j_1)}$, the $(k_1-1)(k_2-1)$ words associated with $0 \le j_i \le k_i-2$ fall into the irreducible representations in the tensor product $V(e_{k_1}) \otimes V(e_{k_2})$, i.e. the decomposition (2.59) applies to both orderings of the non-commutative $e_{k_i}^{(j_i)}$.

As will become clear in later sections, most of the salient features of modular iterated integrals enter their generating series with nested commutators $e_{k_i}^{(j_i)}$ as coefficients. For instance, the key information about $\beta^{\text{eqv}}[e_{k_1}^{(j_1)} e_{k_2}^{(j_2)}]$ and $\beta^{\text{eqv}}[e_{k_1}^{(j_1)} e_{k_2}^{(j_2)}]$ at modular depth two and three will be accompanied by brackets $[e_{k_1}^{(j_1)}, e_{k_2}^{(j_2)}]$ and $[[e_{k_1}^{(j_1)}, e_{k_2}^{(j_2)}], e_{k_3}^{(j_3)}]$, respectively. Accordingly, we will be particularly interested in the tensor-product decomposition (2.59) applied to commutators of $e_{k_i}^{(j_i)}$. In this case, the explicit form of the projectors isolating the lowest-weight vectors in the irreducible representations of (2.59) is given by

$$t_{p,q}^d := t^d(e_p, e_q) := \frac{(d-2)!}{(p-2)!(q-2)!} \sum_{i=0}^{d-2} (-1)^i \frac{(p-2-i)!(q-d+i)!}{i!(d-2-i)!} [e_p^{(i)}, e_q^{(d-2-i)}], \quad (2.60)$$

where we have introduced $d = \frac{1}{2}(p+q-r+2)$ to match the notation in [36, 37] and have fixed an overall normalisation factor. One can check that $[e_0^\vee, t_{p,q}^d] = 0$ and $\text{ad}_{e_0^\vee}^{p+q-2d+1}(t_{p,q}^d) = 0$ such that the vector generates an $(r-1)$-dimensional representation under the action of $\mathfrak{sl}_2$ via $\text{ad}_{e_0^\vee}^j$ with $0 \le j \le r-2$, where $r = p+q-2d+2$.

For $p \ne q$, the commutators $[e_p^{(j_1)}, e_q^{(j_2)}]$ realise all representations on the right-hand side of (2.59) with multiplicity one. The integer parameter $d$ labelling the irreducible representations in (2.60) then covers the full range $2 \le d \le \min(p,q)$. The largest representation of dimension $p+q-3$ that arises in the tensor product corresponds to $d=2$.

For $p=q$, however, the antisymmetrisation in $[e_p^{(j_1)}, e_p^{(j_2)}]$ projects out the irreducible representations $V(e_2), V(e_6), \dots, V(e_{2p-2})$ of dimensions $\in 4\mathbb{N}+1$, and one is left with $V(e_4), V(e_8), \dots, V(e_{2p-4})$ of dimensions $\in 4\mathbb{N}+3$ in the antisymmetrised tensor product of two identical representations $V(e_p)$. Accordingly, the lowest-weight vectors (2.60) at $p=q$ vanish for even $d$, and their non-zero instances are $t_{p,p}^3, t_{p,p}^5, \dots, t_{p,p}^{p-1}$. The letter $e_2$ associated with the singlet of $\mathfrak{sl}_2$ does not enter our generating series (or anywhere else in the construction), though we do encounter one-dimensional representations $V(e_2)$ among the tensor products of two or more $e_{k \ge 4}^{(j)}$.

We can also project to a highest-weight vector proportional to $e_r^{(r-2)}$ in the same module $V(e_r)$ using a formula closely related to (2.60)

$$s_{p,q}^d := s^d(e_p, e_q) := \frac{(d-2)!}{(p-2)!(q-2)!} \sum_{i=0}^{d-2} (-1)^i [e_p^{(p-2-i)}, e_q^{(q-d+i)}]. \quad (2.61)$$

This vector satisfies $[e_0, s_{p,q}^d] = 0$ as well as $\text{ad}_{e_0^\vee}^{p+q-2d+1}(s_{p,q}^d) = 0$ and generates the same $(r-1)$-dimensional representation of $\mathfrak{sl}_2$ as $t_{p,q}^d$ through the repeated action of $\text{ad}_{e_0^\vee}$. This formula is sometimes more convenient than (2.60) since it does not develop singular denominators
when taking some of the parameters outside their standard range $2 \le d \le \min(p, q)$. When taking a $d > \min(p, q)$, one still obtains an element of the tensor product, however, it will not belong to a single irreducible representation $V(e_r)$ but will be a linear combination of vectors of different irreducible representations.

In the ancillary file, numerous main results of this work are presented in terms of the projectors $t_{p,q}^d$ and $s_{p,q}^d$, along with a Mathematica implementation of (2.60), (2.61) and their iterations.

### 2.3.2 Tsunogai's derivation algebra

The above $e_k^{(j)}$ with $0 \le j \le k-2$ are taken to generate a free Lie algebra, i.e. to obey no commutation relations other than $e_k^{(k-1)} = 0$. In this way, they can be later on used to gather all iterated Eisenstein integrals $\mathcal{E}[j_1 \cdots j_\ell]$, $\beta_\pm[j_1 \cdots j_\ell]$ with $0 \le j_i \le k_i-2$ in generating series, without any 'dropouts'. However, the construction of MGFs only involves those iterated Eisenstein integrals whose modular completion (2.44) does not require any cusp forms, or equivalently, which allow for configuration-space representations as iterated integrals over punctures on a torus [75]. The restriction of generating series to this subclass of iterated Eisenstein integrals is attained by sending the letters $e_k$ to certain derivations $\epsilon_k$ dual to holomorphic Eisenstein series. We will refer to the algebra generated by these $\{\epsilon_k, k \in 2\mathbb{N}_0\}$ as Tsunogai's derivation algebra and it has been studied from a multitude of perspectives in the mathematics literature [35, 36, 39-48, 76, 77]. The $sl_2$ algebra is simply mapped to $e_0 \to \epsilon_0$, $e_0^\vee \to \epsilon_0^\vee$, while the derivation $\epsilon_2$ does not enter in our construction.

The derivations $\epsilon_k$ share the $sl_2$ multiplet structure (2.52) and the action of the lowering operator $\epsilon_0^\vee$ of the $e_k$,

$$ \epsilon_k^{(j)} = \mathrm{ad}_{\epsilon_0}^j \epsilon_k = \underbrace{[\epsilon_0, [\epsilon_0, \dots [\epsilon_0, \epsilon_k]\dots]]}_{j} \quad \text{with nilpotency } \epsilon_k^{(k-1)} = 0, \\ (\mathrm{ad}_{\epsilon_0^\vee})^\ell \epsilon_k^{(j)} = \frac{j!(k+\ell-2-j)!}{(j-\ell)!(k-2-j)!} \epsilon_k^{(j-\ell)}, \qquad (2.62) $$

but satisfy a variety of additional commutator relations [36, 75, 76] such as

$$ \begin{aligned} 0 &= [\epsilon_4, \epsilon_{10}] - 3[\epsilon_6, \epsilon_8], \\ 0 &= 80[\epsilon_4^{(1)}, \epsilon_{12}] + 16[\epsilon_{12}^{(1)}, \epsilon_4] - 250[\epsilon_6^{(1)}, \epsilon_{10}] - 125[\epsilon_{10}^{(1)}, \epsilon_6] + 280[\epsilon_8^{(1)}, \epsilon_8] \\ & \quad - 462[\epsilon_4, [\epsilon_4, \epsilon_8]] - 1725[\epsilon_6, [\epsilon_6, \epsilon_4]]. \end{aligned} \qquad (2.63) $$

One way of generating or checking such relations is based on the action of the derivations $\epsilon_k$ on the freely generated Lie algebra $\mathrm{Lie}[a, b]$ of the fundamental group of the once-punctured torus, see (2.74) below. The formal signpost variables $e_k$ by contrast are not realised as derivations of $\mathrm{Lie}[a, b]$.

When replacing $e_k^{(j)} \to \epsilon_k^{(j)}$ in the generating series we will shortly introduce, relations like (2.63) lead to dropouts of the accompanying iterated Eisenstein integrals and project
to their subclass that enters MGFs.<fnref n="15" /> This can for instance be seen from the conjectural matrix representations of $\epsilon_k^{(j)}$ in the differential equations of closed-string integrals [29, 30] that generate all MGFs through their low-energy expansion. When retaining the variables $e_k^{(j)}$ without any relation like (2.63), the generating series accommodate more iterated integrals or modular forms and can be used to construct integral representations such as (2.50) also for those modular invariant $F_{m,k}^{\pm(s)}$ of section 2.2.5 that receive contributions from holomorphic cusp forms [28, 37]. One of the key results in this work is the modular-depth three generalisation of $F_{m,k}^{\pm(s)}$ in sections 3.3 and 3.4, i.e. real-analytic combinations of triple Eisenstein integrals with double integrals of a cusp form and an Eisenstein series in their modular completions.

The relations (2.63) in the derivation algebra are governed by period polynomials of holomorphic cusp forms [36]. Accordingly, the structure of these relations will be later on used to organise the appearance of iterated integrals involving cusp forms such as (2.9) in equivariant iterated Eisenstein integrals. For this purpose, we shall promote the commutators in (2.63) to non-vanishing combinations of free-Lie-algebra generators $e_k^{(j)}$,

$$P_{14}^2 = \frac{1}{49896000} ([e_4, e_{10}] - 3[e_6, e_8]), \qquad (2.64)$$

$$P_{16}^3 = \frac{1}{183883392000} (80[e_4^{(1)}, e_{12}] + 16[e_{12}^{(1)}, e_4] - 250[e_6^{(1)}, e_{10}] - 125[e_{10}^{(1)}, e_6] + 280[e_8^{(1)}, e_8] \\ - 462[e_4, [e_4, e_8]] - 1725[e_6, [e_6, e_4]]),$$

where the normalisation here and below is chosen for later convenience (see section 3.3.1). Relations in the derivation algebra can then be presented as

$$P_w^d|_{e_k^{(j)} \to \epsilon_k^{(j)}} = 0. \qquad (2.65)$$

The non-vanishing combinations $P_w^d$ of $e_k^{(j)}$ that are mapped to zero upon specialisation $e_k^{(j)} \to \epsilon_k^{(j)}$ to Tsunogai's derivations will be referred to as *Pollack combinations* in the rest of this work. Pollack combinations $P_w^d$ are labelled by two integers $w, d$ that account for the expected systematics of $\epsilon_k$ relations: the subscript tracks the sum $w = k_1+k_2+...$ of the labels of the $e_{k_i}$ in each term and will be referred to as *degree* (as it will be directly related to the degree of iterated integrals). The superscript $d$ in turn matches the total number of $e_k$ generators, counting both $e_{k \ge 4}$ and $e_0$ on equal footing. This is different from the counting of $e_{k_i}^{(j_i)}$ or $\epsilon_{k_i}^{(j_i)}$ at $k_i \ge 4$ which is not uniform in the expression (2.64) for $P_{16}^3$. In slight abuse of terminology, we shall refer to the number of $e_{k_i}^{(j_i)}$ and $\epsilon_{k_i}^{(j_i)}$ at nonzero $k_i$ as *modular depth*, i.e. $P_{16}^3$ is a combination of modular-depth-two and modular-depth-three terms.

### 2.3.3 sl₂ structure of $\epsilon_k$ relations and Pollack combinations

In order to compactly represent generalisations of $P_{14}^2, P_{16}^3$ in (2.64) to higher degree, it is convenient to employ iterations of the operation $t_{p,q}^{d_1} = t^{d_1}(e_p, e_q)$ in (2.60). Given that $t_{p,q}^{d_1}$ is

<footnote n="15">¹⁵More details will be provided in a later section, but for the moment as an example of such dropouts we have that the relation $[\epsilon_4, \epsilon_{10}] - 3[\epsilon_6, \epsilon_8] = 0$ among the non-commutative variables in the generating series $J_+$ in (2.84) below implies that not all of the four accompanying integrals $\beta_+[\frac{2}{4} \frac{8}{10}]$, $\beta_+[\frac{8}{10} \frac{2}{4}]$, $\beta_+[\frac{4}{6} \frac{6}{8}]$ and $\beta_+[\frac{6}{8} \frac{4}{6}]$ appear independently. Instead, only three linearly independent combinations of the four integrals remain in the expansion of $J_+$ after eliminating one of $\epsilon_4\epsilon_{10}$, $\epsilon_{10}\epsilon_4$, $\epsilon_6\epsilon_8$ or $\epsilon_8\epsilon_6$ by means of the commutation relation (2.63), see [30, Eq. (3.17)] for more details.</footnote>
the lowest-weight vector of a $(p+q-2d_1+1)$-dimensional $\mathfrak{sl}_2$ module, it can be treated on the same footing as $e_r$ with $r = p+q-2d_1+2$ when entering as an input for another operation $t^{d_2}(e_u, e_r)$ or $s^{d_2}(e_u, e_r)$ in (2.61). We can therefore write for example

$$t^{d_2}(e_u, t_{p,q}^{d_1}) = t^{d_2}(e_u, t^{d_1}(e_p, e_q)), \qquad s^{d_2}(e_u, t_{p,q}^{d_1}) = s^{d_2}(e_u, t^{d_1}(e_p, e_q)). \qquad (2.66)$$

In particular, the contributions $e_r^{(j)}$ to $t^{d_2}(e_u, e_r)$ or $s^{d_2}(e_u, e_r)$ are then promoted to $\mathrm{ad}_{e_0}^j(t_{p,q}^{d_1})$ which can be simplified by means of the Leibniz rule

$$\mathrm{ad}_{e_0}^j[e_p^{(j_1)}, e_q^{(j_2)}] = \sum_{m=0}^{j} \binom{j}{m} [e_p^{(j_1+m)}, e_q^{(j_2+m)}] \qquad (2.67)$$

while discarding contributions outside the contributing $\mathfrak{sl}_2$ multiplets via $e_p^{(p-1)} = e_q^{(q-1)} = 0$. In terms of these iterated $t_{p,q}^d$-operations, we can present the combinations (2.64) encoding the simplest $\epsilon_k$ relations and their generalisations to higher degree $\le 20$ as follows:

* total of $d=2$ letters $e_k$:

$$P_{14}^2 = \frac{4}{5 \cdot 11!} (t_{4,10}^2 - 3t_{6,8}^2), \qquad (2.68)$$

$$P_{18}^2 = \frac{12}{35 \cdot 7! \cdot 11!} (2t_{4,14}^2 - 7t_{6,12}^2 + 11t_{8,10}^2),$$

$$P_{20}^2 = \frac{2}{17!} (-8t_{4,16}^2 + 25t_{6,14}^2 - 26t_{8,12}^2).$$

* total of $d=3$ letters $e_k$:

$$P_{16}^3 = \frac{3}{13820 \cdot 11!} (-160t_{4,12}^3 + 1000t_{6,10}^3 - 840t_{8,8}^3 - 462t^2(e_4, t_{4,8}^2) + 1725t^2(e_6, t_{4,6}^2)),$$

$$P_{20}^3 = \frac{1}{253190 \cdot 7! \cdot 11!} \left( \begin{aligned} & -7560t_{4,16}^3 + 51450t_{6,14}^3 - 113190t_{8,12}^3 + 69300t_{10,10}^3 \\ & + 10970t^2(e_4, t_{4,12}^2) - 166675t^2(e_4, t_{6,10}^2) + 500675t^2(e_6, t_{6,8}^2) + 80388t^2(e_8, t_{8,4}^2) \end{aligned} \right) \qquad (2.69)$$

Upon comparison with the earlier expression for $P_{16}^3$ in (2.64), the commutators $[e_p, e_q^{(1)}]$ and $[e_p^{(1)}, e_q]$ have been combined to $t_{p,q}^3$.

* total of $d=4$ letters $e_k$:

$$P_{18}^4 = \frac{4}{3455 \cdot 13!} (36t_{4,14}^4 - 691t_{6,12}^4 + 2073t_{8,10}^4) \\ + \frac{48}{3455 \cdot 11! \cdot 11!} \left( \begin{aligned} & 26603500t^2(e_6, t_{6,6}^3) - 2404395t^2(e_6, t_{4,8}^3) \\ & - 63679140t^3(e_6, t_{4,8}^3) - 17133660t^3(e_8, t_{4,6}^3) + 86454270t^3(e_8, t_{4,6}^2) \\ & + 2166948t^2(e_4, t_{4,10}^3) + 1805790t^3(e_4, t_{4,10}^2) \end{aligned} \right) \\ + \frac{10771}{39718812672000} [e_4, [e_4, [e_4, e_6]]], \qquad (2.70)$$

where the last commutator can be written as the nested $t_{p,q}^d$-operations $[e_4, [e_4, [e_4, e_6]]] = t^2(e_4, t^2(e_4, t_{4,6}^2))$.
• total of $d = 5$ letters $e_k$:

$$ P_{20}^{5} = \frac{36}{4837 \cdot 13!} (4t_{6,14}^{5} - 25t_{8,12}^{5} + 21t_{10,10}^{5}) + \frac{192}{322887025 \cdot 11! \cdot 13!} \quad (2.71) \\ \times (7106178167028t^{2}(e_{4}, t_{4,12}^{4}) - 4682105275344t^{3}(e_{4}, t_{4,12}^{3}) \\ - 749415645000t^{4}(e_{4}, t_{4,12}^{2}) - 16678946440520t^{2}(e_{6}, t_{4,10}^{4}) \\ + 30606884011392t^{3}(e_{6}, t_{4,10}^{3}) + 16726165060230t^{4}(e_{6}, t_{4,10}^{2}) \\ - 1766591296938t^{2}(e_{10}, t_{4,6}^{4}) + 13691844269480t^{3}(e_{10}, t_{4,6}^{3}) \\ + 15799996899120t^{4}(e_{10}, t_{4,6}^{2}) - 31509458355375t^{2}(e_{6}, t_{6,8}^{4}) \\ + 1515484971000t^{3}(e_{6}, t_{6,8}^{3}) - 11919872175750t^{4}(e_{6}, t_{6,8}^{2}) \\ + 15073861990239t^{2}(e_{8}, t_{4,8}^{4}) - 43428965937084t^{3}(e_{8}, t_{4,8}^{3}) \\ - 20009178181944t^{4}(e_{8}, t_{4,8}^{2})) \\ + \dots $$

The ellipsis in the last line refers to modular-depth-four terms — nested brackets of four $e_{k \ge 4}$ — which can be found in the ancillary files and can be reconstructed from the website [78]. However, in contrast to the presentation of the $\epsilon_k$ relations on the website, all of (2.68) to (2.71) are organised into linear combinations of lowest-weight vectors

$$ [e_0^\vee, P_w^d] = 0 \quad (2.72) $$

which give rise to $(w-2d+1)$-dimensional $sl_2$ multiplets under repeated action of $ad_{e_0}$. For instance, shifting the above $P_{20}^5$ by multiples of $(ad_{e_0})^2 P_{20}^3$ or $(ad_{e_0})^3 P_{20}^2$ as done on the website would preserve the counting associated with the labels $d=5$ and $w=20$ but conflict with the highest-weight property (2.72).

Nevertheless, (2.72) still does not fix a unique form of $P_w^d$ at $w \ge 18$. Both the lowest-weight property and the counting encoded in $d, w$ are preserved by adding $[e_4, P_{w-4}^{d-1}]$ to $P_w^d$, or more general (iterated) brackets of $e_{k \ge 4}$ with $P_{w'}^{d'}$ of suitable lower degree $w' \le w-4$. As we will see in section 3.3.2, this ambiguity in the lowest-weight vectors $P_{w \ge 18}^d$ reflects the freedom of redefining certain new periods by rational multiples of non-critical L-values (2.19). Since each term of the ambiguities $[e_k, P_{w-k}^d]$ has modular depth three or more, these effects kick in with the modular completions of triple Eisenstein integrals $\beta_+ \begin{bmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{bmatrix}$ of degree $k_1+k_2+k_3 \ge 18$ and do not affect the $F_{m,k}^{\pm(s)}$ of modular depth two.

In the same way as the $(k-1)$-dimensional multiplets $e_k^{(j)}$ will be later on associated with the integration kernels $\tau^j G_k$ with $j = 0, 1, \dots, k-2$, the size of the multiplets $(ad_{e_0})^j P_w^d$ singles out the modular weights of holomorphic cusp forms: Since the $sl_2$ module generated by $P_w^d$ has dimension $(w-2d+1)$, it is associated with some $\Delta_{2s}$ at modular weight $2s = w-2d+2$. By the range $d \ge 2$, each cusp form $\Delta_{2s}$ is thereby expected to induce an infinite tower of lowest-weight vectors of degree $2(s+d-1) = 2(s+1)$, $2(s+2)$, $2(s+3)$, ... which yields relations in Tsunogai's derivation algebra under $e_k^{(j)} \to \epsilon_k^{(j)}$. The above examples of $P_w^d$ are
associated with the simplest holomorphic cusp forms $\Delta_{12}, \Delta_{16}, \Delta_{18}$ [36],

$$ \begin{aligned} \Delta_{12} &\leftrightarrow P_{14}^2, P_{16}^3, P_{18}^4, P_{20}^5, \dots & & : 11\text{-dim sl}_2 \text{ multiplets,} \\ \Delta_{16} &\leftrightarrow P_{18}^2, P_{20}^3, \dots & & : 15\text{-dim sl}_2 \text{ multiplets,} \\ \Delta_{18} &\leftrightarrow P_{20}^2, \dots & & : 17\text{-dim sl}_2 \text{ multiplets,} \end{aligned} \quad (2.73) $$

and the dimensions of the multiplets are in one-to-one correspondence to the 11, 15 and 17 choices of integration kernels $\tau^j \Delta_{2s}$ with $0 \le j \le 2s-2$ for $s = 6, 8$ and $9$ in the generating series we will introduce in section 2.5.

Note that, whenever the vector space $S_{2s}$ of cusp forms at weight $2s$ has dimension greater than one, the integration kernels $\tau^j \Delta_{2s}$ run over a basis of Hecke-normalised cusp forms, thus leading to different linearly independent instances of $P_w^d$, one for each basis element. For example, $\dim S_{24} = 2$ leads to two linearly independent instances of $P_{26}^2, P_{28}^3, \dots$, with similar results for the higher-dimensional $S_{2s}$ at $2s \ge 28$.

## 2.4 Zeta generators associated with primitive zeta values

The appearance of primitive zeta values $\zeta_w$ in MGFs for $w \in 2\mathbb{N}+1$ can be understood from derivations on the free Lie algebra on two elements $a$ and $b$. These derivations will be referred to as zeta generators, and the goal of this section is to review their rich interplay with Tsunogai's derivations $\epsilon_k$. We shall also discuss an uplift of these zeta generators to the free Lie algebra of $e_k^{(j)}$ which no longer admits any known description in terms of Lie[$a, b$].

### 2.4.1 Zeta generators $\sigma_w$ for Tsunogai's derivation algebra

Tsunogai's derivations $\epsilon_k$ with $k \in 2\mathbb{N}_0$ are realised through their action on free-Lie-algebra generators $a, b$ of the fundamental group of the once-punctured torus [35, 77, 79, 80]:

$$ \begin{aligned} \epsilon_k([a, b]) &= 0, & \epsilon_k(a) &= \mathrm{ad}_a^{k}b \quad \text{and} \quad \epsilon_k(b) &= \sum_{j=0}^{k/2} (-1)^j [\mathrm{ad}_a^{j}b, \mathrm{ad}_a^{2k-1-j}b] & \text{for } k \ge 2, \\ \epsilon_0(a) &= b, & \epsilon_0(b) &= 0 \quad \text{and} \quad \epsilon_0^{\vee}(a) &= 0, & \epsilon_0^{\vee}(b) &= a. \end{aligned} \quad (2.74) $$

Another infinite family $\sigma_w$ of derivations acting on $a, b$ is associated with odd Riemann zeta values $\zeta_w$ with $w \in 2\mathbb{N}+1$. The explicit form of the action $\sigma_w(a), \sigma_w(b)$ is determined in [45, 46, 80, 81] and the companion paper [34] by the tight interplay between configuration-space integrals at genus zero and at the boundary of moduli space at genus one. Upon comparison of $\sigma_w(a), \sigma_w(b)$ with the action (2.74) of Tsunogai's derivation algebra on $a, b$, the zeta generators can be written as infinite series of nested brackets of $\epsilon_k$, up to a tightly constrained so-called 'arithmetic' or 'non-geometric' contribution $z_w$, e.g. [34]

$$ \begin{aligned} \sigma_3 &= z_3 - \frac{1}{2}\epsilon_4^{(2)} + \frac{1}{480}[\epsilon_4, \epsilon_4^{(1)}] + \sum_{k=6}^{\infty} \mathrm{BF}_k \left( [\epsilon_4^{(1)}, \epsilon_k] - \frac{[\epsilon_4, \epsilon_k^{(1)}]}{k-2} \right) & (2.75) \\ &+ \sum_{m=4}^{\infty} \sum_{r=6}^{\infty} \frac{(m-1)\mathrm{BF}_m\mathrm{BF}_r}{m+r-2} [\epsilon_m, [\epsilon_4, \epsilon_r]] \end{aligned} $$
with $BF_k = B_k/k!$. The arithmetic parts $z_w$ are $sl_2$ singlets and therefore commute with the raising and lowering operators,

$$[z_w, \epsilon_0] = [z_w, \epsilon_0^\vee] = 0. \qquad (2.76)$$

Moreover, they normalise the derivation algebra of the $\epsilon_k$ in the sense that $[z_w, \epsilon_k]$ are expressible in terms of nested brackets of $\epsilon_{k_i}^{(j_i)}$ in a $(k-1)$-dimensional representation of $sl_2$, for instance

$$[z_3, \epsilon_4] = \frac{1}{504} ([\epsilon_6^{(2)}, \epsilon_4] - 3[\epsilon_6^{(1)}, \epsilon_4^{(1)}] + 6[\epsilon_6, \epsilon_4^{(2)}]). \qquad (2.77)$$

As detailed in the companion paper [34], a finite number of contributions to $\sigma_w$ is sufficient to determine both the infinite tower of $\epsilon_{k_i}^{(j_i)}$-brackets and the entirety of commutators $[z_w, \epsilon_k]$:
Both of them can be extracted from the contributions of $\epsilon_{k_i}^{(j_i)}$ to $\sigma_w$ up to and including key degree $\sum_i k_i = 2w$ by using that the $\sigma_w$ commute with

$$N := -\epsilon_0 + \sum_{k=4}^{\infty} (k-1)BF_k \epsilon_k, \quad [N, \sigma_w] = 0. \qquad (2.78)$$

This is equivalent to section 7.1 (iii) of Brown's work [14] and will also be used to pinpoint the modular properties of our generating series in section 4.1.

We note that, starting from $\sigma_7$, there are ambiguities in shifting the arithmetic parts $z_w$ by $sl_2$-invariant nested brackets of $\epsilon_{k_i}^{(j_i)}$ of total degree $2w$, see section 3.2.3 for further details. A preferred choice of $z_w$ resolving this ambiguity is described in section 3.2.3, at the end of section 3.3.4 and the companion paper [34].

### 2.4.2 Zeta generators $\hat{\sigma}_w$ beyond Tsunogai's derivation algebra

In order to use zeta generators for the construction of equivariant iterated Eisenstein integrals beyond MGFs, i.e. with cusp-form contributions, we need to uplift the derivations $\epsilon_k^{(j)}$ in their representations and bracket relations to the free-algebra generators $e_k^{(j)}$. Since the $sl_2$ representation theory of the $\epsilon_k^{(j)}$ and $e_k^{(j)}$ is identical, the only challenge in this uplift stems from the combinations $P_w^d$ at degree $w \ge 14$ that vanish in the $\epsilon_k^{(j)}$-incarnation of zeta generators. Throughout this work, we shall describe guiding principles and (not necessarily canonical) choices on how to reinstate $P_w^d$ into uplifted versions of zeta generators. Still, as will be seen in section 3.3.3, passing from zeta generators for $\epsilon_k^{(j)}$ to their uplifts for $e_k^{(j)}$ generically introduces ambiguities involving $P_w^d$ at modular depth $\ge 3$ and degree $\ge 18$.

We will use the notation $\hat{\sigma}_w, \hat{z}_w$ for the uplifted zeta generators and their arithmetic parts which reduce to the $\sigma_w, z_w$ of the previous section upon the replacement $e_k^{(j)} \to \epsilon_k^{(j)}$,

$$\sigma_w = \hat{\sigma}_w \big|_{e_k^{(j)} \to \epsilon_k^{(j)}}, \quad [z_w, \epsilon_k] = [\hat{z}_w, e_k] \big|_{e_k^{(j)} \to \epsilon_k^{(j)}}. \qquad (2.79)$$

As a first guiding principle, the uplifted zeta generators are still taken to commute with the canonical uplift of $N$, similarly denoted by $\hat{N}$, in (2.78),

$$\hat{N} = -e_0 + \sum_{k=4}^{\infty} (k-1)BF_k e_k, \quad [\hat{N}, \hat{\sigma}_w] = 0. \qquad (2.80)$$
Second, the closed formula of [48] for the modular-depth-two contributions to $[z_w, \epsilon_k]$ is taken to uplift in its simplest representation, i.e.

$$[\hat{z}_w, \epsilon_k] = \frac{\mathrm{BF}_{w+k-1}}{\mathrm{BF}_k} t^{w+1}(\mathrm{e}_{w+1}, \mathrm{e}_{w+k-1}) + \dots, \qquad (2.81)$$

where each term in the ellipsis features modular depth three or higher, see appendix E.1 for examples, and the arithmetic part is once more an $sl_2$ singlet,

$$[\hat{z}_w, \epsilon_0] = [\hat{z}_w, e_0^\vee] = 0. \qquad (2.82)$$

Third, the closed formula for the complete modular-depth-two contributions to $\sigma_w$ determined in [34] is taken to uplift in its simplest representation,

$$\begin{aligned} \hat{\sigma}_w = \hat{z}_w - \frac{1}{(w-1)!} e_{w+1}^{(w-1)} - \frac{1}{2} \sum_{d=3}^{w-2} \frac{\mathrm{BF}_{d-1}}{\mathrm{BF}_{w-d+2}} \sum_{k=d+1}^{w-1} \mathrm{BF}_{k-d+1} \mathrm{BF}_{w-k+1} s_{k,w-k+d}^d \\ - \sum_{d=5}^{w} \mathrm{BF}_{d-1} s_{d-1,w+1}^d - \frac{1}{2} \mathrm{BF}_{w+1} s_{w+1,w+1}^{w+2} + \sum_{k=w+3}^{\infty} \mathrm{BF}_k \sum_{j=0}^{w-2} \frac{(-1)^j \binom{k-2}{j}^{-1}}{j!(w-2-j)!} [e_{w+1}^{(w-2-j)}, e_k^{(j)}] + \dots, \end{aligned} \qquad (2.83)$$

where the terms at $d=3$ in the first line are known from [46] and the ellipsis again gathers all contributions of modular depth three and higher. Setting $w=3$ in (2.81), (2.83) and specialising the generators according to (2.79) reproduces the earlier examples (2.75) and (2.77). As detailed in [34], the closed modular-depth-two formula (2.83) together with the vanishing condition $[\hat{N}, \hat{\sigma}_w] = 0$ determine the modular-depth-three contributions to $\hat{\sigma}_w$ up to highest-weight vectors, see appendix E.2 for examples. Moreover, (2.83) and $[\hat{N}, \hat{\sigma}_w] = 0$ fix the complete modular-depth-three contributions to $[\hat{z}_w, \epsilon_k]$ in the ellipsis of (2.81).

## 2.5 Generating series

We shall finally review generating-series constructions combining the non-commuting variables $\epsilon_k$, $z_w$ relevant to MGFs with the iterated Eisenstein integrals of section 2.1. The uplifts of the series in this subsection to $\epsilon_k$, $\hat{z}_w$ and iterated integrals involving $\Delta_k$ will be the main subject of section 3. Following the conventions of [28], the combinations $\beta_\pm$ of holomorphic iterated Eisenstein integrals in (2.22) entering MGFs are organised into generating series

$$\begin{aligned} \mathcal{J}_\pm(\epsilon_k; \tau) := 1 &+ \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} \frac{(-1)^{j_1}(k_1-1)}{(k_1-2-j_1)!} \beta_\pm \begin{bmatrix} j_1 \\ k_1 \end{bmatrix} \epsilon_{k_1}^{(k_1-j_1-2)} \\ &+ \sum_{k_1=4}^{\infty} \sum_{k_2=4}^{\infty} \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \frac{(-1)^{j_1+j_2}(k_1-1)(k_2-1)}{(k_1-2-j_1)!(k_2-2-j_2)!} \beta_\pm \begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix} \epsilon_{k_2}^{(k_2-j_2-2)} \epsilon_{k_1}^{(k_1-j_1-2)} + \dots \end{aligned} \qquad (2.84)$$

with iterated integrals of modular depth $\ge 3$ in the ellipsis. It will be convenient to introduce a shorthand notation $\epsilon[P]$ for the reoccurring combinations of derivations

$$\mathcal{J}_{\pm}(\epsilon_k; \tau) = \sum_{P \in \begin{Bmatrix} j \\ k \end{Bmatrix}^{\times}} \epsilon[P] \beta_{\pm}[P; \tau], \qquad (2.85)$$

$$\epsilon[P] = \epsilon \begin{bmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{bmatrix} := \left( \prod_{i=1}^{\ell} \frac{(-1)^{j_i} (k_i - 1)}{(k_i - j_i - 2)!} \right) \epsilon_{k_\ell}^{(k_\ell - 2 - j_\ell)} \cdots \epsilon_{k_2}^{(k_2 - 2 - j_2)} \epsilon_{k_1}^{(k_1 - 2 - j_1)},$$
where the summation range $P \in \{j_k\}^\times$ refers to all words in $j_i, k_i$ of length $\ell \in \mathbb{N}_0$ specified in the second line such that $k_i \ge 4$ and $0 \le j_i \le k_i - 2$. Our convention for the empty word at $\ell = 0$ is $\epsilon[\emptyset] = 1$.

Brown described a composition of the generating series $\mathcal{J}_\pm$ whose coefficients are modular forms after invoking the relations among the derivations $\epsilon_k$ [14]: In the notation of [28], Brown's construction of equivariant iterated Eisenstein integrals takes the form

$$ \mathcal{J}^{\text{eqv}}(\epsilon_k; \tau) := \sum_{P \in \{j_k\}^\times} \epsilon[P] \beta^{\text{eqv}}[P; \tau] = \mathcal{J}_+(\epsilon_k; \tau) \mathcal{B}^{\text{sv}}(\epsilon_k; \tau) \phi^{\text{sv}}(\widetilde{\mathcal{J}}_-(\epsilon_k; \tau)), \quad (2.86) $$

where the tilde in the second line reverses the relative order of the derivations and the iterated integrals in $\mathcal{J}_-(\epsilon_k; \tau)$ and can be implemented by replacing $\beta_- \begin{bmatrix} j_1 & j_2 & \cdots & j_\ell \\ k_1 & k_2 & \cdots & k_\ell \end{bmatrix} \to \beta_- \begin{bmatrix} j_\ell & \cdots & j_2 & j_1 \\ k_\ell & \cdots & k_2 & k_1 \end{bmatrix}$ in the expansions (2.84) and (2.85). The coefficients $\beta^{\text{eqv}}[P; \tau]$ of $\epsilon[P]$ are the modular forms with SL(2, $\mathbb{Z}$) weights in (2.45) and whose terms at highest modular depth in (2.44) are generated by the simpler composition $\mathcal{J}_+(\epsilon_k; \tau)\widetilde{\mathcal{J}}_-(\epsilon_k; \tau)$. However, the modularity of $\beta^{\text{eqv}}$ relies on the admixture of MZVs multiplying integrals of lower modular depth as exemplified in (2.41) and (2.43). Brown's prescription to insert appropriate combinations of MZVs is based on two ingredients:

(i) an $\epsilon_k^{(j)}$-valued series $\mathcal{B}^{\text{sv}}$ of single-valued MZVs (see appendix A for the latter notion);

(ii) a change of alphabet $\phi^{\text{sv}}$ for the derivations $\epsilon_k$ multiplying the antiholomorphic integrals $\beta_-$ in the conventions of [28].

### 2.5.1 The series $\mathcal{B}^{\text{sv}}$ of single-valued MZVs

The expansion of $\mathcal{B}^{\text{sv}}$ in words in $\epsilon_k^{(j_i)}$ follows the form of $\mathcal{J}_\pm, \mathcal{J}^{\text{eqv}}$ in (2.85), (2.86), and its components $b^{\text{sv}}[P; \tau]$ only depend rationally on $\tau, \bar{\tau}$:

$$ \mathcal{B}^{\text{sv}}(\epsilon_k; \tau) := \sum_{P \in \{j_k\}^\times} \epsilon[P] b^{\text{sv}}[P; \tau], \quad (2.87) $$

$$ b^{\text{sv}}[\dots; j_k \dots; \tau] := \sum_{p=0}^{k-2-j} \sum_{\ell=0}^{j+p} \binom{k-j-2}{p} \binom{j+p}{\ell} \frac{(-2\pi i \bar{\tau})^\ell}{(4y)^p} c^{\text{sv}}[\dots; \bar{j}-\ell+p \dots]. $$

The double sum over $p, \ell$ is understood to apply separately to each column $j_i$ of $b^{\text{sv}}[\dots; \tau]$. The $\tau$-independent $c^{\text{sv}}[\begin{smallmatrix} j_1 & \cdots & j_\ell \\ k_1 & \cdots & k_\ell \end{smallmatrix}]$ are projected to $\mathbb{Q}$-linear combinations of single-valued MZVs of transcendental weight $\ell+j_1+\dots+j_\ell$ through the relations of the accompanying $\epsilon_k$ in (2.87).$^{16}$ Their modular-depth-one instances take the closed form $c^{\text{sv}}[\begin{smallmatrix} j \\ k \end{smallmatrix}] = -\frac{2\zeta_{k-1}}{k-1} \delta_{j,k-2}$, and they are responsible for the zeta value in the expression (2.41) for $\beta^{\text{eqv}}[\begin{smallmatrix} j \\ k \end{smallmatrix}]$. Numerous examples of $c^{\text{sv}}$ at modular depth two and three can be found in the ancillary files and in section 3.1 of [28].

<footnote>$^{16}$Starting from section 3.1.3, the $c^{\text{sv}}[\begin{smallmatrix} j_1 & \cdots & j_\ell \\ k_1 & \cdots & k_\ell \end{smallmatrix}]$ will be dressed with the free-Lie-algebra generators $e_k^{(j)}$ and therefore be individually accessible. In this setting, the single-valued MZVs of transcendental weight $\ell+j_1+\dots+j_\ell$ in $c^{\text{sv}}[\begin{smallmatrix} j_1 & \cdots & j_\ell \\ k_1 & \cdots & k_\ell \end{smallmatrix}]$ of modular depth $\ell \ge 3$ and degree $\sum_i k_i \ge 18$ will be augmented by new periods to be discussed in section 3.3.3.</footnote>
Note that the constants $c^{\text{sv}}$ give rise to a convenient dictionary between the building blocks $\beta^{\text{sv}}$ for MGFs introduced in [30] and the modular forms $\beta^{\text{eqv}}$ in (2.86) [28]

$$ \beta^{\text{sv}} \begin{bmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{bmatrix} = \sum_{r=0}^{\ell} (-1)^r d^{\text{sv}} \begin{bmatrix} j_r & \dots & j_2 & j_1 \\ k_r & \dots & k_2 & k_1 \end{bmatrix} \beta^{\text{eqv}} \begin{bmatrix} j_{r+1} & \dots & j_\ell \\ k_{r+1} & \dots & k_\ell \end{bmatrix} + \dots, \qquad (2.88) $$

$$ d^{\text{sv}} \begin{bmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{bmatrix} = \sum_{p_1=0}^{k_1-2-j_1} \dots \sum_{p_\ell=0}^{k_\ell-2-j_\ell} \frac{\left( \frac{k_1-2-j_1}{p_1} \right) \left( \frac{k_2-2-j_2}{p_2} \right) \dots \left( \frac{k_\ell-2-j_\ell}{p_\ell} \right)}{(4y)^{p_1+p_2+\dots+p_\ell}} c^{\text{sv}} \begin{bmatrix} j_1+p_1 & j_2+p_2 & \dots & j_\ell+p_\ell \\ k_1 & k_2 & \dots & k_\ell \end{bmatrix}. $$

The ellipsis in the first line refers to terms with at least one holomorphic cusp forms in the integrand which always drop out from the combinations of $\beta^{\text{sv}}$ encountered in MGFs.

### 2.5.2 The change of alphabet $\phi^{\text{sv}}$

While theorem 7.2 of [14] implicitly determines $\mathbb{B}^{\text{sv}}$ and $\phi^{\text{sv}}$ in terms of multiple modular values, the following series in single-valued MZVs gives an explicit characterisation of the change of alphabet in terms of the arithmetic parts $z_w$ of zeta generators in section 2.4.1, see section 4 of [28],

$$ \phi^{\text{sv}}(\mathcal{X}) := \mathbb{M}^{\text{sv}}(z_i) \mathcal{X} \mathbb{M}^{\text{sv}}(z_i)^{-1}, \qquad (2.89) $$

where $\mathcal{X}$ refers to an arbitrary series in $\epsilon_k$ and

$$ \mathbb{M}^{\text{sv}}(z_i) := \sum_{\ell=0}^{\infty} \sum_{\substack{i_1, i_2, \dots, i_\ell \\ \ell \in 2\mathbb{N}+1}} z_{i_1} z_{i_2} \dots z_{i_\ell} \rho^{-1}(\text{sv}(f_{i_1} f_{i_2} \dots f_{i_\ell})) \qquad (2.90) $$

$$ = 1 + 2 \sum_{i_1 \in 2\mathbb{N}+1} z_{i_1} \zeta_{i_1} + 2 \sum_{i_1, i_2 \in 2\mathbb{N}+1} z_{i_1} z_{i_2} \zeta_{i_1} \zeta_{i_2} + \sum_{i_1, i_2, i_3 \in 2\mathbb{N}+1} z_{i_1} z_{i_2} z_{i_3} \rho^{-1}(\text{sv}(f_{i_1} f_{i_2} f_{i_3})) + \dots $$

We stress that formula (2.89) is only applicable to series $\mathcal{X}$ in the generators $\epsilon_k$ and a central part of our work will be to discuss a more general change of alphabet for series $\mathcal{X}$ in the unconstrained $e_k$ in section 3.

Expression (2.90) contains both the isomorphism $\rho^{-1}$ from the $f$-alphabet to real MZVs [32, 33] and the single-valued map on the $f$-alphabet given by [20, 21]

$$ \text{sv}(f_{i_1} f_{i_2} \cdots f_{i_\ell}) = \sum_{j=0}^{\ell} f_{i_j} \cdots f_{i_2} f_{i_1} \sqcup f_{i_{j+1}} \cdots f_{i_\ell}, \quad i_1, \dots, i_\ell \in 2\mathbb{N}+1. \qquad (2.91) $$

In passing to the second line of (2.90), we have used $\rho^{-1}(\text{sv}(f_{i_1})) = 2\zeta_{i_1}$ and $\rho^{-1}(\text{sv}(f_{i_1} f_{i_2})) = 2\zeta_{i_1} \zeta_{i_2}$ for words of length one and two in the $f$-alphabet. Starting from length three, $f$-alphabet expressions such as $\rho^{-1}(\text{sv}(f_{i_1} f_{i_2} f_{i_3}))$ generically comprise (conjecturally) indecomposable single-valued MZVs beyond depth one, see appendix A.2 for examples at weight $\le 15$ and the ancillary file for weight $\le 17$. The expansion (2.90) of $\mathbb{M}^{\text{sv}}(z_i)$ reorganises the conjugation in (2.89) into a series in nested commutators,

$$ \phi^{\text{sv}}(\mathcal{X}) = \sum_{\ell=0}^{\infty} \sum_{\substack{i_1, i_2, \dots, i_\ell \\ \ell \in 2\mathbb{N}+1}} [z_{i_1}, [z_{i_2}, \dots [z_{i_\ell}, \mathcal{X}] \dots]] \rho^{-1}(\text{sv}(f_{i_1} f_{i_2} \dots f_{i_\ell})). \qquad (2.92) $$
Since $[z_w, \epsilon_k]$ is expressible via nested brackets of $\epsilon_k^{(j)}$, the coefficients of each $sv(f_{i_1} \dots f_{i_l})$ take values in Tsunogai's derivation algebra and do not obstruct an extraction of coefficients as in (2.86) after modding out by Pollack-relations. Based on the examples of the uplifted $[\hat{z}_w, e_k]$ in appendix E.1, we find the contributions

$$ 
\begin{aligned}
\phi^{\text{sv}}(\epsilon_4) &= \epsilon_4 + 2\zeta_3[z_3, \epsilon_4] + 2\zeta_5[z_5, \epsilon_4] + 2\zeta_3^2[z_3, [z_3, \epsilon_4]] + \dots \qquad (2.93) \\
&= \epsilon_4 + 2\zeta_3 \frac{\text{BF}_6}{\text{BF}_4} t^4(\epsilon_4, \epsilon_6) + 2\zeta_5 \frac{\text{BF}_8}{\text{BF}_4} t^6(\epsilon_6, \epsilon_8) + \zeta_5 \frac{\text{BF}_6 \text{BF}_2^3}{\text{BF}_4^2} t^4(\epsilon_6, t^3(\epsilon_4, \epsilon_4)) \\
&\quad - \zeta_5 \text{BF}_4 \left\{ \frac{9}{5} t^3(\epsilon_4, t^4(\epsilon_4, \epsilon_6)) + \frac{2}{5} t^4(\epsilon_4, t^3(\epsilon_4, \epsilon_6)) \right\} + 2\zeta_3^2 \frac{\text{BF}_6^2}{\text{BF}_4^2} t^4(t^4(\epsilon_4, \epsilon_6), \epsilon_6) \\
&\quad + 2\zeta_3^2 \frac{\text{BF}_8}{\text{BF}_4} t^4(\epsilon_4, t^4(\epsilon_4, \epsilon_8)) - \frac{9\zeta_3^2}{5} \text{BF}_4 t^4(\epsilon_4, t^2(\epsilon_4, t^3(\epsilon_4, \epsilon_4))) + \dots
\end{aligned}
$$

up to and including transcendental weight six of the accompanying MZVs.

### 3 Generating series of equivariant iterated Eisenstein integrals

In this section, we analyse SL(2, $\mathbb{Z}$)-equivariant iterated Eisenstein integrals in more detail and arrange them into the generating series presented in (2.86). These generating series are expressed in terms of the non-commutative letters $e_k^{(j)}$ introduced in section 2.3. The equivariant integrals can be presented in two different ‘frames’ that we discuss in section 3.1. One of our main results is to identify the way periods (such as multiple zeta values or L-values) enter equivariant expressions and this can be traced back to the zeta generators of section 2.4 as we explain in section 3.2. Since in general iterated integrals of cusp forms are needed to make iterated Eisenstein integrals transform correctly under SL(2, $\mathbb{Z}$), we discuss their appearance in section 3.3, along with two classes of new periods that arise at modular depth three. Moreover, a generating-series description of equivariant iterated Eisenstein integrals at arbitrary modular depth including cusp forms is proposed in (3.74).

The final part 3.4 of this section repackages the information of section 3.3 into solutions of inhomogeneous Laplace equations at modular depth three. In this setting, the two classes of new periods arise as the coefficients of solutions to the associated homogeneous Laplace equations.

#### 3.1 The holomorphic frame

The goal of this section is to rebuild the generating series in (2.86) such that $\mathbb{B}^{\text{sv}}$ no longer depends on $\tau$ and $J_{\pm}$ become (anti-)meromorphic in $\tau$, respectively. As we will see, both properties are accomplished by conjugation with the $SL_2$ transformation

$$ U_{\text{SL}_2}(\tau) := \exp\left(-\frac{e_0^{\vee}}{4y}\right) \exp(2\pi i \bar{\tau} e_0) \in \text{SL}_2. \qquad (3.1) $$

By slight abuse of notation, we employ the same symbol $U_{\text{SL}_2}(\tau)$ for the $SL_2$ transformation $\exp(-\frac{e_0^{\vee}}{4y}) \exp(2\pi i \bar{\tau} e_0)$ acting on Tsunogai's derivations $\epsilon_k^{(j)}$ instead of the free Lie-algebra generators $e_k^{(j)}$. Accordingly, the definitions in this section pave the way for the inclusion of cusp forms into equivariant iterated Eisenstein integrals.
### 3.1.1 Setting up (anti-)holomorphic (1,0) and (0,1)-forms

As a convenient starting point for the construction of (anti-)meromorphic generating series conjugate to $J_\pm$, we introduce the following (1,0)-form in $\tau$ that gathers all the relevant integration kernels

$$ 
\begin{aligned}
A_+(e_k; \tau) &:= \sum_{k=4}^{\infty} \sum_{j=0}^{k-2} (-1)^j \frac{(k-1)}{j!} \left[ \nu\left[\begin{smallmatrix} j \\ k \end{smallmatrix}; \tau\right] e_k^{(j)} + \sum_{\Delta_k \in S_k} \nu\left[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}; \tau\right] e_{\Delta_k^+}^{(j)} \right] \\
&= \sum_{k=4}^{\infty} \sum_{j=0}^{k-2} (-1)^j \frac{(k-1)}{j!} (2\pi i)^{1+j} \tau^j \left[ (2\pi i)^{-k} G_k(\tau) e_k^{(j)} + \sum_{\Delta_k \in S_k} \Delta_k(\tau) e_{\Delta_k^+}^{(j)} \right] d\tau,
\end{aligned} 
\quad (3.2) $$

with $S_k$ the vector space of holomorphic cusp forms of weight $k$, and where $\nu\left[\begin{smallmatrix} j \\ k \end{smallmatrix}; \tau\right]$ and $\nu\left[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}; \tau\right]$ are defined in (2.4) and (2.9), respectively.

As an antiholomorphic analogue of the holomorphic (1,0) form $A_+$ in (3.2), we shall introduce the following (0,1)-form

$$ 
\begin{aligned}
A_-(e_k; \tau) &:= \sum_{k=4}^{\infty} \sum_{j=0}^{k-2} \frac{(k-1)}{j!} \left[ \overline{\nu\left[\begin{smallmatrix} j \\ k \end{smallmatrix}; \tau\right]} e_k^{(j)} + \sum_{\Delta_k \in S_k} \overline{\nu\left[\begin{smallmatrix} j \\ \Delta_k \end{smallmatrix}; \tau\right]} e_{\Delta_k^-}^{(j)} \right] \\
&= - \sum_{k=4}^{\infty} \sum_{j=0}^{k-2} (-1)^j \frac{(k-1)}{j!} (2\pi i)^{1+j} \overline{\tau}^j \left[ (2\pi i)^{-k} \overline{G_k(\tau)} e_k^{(j)} + \sum_{\Delta_k \in S_k} \overline{\Delta_k(\tau)} e_{\Delta_k^-}^{(j)} \right] d\overline{\tau}.
\end{aligned} 
\quad (3.3) $$

We emphasise that $A_-$ is not the complex conjugate of $A_+$ since it differs by a factor $(-1)^j$ in the definition and also by the letters $e_{\Delta_k^+}^{(j)}$ versus $e_{\Delta_k^-}^{(j)}$ accompanying the cusp forms. These letters are non-commutative variables spanning $sl_2$ multiplets according to

$$ e_{\Delta_k^\pm}^{(j)} = ad_{e_0}^j e_{\Delta_k^\pm}, \quad e_{\Delta_k^\pm}^{(k-1)} = 0, \quad ad_{e_0} e_{\Delta_k^\pm} = 0, \quad (3.4) $$

such that $e_{\Delta_k^\pm}$ are lowest-weight vectors. In this work, the letters $e_{\Delta_k^\pm}$ are not independent from the Eisenstein letters $e_k$, but are expressible through nested commutators involving at least one Pollack combination $P_w^d$ of section 2.3, for example

$$ 
\begin{aligned}
e_{\Delta_{12}^{\pm}} = & \frac{1}{\Lambda(\Delta_{12}, 11)} \left( \Lambda(\Delta_{12}, 13) P_{16}^3 + \Lambda(\Delta_{12}, 15) P_{20}^5 \right) \\
& \mp \frac{1}{\Lambda(\Delta_{12}, 10)} \left( \Lambda(\Delta_{12}, 12) P_{14}^2 + \Lambda(\Delta_{12}, 14) P_{18}^4 \right) + \dots,
\end{aligned} 
\quad (3.5) $$

where the ellipsis refers to terms of modular depth $\ge 3$ and degree $\ge 22$, see section 3.3.2 below for details.

In the context of more general classes of iterated integrals, it could be interesting to promote variants of the letters $e_{\Delta_k^\pm}$ to be independent of $e_k$. Upon augmenting the connection (3.2) by weakly holomorphic modular forms, it may be possible to generalise the non-holomorphic modular forms of [82] with poles at the cusp to higher modular depth and to take advantage of independent letters $e_{\Delta_k^\pm}$ in a formulation via generating series.<fnref n="17" />

<footnote n="17">¹⁷Since the letters $e_{\Delta_k^\pm}$ in this work reduce to Eisenstein letters $e_k$, one cannot read off the coefficient of $e_{\Delta_k^\pm}$ from generating series such as (3.22) below. It is tempting to approach higher-depth analogues of the non-holomorphic modular forms of [82] via generating-series methods where the modular completions are obtained by reading off the coefficients of independent letters associated with holomorphic cusp forms.</footnote>
The occurrence of the Pollack combinations $P_w^d$ ensures that the letters $e_{\Delta_k}^{\pm}$ vanish when replacing $e_k$ by $\epsilon_k$. As indicated by the notation on the left-hand sides of (3.2) and (3.3), throughout this paper the symbol $e_{\Delta_k}^{\pm}$ is always understood as a combination of Eisenstein letters $e_k$. We will spell out additional contributions to $e_{\Delta_k}^{\pm}$ at low modular depth in (3.73) below, generalising (3.5) up to and including degree 20.

As will become clear in the following, the distinction between $e_{\Delta_k^+}$ and $e_{\Delta_k^-}$ is necessary for instance to ensure that the modular invariants $F_{m,k}^{+(s)}$ and $F_{m,k}^{-(s)}$ to be extracted in later sections can be even or odd under the upper-half plane involution $\tau \to -\bar{\tau}$. This is in contrast to the holomorphic Eisenstein series $E_k$ in (2.40) and (2.41) that are always even by the relative plus sign between the contributing $\beta_+$ and $\beta_-$, so one has to employ the same letter $e_k$ in $\mathbb{J}_+$ and $\mathbb{J}_-$.

### 3.1.2 Iterated integrals from path-ordered exponentials

Given the (1,0)- and (0,1)-forms $\mathbb{A}_+$ and $\mathbb{A}_-$ in (3.2) and (3.3), respectively, we shall next define generating series of (anti-)holomorphic iterated integrals from their path-ordered exponentials,

$$ \mathbb{I}_+(e_k; \tau) = \mathrm{P-exp} \left( \int_{\tau}^{i\infty} \mathbb{A}_+(e_k; \tau_1) \right) = 1 + \int_{\tau}^{i\infty} \mathbb{A}_+(e_k; \tau_1) + \int_{\tau}^{i\infty} \left( \int_{\tau_1}^{i\infty} \mathbb{A}_+(e_k; \tau_2) \right) \mathbb{A}_+(e_k; \tau_1) + \dots, \qquad (3.6) $$

$$ \tilde{\mathbb{I}}_{-}(e_k; \tau) = \tilde{\mathrm{P-exp}} \left( \int_{\bar{\tau}}^{-i\infty} \mathbb{A}_{-}(e_k; \tau_1) \right) = 1 + \int_{\bar{\tau}}^{-i\infty} \mathbb{A}_{-}(e_k; \tau_1) + \int_{\bar{\tau}}^{-i\infty} \mathbb{A}_{-}(e_k; \tau_1) \int_{\bar{\tau}_1}^{-i\infty} \mathbb{A}_{-}(e_k; \tau_2) + \dots, $$

with integrals over $\ge 3$ powers of $\mathbb{A}_{\pm}$ in the ellipses. In our conventions for P-exp, the integration variable $\tau_1$ closest to the lower endpoint $\tau$ is associated with the rightmost factor of $\mathbb{A}_+$ with respect to the ordering of the $e_k, e_{\Delta_k^{\pm}}$. The extra tilde in the notation $\tilde{\mathrm{P-exp}}$ entering $\tilde{\mathbb{I}}_-$ instructs us to reverse the ordering convention as exemplified in the second-order term $\sim \mathbb{A}_-(\tau_1)\mathbb{A}_-(\tau_2)$. In terms of the iterated Eisenstein integrals (2.7) and their cuspidal analogues such as (2.9), the expansion of the first path-ordered exponential in (3.6) yields

$$ \begin{aligned} \mathbb{I}_+(e_k; \tau) = & 1 + \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} (-1)^{j_1} \frac{(k_1-1)}{j_1!} \left\{ \mathcal{E}\left[\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}; \tau \right] e_{k_1}^{(j_1)} + \sum_{\Delta_{k_1} \in S_{k_1}} \mathcal{E}\left[\begin{smallmatrix} j_1 \\ \Delta_{k_1} \end{smallmatrix}; \tau \right] e_{\Delta_{k_1}^+}^{(j_1)} \right\} \quad (3.7) \\ & + \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} \sum_{k_2=4}^{\infty} \sum_{j_2=0}^{k_2-2} \frac{(-1)^{j_1+j_2}}{j_1!j_2!} \left\{ \mathcal{E}\left[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}; \tau \right] e_{k_1}^{(j_1)} e_{k_2}^{(j_2)} + \sum_{\Delta_{k_2} \in S_{k_2}} \mathcal{E}\left[\begin{smallmatrix} j_1 & j_2 \\ k_1 & \Delta_{k_2} \end{smallmatrix}; \tau \right] e_{k_1}^{(j_1)} e_{\Delta_{k_2}^+}^{(j_2)} \right. \\ & \left. + \sum_{\Delta_{k_1} \in S_{k_1}} \mathcal{E}\left[\begin{smallmatrix} j_1 & j_2 \\ \Delta_{k_1} & k_2 \end{smallmatrix}; \tau \right] e_{\Delta_{k_1}^+}^{(j_1)} e_{k_2}^{(j_2)} + \sum_{\Delta_{k_1} \in S_{k_1}} \sum_{\Delta_{k_2} \in S_{k_2}} \mathcal{E}\left[\begin{smallmatrix} j_1 & j_2 \\ \Delta_{k_1} & \Delta_{k_2} \end{smallmatrix}; \tau \right] e_{\Delta_{k_1}^+}^{(j_1)} e_{\Delta_{k_2}^+}^{(j_2)} \right\} + \dots, \end{aligned} $$
with iterated integrals of modular depth ≥ 3 over holomorphic modular forms in the ellipsis. The analogous expansion of $\tilde{\mathbb{I}}_-$ reads

$$ \tilde{\mathbb{I}}_-(e_k; \tau) = 1 + \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} \frac{(k_1-1)}{j_1!} \left\{ \overline{\mathcal{E}[j_1; \tau]} e_{k_1}^{(j_1)} + \sum_{\Delta_{k_1} \in S_{k_1}} \overline{\mathcal{E}[\Delta_{k_1}; \tau]} e_{\Delta_{k_1}^-}^{(j_1)} \right\} + \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} \sum_{k_2=4}^{\infty} \sum_{j_2=0}^{k_2-2} \frac{(k_1-1)(k_2-1)}{j_1!j_2!} \left\{ \overline{\mathcal{E}[j_2; k_1; \tau]} e_{k_1}^{(j_1)} e_{k_2}^{(j_2)} + \sum_{\Delta_{k_2} \in S_{k_2}} \overline{\mathcal{E}[\Delta_{k_2}; k_1; \tau]} e_{k_1}^{(j_1)} e_{\Delta_{k_2}^-}^{(j_2)} + \sum_{\Delta_{k_1} \in S_{k_1}} \sum_{\Delta_{k_2} \in S_{k_2}} \overline{\mathcal{E}[\Delta_{k_2}; k_1; \tau]} e_{k_1}^{(j_1)} e_{\Delta_{k_2}^-}^{(j_2)} \right\} + \dots, \quad (3.8) $$

where the factors of $(-1)^{j_i}$ seen in (3.7) are absent by our conventions for $\mathbb{A}_-$ in (3.3), and the ellipsis again refers to iterated integrals of modular depth ≥ 3. As mentioned above, our notation $\mathbb{I}_+(e_k; \tau)$ and $\tilde{\mathbb{I}}_-(e_k; \tau)$ for the arguments anticipates that the letters $e_{\Delta_k^\pm}$ will eventually be expressible in terms of nested brackets of $e_k$ (see (3.73) below) and do not enter the construction of this work as independent variables. Nevertheless, the double-integral contributions spelt out in (3.7) and (3.8) illustrate that the reversal from the $\tilde{\mathbb{P}}$-exp operation in (3.6) treats $e_{\Delta_k^\pm}$ as a single letter and is more conveniently thought of as a reversal of the integration order of the $\mathcal{E}[...]$.

### 3.1.3 SL₂ transformation of generating series

We shall now make contact with the generating series $\mathbb{J}_\pm$ and $\mathbb{B}^{sv}$ of iterated Eisenstein integrals $\beta_\pm$ and single-valued MZVs reviewed in section 2.5. Both of them are related to simpler generating series by conjugation with the SL₂ transformation $U_{SL_2}$ in (3.1). The exponentials can be evaluated as a power series

$$ U_{\mathrm{SL}_2}(\tau) \mathbb{X} U_{\mathrm{SL}_2}(\tau)^{-1} = \sum_{m,n=0}^{\infty} \frac{1}{m!n!} \left(-\frac{1}{4y}\right)^m (2\pi i \bar{\tau})^n (\mathrm{ad}_{e_0^\nu})^m (\mathrm{ad}_{e_0})^n \mathbb{X}, \quad (3.9) $$

when acting on $e_k$-valued expressions $\mathbb{X}$. For contributions to $\mathbb{X}$ at fixed degree, the infinite sums in (3.9) truncate to finitely many terms since a given modular-depth-$\ell$ word $e_{k_1}^{(j_1)} \dots e_{k_\ell}^{(j_\ell)}$ is annihilated at the latest by $1+\sum_{i=1}^\ell (k_i-2)$ powers of $\mathrm{ad}_{e_0^\nu}$ or $\mathrm{ad}_{e_0}$.

We prove in appendix B that the conjugation (3.9) by the SL₂ transformation $U_{SL_2}$ relates the combination of kernels $\nu[j_k; \tau_1]$ in the expressions (3.2) and (3.3) for $\mathbb{A}_\pm$ to the alternative kernels $\omega_\pm[j_k; \tau, \tau_1]$ in section 2.1.3,

$$ U_{\mathrm{SL}_2}(\tau) \left( \sum_{j=0}^{k-2} \frac{(-1)^j}{j!} e_k^{(j)} \nu[j_k; \tau_1] \right) U_{\mathrm{SL}_2}(\tau)^{-1} = \sum_{j=0}^{k-2} \frac{(-1)^j}{j!} e_k^{(j)} \omega_+ \left[ \begin{matrix} j \\ k \\ \tau, \tau_1 \end{matrix} \right], \quad (3.10) $$

$$ U_{\mathrm{SL}_2}(\tau) \left( \sum_{j=0}^{k-2} \frac{1}{j!} e_k^{(j)} \overline{\nu \left[ \begin{matrix} j \\ k \\ \tau_1 \end{matrix} \right]} \right) U_{\mathrm{SL}_2}(\tau)^{-1} = \sum_{j=0}^{k-2} \frac{(-1)^j}{j!} e_k^{(j)} \omega_- \left[ \begin{matrix} j \\ k \\ \tau, \tau_1 \end{matrix} \right]. $$

Since the generating series $\mathbb{J}_\pm$ in (2.84) and (2.85) can be written as path-ordered exponentials involving $\omega_\pm$, the conjugations in (3.10) in fact apply to the full series $\mathbb{I}_\pm$,

$$ U_{\mathrm{SL}_2}(\tau) \mathbb{I}_+(e_k; \tau) U_{\mathrm{SL}_2}(\tau)^{-1} = R[\mathbb{J}_+(e_k; \tau)], \quad (3.11) $$

$$ U_{\mathrm{SL}_2}(\tau) \tilde{\mathbb{I}}_-(e_k; \tau) U_{\mathrm{SL}_2}(\tau)^{-1} = R[\tilde{\mathbb{J}}_-(e_k; \tau)]. $$
The reflection operator R on the right-hand sides acts via

$$R[e_{k_1}^{(j_1)} \cdots e_{k_\ell}^{(j_\ell)}] = \left( \prod_{i=1}^{\ell} \frac{j_i!}{(k_i - j_i - 2)!} \right) e_{k_\ell}^{(k_\ell - j_\ell - 2)} \cdots e_{k_1}^{(k_1 - j_1 - 2)}, \quad (3.12)$$

i.e. it reverses the ordering of the letters $e_{k_i}^{(j_i)}$ and acts on the $sl_2$ multiplets through the switch operation<fnref n="18" /> $e_k^{(j)}/j! \to e_k^{(k-j-2)/(k-j-2)!}$.

Comparing the definitions of the two generating series $I_\pm$ and $J_\pm$, we see that for a given coefficient function the order of the letters is opposite in the two series, explaining the reversal part of R. The switch part instead is due to (3.10) and the definition of the coefficients of $J_\pm$ in (2.84) compared with (3.7).

The reflection operator satisfies the following identities when acting on series X, Y in the letters $\epsilon_k^{(j)}$:

$$R[X Y] = R[Y] R[X], \quad (3.13)$$

$$R[\mathbb{M}^{\text{sv}}(z_i)\mathbb{X}(\epsilon_k)\mathbb{M}^{\text{sv}}(z_i)^{-1}] = \mathbb{M}^{\text{sv}}(z_i)^{-1}R[\mathbb{X}(\epsilon_k)]\mathbb{M}^{\text{sv}}(z_i).$$

The reflection property in the second line can be seen by expressing the $\epsilon_k$ produced by $[z_w, \epsilon_m^{(j)}]$ via nested $t^d$-operations (2.60) and observing that the sum of the superscripts d has the right parity to attain the negative sign in $R[[z_w, \epsilon_m^{(j)}]] = -[z_w, R[\epsilon_m^{(j)}]]$. We note that while the first identity in (3.13) holds also for series in $\epsilon_k^{(j)}$, the second one involving the change of alphabet (2.89) is only valid for series X in $\epsilon_k^{(j)}$ and we have emphasised this in the notation. In the later section 3.3, we will extend the change of alphabet to a similar formula involving the uplifted generators $\hat{z}_w$ and $\epsilon_k^{(j)}$, see (3.74), for which the second identity also holds.

It is important to observe that (3.11) introduces an uplift of the series $J_\pm(\epsilon_k; \tau)$ in section 2.5 since we are now employing the free-Lie-algebra generators $\epsilon_k^{(j)}$ instead of $\epsilon_k^{(j)}$. This is consistent with the absence of cusp forms in (2.84) since all the $\epsilon_{\Delta_k^\pm}$ are taken to vanish under the restriction $\epsilon_k^{(j)} \to \epsilon_k^{(j)}$.

The $SL_2$ transformation $U_{SL_2}$ in (3.9) transforms the (anti-)meromorphic generating series $I_\pm$ into path-ordered exponentials $J_\pm$ of the non-holomorphic integration kernels $\omega_\pm$ that have much cleaner modular properties, see (2.23). Accordingly, the series $I_\pm$ will henceforth be referred to as in the holomorphic frame of $sl_2$ whereas $J_\pm$ will be said to be in the modular frame.

The same $SL_2$ transformation and terminology applies to the generating series $\mathbb{B}^{\text{sv}}$ of single-valued MZVs. All the $\tau$-dependence in the definition (2.87) of $\mathbb{B}^{\text{sv}}$ can be identified as an artifact of the $SL_2$ transformation $U_{SL_2}$ in (3.9), and $\mathbb{B}^{\text{sv}}$ turns out to be conjugate (see (3.12) for the reversal operator R)

$$U_{SL_2}(\tau) \mathbb{C}^{\text{sv}}(e_k) U_{SL_2}(\tau)^{-1} = R[\mathbb{B}^{\text{sv}}(e_k; \tau)], \quad (3.14)$$

<footnote n="18">¹⁸This corresponds to the Weyl reflection (2.58) up to sign.</footnote>
to the following series in real constants $c^{\text{sv}}\begin{bmatrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{bmatrix}$:

$$ \begin{aligned} \mathbb{C}^{\text{sv}}(e_k) &:= \sum_{P \in \begin{Bmatrix} j \\ k \end{Bmatrix}^\times} W[P] c^{\text{sv}}[P] \tag{3.15} \\ &= 1 + \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} (-1)^{j_1} \frac{(k_1-1)}{j_1!} c^{\text{sv}}\begin{bmatrix} j_1 \\ k_1 \end{bmatrix} e_k^{(j_1)} \\ &\quad + \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} \sum_{k_2=4}^{\infty} \sum_{j_2=0}^{k_2-2} (-1)^{j_1+j_2} \frac{(k_1-1)(k_2-1)}{j_1!j_2!} c^{\text{sv}}\begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix} e_k^{(j_1)} e_k^{(j_2)} + \dots \end{aligned} $$

The ellipsis refers to contributions of modular depth $\ell \ge 3$, and we have introduced the shorthand notation

$$ W[P] = W\begin{bmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{bmatrix} := (-1)^{j_1+\dots+j_\ell} \frac{(k_1-1)(k_2-1)\dots(k_\ell-1)}{j_1!j_2!\dots j_\ell!} e_{k_1}^{(j_1)} e_{k_2}^{(j_2)} \dots e_{k_\ell}^{(j_\ell)}, \quad (3.16) $$

for words in the $e_k^{(j_i)}$. Note that this notation is related to the one in (2.85) by

$$ R[W[P]]|_{e_k \to \epsilon_k} = \epsilon[P]. \quad (3.17) $$

By analogy with (3.11), the series $\mathbb{C}^{\text{sv}}$ and $\mathbb{B}^{\text{sv}}$ are said to be in the holomorphic and modular frame, respectively. Similar to (2.85), the summation range $P \in \begin{Bmatrix} j \\ k \end{Bmatrix}^\times$ in the first line of (3.15) gathers all words in $j_i, k_i$ subject to $k_i \ge 4$ and $0 \le j_i \le k_i - 2$ with no separate reference to $\Delta_{k_i}$ as in (3.7). Upon dressing with derivations $\epsilon_k^{(j)}$ that satisfy the Pollack relations rather than free generators $e_k^{(j)}$, the real constants $c^{\text{sv}}\begin{bmatrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{bmatrix}$ in (3.15) are projected to single-valued MZVs of [28] according to theorem 7.2 in Brown's work [14]. However, we shall see in section 3.3 that generic $c^{\text{sv}}\begin{bmatrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{bmatrix}$ at modular depth $\ell \ge 3$ and degree $\ge 18$ involve admixtures of new numbers beyond MZVs with combinations of Pollack combinations $P_w^d$ as coefficients.

Note that alternatively, the change of frame via $U_{\text{SL}_2}$ can be performed by introducing commutative bookkeeping variables $X_i, Y_i$ for each $e_k^{(j_i)}$. Passing from coefficients of $X_i^a Y_i^b$ (with $a, b \in \mathbb{N}_0$) to those of $(X_i - \tau Y_i)^a, (X_i - \bar{\tau} Y_i)^b$ as explained in section 7 of [13] introduces the same type of $\tau, \bar{\tau}$ dependent transformations as the exponentials in $U_{\text{SL}_2}$ acting on an $\mathfrak{sl}_2$ multiplet. This mechanism has also been used in section 2.2 of [28] to generate the integration kernels $\omega_\pm$ from an alternative expansion of $(X - \tau_1 Y)^{k-2} G_k(\tau_1)$. A detailed discussion with focus on modular depth three will be given in section 5.

### 3.1.4 Equivariant iterated Eisenstein integrals in the holomorphic frame

Equipped with the previous considerations that apply to generating series in the free Lie algebra generators $e_k$, we shall now consider an uplift of the generating series $\mathbb{J}^{\text{eqv}}(\epsilon_k)$ of (2.86) of equivariant iterated Eisenstein integrals to $\mathbb{J}^{\text{eqv}}(e_k)$ that will also include cusp forms. The starting point of this uplift is to define a generating series $\mathbb{I}^{\text{eqv}}(\epsilon_k)$ in the holomorphic frame

$$ \mathbb{I}^{\text{eqv}}(\epsilon_k; \tau) = (\phi^{\text{sv}})^{-1}(\tilde{\mathbb{I}}_{-}(\epsilon_k; \tau)) \mathbb{C}^{\text{sv}}(\epsilon_k) \mathbb{I}_{+}(\epsilon_k; \tau), \quad (3.18) $$
in terms of (3.7), (3.8) and (3.15) with the specialisation $e_k \to \epsilon_k$. Here, we have used the inverse of the change of alphabet $\phi^{\text{sv}}$ defined in (2.89) that acts on series in $\epsilon_k$ by $(\phi^{\text{sv}})^{-1}(\mathbb{X}) = \mathbb{M}^{\text{sv}}(z_i)^{-1}\mathbb{X}\mathbb{M}^{\text{sv}}(z_i)$. The generating series (3.18) is related to $\mathbb{J}^{\text{eqv}}(\epsilon_k)$ of (2.86) by the change of frame

$$R[\mathbb{J}^{\text{eqv}}(\epsilon_k; \tau)] = U_{\text{SL}_2}(\tau) \mathbb{I}^{\text{eqv}}(\epsilon_k; \tau) U_{\text{SL}_2}(\tau)^{-1}. \quad (3.19)$$

In the above generating series (3.18) and (3.19) all Tsunogai relations are satisfied.

All the objects on the right-hand side of (3.18) have been defined for $e_k$ (see (3.7), (3.8) and (3.15)), with the exception of the change of alphabet $(\phi^{\text{sv}})^{-1}$ that was defined in (2.89) only for series in the generators $\epsilon_k$ satisfying the Tsunogai relations. We shall postulate an extended change of alphabet $\hat{\psi}^{\text{sv}}$ that operates on series in $e_k$:

$$\hat{\psi}^{\text{sv}}(\tilde{\mathbb{I}}_-(e_k; \tau)) := \mathbb{M}^{\text{sv}}(\hat{z}_i)^{-1} \tilde{\mathbb{I}}_-(e_k; \tau) \mathbb{M}^{\text{sv}}(\hat{z}_i) + \dots \quad (3.20)$$

The change of alphabet involves a conjugation by the arithmetic parts $\hat{z}_i$ of the uplifted zeta generators that were introduced in section 2.4.2. However, as will be explained in sections 3.3.4 and 4 below, the modular properties of the uplifted version $\mathbb{I}^{\text{eqv}}(e_k; \tau)$ of (3.18) necessitate terms of modular depth $\ge 4$ in the ellipsis that are also due to new periods, see in particular (3.74). The terms specified on the right-hand side of (3.20) are complete up to and including modular depth three. Similar to the change of alphabet $\phi^{\text{sv}}$ for the $\epsilon_k^{(j)}$ of the antiholomorphic generating series reviewed in section 2.5.2, the $\text{sl}_2$-invariant map $\hat{\psi}^{\text{sv}}$ acts on the free-Lie-algebra generators $e_k^{(j)}$ and we shall postulate that the conjugation property extends implying $\hat{\psi}^{\text{sv}}(\mathbb{X}\mathbb{Y}) = \hat{\psi}^{\text{sv}}(\mathbb{X})\hat{\psi}^{\text{sv}}(\mathbb{Y})$. By (3.13), we know that in the specialisation $e_k^{(j)} \to \epsilon_k^{(j)}$, the change of alphabet (3.20) reduces to the inverse of $\phi^{\text{sv}}$ in (2.89), i.e.

$$\hat{\psi}^{\text{sv}}(\mathbb{X})|_{e_k^{(j)} \to \epsilon_k^{(j)}} = (\phi^{\text{sv}})^{-1}(\mathbb{X}) = \mathbb{M}^{\text{sv}}(z_i)^{-1}\mathbb{X}\mathbb{M}^{\text{sv}}(z_i). \quad (3.21)$$

With this extended change of alphabet $\hat{\psi}^{\text{sv}}$, we can now define

$$\mathbb{I}^{\text{eqv}}(e_k; \tau) = \hat{\psi}^{\text{sv}}(\tilde{\mathbb{I}}_-(e_k; \tau)) \mathbb{C}^{\text{sv}}(e_k) \tilde{\mathbb{I}}_+(e_k; \tau), \quad (3.22)$$

as the uplift of (3.18) in the holomorphic frame as well as

$$R[\mathbb{J}^{\text{eqv}}(e_k; \tau)] = U_{\text{SL}_2}(\tau) \mathbb{I}^{\text{eqv}}(e_k; \tau) U_{\text{SL}_2}(\tau)^{-1}, \quad (3.23)$$

as the uplift of the equivariant iterated Eisenstein integrals in the modular frame. Section 4 is devoted to proving the equivariance of the right-hand side of (3.23) as well as showing how this relates to the generating series of equivariant iterated Eisenstein integrals in (2.86).

The uplifted generating series $\mathbb{I}^{\text{eqv}}(e_k)$ in the holomorphic frame has the following properties (relative to its modular counterpart $\mathbb{J}^{\text{eqv}}(e_k)$):

* $\mathbb{C}^{\text{sv}}$ is a generating series involving only real constants, unlike $\mathbb{B}^{\text{sv}}$ that has a rational dependence on $\tau, \bar{\tau}$;

* the iterated integrals of $\tilde{\mathbb{I}}_+, \tilde{\mathbb{I}}_-$ are (anti-)meromorphic in $\tau$, unlike the $\beta_{\pm}$ entering $\mathbb{J}_{\pm}$.
By analogy with the expansion (2.86) of $\mathbb{J}^{\text{eqv}}$ in terms of modular forms $\beta^{\text{eqv}}$, we expand the analogous series $\mathbb{I}^{\text{eqv}}$ in the holomorphic frame in words (3.16) composed of the independent generators $e_k^{(j)}$,

$$ \mathbb{I}^{\text{eqv}}(e_k; \tau) = \sum_{P \in \{j_k\}^\times} W[P] \mathcal{E}^{\text{eqv}}[P; \tau]. \qquad (3.24) $$

As an initial step towards the study of the expansion coefficients $\mathcal{E}^{\text{eqv}}$, we can at first consider only the contribution to (3.22) coming simply from $\hat{\psi}^{\text{sv}} \to 1$. Restricting to the Eisenstein part this becomes

$$ \mathcal{E}^{\text{eqv}}\left[ \begin{smallmatrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{smallmatrix}; \tau \right] = \sum_{0 \le p \le q \le \ell} (-1)^{j_1+\dots+j_p} \overline{\mathcal{E}\left[ \begin{smallmatrix} j_p & \dots & j_2 \\ k_p & \dots & k_2 \end{smallmatrix}; \tau \right]} c^{\text{sv}}\left[ \begin{smallmatrix} j_{p+1} & \dots & j_q \\ k_{p+1} & \dots & k_q \end{smallmatrix} \right] \mathcal{E}\left[ \begin{smallmatrix} j_{q+1} & \dots & j_\ell \\ k_{q+1} & \dots & k_\ell \end{smallmatrix}; \tau \right] + \dots, \qquad (3.25) $$

where the ellipsis refers to both admixtures of MZVs and certain new periods from $c^{\text{sv}}$, $\hat{\psi}^{\text{sv}}$ and iterated integrals $\mathcal{E}[\dots; \Delta_k]$ involving cusp forms from the expansion (3.7) and (3.8) of $\mathbb{I}_{\pm}$. Examples of these extra terms and further details on their extraction from the generating series will be given in section 3.3 below and appendix C.2.

Since the change of frame between $\mathbb{B}^{\text{sv}} \leftrightarrow \mathbb{C}^{\text{sv}}$ and $\mathbb{J}^{\text{eqv}} \leftrightarrow \mathbb{I}^{\text{eqv}}$ follows the same conjugations (3.14) and (3.23) by $U_{\text{SL}_2}$, the relation (2.87) between the coefficients $b^{\text{sv}} \leftrightarrow c^{\text{sv}}$ can be readily adapted to the equivariant iterated Eisenstein integrals,

$$ \beta^{\text{eqv}}[\dots; j_k \dots] = \sum_{p=0}^{k-2-j} \sum_{\ell=0}^{j+p} \binom{k-j-2}{p} \binom{j+p}{\ell} \frac{(-2\pi i \bar{\tau})^{\ell}}{(4y)^p} \mathcal{E}^{\text{eqv}}[\dots; \begin{smallmatrix} j-\ell+p \\ k \end{smallmatrix}] , \qquad (3.26) $$

where the double sum again applies independently to each column $\begin{smallmatrix} j_i \\ k_i \end{smallmatrix}$. As one can anticipate from the powers of $\bar{\tau}$ and $y^{-1}$ in the expansion coefficients of (3.26), the $\mathcal{E}^{\text{eqv}}$ in the holomorphic frame do not transform as modular forms. Still, they enjoy equivariant transformation properties in the sense that

$$ \mathcal{E}^{\text{eqv}}\left[ \begin{smallmatrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{smallmatrix}; \tau + 1 \right] = \sum_{p_1=0}^{j_1} \dots \sum_{p_\ell=0}^{j_\ell} \left( \prod_{r=1}^{\ell} (2\pi i)^{j_r - p_r} \binom{j_r}{p_r} \right) \mathcal{E}^{\text{eqv}}\left[ \begin{smallmatrix} p_1 & \dots & p_\ell \\ k_1 & \dots & k_\ell \end{smallmatrix}; \tau \right], \qquad (3.27) $$

$$ \mathcal{E}^{\text{eqv}}\left[ \begin{smallmatrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{smallmatrix}; -\frac{1}{\tau} \right] = \left( \prod_{r=1}^{\ell} (-1)^{j_r} (2\pi i)^{2+2j_r-k_r} \right) \mathcal{E}^{\text{eqv}}\left[ \begin{smallmatrix} k_1 & \dots & k_{\ell-2} \\ k_1 & \dots & k_\ell \end{smallmatrix}; \tau \right], $$

is homogeneous in modular depth.<fnref n="19" /> More importantly, equivariant iterated Eisenstein integrals $\mathcal{E}^{\text{eqv}}$ in the holomorphic frame share the modular T and S transformations (2.5) and (2.6) of the kernels $\nu\begin{smallmatrix} j \\ k \end{smallmatrix}; \tau$ in each column and transform in the same way as the formal tensor products

$$ \mathcal{E}^{\text{eqv}}\left[ \begin{smallmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{smallmatrix}; \tau \right] \leftrightarrow \nu\begin{smallmatrix} j_1 \\ k_1 \end{smallmatrix}; \tau \otimes \nu\begin{smallmatrix} j_2 \\ k_2 \end{smallmatrix}; \tau \otimes \dots \otimes \nu\begin{smallmatrix} j_\ell \\ k_\ell \end{smallmatrix}; \tau. \qquad (3.28) $$

In particular, (3.27) takes the same form as the contributions to $\mathcal{E}\begin{smallmatrix} j \\ k \end{smallmatrix}; \tau + 1$ and $\mathcal{E}\begin{smallmatrix} j \\ k \end{smallmatrix}; -\frac{1}{\tau}$ in (2.10) and (2.12) that preserve modular depth one. Following the original idea in Brown's

<footnote n="19">¹⁹After a suitable rescaling, the $\mathcal{E}^{\text{eqv}}$ can be viewed as being in a matrix representation of SL(2, $\mathbb{Z}$) such that they form components of a vector-valued modular form.</footnote>
construction of equivariant iterated Eisenstein integrals [11, 13, 14], it is the joint effect of the series $\mathbb{C}^{\text{sv}}$, the extended change of alphabet $\hat{\psi}^{\text{sv}}$ and the cuspidal iterated integrals in $\mathbb{I}_{\pm}$ in (3.22) that eliminates any term of lower modular depth from the SL(2, $\mathbb{Z}$) transformations (3.27). Note that the $\mathcal{E}^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$ are $\mathbb{Q}[2\pi i]$ multiples of the coefficients of $X_1^{k_1-2-j_1} Y_1^{j_1} X_2^{k_2-2-j_2} Y_2^{j_2}$ in the equivariant double integrals $M_{k_2,k_1}$ in section 9 of [13] whose explicit examples and relations to the modular forms $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$ can be found in section 4.3 of [28]. We will elaborate more on this approach via commutative formal variables $(X_i, Y_i)$ in section 5.

### 3.1.5 Differential equations

The construction of the generating series in this section from path-ordered exponentials results in simple differential equations with respect to $\tau$ and $\bar{\tau}$. The definition (3.6) of $\mathbb{I}_+$ and $\tilde{\mathbb{I}}_-$ together with the (1,0)- and (0,1)-forms $\mathbb{A}_+$ and $\mathbb{A}_-$ in (3.2) and (3.3) readily imply that

$$ 
\begin{aligned}
\partial_{\tau} \mathbb{I}_{+}(e_k; \tau) \, d\tau &= -\mathbb{I}_{+}(e_k; \tau) \mathbb{A}_{+}(e_k; \tau), \qquad (3.29) \\
\partial_{\bar{\tau}} \tilde{\mathbb{I}}_{-}(e_k; \tau) \, d\bar{\tau} &= -\mathbb{A}_{-}(e_k; \tau) \tilde{\mathbb{I}}_{-}(e_k; \tau), \\
\partial_{\bar{\tau}} \mathbb{I}_{+}(e_k; \tau) &= \partial_{\tau} \tilde{\mathbb{I}}_{-}(e_k; \tau) = 0.
\end{aligned}
$$

By virtue of (3.22), the generating series $\mathbb{I}^{\text{eqv}}$ of equivariant iterated Eisenstein integrals in the holomorphic frame share the $\tau$-derivative of (3.29) but involve the $\hat{\psi}^{\text{sv}}$-image of $\mathbb{A}_-$ in the $\bar{\tau}$-derivative:

$$ 
\begin{aligned}
\partial_{\tau} \mathbb{I}^{\text{eqv}}(e_k; \tau) \, d\tau &= -\mathbb{I}^{\text{eqv}}(e_k; \tau) \mathbb{A}_{+}(e_k; \tau), \qquad (3.30) \\
\partial_{\bar{\tau}} \mathbb{I}^{\text{eqv}}(e_k; \tau) \, d\bar{\tau} &= -\hat{\psi}^{\text{sv}}(\mathbb{A}_{-}(e_k; \tau)) \mathbb{I}^{\text{eqv}}(e_k; \tau).
\end{aligned}
$$

For the second equation we have used that $\hat{\psi}^{\text{sv}}$ factors on generating series and have assumed that it does not introduce any extra $\tau$-dependence, so that we can differentiate inside the argument. By the discussion around (3.20), the contributions to $\hat{\psi}^{\text{sv}}(\mathbb{A}_{-}(e_k; \tau))$ of modular depth $\le 3$ coincide with $\mathbb{M}^{\text{sv}}(\hat{z}_i)^{-1}\mathbb{A}_{-}(e_k; \tau)\mathbb{M}^{\text{sv}}(\hat{z}_i)$. Moreover, the restriction $e_k^{(j)} \to \epsilon_k^{(j)}$ of the $\bar{\tau}$-derivative in (3.30) simplifies to

$$ \partial_{\bar{\tau}} \mathbb{I}^{\text{eqv}}(\epsilon_k; \tau) \, d\bar{\tau} = -\mathbb{M}^{\text{sv}}(z_i)^{-1} \mathbb{A}_{-}(\epsilon_k; \tau) \mathbb{M}^{\text{sv}}(z_i) \mathbb{I}^{\text{eqv}}(\epsilon_k; \tau), \qquad (3.31) $$

at arbitrary modular depth, reducing $\hat{\psi}^{\text{sv}}$ to arithmetic parts of zeta generators.

The reflected version of the generating series $\mathbb{J}^{\text{eqv}}$ in the modular frame in (3.23) receives additional contributions to its $\tau$-derivative from the $\text{SL}_2$ transformation in (3.1): As a consequence of $2\pi i(\tau-\bar{\tau})^2\partial_{\tau}\text{U}_{\text{SL}_2}(\tau) = -e_0^\vee\text{U}_{\text{SL}_2}(\tau)$, we have

$$ 
\begin{aligned}
-2\pi i(\tau-\bar{\tau})^2 \partial_{\tau} R[\mathbb{J}^{\text{eqv}}(e_k; \tau)] &= [e_0^\vee, R[\mathbb{J}^{\text{eqv}}(e_k; \tau)]] \\
&\quad + R[\mathbb{J}^{\text{eqv}}(e_k; \tau)] \sum_{m=4}^{\infty} \frac{(m-1)}{(m-2)!} (\tau-\bar{\tau})^m \left\{ G_m(\tau) e_m^{(m-2)} + \sum_{\Delta_m \in S_m} (2\pi i)^m \Delta_m(\tau) e_{\Delta_m^+}^{(m-2)} \right\}.
\end{aligned}
\qquad (3.32) $$

To remove the reflection operator $R$ defined in (3.12) from the left-hand side we simply need to apply it once more on both sides. On the right-hand side this operation replaces in the
first line the commutator with $e_0^\vee$ by $ad_{e_0}$, while in the second line it exchanges highest- and lowest-weight vectors via $\frac{1}{(m-2)!}e_m^{(m-2)} \to e_m$, therefore leading to the simplified result

$$ -2\pi i(\tau - \bar{\tau})^2 \partial_{\tau} \mathcal{J}^{\text{eqv}}(e_k; \tau) = \mathrm{ad}_{e_0} \mathcal{J}^{\text{eqv}}(e_k; \tau) + \sum_{m=4}^{\infty} (m-1)(\tau - \bar{\tau})^m \left\{ \mathrm{G}_m(\tau)e_m + \sum_{\Delta_m \in S_m} (2\pi i)^m \Delta_m(\tau)e_{\Delta_m^+} \right\} \mathcal{J}^{\text{eqv}}(e_k; \tau). \quad (3.33) $$

The modular forms $G_m$ and $\Delta_m$ on the right-hand side of (3.32) and (3.33) illustrate that the construction of $\mathcal{J}_\pm$ and therefore $\mathcal{J}^{\text{eqv}}$ from path-ordered exponentials is consistent with the modular transformations (2.45) of the coefficients $\beta^{\text{eqv}}$.

Since the letters $e_{\Delta^\pm}$ associated with the cusp-form contributions to $\mathcal{J}_\pm(e_k; \tau)$ are built from Pollack combinations, specialisation to $e_k \to \epsilon_k$ results in a holomorphic differential equation with only Eisenstein series [14, 28]:

$$ -2\pi i(\tau - \bar{\tau})^2 \partial_{\tau} \mathcal{J}^{\text{eqv}}(\epsilon_k; \tau) = \mathrm{ad}_{e_0} \mathcal{J}^{\text{eqv}}(\epsilon_k; \tau) + \sum_{m=4}^{\infty} (m-1)(\tau - \bar{\tau})^m G_m(\tau) \epsilon_m \mathcal{J}^{\text{eqv}}(\epsilon_k; \tau). \quad (3.34) $$

The resulting differential equation for the components $\beta^{\text{eqv}}$ reads with $\nabla = 2i(\text{Im } \tau)^2 \partial_\tau$:

$$ -4\pi \nabla \beta^{\text{eqv}} \left[ \begin{matrix} j_1 & j_2 & \cdots & j_\ell \\ k_1 & k_2 & \cdots & k_\ell \end{matrix} ; \tau \right] = \sum_{i=1}^{\ell} (k_i - j_i - 2) \beta^{\text{eqv}} \left[ \begin{matrix} j_1 & \cdots & j_{\ell+1} & \cdots & j_\ell \\ k_1 & \cdots & k_i & \cdots & k_\ell \end{matrix} ; \tau \right] \\ - \delta_{j_\ell, k_\ell - 2} (\tau - \bar{\tau})^{k_\ell} G_{k_\ell}(\tau) \beta^{\text{eqv}} \left[ \begin{matrix} j_1 & j_2 & \cdots & j_{\ell-1} \\ k_1 & k_2 & \cdots & k_{\ell-1} \end{matrix} ; \tau \right] + \cdots, \quad (3.35) $$

where the ellipsis refers to holomorphic cusp forms multiplying $\beta^{\text{eqv}}$ of modular depth $\le \ell-2$. The $\bar{\tau}$-derivative of $\mathcal{J}^{\text{eqv}}(e_k; \tau)$ is not as clean as (3.33) as can be anticipated from (3.30): Apart from a term $\mathrm{ad}_{e_0} \mathcal{J}^{\text{eqv}}(e_k; \tau)$ that is completely analogous to $\mathrm{ad}_{e_0} \mathcal{J}^{\text{eqv}}(e_k; \tau)$ in the holomorphic derivative $\partial_\tau \mathcal{J}^{\text{eqv}}(e_k; \tau)$, the $\bar{\tau}$-derivative also contains contaminations of lower modular depth due to the change of alphabet encoded in $\hat{\psi}^{\text{sv}}$.

## 3.2 Modular graph forms from zeta generators

One of the key results of this work concerns the interplay of the change of alphabet $\hat{\psi}^{\text{sv}}$ and the $\tau$-independent series $\mathbb{C}^{\text{sv}}(e_k)$ within the generating series (3.22) of equivariant iterated Eisenstein integrals. In the specialisation $e_k \to \epsilon_k$ relevant to MGFs, the uplifted zeta generators $\hat{z}_i$, $\hat{\sigma}_i$ reduce to the $z_i$, $\sigma_i$ determined by the methods of the companion paper [34] and reviewed in section 2.4. One of our central claims is that the middle product in the specialisation

$$ \mathcal{I}^{\text{eqv}}(\epsilon_k; \tau) = \mathbb{M}^{\text{sv}}(z_i)^{-1} \tilde{\mathbb{I}}_{-}(\epsilon_k; \tau) \mathbb{M}^{\text{sv}}(z_i) \mathbb{C}^{\text{sv}}(\epsilon_k) \mathbb{I}_{+}(\epsilon_k; \tau), \quad (3.36) $$

of (3.23) can be presented in a unified way

$$ \mathbb{M}^{\text{sv}}(z_i) \mathbb{C}^{\text{sv}}(\epsilon_k) = \mathbb{M}^{\text{sv}}(\sigma_i), \quad (3.37) $$

through the same type of group-like series as in (2.90) with $\sigma_i$ in the place of $z_i$:

$$ \begin{aligned} \mathbb{M}^{\text{sv}}(\sigma_i) &= \sum_{\ell=0}^{\infty} \sum_{\substack{i_1, i_2, \dots, i_\ell \\ \ell \in 2\mathbb{N}+1}} \sigma_{i_1} \sigma_{i_2} \dots \sigma_{i_\ell} \rho^{-1}(\text{sv}(f_{i_1} f_{i_2} \dots f_{i_\ell})) \\ &= 1 + 2 \sum_{i_1 \in 2\mathbb{N}+1} \sigma_{i_1} \zeta_{i_1} + 2 \sum_{i_1, i_2 \in 2\mathbb{N}+1} \sigma_{i_1} \sigma_{i_2} \zeta_{i_1} \zeta_{i_2} + \sum_{i_1, i_2, i_3 \in 2\mathbb{N}+1} \sigma_{i_1} \sigma_{i_2} \sigma_{i_3} \rho^{-1}(\text{sv}(f_{i_1} f_{i_2} f_{i_3})). \end{aligned} \quad (3.38) $$
This leads to our main result for the generating series (3.22) of those equivariant iterated Eisenstein integrals which do not feature any holomorphic cusp forms

$$ \mathbb{I}^{\text{eqv}}(\epsilon_k; \tau) = \mathbb{M}^{\text{sv}}(z_i)^{-1} \tilde{\mathbb{I}}_{-}(\epsilon_k; \tau) \mathbb{M}^{\text{sv}}(\sigma_i) \mathbb{I}_{+}(\epsilon_k; \tau). \qquad (3.39) $$

The resulting expressions for

$$ \begin{gathered} \mathbb{B}^{\text{sv}}(\epsilon_k; \tau) = R[\mathrm{U}_{\text{SL}_2}(\tau) \mathbb{M}^{\text{sv}}(z_i)^{-1} \mathbb{M}^{\text{sv}}(\sigma_i) \mathrm{U}_{\text{SL}_2}(\tau)^{-1}], \qquad (3.40) \\ R[\mathbb{J}^{\text{eqv}}(\epsilon_k; \tau)] = \mathrm{U}_{\text{SL}_2}(\tau) \mathbb{M}^{\text{sv}}(z_i)^{-1} \tilde{\mathbb{I}}_{-}(\epsilon_k; \tau) \mathbb{M}^{\text{sv}}(\sigma_i) \mathbb{I}_{+}(\epsilon_k; \tau) \mathrm{U}_{\text{SL}_2}^{-1}(\tau), \end{gathered} $$

are another major step in making the key ingredients ($\mathbb{B}^{\text{sv}}$, $\phi^{\text{sv}}$) of Brown's construction [14] explicit. With the change of alphabet $\phi^{\text{sv}}$ in (2.92) determined in [28] and the new result (3.40) of this work for $\mathbb{B}^{\text{sv}}$, the equivariant iterated Eisenstein integrals entering MGFs are available to the modular depths and degrees where we have computational control over the zeta generators. There is no conceptual bottleneck in obtaining explicit expansions of $[z_w, \epsilon_k]$ and $\sigma_i$ in terms of $\epsilon_k^{(j_i)}$ to arbitrary degree and modular depth by the methods in the companion paper [34]. Generalisations of the closed-form results (2.81) and (2.83) beyond modular depth two can be found in the reference as well as appendix E and the ancillary files of this work.

Note that the action (3.12) of the reflection operator $R$ in (3.40) is understood to be unchanged under $\epsilon_k^{(j)} \to \epsilon_k^{(j)}$. However, (3.40) cannot be uplifted to the larger generating series $\mathbb{J}^{\text{eqv}}(\epsilon_k; \tau)$ including cuspidal iterated integrals while preserving the form (3.38) of $\mathbb{M}^{\text{sv}}(\sigma_i)$. As we will see in section 3.3, the zeta generators $\sigma_w \to \hat{\sigma}_w$ adapted to the free-Lie-algebra generators $\epsilon_k^{(j)}$ need to be augmented by additional generators $\hat{\sigma}_\omega$ associated with new periods, $\omega$, beyond MZVs. Moreover, modularity requires these $\hat{\sigma}_\omega$ to involve arithmetic parts $\hat{z}_\omega$ which prevent us from reducing $\hat{\psi}^{\text{sv}}(\tilde{\mathbb{I}}_{-}(\epsilon_k; \tau))$ in (3.22) to $\mathbb{M}^{\text{sv}}(\hat{z}_i)^{-1}\tilde{\mathbb{I}}_{-}(\epsilon_k; \tau)\mathbb{M}^{\text{sv}}(\hat{z}_i)$ (though modular depths $\le 3$ are insensitive to the difference). Accordingly, we shall content ourselves with series in Tsunogai's derivations $\epsilon_k^{(j)}$ for the rest of this subsection 3.2 where the modular completion of $\tilde{\mathbb{I}}_{-}(\epsilon_k; \tau)\mathbb{I}_{+}(\epsilon_k; \tau)$ can be found in terms of MZVs.

Similar to our comment below (2.92), it is not manifest term-by-term that the expressions (3.40) for $\mathbb{B}^{\text{sv}}$ and $\mathbb{J}^{\text{eqv}}$ admit an expansion solely in terms of $\epsilon_k^{(j)}$. In both cases, the arithmetic parts of the zeta generators from the left-multiplicative series $\mathbb{M}^{\text{sv}}(z_i)^{-1}$ can be used to "drain out" all the $z_i$-contributions to $\mathbb{M}^{\text{sv}}(\sigma_i)$. As will be detailed in section 3.2.2, this requires iterative use of $\epsilon_k^{(j)} z_w = z_w \epsilon_k^{(j)} - \text{ad}_{\epsilon_0}^j[z_w, \epsilon_k]$ and the fact that the brackets on the right-hand side are expressible via nested commutators of $\epsilon_k^{(j_i)}$, i.e., that the $z_w$ normalise the Tsunogai derivation algebra of the $\epsilon_k$.

### 3.2.1 All information from Riemann zeta values

In the new formulation (3.40), all MZVs entering the generating series of equivariant iterated Eisenstein integrals arise from the group-like series $\mathbb{M}^{\text{sv}}(z_i)$ and $\mathbb{M}^{\text{sv}}(\sigma_i)$ in (2.90) and (3.38). As an important implication of the structure of these $\mathbb{M}^{\text{sv}}$, the operator-valued coefficients $z_w$, $\sigma_w$ of odd Riemann zeta values determine the coefficients of all the remaining (single-valued) MZVs composed of two or more letters in the $f$-alphabet. After reading off the overall
coefficient of $\epsilon_6\epsilon_4\epsilon_4$ from $\mathbb{J}^{\text{eqv}}$, for instance, the appearance of $\zeta_3^2$, $\zeta_3\zeta_5$ and $\zeta_{3,3,5}^{\text{sv}}$ in

$$ \beta^{\text{eqv}}[\begin{smallmatrix} 2 & 2 & 4 \\ 4 & 4 & 6 \end{smallmatrix}]|_{\text{mod depth } 1} = \left( \frac{2(i\pi\bar{\tau})^5\zeta_3}{14175} + \frac{(i\pi\bar{\tau})^3\zeta_5}{675} + \frac{2\zeta_3\zeta_5}{15} \right) \beta_{-[\begin{smallmatrix} 2 \\ 4 \end{smallmatrix}]} + \frac{2\zeta_3^2}{9}\beta_{+[\begin{smallmatrix} 4 \\ 6 \end{smallmatrix}]}, \quad (3.41) $$

$$ \beta^{\text{eqv}}[\begin{smallmatrix} 2 & 2 & 4 \\ 4 & 4 & 6 \end{smallmatrix}]|_{\text{mod depth } 0} = - \frac{(i\pi\bar{\tau})^8\zeta_3}{4536000} - \frac{(i\pi\bar{\tau})^6\zeta_5}{364500} - \frac{2(i\pi\bar{\tau})^5\zeta_3^2}{42525} - \frac{(i\pi\bar{\tau})^3\zeta_3\zeta_5}{2025} - \frac{14573\zeta_{11}}{21600} + \frac{\zeta_{3,3,5}^{\text{sv}}}{225}, $$

can be traced back to the coefficients of $\zeta_3$ and $\zeta_5$ in $\beta^{\text{eqv}}$ of modular depth $\le 2$ which in turn follow from low-order terms in (2.83). Representations of $\zeta_{3,3,5}^{\text{sv}}$ in terms of MZVs and the f-alphabet can be found in (A.14) and (A.15), respectively.

More generally, the appearance of higher-depth MZVs or products of arbitrary MZVs — say $\rho^{-1}(\text{sv}(f_{i_1}f_{i_2}\dots))$ in the f-alphabet — is interlocked with that of $\zeta_w$ through compositions such as $z_{i_1}z_{i_2}\dots$ and $\sigma_{i_1}\sigma_{i_2}\dots$ in (2.90) and (3.38). We reiterate the warning from the end of section 2.2.3 that modular depth zero does not correspond to the Laurent polynomial in $y$ that governs the asymptotics at the cusp.

The same phenomenon was observed in [22] for a basis of disk integrals in tree-level amplitudes of massless open-string states [49]. Instead of derivations $z_w$, $\sigma_w$, the coefficients of Riemann zeta values in the low-energy expansion of disk integrals are square matrices $M_w$ whose entries are degree-$w$ polynomials in certain dimensionless kinematic variables with rational coefficients. The compositions of operators $z_w$, $\sigma_w$ we encounter at genus one are then replaced by matrix products $M_{i_1}M_{i_2}\dots$ in the coefficients of MZVs $\rho^{-1}(f_{i_1}f_{i_2}\dots)$ in the tree-level setting. Similar matrices and the same type of correlations between the coefficients of arbitrary MZVs and those of Riemann zeta values can also be found in more general configuration-space integrals at genus zero [83–85].

Through the single-valued map of MZVs, the sphere integrals in closed-string tree-level amplitudes inherit the group-like series structure (3.38) with the above $M_w$ in the place of $\sigma_w$ [22–27]. The asymptotic expansion as $\tau \to i\infty$ of generic MGFs is not yet available in terms of the known low-energy expansion of sphere integrals, though the solutions for the two-point genus-one examples in terms of four-point tree-level integrals [15, 18, 30] suggest that an explicit map can also be found at higher multiplicity. Nevertheless, (3.38) and (3.40) encode the modular completion of the iterated Eisenstein integrals in all MGFs, regardless of the multiplicity in their configuration-space-integral representation. On these grounds, the results of this section already incorporate the correlations between MZVs in sphere integrals into MGFs, even if their explicit connection is not yet worked out beyond two points.

Still, it remains a pressing problem to deduce the $\tau \to i\infty$ asymptotics of MGFs from low-energy expansions of sphere integrals or other single-valued quantities. It is a notorious gap in the literature to prove the conjecture of [5, 12] that the Laurent polynomials in $y$ describing the $\tau \to i\infty$ asymptotics of MGFs only have single-valued MZVs as coefficients. From the iterated-integral representation of the generating series of MGFs in [30], it is not yet rigorously established that MGFs can only involve $\beta^{\text{eqv}}$ with single-valued MZVs as coefficients. In other words, it remains to rule out that the initial conditions $\tau \to i\infty$ of the generating series in the reference at $n \ge 3$ points yield coefficients outside the ring of single-valued MZVs.
### 3.2.2 Implications for $c^{\text{sv}}$

In order to study the implications of (3.37) for the constant coefficients $c^{\text{sv}}$ of $\mathbb{C}^{\text{sv}}(\epsilon_k)$ in (3.15), we rewrite the equation as

$$ \begin{aligned} \mathbb{C}^{\text{sv}}(\epsilon_k) &= \mathbb{M}^{\text{sv}}(z_i)^{-1} \mathbb{M}^{\text{sv}}(\sigma_i) \\ &= 1 + 2 \sum_{i_1 \in 2\mathbb{N}+1} \zeta_{i_1} \sigma_{i_1}^{\text{g}} + 2 \sum_{i_1, i_2 \in 2\mathbb{N}+1} \zeta_{i_1} \zeta_{i_2} (\sigma_{i_1}^{\text{g}} \sigma_{i_2}^{\text{g}} + [\sigma_{i_1}^{\text{g}}, z_{i_2}]) \\ &\quad + \sum_{i_1, i_2, i_3 \in 2\mathbb{N}+1} \rho^{-1}(\text{sv}(f_{i_1} f_{i_2} f_{i_3})) (\sigma_{i_1}^{\text{g}} \sigma_{i_2}^{\text{g}} \sigma_{i_3}^{\text{g}} + [\sigma_{i_1}^{\text{g}}, \sigma_{i_2}^{\text{g}}, z_{i_3}]) + [\sigma_{i_1}^{\text{g}}, z_{i_2}] \sigma_{i_3}^{\text{g}} + [[\sigma_{i_1}^{\text{g}}, z_{i_2}], z_{i_3}] + \dots, \end{aligned} \quad (3.42) $$

where the geometric contributions to the zeta generators are denoted by

$$ \sigma_w^g = \sigma_w - z_w. \quad (3.43) $$

The geometric part $\sigma_w^g$ is valued in Tsunogai's derivation algebra, i.e., a $\mathbb{Q}$-linear combination of nested brackets of the $\epsilon_k^{(j)}$.

We shall illustrate how different terms in (3.42) led us to anticipate (3.37) or equivalently (3.40) and how (3.42) yields concrete predictions for the MZVs beyond depth one in various $c^{\text{sv}}$.

**First evidence from modular depth one.** A first indication for the representation (3.40) of $\mathbb{B}^{\text{sv}}$ in terms of zeta generators came from the following observation on the coefficients of Riemann zeta values: The coefficients of $\zeta_w$ in the constants $c^{\text{sv}}$ entering $\mathbb{B}^{\text{sv}}$ in (2.87) turn out to match the expansion coefficients of the zeta generators $\sigma_w$ in (2.83)

$$ \begin{aligned} \sigma_w = z_w + & \frac{1}{2} \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} (-1)^{j_1} \frac{(k_1-1)}{j_1!} c^{\text{sv}}\begin{bmatrix} j_1 \\ k_1 \end{bmatrix} \Big|_{\rho^{-1}(f_w)} \epsilon_{k_1}^{(j_1)} \\ & + \frac{1}{2} \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} \sum_{k_2=4}^{\infty} \sum_{j_2=0}^{k_2-2} (-1)^{j_1+j_2} \frac{(k_1-1)(k_2-1)}{j_1!j_2!} c^{\text{sv}}\begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix} \Big|_{\rho^{-1}(f_w)} \epsilon_{k_1}^{(j_1)} \epsilon_{k_2}^{(j_2)} + \dots, \end{aligned} \quad (3.44) $$

where the coefficients of $\frac{1}{2}$ on the right-hand side stem from $\zeta_w^{\text{sv}} = 2\zeta_w$. For the isomorphism $\rho$ from (motivic) MZVs to the $f$-alphabet (see appendix A.1 for a brief review) we employ here the preferred choice described in the companion paper [34] which for instance fixes the ambiguity of $\sigma_{11}$ by rational multiples of $[\sigma_3, [\sigma_3, \sigma_5]]$ and all its higher-weight analogues. We have tested (3.44) to hold for all contributions to $\sigma_{w \le 15}$ subject to $\sum_i (k_i - j_i - 1) \le 5$ and for $\sigma_{17}$ up to $\sum_i (k_i - j_i - 1) = 3$.

**Additional evidence from maximum modular depth.** The next piece of evidence for (3.40) stems from section 4.1.4 of [28] where the instances of $c^{\text{sv}}$ at the maximal value $j_i = k_i - 2$ in each column were observed to line up with

$$ c^{\text{sv}}\begin{bmatrix} k_1-2 & k_2-2 & \dots & k_\ell-2 \\ k_1 & k_2 & \dots & k_\ell \end{bmatrix} = \left( \prod_{i=1}^{\ell} \frac{1}{1-k_i} \right) \rho^{-1}(\text{sv}(f_{k_1-1} f_{k_2-1} \dots f_{k_\ell-1})) + \dots. \quad (3.45) $$

The ellipsis refers to words in the $f$-alphabet with at most $(\ell-2)$ letters as exemplified in the reference, but we will additionally find periods beyond MZVs at degree $k_1 + \dots + k_\ell \ge 18$ in
section 3.3 below. The right-hand side of (3.45) without the sv map also features in Saad's formula for the (motivic version of the) MMVs $\mathfrak{m}[k_1 \ 0 \ 0 \ \dots \ 0 \ k_\ell]$, again modulo contributions from $\le \ell-2$ letters in the $f$-alphabet [86].

Up to and including modular depth three, the terms in (3.45) are readily seen to stem from the geometric parts of the zeta generators in (3.42) by extracting their contribution of lowest modular depth $\sigma_w^g \to -\frac{1}{(w-1)!} \epsilon_{w+1}^{(w-1)}$. At modular depth $\ell = 1$ and $\ell = 2$, the right-hand side of (3.45) in fact gives an exact expression for

$$ \rho(c^{\text{sv}}[k-2]) = \frac{\text{sv}(f_{k-1})}{1-k}, \quad \rho(c^{\text{sv}}[k_1-2 \ k_2-2]) = \frac{\text{sv}(f_{k_1-1}f_{k_2-1})}{(1-k_1)(1-k_2)}. \qquad (3.46) $$

**Predicting MZVs $\zeta_i \zeta_i$ in $c^{\text{sv}}$ at modular depth two.** There are two contributions $\sigma_{i_1}^g \sigma_{i_2}^g$ and $[\sigma_{i_1}^g, z_{i_2}]$ along with the MZVs $\zeta_i \zeta_i$ in the generating series (3.42). While the geometric terms give rise to the expressions $\sim \zeta_{k_1-1} \zeta_{k_2-1}$ for $c^{\text{sv}}[k_1-2 \ k_2-2]$ in (3.46), the commutator multiplying $\zeta_i \zeta_i$ in (3.42) introduces contributions to $c^{\text{sv}}[j_1 \ j_2]$ at non-maximal values $j_i < k_i-2$. The modular-depth-two contributions $[z_w, \epsilon_k] = \frac{\text{BF}_{w+k-1}}{\text{BF}_k} t^{w+1}(\epsilon_{w+1}, \epsilon_{w+k-1}) + \dots$ evaluated using (2.60) and (2.81) give rise to the second line of

$$ \mathbb{C}^{\text{sv}}(\epsilon_k) \Big|_{\text{sv}(f_{i_1} f_{i_2})}^{\epsilon_{k_1}^{(j_1)} \epsilon_{k_2}^{(j_2)}} = \frac{1}{(i_1-1)!(i_2-1)!} \epsilon_{i_1+1}^{(i_1-1)} \epsilon_{i_2+1}^{(i_2-1)} + \frac{\text{BF}_{i_1+i_2}}{(i_1+i_2-2)! \text{BF}_{i_1+1}} \sum_{\ell=0}^{i_2-1} (-1)^\ell [\epsilon_{i_2+1}^{(\ell)}, \epsilon_{i_1+i_2}^{(i_1+i_2-2-\ell)}], \qquad (3.47) $$

which specialises as follows at low values of $w = 3, 5, \dots$ and $k = 4, 6, \dots$:

* the $\zeta_3^2$ terms $c^{\text{sv}}[j_1 \ j_2]$ at $j_1+j_2 = 4$ such as $c^{\text{sv}}[0 \ 4 \ 4 \ 6] = -\frac{1}{315}\zeta_3^2$;

* the $\zeta_3\zeta_5$ terms $c^{\text{sv}}[j_1 \ j_2]$ at $j_1+j_2 = 6$ such as $c^{\text{sv}}[2 \ 4 \ 4 \ 8] = -\frac{1}{6300}\zeta_3\zeta_5$;

* the $\zeta_3\zeta_7$ terms $c^{\text{sv}}[j_1 \ j_2]$ at $j_1+j_2 = 8$ such as $c^{\text{sv}}[0 \ 8 \ 4 \ 10] = -\frac{5}{2673}\zeta_3\zeta_7$;

* the $\zeta_3\zeta_5$ terms $c^{\text{sv}}[j_1 \ j_2]$ at $j_1+j_2 = 6$ such as $c^{\text{sv}}[1 \ 5 \ 6 \ 8] = -\frac{1}{176400}\zeta_3\zeta_5$.

**Predicting MZVs $\rho^{-1}(\text{sv}(f_{i_1} f_{i_2} f_{i_3}))$ in $c^{\text{sv}}$ at modular depth three.** The same reasoning allows us to predict the simplest contributions to the $c^{\text{sv}}$ from irreducible single-valued MZVs of depth greater or equal than three. The contributions from $z_w$ to the coefficients of $\rho^{-1}(\text{sv}(f_{i_1} f_{i_2} f_{i_3}))$ in (3.42) extend (3.45) to non-maximal values $j_i < k_i-2$. The modular-

<footnote>$^{20}$Geometric contributions to (3.42) at higher modular depth or the commutators involving $z_w$ are incompatible with the property $j_i = k_i-2$ of the $c^{\text{sv}}$ in (3.45).</footnote>
depth-two contributions $[z_w, \epsilon_k] \to \frac{\text{BF}_{w+k-1}}{\text{BF}_k} t^{w+1}(\epsilon_{w+1}, \epsilon_{w+k-1})$ already suffice to derive

$$ \begin{aligned} \mathbb{C}^{\text{sv}}(\epsilon_k) \Big|_{\text{sv}(f_{i_1} f_{i_2} f_{i_3})}^{\epsilon_{k_1}^{(j_1)} \epsilon_{k_2}^{(j_2)} \epsilon_{k_3}^{(j_3)}} = & -\frac{1}{(i_1-1)!(i_2-1)!(i_3-1)!} \epsilon_{i_1+1}^{(i_1-1)} \epsilon_{i_2+1}^{(i_2-1)} \epsilon_{i_3+1}^{(i_3-1)} \tag{3.48} \\ & -\frac{\text{BF}_{i_1+i_2}}{(i_1+i_2-2)!\text{BF}_{i_1+1}} \sum_{\ell=0}^{i_2-1} (-1)^\ell [\epsilon_{i_2+1}^{(\ell)}, \epsilon_{i_1+i_2}^{(i_1+i_2-2-\ell)}] \frac{\epsilon_{i_3+1}^{(i_3-1)}}{(i_3-1)!} \\ & -\frac{\text{BF}_{i_1+i_3}}{(i_1+i_3-2)!\text{BF}_{i_1+1}} \sum_{\ell=0}^{i_3-1} (-1)^\ell [\epsilon_{i_3+1}^{(\ell)}, \epsilon_{i_1+i_3}^{(i_1+i_3-2-\ell)}] \frac{\epsilon_{i_2+1}^{(i_2-1)}}{(i_2-1)!} \\ & -\frac{\epsilon_{i_1+1}^{(i_1-1)}}{(i_1-1)!} \frac{\text{BF}_{i_2+i_3}}{(i_2+i_3-2)!\text{BF}_{i_2+1}} \sum_{\ell=0}^{i_3-1} (-1)^\ell [\epsilon_{i_3+1}^{(\ell)}, \epsilon_{i_2+i_3}^{(i_2+i_3-2-\ell)}] \\ & -\frac{\text{BF}_{i_1+i_2}}{(i_1+i_2-2)!\text{BF}_{i_1+1}} \sum_{\ell=0}^{i_2-1} (-1)^\ell [z_{i_3}, [\epsilon_{i_2+1}^{(\ell)}, \epsilon_{i_1+i_2}^{(i_1+i_2-2-\ell)}]] \end{aligned} $$

which for instance predicts the exact combinations of $\zeta_3^2\zeta_5$ and $\zeta_{3,3,5}^{\text{sv}}$ in $c^{\text{sv}}[j_1 \ j_2 \ j_3]$ and $c^{\text{sv}}[j_4 \ j_4 \ j_8]$ with $j_1+j_2+j_3 = 8$ such as $c^{\text{sv}}[0 \ 4 \ 4 \ 4] = \frac{\zeta_{3,3,5}^{\text{sv}}}{15750} + \frac{2\zeta_3^2\zeta_5}{1575} - \frac{232\zeta_{11}}{23625}$ (cf. section 4.1.4 of [28]). Note that the last term in (3.48) is understood to be rewritten in terms of $\epsilon_{k_i}^{(j_i)}$ by means of the Jacobi identity

$$ [z_{i_3}, [\epsilon_{i_2+1}^{(\ell)}, \epsilon_{i_1+i_2}^{(i_1+i_2-2-\ell)}]] = [[z_{i_3}, \epsilon_{i_2+1}^{(\ell)}, \epsilon_{i_1+i_2}^{(i_1+i_2-2-\ell)}]] + [\epsilon_{i_2+1}^{(\ell)}, [z_{i_3}, \epsilon_{i_1+i_2}^{(i_1+i_2-2-\ell)}]], \quad (3.49) $$

followed by another import of the depth-two terms of $[z_w, \epsilon_k] \to \frac{\text{BF}_{w+k-1}}{\text{BF}_k} t^{w+1}(\epsilon_{w+1}, \epsilon_{w+k-1})$.

### 3.2.3 Fixing ambiguities in $\mathbb{B}^{\text{sv}}, \phi^{\text{sv}}$ and $\mathcal{J}^{\text{eqv}}$

As detailed in sections 7 and 8 of [14] as well as section 4.1.2 of [28], the main ingredients $\mathbb{B}^{\text{sv}}, \phi^{\text{sv}}$ of $\mathcal{J}^{\text{eqv}}$ are only well-defined up to

$$ \mathbb{B}^{\text{sv}}(\epsilon_k; \tau) \to \mathbb{B}^{\text{sv}}(\epsilon_k; \tau) \mathcal{O}^{-1}(\epsilon_k), \quad \phi^{\text{sv}}(\mathcal{X}) \to \mathcal{O}(\epsilon_k) \phi^{\text{sv}}(\mathcal{X}) \mathcal{O}^{-1}(\epsilon_k), \quad (3.50) $$

for some $\mathfrak{sl}_2$-invariant and group-like series $\mathcal{O}(\epsilon_k)$ with single-valued MZVs as coefficients in the case of iterated Eisenstein integrals [14]. With the description of $\mathbb{B}^{\text{sv}}$ and $\phi^{\text{sv}}$ via zeta generators, the redefinition (3.50) amounts to

$$ \mathbb{M}^{\text{sv}}(z_i) \to \mathcal{O}(\epsilon_k) \mathbb{M}^{\text{sv}}(z_i), \quad \mathbb{M}^{\text{sv}}(\sigma_i) \to \mathbb{M}^{\text{sv}}(\sigma_i), \quad (3.51) $$

which composes well with the reflection property (3.13) since $R[\mathcal{O}(\epsilon_k)] = \mathcal{O}(\epsilon_k)^{21}$. These redefinitions amount to redistributing $\mathfrak{sl}_2$-invariant nested brackets of $\epsilon_k^{(j)}$ between the two contributions $\sigma_w = \sigma_w^g + z_w$ to the zeta generators. At modular depth three, such $\mathfrak{sl}_2$-invariants are given (up to scalar multiples) by

$$ I_{k_1,k_2,k_3} = \sum_{j=0}^{k_3-2} \sum_{i=0}^{d-2} \sum_{a=0}^{j} (-1)^{i+j} \binom{j}{a} \frac{(k_1-2-i)!(k_2-d+i)!}{i!(d-2-i)!} [\epsilon_{k_1}^{(i+a)}, [\epsilon_{k_2}^{(d-2-i+j-a)}, \epsilon_{k_3}^{(k_3-2-j)}]] \quad (3.52) $$

<footnote>$^{21}$The reflection property $R[\mathcal{O}(\epsilon_k)] = \mathcal{O}(\epsilon_k)$ can be seen by expressing the admissible $\epsilon_k^{(j)}$ in its expansion via nested $t^d$-operations (2.60) and observing that the sum of the superscripts $d$ has the right parity for eigenvalue +1 rather than -1 under $R[\cdot]$.</footnote>