with $d = \frac{1}{2}(k_1+k_2-k_3+2)$ and $[\epsilon_0, I_{k_1,k_2,k_3}] = [\epsilon_0^\vee, I_{k_1,k_2,k_3}] = 0$. The simplest redistribution of terms within $\sigma_w$ compatible with $sl_2$-invariance of $z_w$ is to shift $\sigma_7^g \to \sigma_7^g - qI_{4,4,6}$ and $z_7 \to z_7 + qI_{4,4,6}$ with $q \in \mathbb{Q}$, leaving the total zeta generator $\sigma_7$ unaffected. As detailed in the companion paper [34], we fix a unique choice of $z_w$ by imposing the associated geometric part $\sigma_w^g$ to have no $sl_2$-singlet at key degree $\sum_i k_i = 2w$. Different choices of the above coefficient $q$ of $I_{4,4,6}$ will affect the $sl_2$-singlet of $\sigma_7^g$ at its key degree $k_1+k_2+k_3 = 14$. Hence, the criterion of [34] to select a canonical splitting $\sigma_7 = \sigma_7^g + z_7$ no longer allows for redistributions of $I_{4,4,6}$ between $\sigma_7^g$ and $z_7$.

By the expansion (3.44) of $\sigma_w$, their geometric parts are determined in terms of the constants $c^{\text{sv}}$. The ancillary files of [28] provide the explicit form of $c^{\text{sv}}[\frac{j_1}{4}, \frac{j_2}{4}, \frac{j_3}{6}]$ at $j_1+j_2+j_3 = 4$ and $c^{\text{sv}}[\frac{j_1}{4}, \frac{j_2}{6}, \frac{j_3}{6}]$ at $j_1+j_2+j_3 = 5$ with one-parameter ambiguities $c_{446}, c_{466} \in \mathbb{Q}$. The first one $c_{446}$ accounts for redefinitions of $z_7$ by $I_{4,4,6}$ and is fixed to $c_{446} = 0$ by the choice of [34]. The second parameter in the $c^{\text{sv}}[\frac{j_1}{4}, \frac{j_2}{6}, \frac{j_3}{6}]$ of [28] is already fixed to $c_{466} = 0$ by the group-like form (3.42) of the generating series $\mathbb{C}^{\text{sv}}$.

The redefinitions (3.50) of $\mathbb{B}^{\text{sv}}$ and $\phi^{\text{sv}}$ affect Brown's generating series of equivariant iterated Eisenstein integrals via

$$ \mathcal{J}^{\text{eqv}}(\epsilon_k; \tau) \to \mathcal{J}^{\text{eqv}}(\epsilon_k; \tau)\mathbb{O}^{-1}(\epsilon_k), \qquad (3.53) $$

which shift individual modular forms in its coefficients by products of single-valued MZVs and $\beta^{\text{eqv}}$ of lower modular depth. The description of $\mathbb{B}^{\text{sv}}$ and $\phi^{\text{sv}}$ in terms of zeta generators and the criterion of [34] to fix their arithmetic parts $z_i$ selects the preferred choice of $\mathcal{J}^{\text{eqv}}(\epsilon_k; \tau)$ in (3.40) where the ambiguity (3.53) is resolved.

### 3.2.4 Single-valued iterated Eisenstein integrals and analogies with genus zero

Brown defines single-valued (as opposed to equivariant) iterated Eisenstein integrals through their generating series [14]

$$ \mathcal{J}^{\text{sv}}(\epsilon_k; \tau) := \mathcal{J}^{\text{eqv}}(\epsilon_k; \tau)\mathbb{B}^{\text{sv}}(\epsilon_k; \tau)^{-1} = \mathcal{J}_+(\epsilon_k; \tau)\mathbb{B}^{\text{sv}}(\epsilon_k; \tau)\phi^{\text{sv}}(\tilde{\mathcal{J}}_-(\epsilon_k; \tau))\mathbb{B}^{\text{sv}}(\epsilon_k; \tau)^{-1}, \quad (3.54) $$

which is invariant under the redefinition (3.50), i.e. well defined even without the guiding principles for $z_i$ in section 3.2.3. The invariance of single-valued iterated Eisenstein integrals under (3.50) comes at the cost of their modular properties — components of (3.54) do not transform as modular forms under $\text{SL}(2, \mathbb{Z})$ in any $sl_2$ frame for the $\epsilon_k^{(j)}$.

In view of (3.40), the right-multiplication by $\mathbb{B}^{\text{sv}}(\epsilon_k; \tau)^{-1}$ in (3.54) completes the generating series $\mathbb{M}^{\text{sv}}$ in single-valued MZVs to involve the full zeta generators $\sigma_w$,

$$ R[\mathcal{J}^{\text{sv}}(\epsilon_k; \tau)] = U_{\text{SL}_2}(\tau)\mathbb{M}^{\text{sv}}(\sigma_i)^{-1}\tilde{\mathbb{I}}_-(\epsilon_k; \tau)\mathbb{M}^{\text{sv}}(\sigma_i)\mathbb{I}_+(\epsilon_k; \tau)U_{\text{SL}_2}^{-1}(\tau), \quad (3.55) $$

instead of the standalone appearance of the arithmetic parts $\mathbb{M}^{\text{sv}}(z_i)^{-1}$ in the analogous expression for $R[\mathcal{J}^{\text{eqv}}(\epsilon_k; \tau)]$ in the second line of (3.40).

Converting back to holomorphic frame, (3.55) is equivalent to the following generating series of Brown's single-valued iterated Eisenstein integrals

$$ \mathbb{I}^{\text{sv}}(\epsilon_k; \tau) = \mathbb{M}^{\text{sv}}(\sigma_i)^{-1}\tilde{\mathbb{I}}_-(\epsilon_k; \tau)\mathbb{M}^{\text{sv}}(\sigma_i)\mathbb{I}_+(\epsilon_k; \tau), \quad (3.56) $$
which only differs from its equivariant analogue $\mathbb{I}^{\text{eqv}}(\epsilon_k; \tau)$ in (3.39) by having $M^{\text{sv}}(\sigma_i)^{-1}$ in the place of $M^{\text{sv}}(z_i)^{-1}$ as its leftmost factor on the right-hand side. Accordingly, the antiholomorphic differential equation (3.31) of $\mathbb{I}^{\text{eqv}}(\epsilon_k; \tau)$ carries over to $\mathbb{I}^{\text{sv}}(\epsilon_k; \tau)$ with a modified conjugation of $A_-(\epsilon_k; \tau)$,

$$ \partial_{\tau} \mathbb{I}^{\text{sv}}(\epsilon_k; \tau) d\tau = -\mathbb{I}^{\text{sv}}(\epsilon_k; \tau) A_+(\epsilon_k; \tau), \qquad (3.57) $$

$$ \partial_{\tau} \mathbb{I}^{\text{sv}}(\epsilon_k; \tau) d\bar{\tau} = -M^{\text{sv}}(\sigma_i)^{-1} A_-(\epsilon_k; \tau) M^{\text{sv}}(\sigma_i) \mathbb{I}^{\text{sv}}(\epsilon_k; \tau). $$

While the coefficients of $\mathbb{I}^{\text{eqv}}(\epsilon_k; \tau)$ are denoted by $\beta^{\text{eqv}}$ in (2.86), the coefficients of $\mathbb{I}^{\text{sv}}(\epsilon_k; \tau)$ or $\mathbb{I}^{\text{sv}}(\epsilon_k; \tau)$ do not obey any simple relation to the building blocks $\beta^{\text{sv}}$ for MGFs in (2.88). The arrangement of zeta generators in (3.55) gives rise to non-trivial $T$ transformations at modular depth greater than or equal to two which is incompatible with the $T$-invariance of general $\beta^{\text{sv}}$, also see section 4.2.3 of [28].

As a particular virtue of the representation (3.56) of single-valued iterated Eisenstein integrals, it manifests the close analogy with the construction of single-valued polylogarithms at genus zero [51]. By the description of $\mathbb{B}^{\text{sv}}$ and $\phi^{\text{sv}}$ in terms of zeta generators, (3.56) takes the same form as the generating series [50]

$$ \mathbb{G}_{\{0,1\}}^{\text{sv}}(x_i; z) = M^{\text{sv}}(M_i)^{-1} \overline{\tilde{\mathbb{G}}_{\{0,1\}}(x_i; z)} M^{\text{sv}}(M_i) \mathbb{G}_{\{0,1\}}(x_i; z), \qquad (3.58) $$

of single-valued multiple polylogarithms in one variable: The rightmost series generates meromorphic multiple polylogarithms

$$ \mathbb{G}_{\{0,1\}}(x_i; z) = \sum_{w=0}^{\infty} \sum_{a_1, \dots, a_w=0,1} G(a_1, a_2, \dots, a_w; z) x_{a_w} \dots x_{a_2} x_{a_1}, \qquad (3.59) $$

which are recursively defined by $(a_i, z \in \mathbb{C})$

$$ G(a_1, a_2, \dots, a_w; z) = \int_0^z \frac{dt}{t-a_1} G(a_2, \dots, a_w; t), \qquad (3.60) $$

with $G(\emptyset; z) = 1$ and shuffle regularisation based on $G(0; z) = \log(z)$. The non-commutative $x_0, x_1$ generate the Lie algebra of the fundamental group of $\mathbb{P}^1 \setminus \{0, 1, \infty\}$, i.e. the sphere minus three points, and appear as the braid operators in the one-variable version of the Knizhnik-Zamolodchikov equation

$$ \partial_z \mathbb{G}_{\{0,1\}}(x_i; z) = \mathbb{G}_{\{0,1\}}(x_i; z) \left( \frac{x_0}{z} + \frac{x_1}{z-1} \right), \qquad (3.61) $$

which can be viewed as the genus-zero analogue of the Knizhnik-Zamolodchikov-Bernard equation (3.29) of the series $\mathbb{I}_+(\epsilon_k; \tau)$. Just like the generating series $\tilde{\mathbb{I}}_-(\epsilon_k; \tau)$ of complex-conjugate iterated Eisenstein integrals in (3.56), the antideromorphic polylogarithms in its genus-zero counterpart (3.58) enter the complex-conjugate series $\tilde{\mathbb{G}}_{\{0,1\}}(x_i; z)$ with a reversed integration order $\overline{G}(a_1, \dots, a_w; z) \to \overline{G}(a_w, \dots, a_1; z)$.

Finally, the series (3.58) in single-valued polylogarithms introduces single-valued MZVs through the same type of series $M^{\text{sv}}$ in zeta generators encountered in our genus-one setting. The argument of $M^{\text{sv}}(M_i)$ instructs us to replace the $z_i$ in (2.90) by the genus-zero incarnation
<table><caption>**Table 1.** Dictionary between the constructions of single-valued polylogarithms in one variable (genus zero) and single-valued iterated Eisenstein integrals (genus one).</caption><thead><tr><th>genus zero</th><th>genus one</th></tr></thead><tbody><tr><td>dz / z-a, a &isin; {0, 1}</td><td>&tau;<sup>j</sup>G<sub>k</sub>(&tau;) d&tau;, k &ge; 4, 0 &le; j &le; k-2</td></tr><tr><td>G<sub>{0,1}</sub> series in x<sub>0</sub>, x<sub>1</sub> normalised by M<sub>w</sub></td><td>I<sub>&pm;</sub> series in &epsilon;<sub>k</sub><sup>(j)</sup> normalised by &sigma;<sub>w</sub></td></tr><tr><td>G<sub>{0,1}</sub><sup>sv</sup> = M<sup>sv</sup>(M<sub>i</sub>)<sup>-1</sup> G&#771;<sub>{0,1}</sub> M<sup>sv</sup>(M<sub>i</sub>) G<sub>{0,1}</sub></td><td>I<sup>sv</sup> = M<sup>sv</sup>(&sigma;<sub>i</sub>)<sup>-1</sup> I&#771;<sub>&pm;</sub> M<sup>sv</sup>(&sigma;<sub>i</sub>) I<sub>&pm;</sub></td></tr></tbody></table>

$M_i$ of zeta generators that act as derivations on the braid operators $x_0, x_1$ [52, 53], e.g. $[x_0, M_w] = 0$ and $[x_1, M_3] = [[[x_0, x_1], x_0+x_1], x_1]$. Hence, in the same way as the zeta generators $\sigma_w$ (or their arithmetic parts $z_w$) normalise the $\epsilon_k^{(j)}$, the zeta generators $M_w$ at genus zero normalise the free algebra Lie[$x_0, x_1$]. Accordingly, the stability of the alphabet $\epsilon_k^{(j)}$ of the generating series $\tilde{I}_-(\epsilon_k; \tau)$ under conjugation by $M^{\text{sv}}(\sigma_i)$ or $M^{\text{sv}}(z_i)$ is a genus-one analogue of the fact that $M^{\text{sv}}(M_i)^{-1}\tilde{G}_{\{0,1\}}(x_i; z)M^{\text{sv}}(M_i)$ is expressible in terms of $x_0, x_1$.

The original construction of single-valued polylogarithms [51] introduces single-valued MZVs through the Drinfeld associator, a series in $(x_0, x_1)$. The reformulation in (3.58) relies on the more abstract generating series $M^{\text{sv}}(M_i)$ in zeta generators [50] which no longer exposes the geometric origin of its MZVs from the monodromies of the Knizhnik-Zamolodchikov equation (3.61). Instead, the motivation to employ a Lie-algebraic reformulation of the single-valued map at genus zero stems from the following features:

First, all known $\mathbb{Q}$-relations among single-valued MZVs are automatically incorporated by the use of the $f$-alphabet in (2.90) and $M^{\text{sv}}(M_i)$. Second, the coefficients of the MZVs $\rho^{-1}(\text{sv}(f_{i_1}f_{i_2}\dots f_{i_l}))$ beyond depth one (i.e. for $l \ge 2$) are deduced from those of the primitives $\rho^{-1}(f_w)$ as discussed in section 3.2.1. Third and most importantly, upon comparison of (3.56) with (3.58), the use of zeta generators brings the explicit realisation of the single-valued map into the same form at genus zero and genus one. Single-valued polylogarithms at genus zero in (3.56) are constructed from the same type of conjugation by $M^{\text{sv}}$ of the antiholomorphic contributions as seen in (3.58) for Brown's single-valued iterated Eisenstein integrals at genus one. The dictionary between the respective ingredients at genus zero and genus one is summarised in table 1.

More specifically, the relations among the $\epsilon_k$ imply that the construction applies to those iterated Eisenstein integrals in $I_+(\epsilon_k; \tau)$ which admit a representation via configuration space integrals, so-called elliptic MZVs [75, 87]. In order to relate MGFs in the expansion of $I^{\text{eqv}}(\epsilon_k; \tau)$ to single-valued elliptic MZVs, it remains to adjust the left-multiplicative series in zeta generators. The single-valued map $I_+(\epsilon_k; \tau) \to I^{\text{sv}}(\epsilon_k; \tau)$ to Brown's single-valued iterated Eisenstein integrals differs from the generating series $I^{\text{eqv}}(\epsilon_k; \tau)$ of MGFs by a product of two different series $M^{\text{sv}}$, i.e.

$$I^{\text{sv}}(\epsilon_k; \tau) = M^{\text{sv}}(\sigma_i)^{-1} M^{\text{sv}}(z_i) I^{\text{eqv}}(\epsilon_k; \tau). \qquad (3.62)$$
### 3.3 Reinstating iterated integrals of holomorphic cusp forms

The previous section 3.2 was dedicated to the generating series of MGFs obtained from the specialisation $e_k \to \epsilon_k$ in the generating series $\mathcal{J}^{\text{eqv}}(e_k; \tau)$ or $\mathbb{I}^{\text{eqv}}(e_k; \tau)$ of more general non-holomorphic modular forms. In particular, the specialisation $e_k \to \epsilon_k$ is essential for deriving the analogy with genus zero as discussed in section 3.2.4 where we showed how the constructions of single-valued polylogarithms and single-valued iterated Eisenstein integrals rest on the same type of conjugation by series in single-valued MZVs and zeta generators.

The goal of the present section is to describe new features of the generating series $\mathcal{J}^{\text{eqv}}(e_k; \tau)$, $\mathbb{I}^{\text{eqv}}(e_k; \tau)$ in the free algebra of $e_k^{(j)}$. The Pollack combinations $P_w^d$ of $e_k^{(j)}$ introduced in section 2.3 govern the appearance of holomorphic cusp forms and more general classes of periods beyond MZVs. We shall use these $P_w^d$ to construct the letters $e_{\Delta_k^{\pm}}$ accompanying the cuspidal integration kernels in the one-forms $\mathcal{A}_{\pm}$ of (3.2) and (3.3) that vanish under $e_k \to \epsilon_k$. Both $e_{\Delta_k^{\pm}}$ and the constants $c^{\text{sv}}$ in the expansion (3.42) of the series $\mathbb{C}^{\text{sv}}(e_k)$ in general involve periods beyond MZVs and L-values which drop out upon specialising to $e_k \to \epsilon_k$ and therefore do not appear in the MGF world.

Based on a comprehensive study of non-holomorphic modular forms accompanying all of $e_{k_1}^{(j_1)} e_{k_2}^{(j_2)} e_{k_3}^{(j_3)}$ at degree $k_1+k_2+k_3 \le 20$, we here encounter the following new phenomena at modular depth three:

* in section 3.3.1: (anti-)holomorphic double integrals involving an Eisenstein series and a holomorphic cusp form;

* in section 3.3.2: new periods beyond ratios of L-values accompanying depth-one integrals of holomorphic cusp forms;

* in section 3.3.3: new periods beyond MZVs and L-values entering $c^{\text{sv}}$ and hence $\mathbb{C}^{\text{sv}}(e_k)$.

All of these phenomena are inferred from the modular completion of triple Eisenstein integrals. The Pollack combinations of $e_k^{(j)}$ in their generating-series description are determined by imposing modularity (2.45) of all the coefficients $\beta^{\text{eqv}}[j_1 \ j_2 \ j_3]_{k_1 \ k_2 \ k_3}$ in $\mathcal{J}^{\text{eqv}}(e_k; \tau)$ at degree $k_1+k_2+k_3 \le 20$. Finally, section 3.3.4 provides a qualitative discussion of yet another new feature at modular depth four whose explicit study at the level of non-holomorphic modular form is left for the future.

#### 3.3.1 Double integrals of Eisenstein series and cusp forms

A decomposition of $\mathcal{J}^{\text{eqv}}(e_k; \tau)$ into (anti-)meromorphic iterated integrals of modular forms, a change of alphabet $\hat{\psi}^{\text{sv}}$ and a series $\mathbb{C}^{\text{sv}}(e_k)$ in constants were already given in (3.22) and (3.23). We now focus on the letters $e_{\Delta^{\pm}}$ accompanying the (anti-)holomorphic cusp forms in the one-forms (3.2) and (3.3) for the (anti-)meromorphic series $\mathbb{I}_{\pm}$ and their images $\mathcal{J}_{\pm}$ in the modular frame in (3.11).

At modular depth two, the admixtures of cuspidal integrals $\beta_{\pm}[\Delta_{2s}^j]$ in (2.25) entering the coefficients $\beta^{\text{eqv}}[j_1 \ j_2]_{k_1 \ k_2}$ of $\mathcal{J}^{\text{eqv}}$ are known from [28], see the ancillary files of the reference for all examples up to and including $k_1+k_2=24$. These admixtures are informed by the modular invariants $F_{m,k}^{\pm(s)}$ mentioned in section 2.2.5 and discussed in more detail in section 3.4.
They uniquely specify the appearance of $\beta_{\pm}[\Delta_{2s}]^j$ in the coefficients of $e_{k_1}^{(j_1)}e_{k_2}^{(j_2)}$ of $J_+$ and $\tilde{J}_-$ at modular depth two. Since the $F_{m,k}^{\pm(s)}$ are respectively even and odd under $\tau \to -\bar{\tau}$, one encounters both relative signs between $\beta_+[\Delta_{2s}]^j \pm \beta_-[\Delta_{2s}]^j$ in $J^{\text{eqv}}$ which necessitates two different letters

$$e_{\Delta_{2s}^{+}} = e_{\Delta_{2s}^{\text{ev}}} - e_{\Delta_{2s}^{\text{odd}}}, \quad e_{\Delta_{2s}^{-}} = e_{\Delta_{2s}^{\text{ev}}} + e_{\Delta_{2s}^{\text{odd}}}, \qquad (3.63)$$

in $J_+$ and $\tilde{J}_-$. This is to be contrasted with non-holomorphic Eisenstein series in (2.40) and (2.41) where $\beta_+[\Delta_{2s}]^j$ and $\beta_-[\Delta_{2s}]^j$ only occur with a relative plus sign at modular depth one and therefore descend from the same letter $e_k$ in both of $J_+$ and $\tilde{J}_-$.

As indicated by the superscripts in (3.63), the contributions $e_{k_1}^{(j_1)}e_{k_2}^{(j_2)}$ of modular depth two to $e_{\Delta_{2s}^{\text{ev}}}(e_{\Delta_{2s}^{\text{odd}}})$ are engineered to reproduce the cusp-form contributions to the even functions $F_{m,k}^{+(s)}$ (odd functions $F_{m,k}^{-(s)}$). In both cases, the modular weight of the holomorphic cusp form $\Delta_{2s}$ is twice the Laplace eigenvalue of the underlying $F_{m,k}^{\pm(s)}$, and the degree of the associated contribution to $e_{k_1}^{(j_1)}e_{k_2}^{(j_2)}$ is given by $k_1+k_2 = 2m+2k$.

The simplest two examples $F_{2,5}^{-(6)}$ and $F_{2,6}^{+(6)}$ of the modular invariants with cuspidal contributions [37] for instance imply the following cuspidal parts (denoted by $|_{\Delta_{2s}}$) [28]:

$$\begin{align}
\beta^{\text{eqv}}[\begin{smallmatrix} 0 & 5 \\ 4 & 10 \end{smallmatrix}]|_{\Delta_{2s}} &= \beta^{\text{eqv}}[\begin{smallmatrix} 1 & 4 \\ 4 & 10 \end{smallmatrix}]|_{\Delta_{2s}} = \beta^{\text{eqv}}[\begin{smallmatrix} 2 & 3 \\ 4 & 10 \end{smallmatrix}]|_{\Delta_{2s}} = \frac{\Lambda(\Delta_{12}, 12)(\beta_-[\begin{smallmatrix} 5 \\ \Delta_{12} \end{smallmatrix}] - \beta_+[\begin{smallmatrix} 5 \\ \Delta_{12} \end{smallmatrix}])}{122472000 \Lambda(\Delta_{12}, 10)}, \nonumber \\
\beta^{\text{eqv}}[\begin{smallmatrix} 0 & 6 \\ 4 & 12 \end{smallmatrix}]|_{\Delta_{2s}} &= -\beta^{\text{eqv}}[\begin{smallmatrix} 2 & 4 \\ 4 & 12 \end{smallmatrix}]|_{\Delta_{2s}} = \frac{\Lambda(\Delta_{12}, 13)(\beta_+[\begin{smallmatrix} 5 \\ \Delta_{12} \end{smallmatrix}] + \beta_-[\begin{smallmatrix} 5 \\ \Delta_{12} \end{smallmatrix}])}{5746356000 \Lambda(\Delta_{12}, 11)}, \nonumber \\
\beta^{\text{eqv}}[\begin{smallmatrix} 1 & 5 \\ 4 & 12 \end{smallmatrix}]|_{\Delta_{2s}} &= 0, \tag{3.64}
\end{align}$$

with similar results on $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 \\ 6 & 8 \end{smallmatrix}]|_{\Delta_{2s}}$ from $F_{3,4}^{-(6)}$ and $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 \\ 6 & 10 \end{smallmatrix}]|_{\Delta_{2s}}$, $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 \\ 8 & 8 \end{smallmatrix}]|_{\Delta_{2s}}$ from $F_{3,5}^{+(6)}, F_{4,4}^{+(6)}$. This needs to be matched with the contributions $\mathcal{E}[\begin{smallmatrix} j \\ \Delta_{12} \end{smallmatrix}]e_{\Delta_{12}^{+}}^{(j)}$ to (3.7) and $\overline{\mathcal{E}[\begin{smallmatrix} j \\ \Delta_{12} \end{smallmatrix}]}e_{\Delta_{12}^{-}}^{(j)}$ to (3.8) translated into the modular-frame series $J_{\pm}(e_k; \tau)$ via (3.11).

Since MGFs obtained from the specialisation $e_k \to \epsilon_k$ are free of cusp forms, the contributions to $e_{\Delta_{12}^{\text{odd}}}$ and $e_{\Delta_{12}^{\text{even}}}$ encoded in (3.64) have to be proportional to the simplest Pollack combinations $P_w^d$ of section 2.3,

$$e_{\Delta_{12}^{\text{odd}}} = \frac{\Lambda(\Delta_{12}, 12)}{\Lambda(\Delta_{12}, 10)} P_{14}^2 + \dots, \quad e_{\Delta_{12}^{\text{ev}}} = \frac{\Lambda(\Delta_{12}, 13)}{\Lambda(\Delta_{12}, 11)} P_{16}^3 + \dots, \qquad (3.65)$$

with an infinite series of nested brackets of $e_k^{(j)}$ of degree $\ge 18$ and modular depth $\ge 3$ in the ellipsis. Here and below, we fix normalisation conventions for $P_w^d$ such that they enter $e_{\Delta_{2s}^{\text{odd}}}$ and $e_{\Delta_{2s}^{\text{even}}}$ in (3.65) with unit coefficients besides the ratios of L-values (2.19). This is always possible since the defining property $P_w^d(\epsilon_k) = 0$ holds in any normalisation, and each $P_w^d$ is uniquely associated with cusp forms $\Delta_{w-2d+2}$ and degree $w$. The large integers in the denominators of (3.64) are then responsible for the similarly unwieldy normalisation factors in the expressions (2.68) and (2.69) for $P_{14}^2$ and $P_{16}^3$.
In fact, inspection of the cuspidal contributions to $\beta^{\text{eqv}}[\begin{matrix} 0 & 5 \\ 4 & 10 \end{matrix}]$ and $\beta^{\text{eqv}}[\begin{matrix} 0 & 6 \\ 4 & 12 \end{matrix}]$ is already enough to determine these normalisations. The appearance of the cuspidal $\beta_{\pm}[\begin{matrix} 5 \\ \Delta_{12} \end{matrix}]$ in $\beta^{\text{eqv}}[\begin{matrix} 4 & 4 \\ 1 & 10 \end{matrix}]$, $\beta^{\text{eqv}}[\begin{matrix} 2 & 3 \\ 4 & 10 \end{matrix}]$ and $\beta^{\text{eqv}}[\begin{matrix} 1 & 5 \\ 4 & 12 \end{matrix}]$, $\beta^{\text{eqv}}[\begin{matrix} 2 & 4 \\ 4 & 12 \end{matrix}]$ in (3.64) is interlocked with that in $\beta^{\text{eqv}}[\begin{matrix} 0 & 5 \\ 4 & 10 \end{matrix}]$ and $\beta^{\text{eqv}}[\begin{matrix} 0 & 6 \\ 4 & 12 \end{matrix}]$ by the rewriting of $\text{ad}_{e_0}^j(P_{14}^2)$ and $\text{ad}_{e_0}^j(P_{16}^3)$ in the cuspidal sector of $\mathcal{J}_{\pm}$ in terms of $e_{k_1}^{(j_1)}e_{k_2}^{(j_2)}$. Similarly, the cuspidal contributions to $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ 6 & 8 \end{matrix}]$, $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ 6 & 10 \end{matrix}]$ and $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ 8 & 8 \end{matrix}]$ are completely fixed by the results on $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ 4 & 10 \end{matrix}]$ and $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ 4 & 12 \end{matrix}]$ in (3.64) since they follow from the known relative factors of $t_{4,10}^2, t_{6,8}^2$ in $P_{14}^2$ and $t_{4,12}^3, t_{6,10}^3, t_{8,8}^3$ in $P_{16}^3$ (see (2.68) and (2.69)) which are in fact determined by the period polynomial of $\Delta_{12}$ [36].

By the same logic, the $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$ informed by $F_{m,k}^{\pm(s)}$ at higher Laplace eigenvalues and higher degrees $k_1+k_2 = 2m+2k$ determine the ratios of L-values in

$$e_{\Delta_{12}^{\text{odd}}} = \frac{\Lambda(\Delta_{12}, 12)}{\Lambda(\Delta_{12}, 10)} P_{14}^2 + \frac{\Lambda(\Delta_{12}, 14)}{\Lambda(\Delta_{12}, 10)} P_{18}^4 + \dots, \quad e_{\Delta_{16}^{\text{odd}}} = \frac{\Lambda(\Delta_{16}, 16)}{\Lambda(\Delta_{16}, 14)} P_{18}^2 + \dots, \quad (3.66)$$

$$e_{\Delta_{12}^{\text{ev}}} = \frac{\Lambda(\Delta_{12}, 13)}{\Lambda(\Delta_{12}, 11)} P_{16}^3 + \frac{\Lambda(\Delta_{12}, 15)}{\Lambda(\Delta_{12}, 11)} P_{20}^5 + \dots, \quad e_{\Delta_{16}^{\text{ev}}} = \frac{\Lambda(\Delta_{16}, 17)}{\Lambda(\Delta_{16}, 15)} P_{20}^3 + \dots,$$

$$e_{\Delta_{18}^{\text{odd}}} = \frac{\Lambda(\Delta_{18}, 18)}{\Lambda(\Delta_{18}, 16)} P_{20}^2 + \dots,$$

with degree $\ge 18$ and modular depth $\ge 3$ in the ellipsis. Moreover, the detailed prefactors of the $\beta_{\pm}[\begin{matrix} j \\ \Delta_{2s} \end{matrix}]$ in these $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$ fix the normalisations of the $P_w^d$ in (2.68) to (2.71) that are compatible with unit coefficients in (3.66).

By the construction of the generating series $\mathcal{J}^{\text{eqv}}$ from path-ordered exponentials in (3.22) and (3.23), our systematic handle on the modular-depth-two contributions to $e_{\Delta_{2s}^{\pm}}$ also has implications for $\beta^{\text{eqv}}$ at modular depth three. More specifically, expressions like (3.66) at modular depth two translate into modular-depth-three contributions $\mathcal{E}[\begin{matrix} j_1 & j_2 \\ k_1 & \Delta_{k_2} \end{matrix}] e_{k_1}^{(j_1)} e_{\Delta_{k_2}^{+}}^{(j_2)}$ and $\mathcal{E}[\begin{matrix} j_1 & j_2 \\ \Delta_{k_1} & k_2 \end{matrix}] e_{k_1}^{(j_1)} e_{\Delta_{k_2}^{-}}^{(j_2)}$ to (3.7) as well as $\overline{\mathcal{E}[\begin{matrix} j_2 & j_1 \\ \Delta_{k_2} & k_1 \end{matrix}]} e_{k_1}^{(j_1)} e_{\Delta_{k_2}^{-}}^{(j_2)}$ and $\overline{\mathcal{E}[\begin{matrix} j_2 & j_1 \\ k_2 & \Delta_{k_1} \end{matrix}]} e_{\Delta_{k_1}^{-}}^{(j_1)} e_{k_2}^{(j_2)}$ to (3.8). These terms inject double integrals involving one holomorphic Eisenstein series and one cusp forms into various $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$. Similarly, contributions like (3.66) to $e_{\Delta_{2s}^{\pm}}$ determine the products of holomorphic and antiholomorphic integrals of modular depth one involving one Eisenstein series and one cusp form that enter generic $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$. These mechanisms lead to terms of the schematic form

$$\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] = \dots + \left\{ \frac{\Lambda(\Delta_{2s}, 2s+2n)}{\Lambda(\Delta_{2s}, 2s-2)} \text{ or } \frac{\Lambda(\Delta_{2s}, 2s+2n+1)}{\Lambda(\Delta_{2s}, 2s-1)} \right\} \quad (3.67) \\ \times \left\{ \beta_{\pm}[\begin{matrix} \ell_1 & \ell_2 \\ k & \Delta_{2s} \end{matrix}] \text{ or } \beta_{\pm}[\begin{matrix} \ell_1 & \ell_2 \\ \Delta_{2s} & k \end{matrix}] \text{ or } \beta_{+}[\begin{matrix} \ell_1 \\ k \end{matrix}] \beta_{-}[\begin{matrix} \ell_2 \\ \Delta_{2s} \end{matrix}] \text{ or } \beta_{-}[\begin{matrix} \ell_1 \\ k \end{matrix}] \beta_{+}[\begin{matrix} \ell_2 \\ \Delta_{2s} \end{matrix}] \right\},$$

with $n \in \mathbb{N}_0$ and $\ell_1, \ell_2$ distinct from $j_1, j_2, j_3$. One can see from the differential equations in $\tau$ that the contributions to $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$ in (3.67) are necessary to attain the desired modular properties (2.45). For instance, translating (3.33) into components identifies contributions

$$2\pi i(\tau - \bar{\tau})^2 \partial_{\tau} \beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}] = -\delta_{j_3, k_3-2}(\tau - \bar{\tau})^{k_3} G_{k_3}(\tau) \beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}] + \dots, \quad (3.68)$$

where the modular completions $\beta_{\pm}[\begin{matrix} j \\ \Delta_{2s} \end{matrix}]$ of the double Eisenstein integrals present in the $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 \\ k_1 & k_2 \end{matrix}]$ on the right-hand side, can only come from terms of the form $\beta_{+}[\begin{matrix} j & k_3-2 \\ \Delta_{2s} & k_3 \end{matrix}]$ and $\beta_{+}[\begin{matrix} k_3-2 \\ k_3 \end{matrix}] \beta_{-}[\begin{matrix} j \\ \Delta_{2s} \end{matrix}]$ in the $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$ on the left-hand side.
Since the expansion of $e_{\Delta_{2s}^{\pm}}$ in terms of brackets of $e_k^{(j)}$ starts at degree $2s+2$ (i.e. at degrees $\ge 14$), the mixing (3.67) of Eisenstein and cuspidal integrals in $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$ begins at degree $k_1+k_2+k_3 = 18$, for instance

$$ \beta^{\text{eqv}}[\begin{matrix} 0 & 0 & 0 \\ 4 & 4 & 10 \end{matrix}]|_{\Delta_{2s}} = \frac{\Lambda(\Delta_{12}, 12)}{122472000\Lambda(\Delta_{12}, 10)} \left( \beta_-[\begin{matrix} 0 & 0 \\ 4 & \Delta_{12} \end{matrix}] - \beta_+[\begin{matrix} 0 & 0 \\ 4 & \Delta_{12} \end{matrix}] \right. \\ \left. - \beta_-[\begin{matrix} 0 \\ 4 \end{matrix}]\beta_+[\begin{matrix} 0 \\ \Delta_{12} \end{matrix}] + \frac{\zeta_3}{24y^2}\beta_+[\begin{matrix} 0 \\ \Delta_{12} \end{matrix}] \right). \quad (3.69) $$

### 3.3.2 New periods and depth-one integrals of cusp form

Apart from the contributions (3.67) mixing holomorphic Eisenstein series and cusp forms, generic $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$ will involve additional terms $\beta_{\pm}[\begin{matrix} j \\ \Delta_{2s} \end{matrix}]$ without any admixtures of Eisenstein series. A simple class of cuspidal contributions with coefficients $\sim \zeta_{k-1} \frac{\Lambda(\Delta_{2s}, 2s+2n)}{\Lambda(\Delta_{2s}, 2s-2)}$ or $\sim \zeta_{k-1} \frac{\Lambda(\Delta_{2s}, 2s+2n+1)}{\Lambda(\Delta_{2s}, 2s-1)}$ exemplified by the last term of (3.69) arises from combining the coefficients $\sim \zeta_{k-1}$ of $e_k^{(j)}$ in $\mathbb{B}^{\text{sv}}$ with terms of modular depth two in the $e_{\Delta_{2s}^{\pm}}$ of $\mathcal{J}_{\pm}$.<fnref n="22" /> However, our main interest in this section is on the terms of schematic form $[e_{k_1}^{(j_1)}, [e_{k_2}^{(j_2)}, e_{k_3}^{(j_3)}]]$ entering the letters $e_{\Delta_{2s}^{\pm}}$ whose contributions to $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$ involve neither an Eisenstein series nor any MZVs.

Contributions to $e_{\Delta_{2s}^{\pm}}$ of modular depth three can be anticipated from the fact that generic Pollack combinations $P_w^d$ at $d \ge 3$ mix nested brackets of $e_k^{(j)}$ of different modular depth 2, 3..., $d$. In the simplest case of $P_{16}^3$, terms of modular depth two such as $t_{4,12}^3, t_{6,10}^3, t_{8,8}^3$ coexist with $[e_4, [e_4, e_8]]$ and $[e_6, [e_6, e_4]]$ of modular depth three, see (2.69) for their coefficients. The contributions $[e_4, [e_4, e_8]]$ and $[e_6, [e_6, e_4]]$ to $P_{16}^3$ are the unique modular-depth-three completion of $t_{4,12}^3, t_{6,10}^3, t_{8,8}^3$ to attain a vanishing combination in Tsunogai's derivation algebra. In this way, the expansion of (3.23) together with (3.65) predicts contributions $\sim \beta_+[\begin{matrix} j \\ \Delta_{12} \end{matrix}] + \beta_-[\begin{matrix} j \\ \Delta_{12} \end{matrix}]$ to various $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ 4 & 6 & 6 \end{matrix}]$ and $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ 4 & 4 & 8 \end{matrix}]$ with $j_1+j_2+j_3 = j$, for instance

$$ \beta^{\text{eqv}}[\begin{matrix} 0 & 1 & 2 \\ 4 & 4 & 8 \end{matrix}]|_{\Delta_{2s}} = -\frac{11\Lambda(\Delta_{12}, 13)}{25075008000\Lambda(\Delta_{12}, 11)} \left( \beta_+[\begin{matrix} 3 \\ \Delta_{12} \end{matrix}] + \beta_-[\begin{matrix} 3 \\ \Delta_{12} \end{matrix}] \right), \quad (3.70) $$

$$ \beta^{\text{eqv}}[\begin{matrix} 2 & 3 & 2 \\ 4 & 6 & 6 \end{matrix}]|_{\Delta_{2s}} = -\frac{23\Lambda(\Delta_{12}, 13)}{16716672000\Lambda(\Delta_{12}, 11)} \left( \beta_+[\begin{matrix} 7 \\ \Delta_{12} \end{matrix}] + \beta_-[\begin{matrix} 7 \\ \Delta_{12} \end{matrix}] \right). $$

However, the analogous completion of generic $P_w^d$ at $d \ge 3$ and $w \ge 18$ is ambiguous. Only the terms $t_{p,q}^d$ of modular depth two and degree $p+q = w$ are canonical and determined by period polynomials of $\Delta_{2s}$. The additional nested brackets at modular depth $\ge 3$ such as $[e_{k_1}^{(j_1)}, [e_{k_2}^{(j_2)}, e_{k_3}^{(j_3)}]]$ needed to attain $P_w^d \to 0$ as $e_k^{(j)} \to \epsilon_k^{(j)}$ can typically be shifted by brackets of $e_k^{(j)}$ with simpler Pollack combinations while maintaining their defining properties.

$$ \mathbb{B}^{\text{sv}}(e_k; \tau) = 1 - 2 \sum_{k=4}^{\infty} \sum_{j=0}^{k-2} \frac{(-1)^j \zeta_{k-1} e_k^{(j)}}{j!(4y)^j} + \dots $$

<footnote n="22">²²More specifically, the relevant terms of modular depth one are given by</footnote>
For instance, $P_{18}^4$ in (2.70) can be shifted by multiples of $t^3(e_4, P_{14}^2) \sim t^3(e_4, t^2(e_4, e_{10})) - 3t^3(e_4, t^2(e_6, e_8))$ while preserving

(i) the lowest-weight-vector condition $[e_0^\vee, P_{18}^4] = 0$, see (2.72);

(ii) degree 18;

(iii) total number 4 of $e_{k \ge 0}$.

Similar ambiguities arise for other $P_w^d$ in (2.69) to (2.71) with one free parameter for $P_{20}^3$ and three free parameters for $P_{20}^5$,

$$ P_{18}^4 \leftrightarrow t^3(e_4, P_{14}^2), \quad P_{20}^3 \leftrightarrow t^2(e_6, P_{14}^2), \\ P_{20}^5 \leftrightarrow t^4(e_6, P_{14}^2), t^3(e_4, P_{16}^3), [\hat{z}_3, P_{14}^2], \qquad (3.71) $$

all of which preserve the vanishing of $P_w^d$ under $e_k^{(j)} \to \epsilon_k^{(j)}$, the lowest-weight vector property $[e_0^\vee, P_w^d] = 0$, degree $w$ and the total number $d$ of $e_{k \ge 0}$.

Hence, it is a priori unclear how to generalise the contribution $\sim \frac{\Lambda(\Delta_{12}, 14)}{\Lambda(\Delta_{12}, 10)} P_{18}^4$ generating the modular-depth-two terms of $e_{\Delta_{12}^{\text{odd}}}$ in (3.66) to modular depth three. We consider a one-parameter ansatz of possible deformations $P_{18}^4 \to P_{18}^4 + c t^3(e_4, t^2(e_4, e_{10})) - 3c t^3(e_4, t^2(e_6, e_8))$ to $e_{\Delta_{12}^{\text{odd}}}$ and then impose the modular properties (2.45) on the resulting $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ 4 & 4 & 10 \end{smallmatrix}]$ and $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ 4 & 6 & 8 \end{smallmatrix}]$ to determine $c \in \mathbb{R}$. Remarkably, there is no rational solution for $c$. Instead, we obtain the expression (with modular depth $\ge 4$ and degree $\ge 22$ in the ellipsis)

$$ e_{\Delta_{12}^{\text{odd}}} = \frac{1}{\Lambda(\Delta_{12}, 10)} \left\{ \Lambda(\Delta_{12}, 12) P_{14}^2 + \Lambda(\Delta_{12}, 14) P_{18}^4 + \Lambda_{4,14}^{3,2} t^3(e_4, P_{14}^2) \right\} + \dots, \qquad (3.72) $$

involving a new period $\Lambda_{4,14}^{3,2} \in \mathbb{R}$ that we did not manage to express in terms of MZVs and (critical or non-critical) L-values (2.19). By numerical and analytical studies of the modularity properties of $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ at $k_1+k_2+k_3 = 18$, the new number is available both as an approximation $\Lambda_{4,14}^{3,2} = 0.0001108864...$ to 300 digits and as a combination of MMVs (2.14) and (2.17) in an ancillary file.

More generally, we insert free parameters along with the ambiguities in (3.71) and impose the modular properties (2.45) on all $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ at $k_1+k_2+k_3 \le 20$. This process gives rise to additional new periods $\Lambda_{k,w}^{d_1, d_2} \in \mathbb{R}$ that cannot be expressed in terms of MZVs or L-values and that we label according to the accompanying $t^{d_1}(e_k, P_w^{d_2})$ in

$$ \begin{aligned} e_{\Delta_{12}^{\text{odd}}} &= \frac{1}{\Lambda(\Delta_{12}, 10)} \left\{ \Lambda(\Delta_{12}, 12) P_{14}^2 + \Lambda(\Delta_{12}, 14) P_{18}^4 + \Lambda_{4,14}^{3,2} t^3(e_4, P_{14}^2) \right\} + \dots, \\ e_{\Delta_{12}^{\text{ev}}} &= \frac{1}{\Lambda(\Delta_{12}, 11)} \left\{ \Lambda(\Delta_{12}, 13) P_{16}^3 + \Lambda(\Delta_{12}, 15) P_{20}^5 \right. \\ & \qquad \left. + \Lambda_{6,14}^{4,2} (t^4(e_6, P_{14}^2) - \frac{364}{15} [\hat{z}_3, P_{14}^2]) + \Lambda_{4,16}^{3,3} t^3(e_4, P_{16}^3) \right\} + \dots, \\ e_{\Delta_{16}^{\text{odd}}} &= \frac{\Lambda(\Delta_{16}, 16)}{\Lambda(\Delta_{16}, 14)} P_{18}^2 + \dots, \\ e_{\Delta_{16}^{\text{ev}}} &= \frac{1}{\Lambda(\Delta_{16}, 15)} \left\{ \Lambda(\Delta_{16}, 17) P_{20}^3 + \Lambda_{6,14}^{2,2} t^2(e_6, P_{14}^2) \right\} + \dots, \\ e_{\Delta_{18}^{\text{odd}}} &= \frac{\Lambda(\Delta_{18}, 18)}{\Lambda(\Delta_{18}, 16)} P_{20}^2 + \dots. \end{aligned} \qquad (3.73) $$
<table><caption>**Table 2.** New periods $\Lambda_{k,w}^{d_1,d_2}$ encountered in the contributions to $e_{\Delta_{2s}^{\text{even}}}, e_{\Delta_{2s}^{\text{odd}}}$ at modular depth three and degree $k_1+k_2+k_3 = 18, 20$.</caption><thead><tr><th>new period</th><th>well defined up to</th><th>transcendental weight</th><th>approx. numerical value</th></tr></thead><tbody><tr><td>Λ<sub>4,14</sub><sup>3,2</sup></td><td>Λ(Δ<sub>12</sub>, 14)</td><td>3</td><td>0.0001108864...</td></tr><tr><td>Λ<sub>6,14</sub><sup>4,2</sup></td><td>Λ(Δ<sub>12</sub>, 15)</td><td>4</td><td>-0.0001258966...</td></tr><tr><td>Λ<sub>4,16</sub><sup>3,3</sup></td><td>Λ(Δ<sub>12</sub>, 15)</td><td>4</td><td>-0.0009720209...</td></tr><tr><td>Λ<sub>6,14</sub><sup>2,2</sup></td><td>Λ(Δ<sub>16</sub>, 17)</td><td>2</td><td>0.00006694828...</td></tr></tbody></table>

The ellipsis refers to $e_k$ of degree $\ge 22$, and the expansions of $e_{\Delta_{18}^{\text{ev}}}$ as well as $e_{\Delta_{2s}^{\text{even}}}, e_{\Delta_{2s}^{\text{odd}}}$ with $2s \ge 20$ start at degree $\ge 22$. We have normalised the new periods $\Lambda_{k,w}^{d_1,d_2}$ such that their products with $t^{d_1}(e_k, P_w^{d_2})$ enter the curly brackets of (3.73) with unit coefficients. The coefficients of $[\hat{z}_3, P_{14}^2]$ in our ansatz for ambiguities of $P_{20}^5$ in $e_{\Delta_{12}^{\text{ev}}}$ turned out to be a rational multiple of $\Lambda_{6,14}^{4,2}$ along with $t^4(e_6, P_{14}^2)$, hence, there was no need to extend the notation beyond $\Lambda_{k,w}^{d_1,d_2} \leftrightarrow t^{d_1}(e_k, P_w^{d_2})$. It would be interesting to investigate the origin of the ratio $-\frac{364}{15}$ in this expression for $e_{\Delta_{12}^{\text{ev}}}$.

The new period $\Lambda_{4,14}^{3,2}$ at degree $k_1+k_2+k_3 = 18$ and the new periods $\Lambda_{6,14}^{4,2}, \Lambda_{4,16}^{3,3}, \Lambda_{6,14}^{2,2}$ at $k_1+k_2+k_3 = 20$ are only well defined up to rational multiples of certain non-critical L-values. For instance, in the combination $\Lambda(\Delta_{12}, 14)P_{18}^4 + \Lambda_{4,14}^{3,2}t^3(e_4, P_{14}^2)$ entering $e_{\Delta_{12}^{\text{odd}}}$ in (3.72), redefining $P_{18}^4 \rightarrow P_{18}^4 + ct^3(e_4, P_{14}^2)$ with $c \in \mathbb{Q}$ amounts to the redefinition $\Lambda_{4,14}^{3,2} \rightarrow \Lambda_{4,14}^{3,2} + c\Lambda(\Delta_{12}, 14)$. Accordingly, we propose to assign the transcendental weight $n+1$ of the non-critical L-value $\Lambda(\Delta_{2s}, 2s+n)$ in the potential redefinition to each $\Lambda_{k,w}^{d_1,d_2}$. Table 2 summarises the ambiguity, transcendental weight and numerical value of the new periods encountered in (3.73) up to $k_1+k_2+k_3 = 20$, also see an ancillary file for representations of $\Lambda_{k,w}^{d_1,d_2}$ in terms of MMVs.

### 3.3.3 New periods in $c^{\text{sv}}$ and Laurent polynomials

Upon specialising $e_k \rightarrow \epsilon_k$, the zeta generators $\sigma_i$ provided a unified description (3.37) of the ingredients $M^{\text{sv}}(z_i)$ and $\mathbb{C}^{\text{sv}}(\epsilon_k)$ in the expression (3.22) for $\mathbb{I}^{\text{eqv}}(\epsilon_k; \tau)$ where all coefficients are $\mathbb{Q}$-linear combinations of (single-valued) MZVs. Also in the free algebra of $e_k^{(j)}$, the MZV-contributions to $c^{\text{sv}}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ at $k_1+k_2+k_3 \le 20$ are still governed by the group-like series $M^{\text{sv}}(\hat{\sigma}_i)$ in uplifted zeta generators given by (2.90) with $\hat{\sigma}_i$ in place of $z_i$. We have performed comprehensive checks up to degree 20 that the coefficients of single-valued MZVs $\zeta_{i_1}\zeta_{i_2}$ and $\rho^{-1}(\text{sv}(f_{i_1}f_{i_2}f_{i_3}))$ in $\mathbb{C}^{\text{sv}}(e_k)$ follow from the uplift $(\sigma_i, z_i) \rightarrow (\hat{\sigma}_i, \hat{z}_i)$ of (3.42). It is important to use the form of zeta generators in (2.83) and the data in appendix E.2 as well as the brackets $[\hat{z}_w, e_k]$ of appendix E.1 without adding any Pollack combinations.

However, in the free algebra of $e_k^{(j)}$, the modular completions of iterated Eisenstein integrals to several $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ at $k_1+k_2+k_3 \ge 18$ give rise to periods beyond MZVs and L-values. A first class of new numbers $\Lambda_{k,w}^{d_1,d_2}$ entering the letters $e_{\Delta^{\pm}}$ for cusp forms was discussed in the previous section. We shall here describe a second class of new periods $\varpi_{k,w}^{d_1,d_2} \in \mathbb{R}$ with similar labels $d_1, d_2, k, w \in \mathbb{N}$ to be explained below. As a defining property
of the new periods $\bar{\omega}_{k,w}^{d_1,d_2}$, they appear in the constants $c^{\text{sv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$ that enter $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$ of degree $k_1+k_2+k_3 \ge 18$ without any accompanying iterated integrals.

At the level of generating series, the new periods $\bar{\omega}_{k,w}^{d_1,d_2}$ from $c^{\text{sv}}$ enter $\mathbb{J}^{\text{eqv}}(e_k; \tau)$ on a similar footing as the MZVs in $\mathbb{M}^{\text{sv}}(\hat{\sigma}_i)$. This leads us to associate additional generators $\hat{\sigma}_{\bar{\omega}} = \hat{\sigma}_{k,w}^{d_1,d_2}$ which aim to generalise the way that $\hat{\sigma}_m$ track the appearance of $\zeta_m$ in the series $\mathbb{M}^{\text{sv}}(\hat{\sigma}_i)$. Following this analogy, the generators $\hat{\sigma}_{k,w}^{d_1,d_2}$ are taken to have arithmetic terms $\hat{z}_{\bar{\omega}} = \hat{z}_{k,w}^{d_1,d_2}$ similar to $\hat{\sigma}_i$. Generators of the type $\hat{\sigma}_{\bar{\omega}}$ have also appeared in [38] as modular elements associated with primitive periods of the moduli space $\mathcal{M}_{1,1}$ beyond MZVs, starting with non-critical L-values of holomorphic cusp forms. The modular elements of the reference have a similar decomposition into geometric and arithmetic parts.

In this setting, we propose the following uplift of (3.37) and (3.40) to the free algebra of $e_k^{(j)}$ instead of the $\epsilon_k^{(j)}$,

$$ \begin{gathered} \hat{\psi}^{\text{sv}}(\mathbb{X}) = \mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_{\bar{\omega}})^{-1} \mathbb{X} \mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_{\bar{\omega}}), \quad \mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_{\bar{\omega}})\mathbb{C}^{\text{sv}}(e_k) = \mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}}), \quad (3.74) \\ R[\mathbb{J}^{\text{eqv}}(e_k; \tau)] = U_{\text{SL}_2}(\tau)\mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_{\bar{\omega}})^{-1}\tilde{\mathbb{I}}_{-}(e_k; \tau)\mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}})\mathbb{I}_{+}(e_k; \tau)U_{\text{SL}_2}^{-1}(\tau), \end{gathered} $$

see (3.22) and (3.20) for the change of alphabet $\hat{\psi}^{\text{sv}}$. Both of

$$ \mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}}) = \mathbb{M}^{\text{sv}}(\hat{\sigma}_i) + \dots, \quad \mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_{\bar{\omega}}) = \mathbb{M}^{\text{sv}}(\hat{z}_i) + \dots \quad (3.75) $$

refer to extensions of the series (2.90) beyond MZVs starting with pairs $\bar{\omega}_{k,w}^{d_1,d_2}$ and $\bar{\omega}_{k,w}^{d_1,d_2} \hat{z}_{k,w}^{d_1,d_2}$. The first periods beyond MZVs in $\mathbb{C}^{\text{sv}}(e_k)$ and in $\mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}})$ are encountered at modular depth three and degree $k_1+k_2+k_3 = 18$ and 20. After a comprehensive study of the relevant $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$, we find two and four contributions at leading and subleading order to the uplift of $\mathbb{M}^{\text{sv}}(\hat{\sigma}_i)$ in the generating series (3.74),

$$ \begin{aligned} \mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}}) = & \mathbb{M}^{\text{sv}}(\hat{\sigma}_i) + \bar{\omega}_{4,14}^{2,2} \hat{\sigma}_{4,14}^{2,2} + \bar{\omega}_{4,14}^{4,2} \hat{\sigma}_{4,14}^{4,2} \\ & + \bar{\omega}_{6,14}^{2,2} \hat{\sigma}_{6,14}^{2,2} + \bar{\omega}_{6,14}^{4,2} \hat{\sigma}_{6,14}^{4,2} + \bar{\omega}_{6,14}^{6,2} \hat{\sigma}_{6,14}^{6,2} + \bar{\omega}_{4,16}^{3,3} \hat{\sigma}_{4,16}^{3,3} + \dots \end{aligned} \quad (3.76) $$

Just like the zeta generators, the new generators $\hat{\sigma}_{\bar{\omega}} \to \hat{\sigma}_{k,w}^{d_1,d_2}$ are expected to involve infinite series in nested brackets of $e_k^{(j)}$, starting with the following highest-weight vectors identified by our computations at modular depth three,

$$ \begin{align} \hat{\sigma}_{4,14}^{2,2} &= s^2(e_4, P_{14}^2) + \dots, & \hat{\sigma}_{4,14}^{4,2} &= s^4(e_4, P_{14}^2) + \dots, \tag{3.77} \\ \hat{\sigma}_{6,14}^{2,2} &= s^2(e_6, P_{14}^2) + \dots, & \hat{\sigma}_{6,14}^{4,2} &= s^4(e_6, P_{14}^2) + \dots, \nonumber \\ \hat{\sigma}_{6,14}^{6,2} &= s^6(e_6, P_{14}^2) + \dots, & \hat{\sigma}_{4,16}^{3,3} &= s^3(e_4, P_{16}^3) + \dots. \nonumber \end{align} $$

In both (3.76) and (3.77) the ellipses refer to terms of degree $\ge 22$ or modular depth $\ge 4$. While the notation for the new periods $\Lambda_{k,w}^{d_1,d_2}$ in the letters $e_{\Delta\pm}$ is adapted to the associated lowest-weight vectors $t^{d_1}(e_k, P_w^{d_2})$, the labels of the new periods $\bar{\omega}_{k,w}^{d_1,d_2}$ in this section refer to highest-weight vectors $s^{d_1}(e_k, P_w^{d_2})$ in (3.77) describing their lowest-order occurrence in $\mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\bar{\omega}})$ or $\mathbb{C}^{\text{sv}}(e_k)$ (see (3.74)).
An independent realisation of the six new periods in (3.76) is provided by

$$
\begin{align}
c^{\text{sv}}[\begin{smallmatrix} 2 & 2 & 8 \\ 4 & 4 & 10 \end{smallmatrix}] &= -\frac{2}{81} \rho^{-1}(2f_3 f_3 f_9 + f_3 f_9 f_3 + f_9 f_3 f_3) + \frac{1}{4041576000} \omega_{4,14}^{2,2} \nonumber \\
&\quad + \frac{26869796704014139979194459442197511}{21407683345986402107516651097196800} \zeta_{15}, \\
c^{\text{sv}}[\begin{smallmatrix} 2 & 2 & 6 \\ 4 & 4 & 10 \end{smallmatrix}] &= \frac{5}{112266} \rho^{-1}(f_3 f_7 f_3 + f_7 f_3 f_3) + \frac{1}{181870920000} \omega_{4,14}^{4,2} - \frac{12816078754315007321}{2004437855486547204570000} \zeta_{13}, \\
c^{\text{sv}}[\begin{smallmatrix} 2 & 2 & 8 \\ 4 & 4 & 12 \end{smallmatrix}] &= \frac{691}{30405375} \rho^{-1}(f_3 f_9 f_3 + f_9 f_3 f_3) + \frac{1}{5688892440000} \omega_{4,16}^{3,3} \nonumber \\
&\quad + \frac{9052954611948991353652521347483}{3794998411333953100877951785412160000} \zeta_{15}, \\
c^{\text{sv}}[\begin{smallmatrix} 4 & 4 & 6 \\ 6 & 6 & 8 \end{smallmatrix}] &= -\frac{2}{175} \rho^{-1}(2f_5 f_5 f_7 + f_5 f_7 f_5 + f_7 f_5 f_5) - \frac{1}{29106000000} \omega_{6,14}^{2,2} \nonumber \\
&\quad - \frac{957793720722761645418983400734693368126541316174567357528329392131303221923951350785293}{46407385633150935787854910109345468165086540982881530827257255703301703974682993546875} \zeta_{17}, \\
c^{\text{sv}}[\begin{smallmatrix} 4 & 4 & 4 \\ 6 & 6 & 8 \end{smallmatrix}] &= -\frac{1}{130977000000} \omega_{6,14}^{4,2} - \frac{4199}{1343304000} \zeta_{15}, \\
c^{\text{sv}}[\begin{smallmatrix} 4 & 4 & 2 \\ 6 & 6 & 8 \end{smallmatrix}] &= -\frac{1}{1102500} \rho^{-1}(f_3 f_5 f_5 + f_5 f_3 f_5) - \frac{1}{611226000000} \omega_{6,14}^{6,2} \nonumber \\
&\quad + \frac{108119242521250682513}{166672044710760167555760000} \zeta_{13},
\end{align}
$$

and we have determined both numerical values of these $\omega_{k,w}^{d_1,d_2}$ (typically to 300 digits) and analytical representations in terms of MMVs by imposing the modular properties (2.45) on the associated $\beta^{\text{eqv}}[j_1 \ j_2 \ j_3]$, see also the ancillary file for the representation of these numbers in terms of MMVs. The increasingly large rational coefficients of $\zeta_w$ are due to our choice of f-alphabet discussed in appendix A and [34].

From the $c^{\text{sv}}[j_1 \ j_2 \ j_3]$ that are expressible in terms of MZVs we can read directly that the transcendental weight is $j_1+j_2+j_3+3$. Hence, (3.78) leads us to assign transcendental weight 13 to $\omega_{4,14}^{4,2}$, $\omega_{6,14}^{6,2}$, weight 15 to $\omega_{4,14}^{2,2}$, $\omega_{6,14}^{4,2}$, $\omega_{4,16}^{3,3}$ and 17 to $\omega_{6,14}^{2,2}$. This is consistent with the total number of $e_k \ge 0$ in the associated $\hat{\sigma}_{k,w}^{d_1,d_2}$ in (3.76) and (3.77) which more generally assigns transcendental weight $k+w-d_1-d_2+1$ to $\omega_{k,w}^{d_1,d_2}$.

However, the new periods $\omega_{k,w}^{d_1,d_2}$ are only well-defined up to adding rational multiples of $\zeta_{k+w-d_1-d_2+1}$. Such redefinitions simply shift the associated zeta generator $\hat{\sigma}_{k+w-d_1-d_2+1}$ by rational multiples of the highest-weight vectors $s^{d_1}(e_k, P_w^{d_2})$ at the leading order of the new generators $\hat{\sigma}_{k,w}^{d_1,d_2}$ in (3.77). At modular depth three, these shifts by $s^{d_1}(e_k, P_w^{d_2})$ are ambiguities in the zeta generators $\hat{\sigma}_{k+w-d_1-d_2+1}$ since

(i) as highest-weight vectors, the $s^{d_1}(e_k, P_w^{d_2})$ preserve the key property $[\hat{N}, \hat{\sigma}_m] = 0$ at modular depth three, with $\hat{N}$ given by (2.80),

(ii) the Pollack combination $P_w^{d_2}$ vanishes under $e_k \to \epsilon_k$, so the image of $s^{d_1}(e_k, P_w^{d_2})$ in the Tsunogai algebra annihilates the free-Lie-algebra generators a, b of (2.74) and cannot be detected from the methods in [34].

Table 3 summarises the possible redefinition, transcendental weight and numerical value of the new periods encountered in (3.76).

Eventually, the modular completion of iterated Eisenstein integrals to $\beta^{\text{eqv}}[j_1 \ j_2 \ \dots \ j_\ell]$
at modular depth $\ell \ge 4$ should inform the general structure of the extension $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\omega) = M^{\text{sv}}(\hat{\sigma}_i) + \dots$ involving words in both zeta generators and (one or more) $\hat{\sigma}_{k,w}^{d_1,d_2}$. The wealth of new periods expected as the coefficients of such words in $\hat{\sigma}_i$ and $\hat{\sigma}_{k,w}^{d_1,d_2}$ may admit a classification into primitive ones (akin to $\zeta_k$) and 'composite' ones (akin to MZVs $\rho^{-1}(\text{sv}(f_{i_1}f_{i_2}\dots f_{i_\ell}))$
<table><caption>**Table 3.** New periods $\overline{\omega}_{k,w}^{d_1,d_2}$ encountered in the constants $c^{\text{sv}}[...]$ at $k_1+k_2+k_3 = 18, 20$ and hence in the simplest extensions $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\overline{\omega}})$ of the series in zeta generators in (3.76).</caption><thead><tr><th>new period</th><th>well defined up to</th><th>transcendental weight</th><th>approx. numerical value</th></tr></thead><tbody><tr><td>ω<sup>4,2</sup><sub>4,14</sub></td><td>ζ<sub>13</sub></td><td>13</td><td>-3.370063... × 10<sup>6</sup></td></tr><tr><td>ω<sup>2,2</sup><sub>4,14</sub></td><td>ζ<sub>15</sub></td><td>15</td><td>-5.066084... × 10<sup>9</sup></td></tr><tr><td>ω<sup>6,2</sup><sub>6,14</sub></td><td>ζ<sub>13</sub></td><td>13</td><td>-19555.45...</td></tr><tr><td>ω<sup>4,2</sup><sub>6,14</sub></td><td>ζ<sub>15</sub></td><td>15</td><td>506391.8...</td></tr><tr><td>ω<sup>3,3</sup><sub>4,16</sub></td><td>ζ<sub>15</sub></td><td>15</td><td>-8.585493... × 10<sup>7</sup></td></tr><tr><td>ω<sup>2,2</sup><sub>6,14</sub></td><td>ζ<sub>17</sub></td><td>17</td><td>-6.007268... × 10<sup>10</sup></td></tr></tbody></table>

at $\ell \ge 2$). The systematics of $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\overline{\omega}})$ is beyond the reach of the studies in this work, and the mathematical understanding of the vast system of periods and associated $(\hat{\sigma}_i, \hat{\sigma}_{\overline{\omega}})$-generators in $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\overline{\omega}})$ is expected to crucially build upon Brown's work [38].

### 3.3.4 Arithmetic parts of the new generators

On the one hand, the arithmetic parts $\hat{z}_{k,w}^{d_1,d_2}$ expected for the new generators $\hat{\sigma}_{k,w}^{d_1,d_2}$ are inaccessible to our explicit studies of $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ at modular depth $\le 3$. On the other hand, the necessity for $\hat{z}_{k,w}^{d_1,d_2}$ can be easily inferred from modular $T$ invariance at modular depth $\ge 4$. We will see in the later discussion around (4.21) that the series $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_{\overline{\omega}})$ has to commute with $\hat{N}$ in (2.80) which implies

$$0 = [\hat{N}, \hat{\sigma}_{k,w}^{d_1,d_2}] = -[e_0, \hat{\sigma}_{k,w}^{d_1,d_2}] + \sum_{\ell=4}^{\infty} (\ell-1) \text{BF}_{\ell}[e_{\ell}, \hat{\sigma}_{k,w}^{d_1,d_2}]. \quad (3.79)$$

This condition will be studied in a decomposition $\hat{\sigma}_{k,w}^{d_1,d_2} = \hat{z}_{k,w}^{d_1,d_2} + s^{d_1}(e_k, P_w^{d_2}) + \hat{\rho}_{k,w}^{d_1,d_2}$ with $\hat{\rho}_{k,w}^{d_1,d_2}$ referring to all geometric modular-depth $\ge 4$ terms in the new generator. The contribution to (3.79) at modular depth three vanishes by the highest-weight-vector condition $[e_0, s^{d_1}(e_k, P_w^{d_2})] = 0$, assuming that the expansion of $[\hat{N}, \hat{z}_{k,w}^{d_1,d_2}]$ with $\mathfrak{sl}_2$ invariant $\hat{z}_{k,w}^{d_1,d_2}$ starts at modular depth four. Validity of (3.79) at modular depth four in turn imposes

$$[e_0, \hat{\rho}_{k,w}^{d_1,d_2}] = \sum_{\ell=4}^{\infty} (\ell-1) \text{BF}_{\ell} \left( [e_{\ell}, s^{d_1}(e_k, P_w^{d_2})] + [e_{\ell}, \hat{z}_{k,w}^{d_1,d_2}] \right) + \dots, \quad (3.80)$$

where the ellipsis instructs us to discard terms of modular depth $\ge 5$. In absence of $\hat{z}_{k,w}^{d_1,d_2}$, (3.80) could only hold if $[e_{\ell}, s^{d_1}(e_k, P_w^{d_2})]$ was expressible as an $\text{ad}_{e_0}$ image for all $\ell \ge 4$.

However, as soon as the $\mathfrak{sl}_2$-module with $e_{\ell}$ as its lowest-weight vector is bigger than the $\mathfrak{sl}_2$-module with $s^{d_1}(e_k, P_w^{d_2})$ as its highest-weight vector, this is impossible.<fnref n="23" /> The reason is that $\text{ad}_{e_0}$ raises the $\text{ad}_h$-eigenvvalue determined by (2.53), and the one of $[e_{\ell}, s^{d_1}(e_k, P_w^{d_2})]$ is negative for sufficiently large $\ell$. However, the dimension of the $\text{ad}_h$-eigenspaces in the

<footnote n="23">²³Note that $P_w^{d_2}$ is the lowest-weight vector in a $(w-2d_2+1)$-dimensional $\mathfrak{sl}_2$-module and $s^{d_1}(e_k, P_w^{d_2})$ is the highest-weight vector in a $(k+w-2d_1-2d_2+3)$-dimensional $\mathfrak{sl}_2$-module.</footnote>
tensor product of the two representations grows strictly monotonically with the eigenvalue for negative eigenvalues. Therefore $\mathrm{ad}_{e_0}$ cannot be surjective in this range of $\mathrm{ad}_h$-eigenvalues and it is a quick check that $[e_\ell, s^{d_1}(e_k, P^{d_2}_w)]$ for sufficiently large $\ell$ is not in the image since $e_\ell$ is a lowest-weight vector of the bigger representation. Hence, for an infinity of values $\ell > k+w-2d_1-2d_2+4$, the arithmetic terms in $[e_\ell, \hat{z}^{d_1,d_2}_{k,w}]$ in (3.80) have to contribute non-trivially such that the right-hand side is expressible as $[e_0, \hat{\rho}^{d_1,d_2}_{k,w}]$ for a suitable choice of the modular-depth-four contributions to $\hat{\rho}^{d_1,d_2}_{k,w}$.

The $sl_2$-invariance $[e_0, \hat{z}^{d_1,d_2}_{k,w}] = 0$ of the arithmetic parts used in the above arguments will be justified when analysing the modular properties of $\mathcal{J}^{\mathrm{eqv}}(e_k; \tau)$ in section 4. As we will soon show, $SL_2$-equivariance of the expression (3.74) for $R[\mathcal{J}^{\mathrm{eqv}}(e_k; \tau)]$ hinges on

$$[e_0, \mathbb{M}^{\mathrm{sv}}(\hat{z}_i, \hat{z}_\varpi)] = [e_0^\vee, \mathbb{M}^{\mathrm{sv}}(\hat{z}_i, \hat{z}_\varpi)] = 0. \qquad (3.81)$$

Similar to the discussion of zeta generators in section 3.2.3, the arithmetic parts $\hat{z}_\varpi$ in the decomposition $\hat{\sigma}_\varpi = \hat{z}_\varpi + \sigma_\varpi^g$ (with geometric terms $\sigma_\varpi^g$ built from nested brackets of $e_k^{(j)}$) are not yet fixed by (3.81). One could still redistribute $sl_2$ invariant terms between $\hat{z}_\varpi$ and $\sigma_\varpi^g$ without altering the overall generator $\hat{\sigma}_\varpi$. By (3.74), the net effect of such redefinitions of $\hat{z}_\varpi$ is to add $\mathbb{Q}$ multiples of the associated period $\varpi$ to some of the $c^{\mathrm{sv}}$ and $\beta^{\mathrm{eqv}}$ that fall into singlet representations of $sl_2$.

In order to fix these ambiguities, we follow the logic applied to zeta generators in section 3.2.3. We single out a preferred choice of $\hat{z}_\varpi$ by imposing that $\sigma_\varpi^g$ does not contain any $sl_2$ singlet. In this way, $\hat{z}_\varpi$ captures the entire $sl_2$ singlet of $\hat{\sigma}_\varpi$, and one is led to canonically defined $c^{\mathrm{sv}}$ and $\beta^{\mathrm{eqv}}$ from the coefficients in (3.74). However, the preferred choices of $\hat{z}_i$ and $\hat{z}_\varpi$ as the $sl_2$ singlets of $\hat{\sigma}_i$ and $\hat{\sigma}_\varpi$ do not eliminate the possibility to redefine $\varpi^{d_1,d_2}_{k,w}$ by adding rational multiples of $\zeta_{k+w-d_1-d_2+1}$ which amounts to shifting $\hat{\sigma}_{k+w-d_1-d_2+1}$ by $\hat{\sigma}^{d_1,d_2}_{k,w}$.

## 3.4 Reformulation in terms of Laplace equations

At modular depth two, the space of modular invariants $\beta^{\mathrm{eqv}} \begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix}$ can be re-expressed through irreducible modular-depth-two functions and products of modular-depth-one functions [37, 74]. More specifically, the modular invariant functions satisfy $2j_1+2j_2 = k_1+k_2-4$ and can be expressed through solutions $F^{\pm(s)}_{m,k}$ to the inhomogeneous Laplace equations (with $(2m, 2k) = (k_1, k_2)$)

$$(\Delta - s(s-1))F_{m,k}^{+(s)} = E_m E_k, \qquad (3.82)$$

$$(\Delta - s(s-1))F_{m,k}^{-(s)} = \frac{(\nabla E_m)(\overline{\nabla E}_k) - (\nabla E_k)(\overline{\nabla E}_m)}{2(\mathrm{Im} \, \tau)^2}.$$

Assuming without loss of generality that $m \le k$, the spectrum of the irreducible modular-depth-two functions is given by

$$F_{m,k}^{+(s)}: \qquad s \in \{k-m+2, k-m+4, \dots, k+m-4, k+m-2\}, \\ F_{m,k}^{-(s)}: \qquad s \in \{k-m+1, k-m+3, \dots, k+m-3, k+m-1\}, \qquad (3.83)$$

with multiplicity one. The superscripts $+/-$ denote whether the functions are even/odd under the involution $\tau \mapsto -\bar{\tau}$.
We can use this basis to count the number of modular invariants functions $\beta^{\text{eqv}}\begin{bmatrix} j_1 & j_2 \\ 2m & 2k \end{bmatrix}$ at given $m$ and $k$. For $m < k$, there are $2m-1$ modular invariants among $\beta^{\text{eqv}}\begin{bmatrix} j_1 & j_2 \\ 2m & 2k \end{bmatrix}$, which ties in with the $2m-1$ irreducible functions $F_{m,k}^{\pm(s)}$ due to the counting in (3.83). Products $\nabla^\ell E_m \bar{\nabla}^\ell E_k$ or $\bar{\nabla}^\ell E_m \nabla^\ell E_k$ with $\ell < m$ in turn exhaust the symmetric combinations or shuffles

$$ \beta^{\text{eqv}}\begin{bmatrix} j_1 & j_2 \\ 2m & 2k \end{bmatrix} + \beta^{\text{eqv}}\begin{bmatrix} j_2 & j_1 \\ 2k & 2m \end{bmatrix} = \beta^{\text{eqv}}\begin{bmatrix} j_1 \\ 2m \end{bmatrix} \beta^{\text{eqv}}\begin{bmatrix} j_2 \\ 2k \end{bmatrix}, \quad (3.84) $$

and thus the reversely ordered $\beta^{\text{eqv}}\begin{bmatrix} j_2 & j_1 \\ 2k & 2m \end{bmatrix}$. Note that, up to shuffles, the functions $F_{m,k}^{\pm(s)}$ belong to $sl_2$ multiplets of dimensions $2s-1$ which also agrees with the tensor-product decomposition (2.59). For $m=k$, there are no odd functions $F_{m,m}^{-(s)}$ and the number of modular invariants reduces to $m-1$ up to shuffles which is in agreement with antisymmetric part of the tensor product (2.59).

The $sl_2$ multiplet here is defined by the number of $\nabla$ or $\bar{\nabla}$ derivatives that can be applied to $F_{m,k}^{\pm(s)}$ before the Cauchy-Riemann equation becomes a sum of products of modular-depth-one functions [37, 74]. This notion of $sl_2$ multiplets in terms of $\nabla$ and $\bar{\nabla}$ is related to the one in terms of the generators $ad_{e_0}$ and $ad_{e_0^\vee}$ introduced in section 2.3 as follows: In the holomorphic derivative $\nabla J^{\text{eqv}}(e_k; \tau)$ there is a contribution $\sim ad_{e_0} J^{\text{eqv}}(e_k; \tau)$ from the differential equation (3.33) such that the $sl_2$ raising operator $e_0$ on the generating series captures the action of $\nabla$ on its components $\beta^{\text{eqv}}$ in (3.35). There is an analogous structure in the antiholomorphic derivative $\bar{\nabla} J^{\text{eqv}}(e_k; \tau)$ with $e_0^\vee$ such that $\bar{\nabla}$ takes the role of the lowering operator of $sl_2$.<fnref n="24" /> The leftover terms in (3.33) involve factors of $G_m$ or $\Delta_m$ that are not relevant to the $sl_2$ multiplet structure and similarly for the antiholomorphic derivative. The interplay of Laplace equations, Cauchy-Riemann equations and $sl_2$ multiplets at modular depth three is further discussed in section 3.4.3 below.

By abuse of terminology, we will refer to the functions in (3.82) as Laplace eigenfunctions and $s$ as their eigenvalue, even though the equation they satisfy is inhomogeneous. Given the usefulness of the equivalent basis of Laplace eigenfunctions at modular depth two [37, 74], we will now discuss to what extent an analogous basis can be set up at modular depth three.

### 3.4.1 Counting of irreducible functions at modular depth three

In the quest for constructing similar modular invariants $F_{m,k,\ell}^{\pm(s)}$ at modular depth three, a first step is to investigate what source terms can be expected on the right-hand sides of the candidate inhomogeneous Laplace equations. Let us consider first integers $2 \le m < k < \ell$, so that all subscripts are different. Up to shuffles there are the two independent arrangements of columns in the equivariant modular-depth-three Eisenstein integrals, for instance

$$ \beta^{\text{eqv}}\begin{bmatrix} j_1 & j_2 & j_3 \\ 2m & 2k & 2\ell \end{bmatrix} \quad \text{and} \quad \beta^{\text{eqv}}\begin{bmatrix} j_2 & j_1 & j_3 \\ 2k & 2m & 2\ell \end{bmatrix}. \quad (3.85) $$

The counting of the number of modular invariants is more intricate in this case, and we begin with analysing the triple tensor product using associativity and the notation of (2.59),

$$ V(e_{2m}) \otimes V(e_{2k}) \otimes V(e_{2\ell}) = V(e_{2m}) \otimes \bigoplus_{p=\ell-k+1}^{\ell+k-1} V(e_{2p}) = \bigoplus_{p=\ell-k+1}^{\ell+k-1} \bigoplus_{q=|m-p|+1}^{m+p-1} V(e_{2q}). \quad (3.86) $$

<footnote n="24">²⁴Strictly speaking, the lowering operator on modular forms $\beta^{\text{eqv}}$ of modular weight $(0, \bar{w})$ is $y^{-\bar{w}} \bar{\nabla} y^{\bar{w}}$ as done in (3.93) with $\bar{w} = -2s$.</footnote>
Each of the $(2q-1)$-dimensional representations $V(e_{2q})$ above contains one modular invariant and, in view of (3.85), they occur with multiplicity two for the shuffle-irreducible modular-depth-three functions with $2 \le m < k < \ell$.

A main difference to the modular-depth-two system (3.82) is that there are now multiplicities involved in the tensor product, e.g. for $(m, k, \ell) = (2, 3, 5)$:

$$V(e_4) \otimes V(e_6) \otimes V(e_{10}) = V(e_4) \oplus 2 \times V(e_6) \oplus 3 \times V(e_8) \oplus 3 \times V(e_{10}) \\ \oplus 3 \times V(e_{12}) \oplus 2 \times V(e_{14}) \oplus V(e_{16}), \qquad (3.87)$$

where we recall that $V(e_{2m})$ denotes the $(2m-1)$-dimensional representation of $sl_2$ and that it has Casimir eigenvalue $m(m-1)$, see (2.57).

For $\ell \ge m+k$, the number of invariants is $2 \times (2m-1)(2k-1)$, but for lower values of $\ell$ the counting is more restricted. We also note that, at modular depth three, the one-dimensional representation $V(e_2)$ can appear in the tensor product even after having removed shuffles. The phenomenon happens for instance for $\ell = m+k-1$, and the explicit form of the $sl_2$ singlets can be obtained from the uplift $\epsilon_k^{(j)} \to e_k^{(j)}$ of the $I_{k_1,k_2,k_3}$ given by (3.52).

When some of $(m, k, \ell)$ coincide, the number of shuffle irreducibles, giving rise to new irreducible modular-depth-three functions, has to be adapted. The general pattern is

* All three of $(m, k, \ell)$ distinct. The shuffle irreducibles are given by twice the triple tensor product $V(e_{2m}) \otimes V(e_{2k}) \otimes V(e_{2\ell})$ as detailed above.

* Exactly two of $(m, k, \ell)$ are identical. The shuffle irreducibles are given by the triple tensor product $V(e_{2m}) \otimes V(e_{2k}) \otimes V(e_{2\ell})$ (i.e. without the doubling of the previous case), for instance

$$V(e_4) \otimes V(e_4) \otimes V(e_6) = V(e_2) \oplus 2 \times V(e_4) \oplus 3 \times V(e_6) \oplus 2 \times V(e_8) \oplus V(e_{10}), \qquad (3.88)$$

$$V(e_4) \otimes V(e_4) \otimes V(e_8) = V(e_4) \oplus 2 \times V(e_6) \oplus 3 \times V(e_8) \oplus 2 \times V(e_{10}) \oplus V(e_{12}),$$

$$V(e_4) \otimes V(e_6) \otimes V(e_6) = V(e_2) \oplus 3 \times V(e_4) \oplus 3 \times V(e_6) \oplus 3 \times V(e_8) \oplus 2 \times V(e_{10}) \oplus V(e_{12}),$$

where the $sl_2$ singlet $V(e_2)$ in the first line is realized through the quantity $I_{4,4,6}$ discussed below (3.52).

* All three of $(m, k, \ell)$ are the same. The shuffle irreducibles are given by half of the 'mixed symmetry projection' of the triple tensor product:

$$\frac{1}{2} [(V(e_{2m}) \otimes V(e_{2m}) \otimes V(e_{2m})) \ominus \mathrm{Sym}^3(V(e_{2m})) \ominus \mathrm{Alt}^3(V(e_{2m}))]. \qquad (3.89)$$

Here, $\mathrm{Sym}^3$ and $\mathrm{Alt}^3$ denote the fully symmetrised and antisymmetrised triple tensor product of representations. The removal of $\mathrm{Alt}^3(V(e_{2m}))$ can be understood from the vanishing of totally antisymmetrised nested brackets $[e_{2m}^{(j_1)}, [e_{2m}^{(j_2)}, e_{2m}^{(j_3)}]]$ by Jacobi identities. The factor of $1/2$ is meaningful since all representations appear with even multiplicity. In the simplest examples, (3.89) gives rise to shuffle irreducibles in

$$(m, k, \ell) = (2, 2, 2) \Rightarrow V(e_4) \oplus V(e_6), \qquad (3.90)$$

$$(m, k, \ell) = (3, 3, 3) \Rightarrow V(e_4) \oplus 2 \times V(e_6) \oplus V(e_8) \oplus V(e_{10}) \oplus V(e_{12}),$$

after for instance removing the totally antisymmetric and symmetric parts $V(e_2)$ and $V(e_8) \oplus V(e_4)$ from $V(e_4) \otimes V(e_4) \otimes V(e_4) = V(e_2) \oplus 3 \times V(e_4) \oplus 2 \times V(e_6) \oplus V(e_8)$.
### 3.4.2 Examples of Laplace equations at modular depth three

The tensor-product decomposition (3.86) also suggests the types of sources that can arise in the Laplace equations for a given eigenvalue $s$, corresponding to the various factors in the decomposition. These source terms can be constructed out of products of modular-depth-two functions with modular-depth-one functions (and up to first derivatives) as well as triple products of modular-depth-one functions.

Representative equations are

$$ (\Delta - 2)F_{2,2,2}^{+(2)} = \frac{1}{6}E_2^3 + E_2F_{2,2}^{+(2)}, \qquad (3.91) $$

$$ (\Delta - 6)F_{2,2,2}^{-(3)} = \frac{(\nabla E_2)\bar{\nabla}F_{2,2}^{+(2)} - (\bar{\nabla}E_2)\nabla F_{2,2}^{+(2)}}{2(\mathrm{Im} \, \tau)^2}, $$

$$ (\Delta - 2)F_{2,2,3}^{-(2a)} = E_2F_{2,3}^{-(2)} + \frac{(\nabla E_2)\bar{\nabla}F_{2,3}^{+(3)} - (\bar{\nabla}E_2)\nabla F_{2,3}^{+(3)}}{2(\mathrm{Im} \, \tau)^2}, $$

$$ (\Delta - 2)F_{2,2,3}^{-(2b)} = 2E_2F_{2,3}^{-(2)} + \frac{(\nabla E_3)\bar{\nabla}F_{2,2}^{+(2)} - (\bar{\nabla}E_3)\nabla F_{2,2}^{+(2)}}{2(\mathrm{Im} \, \tau)^2}, $$

$$ (\Delta - 6)F_{2,2,3}^{+(3a)} = E_2^2 E_3 + 2E_3F_{2,2}^{+(2)} + 12E_2F_{2,3}^{+(3)}, $$

$$ (\Delta - 30)F_{2,2,5}^{-(6a)} = E_2F_{2,5}^{-(6)} - \frac{(\nabla E_5)\bar{\nabla}F_{2,2}^{+(2)} - (\bar{\nabla}E_5)\nabla F_{2,2}^{+(2)}}{10(\mathrm{Im} \, \tau)^2}, $$

$$ (\Delta - 42)F_{2,2,5}^{+(7)} = 2E_2^2 E_5 + \frac{(\nabla E_2)\bar{\nabla}F_{2,5}^{-(6)} - (\bar{\nabla}E_2)\nabla F_{2,5}^{-(6)}}{2(\mathrm{Im} \, \tau)^2} - \frac{E_2(\nabla E_2)\bar{\nabla}E_5 + E_2(\bar{\nabla}E_2)\nabla E_5}{4(\mathrm{Im} \, \tau)^2}. $$

The examples above were chosen to illustrate the possible types of source terms. We will comment below what fixes the right-hand sides of these equations and the relative factors therein. When there is a multiplicity to an eigenvalue, we have made a choice of basis and labelled the corresponding eigenfunctions by $a, b, \dots$ along with the superscript $(s)$. The ancillary file contains a list of all the irreducible functions $F_{m,k,l}^{\pm(s)}$ of degree $2m + 2k + 2\ell \le 20$ in terms of $\beta^{\text{eqv}}$ along with an explanation of the choice of basis $a, b, \dots$.

### 3.4.3 sl₂ structure of the Laplace eigenfunctions

The form of the Laplace equations (3.91) at modular depth three is more complicated compared to the modular-depth-two case (3.82). The precise form of the Laplace equations is dictated by the fact that, up to shuffles, the modular invariant functions with eigenvalue $s(s-1)$ belong to an sl₂ multiplet of dimension $2s-1$. The association of $F_{m,k,l}^{\pm(s)}$ with $(2s-1)$-dimensional multiplets follows from their Cauchy-Riemann equations: By the differential equations discussed in section 3.1.5 and the comments just before section 3.4.1, the Maaß operators $\nabla$ and $\bar{\nabla}$ in (2.33) can be thought of as raising and lowering operators, dropping contributions proportional to holomorphic modular forms. While $\nabla^s F^{\pm(s)}$ and $\bar{\nabla}^s F^{\pm(s)}$ of order $s$ simplify to quantities of lower modular depth, their lower-order Cauchy-Riemann derivatives $(\nabla^\ell F^{\pm(s)})$ and $\bar{\nabla}^\ell F^{\pm(s)}$ with $\ell < s$) are still indecomposable modular-depth-three
objects. For example,

$$ \nabla^2 F_{2,2,2}^{+(2)} = \frac{1}{2} (\nabla E_2) (\nabla F_{2,2}^{+(2)}), \qquad (3.92) $$

$$ \nabla^3 F_{2,2,2}^{-(3)} = -\frac{1}{2} E_2(\nabla E_2)(\nabla^2 E_2) + \frac{1}{2}(\nabla^3 E_2)F_{2,2}^{+(2)} + 2(\nabla^2 E_2)(\nabla F_{2,2}^{+(2)}) + \frac{21\zeta_3}{200}(\nabla^3 E_3), $$

leave a triplet $(\nabla F_{2,2,2}^{+(2)}, F_{2,2,2}^{+(2)}, \bar{\nabla} F_{2,2,2}^{+(2)})$ of quantities which cannot be expressed in terms of $E_k$, $F_{m,k}^{\pm(s)}$ and their Cauchy-Riemann derivatives. In $\nabla^3 F_{2,2,2}^{-(3)}$, the occurrence of the last source term with an explicit $\zeta_3$ might appear surprising, but is in line with similar homogeneous terms arising at modular depth two [74] and it can be traced back to the term $t^2(e_4, t^3(e_4, e_4))$ in the expression (E.1) for $[\hat{z}_3, e_6]$.

The fact that the $s^{th}$ Cauchy-Riemann derivative of $F^{\pm(s)}$ can be expressed through product functions is tantamount to being outside of the $sl_2$ multiplet of irreducible functions. More generally, the interpretation of $\nabla$ and $\bar{\nabla}$ as raising and lowering operators is useful to group generic $\beta^{eqv}$ subject to the Cauchy-Riemann equations (3.35) into multiplets of $sl_2$ and to make contact with the functions $F_{m,k,\ell}^{\pm(s)}$ at leading modular depth. At depth one, for instance, (2.34) relates $E_k$ to a multiplet of dimension $2k-1$ since $(\mathrm{Im} \, \tau)^{2k} G_{2k}$ (and its complex conjugate) generate submodules of the infinite-dimensional non-unitary principal series of $SL_2(\mathbb{R})$ with spherical vector $E_k$ whose quotient space is of dimension $2k-1$, see for instance [88]. At higher modular depth, one finds submodules spanned by shuffles, i.e. the functions appearing on the right-hand sides of (3.92) are not lowest-weight vectors of $sl_2$ but they do not yield any shuffle irreducible functions at modular depth three under the action with a lowering operator $\bar{\nabla}$.

Equipped with this understanding of the $sl_2$-multiplets at higher modular depth, we can now explain how the interplay between the Laplace equation and the Cauchy-Riemann equation fixes the right-hand sides in (3.91). The fact that the $s^{th}$ Cauchy-Riemann derivative must be expressible through (products of) derivatives of functions of lower modular depth leads to an ansatz for $\nabla^s F_{m,k,\ell}^{\pm(s)}$ in terms of known objects by selecting objects built from the constituent $sl_2$ representations as well as homogeneous terms (such as the last term in the second line of (3.92)). Furthermore, the identity

$$ \pi \bar{\nabla} (y^{-2s} (\pi \nabla)^s F_{m,k,\ell}^{\pm(s)}) = y^{-2(s-1)} (\pi \nabla)^{s-1} (\Delta - s(s-1)) F_{m,k,\ell}^{\pm(s)}, \qquad (3.93) $$

imposes a non-trivial condition on the ansatz for the Laplace equation of $F_{m,k,\ell}^{\pm(s)}$. Solving this equation constrains the Laplace equation and the Cauchy-Riemann equation at the same time and leads to solution spaces of the dimension discussed after (3.87).<fnref n="25" /> In particular, this fixes the linear combination $\frac{1}{6}E_2^3 + E_2 F_{2,2}^{+(2)}$ on the right-hand side of the first equation in (3.91) since for no other linear combination of these terms (excluding an overall scaling) can one find a Cauchy-Riemann equation — corresponding to a triplet of $sl_2$ — that is consistent with the Laplace equation in the sense of (3.93).

As a final comment, we emphasise that it is possible to obtain modular invariant solutions to inhomogeneous Laplace eigenvalue problems more general than (3.82) and (3.91). In

<footnote n="25">²⁵Both ansätze can be simplified by considering the effect of redefining $F_{m,k,\ell}^{\pm(s)}$ itself by products of functions of lower modular depth.</footnote>
particular, using spectral analysis for square-integrable functions on the upper half-plane quotiented by SL(2, $\mathbb{Z}$) [89], we can construct solutions to (3.82) whose eigenvalues lie outside of the spectrum (3.83) or possess source terms given by $E_m E_k$ with $m, k \in \mathbb{C}$. Particularly important is the case where the source terms involve half-integral non-holomorphic Eisenstein series, see e.g. [90–92]. Although these more general inhomogeneous Laplace equations do give rise to well-defined modular invariant functions, these objects will not be expressible in terms of the equivariant iterated integrals considered here since their Cauchy-Riemann equations will not have the proper factorised form discussed above.

### 3.4.4 Laplace equations modulo lower modular depth

As another method to determine the admissible source terms in (3.91), one can evaluate the Laplacian of the modular-depth-three contributions to modular invariant $\beta^{\text{eqv}} \begin{bmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{bmatrix}$ in closed form: Based on the first-order equations (2.27) and (2.28) of $\beta_{\pm}$ in the terms (2.44) of leading modular depth, it is not hard to derive (see section 3.1 of [74])

$$ \begin{aligned} \Delta \beta^{\text{eqv}} \begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix} \bigg|_{j_1+j_2=\frac{1}{2}(k_1+k_2-4)} = & ((k_1-j_1-2)(j_1+1) + (k_2-j_2-2)(j_2+1)) \beta^{\text{eqv}} \begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix} \\ & + j_1(k_2-j_2-2)\beta^{\text{eqv}} \begin{bmatrix} j_1-1 & j_2+1 \\ k_1 & k_2 \end{bmatrix} + j_2(k_1-j_1-2)\beta^{\text{eqv}} \begin{bmatrix} j_1+1 & j_2-1 \\ k_1 & k_2 \end{bmatrix} \\ & - \delta_{j_2,k_2-2}(\tau-\bar{\tau})^{k_2} G_{k_2} j_1 \beta^{\text{eqv}} \begin{bmatrix} j_1-1 \\ k_1 \end{bmatrix} - \delta_{j_1,0} \frac{\overline{G_{k_1}}}{(2\pi i)^{k_1}} (k_2-j_2-2)\beta^{\text{eqv}} \begin{bmatrix} j_2+1 \\ k_2 \end{bmatrix} \\ & + \delta_{j_1,0} \delta_{j_2,k_2-2}(\tau-\bar{\tau})^{k_2} G_{k_2} \frac{\overline{G_{k_1}}}{(2\pi i)^{k_1}} \mod \text{modular depth} \le 1, \end{aligned} \quad (3.94) $$

and its generalisation to modular depth three,

$$ \begin{aligned} \Delta \beta^{\text{eqv}} \begin{bmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{bmatrix} \bigg|_{j_1+j_2+j_3=\frac{1}{2}(k_1+k_2+k_3-6)} = & \left( \sum_{i=1}^{3} (k_i-j_i-2)(j_i+1) \right) \beta^{\text{eqv}} \begin{bmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{bmatrix} \\ & + j_1(k_2-j_2-2)\beta^{\text{eqv}} \begin{bmatrix} j_1-1 & j_2+1 & j_3 \\ k_1 & k_2 & k_3 \end{bmatrix} + j_2(k_1-j_1-2)\beta^{\text{eqv}} \begin{bmatrix} j_1+1 & j_2-1 & j_3 \\ k_1 & k_2 & k_3 \end{bmatrix} \\ & + j_1(k_3-j_3-2)\beta^{\text{eqv}} \begin{bmatrix} j_1-1 & j_2 & j_3+1 \\ k_1 & k_2 & k_3 \end{bmatrix} + j_3(k_1-j_1-2)\beta^{\text{eqv}} \begin{bmatrix} j_1+1 & j_2 & j_3-1 \\ k_1 & k_2 & k_3 \end{bmatrix} \\ & + j_2(k_3-j_3-2)\beta^{\text{eqv}} \begin{bmatrix} j_1 & j_2-1 & j_3+1 \\ k_1 & k_2 & k_3 \end{bmatrix} + j_3(k_2-j_2-2)\beta^{\text{eqv}} \begin{bmatrix} j_1 & j_2+1 & j_3-1 \\ k_1 & k_2 & k_3 \end{bmatrix} \\ & - \delta_{j_3,k_3-2}(\tau-\bar{\tau})^{k_3} G_{k_3} (j_1\beta^{\text{eqv}} \begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix} + j_2\beta^{\text{eqv}} \begin{bmatrix} j_1 & j_2-1 \\ k_1 & k_2 \end{bmatrix}) \\ & - \delta_{j_1,0} \frac{\overline{G_{k_1}}}{(2\pi i)^{k_1}} ((k_2-j_2-2)\beta^{\text{eqv}} \begin{bmatrix} j_2+1 & j_3 \\ k_2 & k_3 \end{bmatrix} + (k_3-j_3-2)\beta^{\text{eqv}} \begin{bmatrix} j_2 & j_3+1 \\ k_2 & k_3 \end{bmatrix}) \\ & + \delta_{j_1,0} \delta_{j_3,k_3-2}(\tau-\bar{\tau})^{k_3} G_{k_3} \frac{\overline{G_{k_1}}}{(2\pi i)^{k_1}} \beta^{\text{eqv}} \begin{bmatrix} j_2 \\ k_2 \end{bmatrix} \mod \text{modular depth} \le 2. \end{aligned} \quad (3.95) $$

In both of (3.94) and (3.95), the disclaimer mod modular depth $\le \ell$ indicates that $\beta_{\pm}$ of modular depth $\le \ell$ are discarded. One can easily check via (3.95) as well as $E_2 = -6\beta^{\text{eqv}} \begin{bmatrix} 1 \\ 4 \end{bmatrix}$ and $F_{2,2}^{+(2)} = 18\beta^{\text{eqv}} \begin{bmatrix} 2 & 0 \\ 4 & 4 \end{bmatrix}$ that the expressions

$$ \begin{aligned} F_{2,2,2}^{+(2)} &= -54\beta^{\text{eqv}} \begin{bmatrix} 2 & 1 & 0 \\ 4 & 4 & 4 \end{bmatrix}, & (3.96) \\ F_{2,2,2}^{-(3)} &= 54(\beta^{\text{eqv}} \begin{bmatrix} 1 & 2 & 0 \\ 4 & 4 & 4 \end{bmatrix} - \beta^{\text{eqv}} \begin{bmatrix} 2 & 0 & 1 \\ 4 & 4 & 4 \end{bmatrix}) - \frac{63}{20}\zeta_3 \beta^{\text{eqv}} \begin{bmatrix} 2 \\ 6 \end{bmatrix}, & \end{aligned} $$
are consistent with the first two Laplace equations in (3.91) (the last contribution ~ $\zeta_3\beta^{\text{eqv}}[\begin{matrix} 2 \\ 6 \end{matrix}]$ can be fixed from the Cauchy-Riemann equation (3.92)). Note that the even modular function in (3.96) is related to the combination $E_{2,2,2} = -C_{2,2,1,1} + \dots$ of MGFs (with a three-loop MGF $C_{2,2,1,1} = (\frac{\text{Im}\tau}{\pi})^6 \sum_{p_1,p_2,p_3,p_4 \neq 0} \frac{\delta(p_1+p_2+p_3+p_4)}{|p_1|^4|p_2|^4|p_3|^2|p_4|^2}$ and graphs involving $\le 2$ loops in the ellipsis) via $F_{2,2,2}^{+(2)} = \frac{1}{4}E_{2,2,2}$, see [9, 30] for different integral representations of $E_{2,2,2}$.

Similar to the construction of $F_{m,k}^{\pm(s)}$ in [74], the guiding principle for $\beta^{\text{eqv}}[\begin{matrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{matrix}]$-representations of $F_{m,k,\ell}^{\pm(s)}$ is to minimize the value of $j_3$ and to maximize the value of $j_1$ while preserving the modular-invariance condition $j_1+j_2+j_3 = \frac{1}{2}(k_1+k_2+k_3-6)$. In this way, the occurrence of holomorphic Eisenstein series $G_{k_3}$ and $\overline{G}_{k_1}$ in Laplace or Cauchy-Riemann equations is delayed, and the source terms in the Laplace equations involve at most one $\nabla, \bar{\nabla}$-derivative of $E_s$ or $F_{m,k}^{\pm(s)}$. By the same reasoning, $F_{2,2}^{+(2)}$ is constructed from a single-term $\beta^{\text{eqv}}[\begin{matrix} 2 & 0 \\ 4 & 4 \end{matrix}]$ and not from the combination $\frac{1}{2}(\beta^{\text{eqv}}[\begin{matrix} 2 & 0 \\ 4 & 4 \end{matrix}] - \beta^{\text{eqv}}[\begin{matrix} 0 & 2 \\ 4 & 4 \end{matrix}])$ which is annihilated by $\nabla^2$ modulo terms involving $G_4$.

We reiterate that, thinking of $\nabla$ and $\bar{\nabla}$ as raising and lowering operators and dropping contributions proportional to (anti-)holomorphic modular forms, the differential equation (3.35) can be used to group the $\beta^{\text{eqv}}$ into multiplets of $sl_2$. These multiplets are useful for relating the functions $F_{m,k,\ell}^{\pm(s)}$ to the $\beta^{\text{eqv}}$ at leading modular depth.

Note that the involution $\tau \mapsto -\bar{\tau}$ that maps $F_{m,k}^{\pm(s)} \mapsto \pm F_{m,k}^{\pm(s)}$ and $F_{m,k,\ell}^{\pm(s)} \mapsto \pm F_{m,k,\ell}^{\pm(s)}$ acts on the leading-depth contributions to the modular forms $\beta^{\text{eqv}}$ via

$$ \beta^{\text{eqv}} \begin{bmatrix} j_1 & j_2 & \dots & j_\ell \\ k_1 & k_2 & \dots & k_\ell \end{bmatrix} \Big|_{\tau \mapsto -\bar{\tau}} = (4y)^{2\ell+2}(j_1+j_2+\dots+j_\ell) - k_1 - k_2 - \dots - k_\ell \quad (3.97) \\ \times \beta^{\text{eqv}} \begin{bmatrix} k_\ell - j_\ell - 2 & \dots & k_2 - j_2 - 2 & k_1 - j_1 - 2 \\ & \ddots & & \\ & & k_\ell & k_1 - j_1 - 2 \end{bmatrix} \quad \text{mod modular depth} \le \ell-1, $$

which explains the two-term combination in the second line of (3.96).

### 3.4.5 New periods in Laurent polynomials

The new periods $\varpi_{k,w}^{d_1,d_2}$ of section 3.3.3 appear in the Laurent polynomials of the irreducible modular-depth-three functions $F_{m,k,\ell}^{+(s)}$ as coefficients of the term $y^{1-s}$ in the Laurent polynomial. This homogeneous solution to the Laplace equation is the only power where periods different from those already contained in the source terms on the right-hand sides of the Laplace equation can arise. The coefficients of this homogeneous solution $y^{1-s}$ cannot be fixed from the differential equation but are fixed by modularity.

Examples covering all six new periods listed in table 3 are

$$ F_{2,2,5}^{+(5c)}|_{\text{LP},y^{-4}} = \left( \frac{\varpi_{4,14}^{4,2}}{41472000} + \frac{235431\zeta_{13}}{1408000} \right) y^{-4}, $$

$$ F_{2,2,5}^{+(7)}|_{\text{LP},y^{-6}} = \left( -\frac{\varpi_{4,14}^{2,2}}{221184000} + \frac{35}{512}\rho^{-1}(\text{sv}(f_3f_3f_9)) - \frac{161}{2048}\rho^{-1}(\text{sv}(f_3f_9f_3)) \right. \\ \left. - \frac{26869796704014139979194459442197511}{1171581836689117404633480344371200}\zeta_{15} \right) y^{-6}, $$

$$ F_{2,2,6}^{+(6a)}|_{\text{LP},y^{-5}} = \left( -\frac{\varpi_{4,16}^{3,3}}{116751360000} + \frac{6973601\zeta_{15}}{31841280000} \right) y^{-5}, $$
$$
\begin{align}
F_{3,3,4}^{+(8)}|_{\text{LP},y^{-7}} &= \left( \frac{13\varpi_{6,14}^{2,2}}{1032192000} + \frac{125}{2048}\rho^{-1}(\text{sv}(f_5f_5f_7)) - \frac{19}{2048}\rho^{-1}(\text{sv}(f_5f_7f_5)) \right. \nonumber \\
&\qquad \left. + \frac{12451318369395901390446784209551013785645037110269375647868282097706941885011367560208809}{16457545589037769089788200156526323602094759468907597425839456221707686535091042560000} \zeta_{17} \right) y^{-7}, \\
F_{2,3,5}^{+(6e)}|_{\text{LP},y^{-5}} &= \left( \frac{13\varpi_{6,14}^{4,2}}{1013760000} - \frac{615627961\zeta_{15}}{11208130560} \right) y^{-5}, \\
F_{2,3,5}^{+(4e)}|_{\text{LP},y^{-3}} &= \left( -\frac{\varpi_{6,14}^{6,2}}{8064000} + \frac{21240581\zeta_{13}}{1341204480} \right) y^{-3}, \tag{3.98}
\end{align}
$$

where the letters in the superscripts of $F_{2,2,5}^{+(5c)}$ and similar functions are again due to the non-trivial multiplicities of the relevant Laplace eigenspaces. In these equations we have restricted ourselves to writing the term in the Laurent polynomials which corresponds to the homogeneous solution $y^{1-s}$, as all remaining terms can be uniquely fixed from the Laplace equations (3.91).

Similarly, indecomposable single-valued MZVs beyond depth one firstly appear through contributions of the homogeneous solution $y^{1-s}$ to Laurent polynomials. For instance, the Laurent polynomial of the modular invariant $F_{2,2,3}^{+(5)}$ in the largest $sl_2$-module $V(e_{10})$ of the relevant tensor product in the first line of (3.88) has a contribution $\sim \zeta_{3,3,5}^{\text{sv}}/y^4$.

Finally, although as stressed the coefficient of the homogeneous term $y^{1-s}$ cannot be fixed directly from the Laplace equation, this can nonetheless be reconstructed by analysing the behaviour as $y \to 0$ of the series of contributions $(q\bar{q})^n = \exp(-4\pi ny)$, with $n \in \mathbb{N}$, exponentially suppressed at the cusp $y \gg 1$. This fact is a consequence of modularity which deeply intertwines the two expansion regimes $y \gg 1$ and $y \to 0$ for any modular invariant function. In [93, 94], this phenomenon was manifested for the modular-depth-two functions $F_{m,k}^{+(s)}$ defined in (3.82) by exploiting a Mellin transform argument and spectral theory. It would be extremely interesting to understand how these new periods appearing in the Laurent polynomials of $F_{m,k,l}^{\pm(s)}$ can be recovered from the small-$y$ behaviour of the series of contributions $(q\bar{q})^n$.

### 3.4.6 New periods multiplying cusp forms

In the same way that the Laurent polynomials contain the new periods of table 3, the new periods $\Lambda_{k,w}^{d_1,d_2}$ of section 3.3.2 and table 2 appear as the coefficients of iterated integrals of cusp forms. For the irreducible modular-depth-three functions this can happen when the eigenvalue parameter $s$ is half the modular weight of a holomorphic cusp form, $\Delta_{2s}$, since in this case the accompanying $\beta_{\pm}[\Delta_{2s}^{s-1}]$ are solutions of the homogeneous Laplace equation.

Four explicit examples covering all four new periods of table 2 are

$$
\begin{align*}
F_{2,2,5}^{-(6)}|_{\text{mod depth } 1} &= \left( -\frac{8599\Lambda(\Delta_{12}, 14)}{10945440\Lambda(\Delta_{12}, 10)} - \frac{\Lambda_{4,14}^{3,2}}{\Lambda(\Delta_{12}, 10)} \right) \frac{\beta_+ [\Delta_{12}^5] - \beta_- [\Delta_{12}^5]}{90000} + \dots, \\
F_{2,2,6}^{+(6a)}|_{\text{mod depth } 1} &= \left( \frac{13934837129\Lambda(\Delta_{12}, 15)}{1082489265000\Lambda(\Delta_{12}, 11)} + \frac{\Lambda_{4,16}^{3,3}}{\Lambda(\Delta_{12}, 11)} \right) \frac{\beta_+ [\Delta_{12}^5] + \beta_- [\Delta_{12}^5]}{10365000} + \dots,
\end{align*}
$$
$$
\begin{align}
F_{2,3,5}^{+(8)}|_{\text{mod depth } 1}^{\Delta_{2s}} &= -\frac{13\Lambda_{6,14}^{2,2}}{12600\Lambda(\Delta_{16}, 15)} (\beta_+[\Delta_{16}^7]+\beta_-[\Delta_{16}^7])+\dots, \\
F_{2,3,5}^{+(6)}|_{\text{mod depth } 1}^{\Delta_{2s}} &= \left(\frac{13929\Lambda(\Delta_{12}, 15)}{425656\Lambda(\Delta_{12}, 11)} + \frac{13\Lambda_{6,14}^{4,2}}{\Lambda(\Delta_{12}, 11)}\right) \frac{\beta_+[\Delta_{12}^5]+\beta_-[\Delta_{12}^5]}{90000} +\dots \tag{3.99}
\end{align}
$$

The notation $|_{\text{mod depth } 1}^{\Delta_{2s}}$ indicates that we are only selecting the cuspidal contributions at modular depth one, while mixed integrals $\beta_{\pm}[\Delta_{2s}^{j_1 j_2}]$, $\beta_{\pm}[\Delta_{2s}^{j_1 j_2}]$ or $\beta_{\mp}[\Delta_{2s}^{j_2}]$ are not tracked on the right-hand sides.

4 Modular properties

In this section we show that the generating series J$^{eqv}$ introduced in (3.74) indeed gener-
ates equivariant iterated integrals. Given the modular properties (2.45) satisfied by $\beta^{eqv}$,
equivariance can be recast at the level of the generating series (3.74) as

$$
R[\mathcal{J}^{\text{eqv}}(e_k; \tau+1)] = R[\mathcal{J}^{\text{eqv}}(e_k; \tau)], \quad (4.1)
$$

$$
R[\mathcal{J}^{\text{eqv}}(e_k; -\frac{1}{\tau})] = \bar{\tau}^{-[e_0, e_0^{\vee}]} R[\mathcal{J}^{\text{eqv}}(e_k; \tau)] \bar{\tau}^{[e_0, e_0^{\vee}]}. \quad (4.2)
$$

This can be easily seen by realising that the desired modular properties (2.45) imply invariance
under T-transformation, $T \cdot \tau := \tau+1$, of the corresponding generating series. While under
S-transformation, $S \cdot \tau := -1/\tau$, we note that given a word $P = [j_1 \dots j_\ell]$
$[k_1 \dots k_\ell]$ equation (2.45)
is equivalent to

$$
R(e[P]\beta^{\text{eqv}}[P; -\frac{1}{\tau}]) = \bar{\tau}^{-[e_0, e_0^{\vee}]} R(e[P]\beta^{\text{eqv}}[P; \tau]) \bar{\tau}^{[e_0, e_0^{\vee}]}, \quad (4.3)
$$

where we used the definition (2.85), adapted to free-algebra generators

$$
e[P] := \left( \prod_{i=1}^{\ell} \frac{(-1)^{j_i} (k_i - 1)}{(k_i - j_i - 2)!} \right) e_{k_{\ell}}^{(k_{\ell} - 2 - j_{\ell})} \cdots e_{k_2}^{(k_2 - 2 - j_2)} e_{k_1}^{(k_1 - 2 - j_1)}, \quad (4.4)
$$

combined with the *R*-operation defined in (3.12), and the following eigenvalues under the
SL₂ Cartan element [*e*₀, *e*₀<sup>√</sup>]<sub>*</sub>

$$
[e_0, e_0^\vee] e_k^{(j)} = (2+2j-k) e_k^{(j)} \Rightarrow [e_0, e_0^\vee] e \left[ \begin{smallmatrix} j_1 & \cdots & j_\ell \\ k_1 & \cdots & k_\ell \end{smallmatrix} \right] = \sum_{i=1}^\ell (2+2j_i-k_i) e \left[ \begin{smallmatrix} j_1 & \cdots & j_\ell \\ k_1 & \cdots & k_\ell \end{smallmatrix} \right]. \quad (4.5)
$$

Note that for the rest of the section we will consider general words *P*, where the entries can
be both Eisenstein series and holomorphic cusp forms. For this reason we keep the derivations
*e*<sub>*k*</sub> and *e*<sub>*Δ±*</sub> as independent symbols for part of the computation and eventually substitute
the relations (3.73) expressing *e*<sub>*Δ±*</sub> in terms of *e*<sub>*k*</sub>. Along the road, we will comment on
the key differences between this more general case and the special case where we restrict
to Tsunogai’s derivations, i.e. *e*<sub>*k*</sub> → *ε*<sub>*k*</sub> and *e*<sub>*Δ±*</sub> → 0.

The remainder of this section is devoted to showing that the right-hand side of (3.74)
satisfies the equivariant properties (4.1). To this end let us show separately *T*-equivariance
and *S*-equivariance.
## 4.1 T-equivariance

We want to show that the right-hand side of (3.74) is equivariant under T-transformation $T \cdot \tau = \tau+1$. Given the definition (3.1) of the $SL_2$ transformation $U_{SL_2}(\tau)$, it is immediate to conclude that

$$ U_{SL_2}(\tau+1) = U_{SL_2}(\tau)e^{2\pi i e_0}. \qquad (4.6) $$

Let us now turn our attention towards obtaining the T-transformation for the generating series $\mathbb{I}_{\pm}$ using their representations via path-ordered exponentials (3.6) constructed using the (1,0)-form $\mathbb{A}_{+}(e_k; \tau_1)$ and the (0,1)-form $\mathbb{A}_{-}(e_k; \tau_1)$, respectively.

Starting with $\mathbb{I}_{+}$, a simple application of the T-transformation (2.5) for the $\nu$-kernels combined with the $sl_2$ relations (2.53) shows that $\mathbb{A}_{+}(e_k; \tau_1)$ defined in (3.2) transforms as

$$ \mathbb{A}_{+}(e_k; \tau_1+1) = e^{-2\pi i e_0} \mathbb{A}_{+}(e_k; \tau_1) e^{2\pi i e_0}. \qquad (4.7) $$

The path-ordered exponential then transforms both through the T-action (4.7) on the integrand and through the transformation of the integration domain

$$ \begin{aligned} \mathbb{I}_{+}(e_k; \tau+1) &= \mathrm{P}\text{-exp}\left(\int_{\tau+1}^{i\infty} \mathbb{A}_{+}(e_k; \tau_1)\right) = \mathrm{P}\text{-exp}\left(\int_{\tau}^{i\infty-1} \mathbb{A}_{+}(e_k; \tau_1+1)\right) \quad (4.8) \\ &= \mathrm{P}\text{-exp}\left(\int_{i\infty}^{i\infty-1} \mathbb{A}_{+}(e_k; \tau_1+1)\right) \mathrm{P}\text{-exp}\left(\int_{\tau}^{i\infty} \mathbb{A}_{+}(e_k; \tau_1+1)\right) \\ &= e^{-2\pi i e_0} \mathrm{P}\text{-exp}\left(\int_{i\infty}^{i\infty-1} \mathbb{A}_{+}(e_k; \tau_1)\right) \mathbb{I}_{+}(e_k; \tau) e^{2\pi i e_0} \\ &= e^{-2\pi i e_0} \mathbb{I}_{+}(e_k)^{-1} \mathbb{I}_{+}(e_k; \tau) e^{2\pi i e_0}. \end{aligned} $$

It remains to simplify the $\tau$-independent path-ordered exponential

$$ \mathbb{T}_{+}(e_k) := \mathrm{P}\text{-exp}\left(\int_{i\infty-1}^{i\infty} \mathbb{A}_{+}(e_k; \tau_1)\right), \qquad (4.9) $$

i.e. the T-cocycle at infinity. This integral can be evaluated via tangential-basepoint regularisation (see in particular section 6 of [11]) effectively selecting contributions only from Eisenstein series and in particular from their zero-mode $G_k(\tau) = 2\zeta_k + O(e^{2\pi i \tau})$. This means that to evaluate the T-cocycle at infinity we can just focus on the zero-mode part of $\mathbb{A}_{+}(e_k; \tau_1)$ which, using the identity $2\zeta_k = -\mathrm{BF}_k(2\pi i)^k$ with $\mathrm{BF}_k = B_k/k!$, reduces to

$$ \mathbb{A}_{+}^{i\infty}(e_k; \tau_1) = -2\pi i \sum_{k=4}^{\infty} (k-1) \mathrm{BF}_k \sum_{j=0}^{k-2} \frac{1}{j!} (-2\pi i \tau_1)^j e_k^{(j)}. \qquad (4.10) $$

The inverse T-cocycle at infinity can then be computed from the zero modes as

$$ \begin{aligned} \mathbb{T}_{+}(e_k)^{-1} &= \mathrm{P}\text{-exp}\left(\int_{i\infty}^{i\infty-1} \mathbb{A}_{+}^{i\infty}(e_k; \tau_1)\right) \qquad (4.11) \\ &= 1 + \sum_{r=1}^{\infty} \sum_{j_1, j_2, \dots, j_r=0}^{\infty} \left(\prod_{i=1}^{r} \frac{(2\pi i)^{j_i+1}}{j_i! \sum_{m=i}^{r} (j_m+1)}\right) \hat{N}_{+}^{(j_1)} \hat{N}_{+}^{(j_2)} \dots \hat{N}_{+}^{(j_r)}, \end{aligned} $$
where we introduced the variant

$$ \hat{N}_{+} := \sum_{k=4}^{\infty} (k-1) \mathrm{BF}_k e_k, \qquad \hat{N} = \hat{N}_{+} - e_0, \qquad (4.12) $$

of the generator $\hat{N}$ in (2.80) subject to $[\hat{N}, \hat{\sigma}_w] = 0$.

It is particularly useful to express the T-cocycle at infinity via (4.11) as it allows us to prove the following lemma which should be equivalent to the results in section 6 of [11].

**Lemma 2** The expression (4.11) for the inverse T-cocycle at infinity admits the generating-series representation

$$ \mathbb{T}_+(e_k)^{-1} = e^{2\pi i e_0} e^{2\pi i (\hat{N}_+-e_0)} = e^{2\pi i e_0} e^{2\pi i \hat{N}}. \qquad (4.13) $$

The proof of this lemma can be found in appendix D.

Finally, by combining (4.8) with (4.13) we arrive at the modular T-transformation

$$ \mathbb{I}_+(e_k; \tau+1) = e^{2\pi i \hat{N}} \mathbb{I}_+(e_k; \tau) e^{2\pi i e_0}. \qquad (4.14) $$

Note that, since cusp forms do not contribute to the $\tau_1 \to i\infty$ regime in (4.10), the T-cocycle at infinity (4.11) takes the same form upon restriction to Eisenstein series and Tsunogai's derivations, i.e. $e_k \to \epsilon_k$, $e_{\Delta\pm} \to 0$. Hence the same modular T-transformation (4.14) holds for $\mathbb{I}_+(\epsilon_k; \tau)$, i.e.

$$ \mathbb{I}_+(\epsilon_k; \tau+1) = e^{2\pi i N} \mathbb{I}_+(\epsilon_k; \tau) e^{2\pi i e_0}, \qquad (4.15) $$

where $N$ in (2.78) is the specialisation of $\hat{N}$ to $e_k \to \epsilon_k$.

A similar argument can be repeated for $\tilde{\mathbb{I}}_-$ starting from the path-ordered exponential representation (3.6) in terms of the (0, 1)-form $\mathbb{A}_-(e_k; \tau_1)$ defined in (3.3). Given the immediate analogue of (4.7),

$$ \mathbb{A}_-(e_k; \tau_1+1) = e^{-2\pi i e_0} \mathbb{A}_-(e_k; \tau_1) e^{2\pi i e_0}, \qquad (4.16) $$

the steps in (4.8) can be adapted with the reversal prescription to arrive at

$$ \tilde{\mathbb{I}}_-(e_k; \tau+1) = e^{-2\pi i e_0} \tilde{\mathbb{I}}_-(e_k; \tau) \tilde{\mathrm{P}}\text{-exp} \left( \int_{-i\infty}^{-i\infty-1} \mathbb{A}_-(e_k; \tau_1) \right) e^{2\pi i e_0}. \qquad (4.17) $$

A computation similar to (4.11) yields the inverse T-cocycle at infinity

$$ \tilde{\mathbb{T}}_-(e_k)^{-1} := \tilde{\mathrm{P}}\text{-exp} \left( \int_{-i\infty}^{-i\infty-1} \mathbb{A}_-(e_k; \tau_1) \right) = e^{-2\pi i (\hat{N}_+-e_0)} e^{-2\pi i e_0} = e^{-2\pi i \hat{N}} e^{-2\pi i e_0}. \qquad (4.18) $$

Inserting this expression in (4.17), we conclude that

$$ \tilde{\mathbb{I}}_-(e_k; \tau+1) = e^{-2\pi i e_0} \tilde{\mathbb{I}}_-(e_k; \tau) e^{-2\pi i \hat{N}}. \qquad (4.19) $$

The same modular T-transformation with $e_0 \to \epsilon_0$ and $\hat{N} \to N$ applies to the generating series of Tsunogai's derivations, i.e. $e_k \to \epsilon_k$, $e_{\Delta\pm} \to 0$, by the arguments below (4.14).
We are now in position of computing the T-transformation of (3.74). By inserting the T-transformations (4.6), (4.14) and (4.19) into that of $R[J^{\text{eqv}}]$ in (3.74), it follows that

$$ 
\begin{aligned}
R[J^{\text{eqv}}(e_k; \tau+1)] &= U_{\text{SL}_2}(\tau+1)\mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_\varpi)^{-1}\tilde{\mathbb{I}}_{-}(e_k; \tau+1)\mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\varpi)\mathbb{I}_{+}(e_k; \tau+1)U_{\text{SL}_2}^{-1}(\tau+1) \\
&= U_{\text{SL}_2}(\tau)e^{2\pi i e_0}\mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_\varpi)^{-1}e^{-2\pi i e_0}\tilde{\mathbb{I}}_{-}(e_k; \tau)e^{-2\pi i \hat{N}}\mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\varpi)e^{2\pi i \hat{N}}\mathbb{I}_{+}(e_k; \tau)U_{\text{SL}_2}^{-1}(\tau),
\end{aligned} 
\quad (4.20) $$

which manifests that T-invariance of $J^{\text{eqv}}(e_k; \tau)$ is ensured by the following conditions:

$$ e^{2\pi i e_0} \mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_\varpi)^{-1} e^{-2\pi i e_0} = \mathbb{M}^{\text{sv}}(\hat{z}_i, \hat{z}_\varpi)^{-1}, \quad e^{-2\pi i \hat{N}} \mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\varpi) e^{2\pi i \hat{N}} = \mathbb{M}^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\varpi). \quad (4.21) $$

The first equation is implied by imposing SL₂-invariance of $\hat{z}_i, \hat{z}_\varpi$ as already anticipated in (3.81). The second equation of (4.21) in turn is satisfied if $[\hat{N}, \hat{\sigma}_i] = [\hat{N}, \hat{\sigma}_\varpi] = 0$.

We conclude by noting again that the above relations are still valid after we impose Tsunogai's relations, i.e. even after having substituted $e_k \to \epsilon_k$ and having set to zero all cusp form contributions, $e_{\Delta\pm} \to 0$ and $\hat{\sigma}_\varpi, \hat{z}_\varpi \to 0$. In case of the single-valued series $R[J^{\text{sv}}(\epsilon_k; \tau)]$ in (3.55), however, the analogue of (4.20)

$$ R[J^{\text{sv}}(\epsilon_k; \tau+1)] = U_{\text{SL}_2}(\tau)e^{2\pi i \epsilon_0} \mathbb{M}^{\text{sv}}(\sigma_i)^{-1} e^{-2\pi i \epsilon_0} \tilde{\mathbb{I}}_{-}(\epsilon_k; \tau) e^{-2\pi i N} \mathbb{M}^{\text{sv}}(\sigma_i) e^{2\pi i N} \mathbb{I}_{+}(\epsilon_k; \tau) U_{\text{SL}_2}^{-1}(\tau) \quad (4.22) $$

does not produce $R[J^{\text{sv}}(\epsilon_k; \tau)]$ on the right-hand side since $e^{2\pi i \epsilon_0} \mathbb{M}^{\text{sv}}(\sigma_i)^{-1} e^{-2\pi i \epsilon_0} \neq \mathbb{M}^{\text{sv}}(\sigma_i)^{-1}$. That is why the coefficients of $R[J^{\text{sv}}(\epsilon_k; \tau)]$ cannot possibly match the T-invariant $\beta^{\text{sv}}$ as discussed below (3.57).

## 4.2 S-equivariance

The analysis of the modular S properties of the constituents of $\mathbb{I}^{\text{eqv}}$, $J^{\text{eqv}}$ proceeds in a similar fashion. We start from the definition (3.1) to conclude that

$$ U_{\text{SL}_2}(-\frac{1}{\tau}) = \bar{\tau}^{-[e_0, e_0^\vee]} U_{\text{SL}_2}(\tau) w (2\pi i)^{-[e_0, e_0^\vee]}, \quad (4.23) $$

where w denotes the Weyl reflection $w = e_0^\vee e^{-e_0} e_0^\vee$. This equation can be checked more easily in any faithful representation of SL₂, e.g. by considering the case $k=4$.

Upon conjugation by $U_{\text{SL}_2}(-\frac{1}{\tau})$, we see that it is sensible to define the “S-modular” transformation on the derivation $e_k^{(j)}$ as

$$ e_k^{(j)}|_S := w \left[ (2\pi i)^{-[e_0, e_0^\vee]} e_k^{(j)} (2\pi i)^{[e_0, e_0^\vee]} \right] w^{-1}, \quad (4.24) $$

which, given the Weyl reflection (2.58) and the sl₂ relations (2.53), can be rewritten as

$$ e_k^{(j)}|_S = (-1)^j (2\pi i)^{k-2-2j} \frac{j!}{(k-j-2)!} e_k^{(k-j-2)}. \quad (4.25) $$

Note that since the above relations are only sensitive to the sl₂ structure of the derivations $e_k$, (4.25) holds in identical form for the cuspidal variables $e_{\Delta\pm}$ and for Tsunogai's derivations, i.e. if we substitute $e_0 \to \epsilon_0$, $e_0^\vee \to \epsilon_0^\vee$ and $e_k^{(j)} \to \epsilon_k^{(j)}$.
We can now turn our attention towards the S-transformations of the generating series $\mathbb{I}_{\pm}$. Starting again with $\mathbb{I}_{+}$, we can use the S-transformation of the $\nu$ kernels (2.6) and the definition (3.2) of $\mathbb{A}_{+}$ to deduce

$$ \mathbb{A}_{+}(e_k; -\frac{1}{\tau}) = \mathbb{A}_{+}(e_k; \tau)|_{S}, \qquad (4.26) $$

where the S-transformation on the right-hand side only acts on the derivations $e_k^{(j)}$, as given in (4.25) or equivalently (4.24).

Computing the S-transformation of the path-ordered exponential (3.6) yields

$$ \begin{aligned} \mathbb{I}_{+}(e_k; -\frac{1}{\tau}) &= \mathrm{P}\text{-exp}\left(\int_{-\frac{1}{\tau}}^{i\infty} \mathbb{A}_{+}(e_k; \tau_1)\right) = \mathrm{P}\text{-exp}\left(\int_{\tau}^{0} \mathbb{A}_{+}(e_k; -\frac{1}{\tau_1})\right) \\ &= \mathrm{P}\text{-exp}\left(\int_{i\infty}^{0} \mathbb{A}_{+}(e_k; -\frac{1}{\tau_1})\right)\mathrm{P}\text{-exp}\left(\int_{\tau}^{i\infty} \mathbb{A}_{+}(e_k; -\frac{1}{\tau_1})\right) \\ &= (\mathbb{S}_{+}(e_k)^{-1}\mathbb{I}_{+}(e_k; \tau))|_{S}, \end{aligned} \qquad (4.27) $$

where we defined the tangentially regulated S-cocycle at infinity as

$$ \mathbb{S}_{+}(e_k) := \mathrm{P}\text{-exp}\left(\int_{0}^{i\infty} \mathbb{A}_{+}(e_k; \tau_1)\right), \qquad (4.28) $$

or equivalently the generating series of all MMVs (2.14) and (2.17)

$$ \begin{aligned} \mathbb{S}_+(e_k) = &+ \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} (-1)^{j_1} \frac{(k_1-1)}{j_1!} (2\pi i)^{j_1+1-k_1} \left\{ \mathbf{m}\begin{bmatrix} j_1 \\ k_1 \end{bmatrix} e_{k_1}^{(j_1)} + \sum_{\Delta_{k_1} \in S_{k_1}} \mathbf{m}\begin{bmatrix} j_1 \\ \Delta_{k_1} \end{bmatrix} e_{\Delta_{k_1}^{+}}^{(j_1)} \right\} \quad (4.29) \\ &+ \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} \sum_{k_2=4}^{\infty} \sum_{j_2=0}^{k_2-2} (-1)^{j_1+j_2} \frac{(k_1-1)(k_2-1)}{j_1!j_2!} (2\pi i)^{j_1+j_2+2-k_1-k_2} \left\{ \mathbf{m}\begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix} e_{k_1}^{(j_1)} e_{k_2}^{(j_2)} \right. \\ & \qquad \left. + \sum_{\Delta_{k_2} \in S_{k_2}} \mathbf{m}\begin{bmatrix} j_1 & j_2 \\ k_1 & \Delta_{k_2} \end{bmatrix} e_{k_1}^{(j_1)} e_{\Delta_{k_2}^{+}}^{(j_2)} + \sum_{\Delta_{k_1} \in S_{k_1}} \mathbf{m}\begin{bmatrix} j_1 & j_2 \\ \Delta_{k_1} & k_2 \end{bmatrix} e_{\Delta_{k_1}^{+}}^{(j_1)} e_{k_2}^{(j_2)} \right. \\ & \qquad \left. + \sum_{\Delta_{k_1} \in S_{k_1}} \sum_{\Delta_{k_2} \in S_{k_2}} \mathbf{m}\begin{bmatrix} j_1 & j_2 \\ \Delta_{k_1} & \Delta_{k_2} \end{bmatrix} e_{\Delta_{k_1}^{+}}^{(j_1)} e_{\Delta_{k_2}^{+}}^{(j_2)} \right\} + \dots, \end{aligned} $$

with terms of modular depth greater than or equal to three in the ellipsis. In contrast to the T-cocycle at infinity in (4.11), the S-cocycle at infinity receives contributions from both Eisenstein series and cusp forms and considerably simplifies when we specialise to Tsunogai's derivations, i.e. $e_k \to \epsilon_k$ and $e_{\Delta_{\pm}} \to 0$. In this case, the S-cocycle at infinity reduces to the generating series of the MMVs (2.14) from Eisenstein series,

$$ \begin{aligned} \mathbb{S}_+(\epsilon_k) = &+ \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} \frac{(-1)^{j_1}(k_1-1)}{j_1!} (2\pi i)^{j_1+1-k_1} \mathbf{m}\begin{bmatrix} j_1 \\ k_1 \end{bmatrix} \epsilon_{k_1}^{(j_1)} \quad (4.30) \\ &+ \sum_{k_1,k_2=4}^{\infty} \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \frac{(-1)^{j_1+j_2}(k_1-1)(k_2-1)}{j_1!j_2!} (2\pi i)^{j_1+j_2+2-k_1-k_2} \mathbf{m}\begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix} \epsilon_{k_1}^{(j_1)} \epsilon_{k_2}^{(j_2)} \\ &+ \dots \end{aligned} $$
Note that we can find an alternative representation of (4.27) using the identity

$$ \mathbf{m}^{\begin{bmatrix} k_1-j_1-2 & k_2-j_2-2 & \dots & k_\ell-j_\ell-2 \\ k_1 & k_2 & \dots & k_\ell \end{bmatrix}} = (-1)^{\sum_{i=1}^\ell (j_i+1)} \mathbf{m}^{\begin{bmatrix} j_\ell & \dots & j_2 & j_1 \\ k_\ell & \dots & k_2 & k_1 \end{bmatrix}}, \quad (4.31) $$

and similar generalisations upon replacing Eisenstein integration kernels with holomorphic cusp forms. This identity translates into

$$ \mathbb{S}_+(e_k)|_S = \mathbb{S}_+(e_k)^{-1}, \quad (4.32) $$

at the level of generating series which leads to the equivalent form for the S-transformation (4.27) of $\mathbb{I}_+$ given by

$$ \mathbb{I}_+(e_k; -\frac{1}{\tau}) = \mathbb{S}_+(e_k)(\mathbb{I}_+(e_k; \tau)|_S). \quad (4.33) $$

Similar arguments apply to the S-transformation of $\tilde{\mathbb{I}}_-$ Firstly, from (2.6) and the definition (3.3), we deduce

$$ \mathbb{A}_-(e_k; -\frac{1}{\tau}) = \mathbb{A}_-(e_k; \tau)|_S. \quad (4.34) $$

From this equation we can proceed as above and write

$$ \begin{aligned} \tilde{\mathbb{I}}_-(e_k; -\frac{1}{\tau}) &= \left( \tilde{\mathbb{I}}_-(e_k; \tau) \tilde{\mathbf{P}}\text{-exp}\left( \int_{-i\infty}^0 \mathbb{A}_-(e_k; \tau_1) \right) \right)|_S \\ &= \left( \tilde{\mathbb{I}}_-(e_k; \tau) \mathbb{S}_-(e_k) \right)|_S, \end{aligned} \quad (4.35) $$

where in the last line we introduced the S-cocycle at infinity<fnref n="26" />

$$ \mathbb{S}_-(e_k) := \tilde{\mathbf{P}}\text{-exp}\left(\int_{-i\infty}^{0} \mathbb{A}_-(e_k; \tau_1)\right). \quad (4.36) $$

Based on tangential-basepoint regularisation and the reality properties

$$ \overline{\mathbf{m}^{\begin{bmatrix} j_1 & j_2 & \dots & j_\ell \end{bmatrix}}} = (-1)^{\sum_{i=1}^\ell (j_i+1)} \mathbf{m}^{\begin{bmatrix} j_1 & j_2 & \dots & j_\ell \end{bmatrix}}, \quad (4.37) $$

of MMVs (2.14) and (2.17) (which hold in identical form for $k_i \to \Delta_{k_i}$), the expansion of (4.36) takes a form analogous to (4.29),

$$ \begin{aligned} \mathbb{S}_-(e_k) = &+ \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} (-1)^{j_1} \frac{(k_1-1)}{j_1!} (-2\pi i)^{j_1+1-k_1} \left\{ \mathbf{m}^{\begin{bmatrix} j_1 \\ k_1 \end{bmatrix}} e_{k_1}^{(j_1)} + \sum_{\Delta_{k_1} \in \mathcal{S}_{k_1}} \mathbf{m}^{\begin{bmatrix} j_1 \\ \Delta_{k_1} \end{bmatrix}} e_{\Delta_{k_1}^{-}}^{(j_1)} \right\} \\ &+ \sum_{k_1=4}^{\infty} \sum_{j_1=0}^{k_1-2} \sum_{k_2=4}^{\infty} \sum_{j_2=0}^{k_2-2} (-1)^{j_1+j_2} \frac{(k_1-1)(k_2-1)}{j_1!j_2!} (-2\pi i)^{j_1+j_2+2-k_1-k_2} \left\{ \mathbf{m}^{\begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix}} e_{k_1}^{(j_1)} e_{k_2}^{(j_2)} \right. \\ &+ \sum_{\Delta_{k_2} \in \mathcal{S}_{k_2}} \mathbf{m}^{\begin{bmatrix} j_1 & j_2 \\ k_1 & \Delta_{k_2} \end{bmatrix}} e_{k_1}^{(j_1)} e_{\Delta_{k_2}^{-}}^{(j_2)} + \sum_{\Delta_{k_1} \in \mathcal{S}_{k_1}} \mathbf{m}^{\begin{bmatrix} j_1 & j_2 \\ \Delta_{k_1} & k_2 \end{bmatrix}} e_{\Delta_{k_1}^{-}}^{(j_1)} e_{k_2}^{(j_2)} \\ &\left. + \sum_{\Delta_{k_1} \in \mathcal{S}_{k_1}} \sum_{\Delta_{k_2} \in \mathcal{S}_{k_2}} \mathbf{m}^{\begin{bmatrix} j_1 & j_2 \\ \Delta_{k_1} & \Delta_{k_2} \end{bmatrix}} e_{\Delta_{k_1}^{-}}^{(j_1)} e_{\Delta_{k_2}^{-}}^{(j_2)} \right\} + \dots \end{aligned} \quad (4.38) $$

<footnote n="26">²⁶The careful reader may be confused as to why our definition (4.27) for $\mathbb{S}_+$ involves the path ordered exponential of $\int_0^{i\infty} \mathbb{A}_+(e_k; \tau_1)$ while the present definition (4.35) for $\mathbb{S}_-$ is written in terms of $\int_{-i\infty}^0 \mathbb{A}_-(e_k; \tau_1)$. The reason for this apparent discrepancy lies in the two different conventions for the path ordered exponentials, P-exp and the reverse ordering $\tilde{\mathbf{P}}$-exp, as defined in (3.6). Upon expanding (4.35), the coefficient of a generic term $e_{k_1}^{(j_1)} \dots e_{k_\ell}^{(j_\ell)}$ is given by an iterated integral over the domain $0 < \bar{\tau}_\ell < \dots < \bar{\tau}_1 < -i\infty$. From (2.14) we see that this is precisely the correct structure to produce the combination $e_{k_1}^{(j_1)} \dots e_{k_\ell}^{(j_\ell)} \overline{\mathbf{m}^{\begin{bmatrix} j_1 & \dots & j_\ell \\ k_1 & \dots & k_\ell \end{bmatrix}}}$.</footnote>
## 5 Reformulation in terms of commutative $X_i, Y_i$

In this section, we combine various iterated integrals with the help of auxiliary commutative variables $(X_i, Y_i)$ for each $1 \le i \le r$, where $r$ is the modular depth. Each pair of commutative variables transforms as a doublet under SL(2, $\mathbb{Z}$):

$$ \begin{pmatrix} X_i \\ Y_i \end{pmatrix} \mapsto \begin{pmatrix} aX_i + bY_i \\ cX_i + dY_i \end{pmatrix} \quad \text{with} \quad \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}(2, \mathbb{Z}). \qquad (5.1) $$

Thus the combination

$$ X_i - \tau Y_i \mapsto (c\tau + d)^{-1}(X_i - \tau Y_i), \qquad (5.2) $$

transforms with modular weight $(-1, 0)$. Combining powers of $(X_i - \tau Y_i)$ with holomorphic modular forms in the integration kernels will then produce equivariant expressions, i.e. modular invariants after taking the transformation (5.1) of $X_i$ and $Y_i$ into account. These are the building blocks of the equivariant integrals introduced by Brown [13, 14] (see also [16, 28]) and that represent a different organisation of the same iterated integrals introduced in (2.7).

Factors $(X_i - \tau Y_i)^{k_i - 2}$ of weight $(2 - k_i, 0)$ are homogeneous polynomials in the $X_i$ and $Y_i$ of degree $k_i - 2$. Since the variables $(X_i, Y_i)$ are commuting and transform in a doublet, the transformation of $(X_i, Y_i)$ in $(X_i - \tau Y_i)^{k_i - 2}$ under SL(2, $\mathbb{Z}$) is that of a $(k-1)$-dimensional representation isomorphic to $V(e_{k_i})$ in section 2.3.1. Products $(X_i - \tau Y_i)^{k_i - 2} (X_j - \tau Y_j)^{k_j - 2}$ with distinct bookkeeping variables $(X_i, Y_i)$ and $(X_j, Y_j)$ then yield polynomials transforming in the tensor product $V(e_{k_i}) \otimes V(e_{k_j})$. The projection to the various irreducible components of the tensor product can be achieved by applying certain differential operators in $(X_i, Y_i)$ and $(X_j, Y_j)$ [13] that are reviewed below.

In this section we review such a construction at modular depths up to three and show in particular in section 5.2 that a completion of elementary integrals leads to a completely equivalent description of the modular forms discussed in the previous sections. Moreover, we will argue in section 5.3 that this construction generalises to arbitrary modular depth.

### 5.1 Equivariant iterated integrals at modular depths one and two

In this section, we will explain the connection between the work of the previous sections and equivariant triple integrals. The construction of our triple integrals will involve functions produced from the double-integral case and follows a similar procedure (originally given by Brown in [13] and also recapped in [28]).

#### 5.1.1 Modular depth one

A first step is to combine the integration kernels $\nu[j_k; \tau]$ defined in (2.4) to the (1, 0)-form valued polynomial in the commutative bookkeeping variables $X$ and $Y$,

$$ \begin{align} \underline{G}_k[X, Y; \tau] &:= \frac{(k-1)!}{2(2\pi i)^{k-1}} (X - \tau Y)^{k-2} \underline{G}_k(\tau) \, d\tau \tag{5.3} \\ &= \frac{1}{2} (k-1)! \sum_{j=0}^{k-2} (-1)^j \binom{k-2}{j} X^{k-2-j} \left(\frac{Y}{2\pi i}\right)^j \nu[j_k; \tau]. \nonumber \end{align} $$
In view of (5.2), this (1,0)-form is equivariant under SL(2, Z), i.e. invariant when transforming both (X, Y) and $\tau$.

We wish to use this (1,0)-form and its complex conjugate as integration kernels for a line integral. In order for the integral to be path-independent, the forms must be closed. For a one-form of the type $\varphi(\tau) := \varphi^+(\tau) + \varphi^-(\tau) := \varphi^\tau(\tau)d\tau + \varphi^{\bar{\tau}}(\tau)d\bar{\tau}$, the closure condition $d\varphi = 0$ translates into the following partial differential equations for its components $\varphi^\tau, \varphi^{\bar{\tau}}$ or constituent (1,0)- and (0,1)-forms $\varphi^\pm$:

$$ \partial_{\bar{\tau}} \varphi^{\tau} = \partial_{\tau} \varphi^{\bar{\tau}} \quad \Leftrightarrow \quad \partial_{\bar{\tau}} \varphi^{+} \wedge 2d\bar{\tau} = 2d\tau \wedge \partial_{\tau} \varphi^{-}. \qquad (5.4) $$

Since $\mathcal{G}_k[X, Y; \tau]$ is holomorphic, closure is trivially satisfied, and this form and its complex conjugate can be used to reorganise the depth-one modular forms $\beta^{\text{eqv}}$ in (2.40) and (2.41) as follows:

$$ \begin{aligned} M_k[X, Y; \tau] &:= -\frac{1}{2} \int_{\tau}^{i\infty} \mathcal{G}_k[X, Y; \tau_1] - \frac{1}{2} \int_{\bar{\tau}}^{-i\infty} \overline{\mathcal{G}_k[X, Y; \tau_1]} + \frac{(k-2)! \zeta_{k-1}}{2(2\pi i)^{k-2}} Y^{k-2} \\ &= -\frac{1}{4}(k-1)! \sum_{j=0}^{k-2} \frac{\binom{k-2}{j}}{(-4y)^j} \beta^{\text{eqv}}[j; \tau] (X-\tau Y)^j (X-\bar{\tau} Y)^{k-2-j}. \end{aligned} \qquad (5.5) $$

The last term $\sim \zeta_{k-1}Y^{k-2}$ of the first line is engineered to attain the equivariant transformation law under $(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}) \in \text{SL}(2, \mathbb{Z})$,

$$ M_k[aX+bY, cX+dY; \frac{a\tau+b}{c\tau+d}] = M_k[X, Y; \tau], \qquad (5.6) $$

i.e. to cancel the cocycle under the modular S-transformation of the first two terms of (5.5). By the modular weights $(-1, 0)$ and $(0, -1)$ of $(X-\tau Y)$ and $(X-\bar{\tau} Y)$, the equivariance property (5.6) identifies the coefficients proportional to $\beta^{\text{eqv}}[\begin{smallmatrix} j \\ k \end{smallmatrix}; \tau]/y^j$ of $(X-\tau Y)^j(X-\bar{\tau} Y)^{k-2-j}$ as modular forms of weight $(j, k-j-2)$, reproducing the known modular properties of $\beta^{\text{eqv}}[\begin{smallmatrix} j \\ k \end{smallmatrix}; \tau]$, namely $\beta^{\text{eqv}}[\begin{smallmatrix} j \\ k \end{smallmatrix}; \frac{a\tau+b}{c\tau+d}] = (c\bar{\tau}+d)^{k-2j-2}\beta^{\text{eqv}}[\begin{smallmatrix} j \\ k \end{smallmatrix}; \tau]$. From expression (5.5) it is straightforward to check the differential equations

$$ 2 \partial_{\tau} M_k[\tau] d\tau = \mathcal{G}_k[\tau], \qquad 2 \partial_{\bar{\tau}} M_k[\tau] d\bar{\tau} = \overline{\mathcal{G}_k[\tau]}. \qquad (5.7) $$

In the remainder of this section, we will use the analogous fact that equivariant functions in several pairs of commutative variables $(X_i, Y_i)$, $i = 1, 2, \dots, r$ yield modular forms of weights $\sum_{i=1}^r (j_i, k_i-j_i-2)$ upon rewriting

$$ X = \frac{\tau(X-\bar{\tau}Y)}{\tau-\bar{\tau}} - \frac{\bar{\tau}(X-\tau Y)}{\tau-\bar{\tau}}, \qquad Y = \frac{(X-\bar{\tau}Y)}{\tau-\bar{\tau}} - \frac{(X-\tau Y)}{\tau-\bar{\tau}}, \qquad (5.8) $$

and taking their coefficients of $\prod_{i=1}^r (X_i-\tau Y_i)^{j_i} (X_i-\bar{\tau} Y_i)^{k_i-2-j_i}$ [13]. In the language of section 3.1, converting the coefficients of $Y_i^{j_i} X_i^{k_i-j_i-2}$ to those of $(X_i-\tau Y_i)^{j_i} (X_i-\bar{\tau} Y_i)^{k_i-2-j_i}$ amounts to passing from the holomorphic frame to the modular frame. In other words, the rewriting (5.8) of expansion variables is an alternative way of implementing the SL$_2$ transformation $U_{\text{SL}_2}(\tau)$ given by (3.1).
### 5.1.2 Towards equivariant integrals at modular depth two

In [13], a crucial ingredient for the construction of the equivariant double integrals was an equivariant closed one-form of the type

$$D_{k_1,k_2}[X_1, Y_1, X_2, Y_2; \tau] := -\frac{1}{2} \left( \underline{G}_{k_1}[X_1, Y_1; \tau] M_{k_2}[X_2, Y_2; \tau] + M_{k_1}[X_1, Y_1; \tau] \overline{G}_{k_2}[X_2, Y_2; \tau] \right), \quad (5.9)$$

which now depends on four commutative bookkeeping variables $X_1, Y_1, X_2, Y_2$, and where $\underline{G}_{k_i}[X_i, Y_i; \tau]$ and $M_{k_j}[X_j, Y_j; \tau]$ are given by (5.3) and (5.5), respectively. Closure of $D_{k_1,k_2}$ in the sense of (5.4) can be verified using (5.3) and (5.7). As a closed one-form, $D_{k_1,k_2}$ in (5.9) yields a well-defined function of $\tau$ upon integration,

$$K_{k_1,k_2}[\tau] := \frac{1}{4} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \int_{\tau_1}^{i\infty} \underline{G}_{k_2}[\tau_2] + \frac{1}{4} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \times \int_{\bar{\tau}}^{-i\infty} \overline{G}_{k_2}[\tau_2] \quad (5.10) \\ + \frac{1}{4} \int_{\bar{\tau}}^{-i\infty} \overline{G}_{k_2}[\tau_2] \int_{\bar{\tau}_2}^{-i\infty} \overline{G}_{k_1}[\tau_1] + \frac{1}{2} c_{k_2} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] + \frac{1}{2} c_{k_1} \int_{\bar{\tau}}^{-i\infty} \overline{G}_{k_2}[\tau_2],$$

where we employ the following shorthand in the last two terms,

$$c_{k_i} = c_{k_i}[Y_i] := - \frac{(k_i-2)! \zeta_{k_i-1}}{2(2\pi i)^{k_i-2}} Y_i^{k_i-2}. \quad (5.11)$$

To avoid cluttering (especially in later parts), from now on we will no longer display the dependence of $K_{k_1,k_2}$ and related objects on the commutative bookkeeping variables $X_i, Y_i$. We choose to keep the square brackets, however, to emphasise the connection to these variables. Furthermore, whenever this notation is used, the correct $X_i, Y_i$ can always be inferred by the values assigned to the subscript $k_i$, for example, $K_{k_1,k_2}[\tau] = K_{k_1,k_2}[X_1, Y_1, X_2, Y_2; \tau]$.

By the alternative expansions

$$\underline{G}_k[X, Y; \tau_1] = \frac{1}{2}(k-1)! \sum_{j=0}^{k-2} \binom{k-2}{j} \frac{1}{(-4y)^j} (X-\tau Y)^j (X-\bar{\tau} Y)^{k-2-j} \omega_+ \begin{bmatrix} j \\ k \end{bmatrix}; \tau, \tau_1, \quad (5.12) \\ \overline{G}_k[X, Y; \tau_1] = \frac{1}{2}(k-1)! \sum_{j=0}^{k-2} \binom{k-2}{j} \frac{1}{(-4y)^j} (X-\tau Y)^j (X-\bar{\tau} Y)^{k-2-j} \omega_- \begin{bmatrix} j \\ k \end{bmatrix}; \tau, \tau_1, \quad (5.13)$$

of the holomorphic (1,0)-form (5.3) and its complex conjugate, the expression (5.10) for $K_{k_1,k_2}$ can be rewritten in terms of the iterated integrals $\beta_\pm$ in (2.22) [28]$^{27}$

$$K_{k_1,k_2}[\tau] = \frac{(k_1-1)!(k_2-1)!}{16} \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \frac{\binom{k_1-2}{j_1} \binom{k_2-2}{j_2}}{(-4y)^{j_1+j_2}} \left( \beta_+ \begin{bmatrix} j_2 & j_1 \\ k_2 & k_1 \end{bmatrix} + \beta_+ \begin{bmatrix} j_1 \\ k_1 \end{bmatrix} \beta_- \begin{bmatrix} j_2 \\ k_2 \end{bmatrix} + \beta_- \begin{bmatrix} j_1 & j_2 \\ k_1 & k_2 \end{bmatrix} \right) \\ \times (X_1 - \tau Y_1)^{j_1} (X_1 - \bar{\tau} Y_1)^{k_1-2-j_1} (X_2 - \tau Y_2)^{j_2} (X_2 - \bar{\tau} Y_2)^{k_2-2-j_2} \quad (5.13) \\ - \frac{(k_2-2)!(k_1-1)!\zeta_{k_2-1}}{8(2\pi i)^{k_2-2}} \sum_{j=0}^{k_1-2} \binom{k_1-2}{j} \frac{(-1)^j}{(4y)^j} \beta_+ \begin{bmatrix} j \\ k_1 \end{bmatrix} (X_1 - \tau Y_1)^j (X_1 - \bar{\tau} Y_1)^{k_1-2-j} Y_2^{k_2-2} \\ - \frac{(k_1-2)!(k_2-1)!\zeta_{k_1-1}}{8(2\pi i)^{k_1-2}} \sum_{j=0}^{k_2-2} \binom{k_2-2}{j} \frac{(-1)^j}{(4y)^j} \beta_- \begin{bmatrix} j \\ k_2 \end{bmatrix} Y_1^{k_1-2} (X_2 - \tau Y_2)^j (X_2 - \bar{\tau} Y_2)^{k_2-2-j}.$$

<footnote>$^{27}$In the first line of (5.13), we fixed a minus-sign mistake concerning the coefficient of $\beta_+ \begin{bmatrix} j_1 \\ k_1 \end{bmatrix} \beta_- \begin{bmatrix} j_2 \\ k_2 \end{bmatrix}$ in (4.45) of [28].</footnote>
### 5.1.3 Equivariant completion at modular depth two

In spite of its equivariant total differential (5.9), the function (5.13) itself fails to be equivariant by a cocycle: evaluating the SL(2, $\mathbb{Z}$) transformation

$$ (X_i, Y_i, \tau) \to (aX_i+bY_i, cX_i+dY_i, \frac{a\tau+b}{c\tau+d}) \qquad (5.14) $$

of $K_{k_1,k_2}[\tau]$ introduces an inhomogeneous term independent on $\tau$ and polynomial in $X_i, Y_i$. Still, there is a method to restore equivariance by adding simple classes of terms [13] with compensating cocycles. We start by introducing the equivariant projector

$$ \delta^{\ell} := m \circ \left( \frac{\partial}{\partial X_1} \otimes \frac{\partial}{\partial Y_2} - \frac{\partial}{\partial Y_1} \otimes \frac{\partial}{\partial X_2} \right)^{\ell}, \qquad (5.15) $$

defined for $\ell \ge 0$, and where $m: \mathbb{Q}[X_1, Y_1] \otimes \mathbb{Q}[X_2, Y_2] \to \mathbb{Q}[X_1, Y_1]$ is the multiplication map setting $X_2 = X_1$ and $Y_2 = Y_1$ after evaluating the derivatives in (5.15). We then define the function

$$ K_{k_1,k_2}^{(\ell)}[X_1, Y_1; \tau] := \frac{(i\pi)^{\ell}}{(\ell!)^2} \delta^{\ell}(K_{k_1,k_2}[\tau]), \qquad (5.16) $$

in the normalisation conventions of [13] which is equivariant up to a cocycle that only depends on one pair $X_1, Y_1$ of bookkeeping variables. It transforms in an sl$_2$-representation of dimension $k_1+k_2-2\ell-3$, showing that only a finite range of $\ell$-values have to be considered.

By the Eichler-Shimura theorem [95, 96] we can express cocycles of $K_{k_1,k_2}^{(\ell)}$ in terms of a coboundary and linear combinations of cocycles of modular forms. In other words, there is a systematic completion $M_{k_1,k_2}^{(\ell)}$ of $K_{k_1,k_2}^{(\ell)}$ which produces an equivariant function of the form

$$ M_{k_1,k_2}^{(\ell)}[X_1, Y_1; \tau] = K_{k_1,k_2}^{(\ell)}[X_1, Y_1; \tau] - c_{k_1,k_2}^{(\ell)}[X_1, Y_1] \\ - \frac{1}{2} \left\{ \int_{\tau}^{i\infty} \mathfrak{f}_{k_1,k_2}^{(\ell)}[X_1, Y_1; \tau_1] + \int_{\bar{\tau}}^{-i\infty} \mathfrak{g}_{k_1,k_2}^{(\ell)}[X_1, Y_1; \tau_1] \right\}, \qquad (5.17) $$

where the $\tau$-independent quantity $c_{k_1,k_2}^{(\ell)}[X_1, Y_1]$ in the first line is a polynomial in $X_1, Y_1$ of homogeneity degree $k_1+k_2-2\ell-4$. In the second line of (5.17), $\mathfrak{f}_{k_1,k_2}^{(\ell)}[X_1, Y_1; \tau_1]$ is an equivariant (1,0)-form composed of a holomorphic cusp form of weight $k_1+k_2-2\ell-2$ multiplying $(X_1-\tau_1 Y_1)^{k_1+k_2-2\ell-4}$, and $\mathfrak{g}_{k_1,k_2}^{(\ell)}[X_1, Y_1; \tau_1]$ is an equivariant (0,1)-form composed of an antiholomorphic modular form of weight $k_1+k_2-2\ell-2$ multiplying $(X_1-\bar{\tau}_1 Y_1)^{k_1+k_2-2\ell-4}$. As a consequence, holomorphic cusp forms of modular weight 2s can occur in $\mathfrak{f}_{k_1,k_2}^{(\ell)}$, $\mathfrak{g}_{k_1,k_2}^{(\ell)}$ at degrees $k_1+k_2 \ge 2s+2$ and values $\ell = \frac{1}{2}(k_1+k_2)-s-1$. We note that while $\mathfrak{f}_{k_1,k_2}^{(\ell)}$ involves a cusp form, the form entering $\mathfrak{g}_{k_1,k_2}^{(\ell)}$ is a general holomorphic modular form and so can be a combination of a cuspidal and Eisenstein part.

We note that the application of the Eichler-Shimura theorem relies on the presence of only one pair of bookkeeping variables $X_i, Y_i$, which explains the need to introduce the projector $\delta^{\ell}$. Fortunately, we can study $M_{k_1,k_2}^{(\ell)}$ and their components at each level $\ell = 0, 1, \dots, \min(k_1, k_2)-2$ and, as was demonstrated in [28], reconstruct equivariant functions $M_{k_1,k_2}[\tau] = M_{k_1,k_2}[X_1, Y_1, X_2, Y_2; \tau]$. Such functions can be represented in the same format
as in (5.17), but they can also be described simply in terms of modular-depth-two $\beta^{\text{eqv}}$ (see appendix C.3 for a detailed comparison of the two representations of $M_{k_1,k_2}[\tau]$ in (5.18)):

$$ M_{k_1,k_2}[\tau] := K_{k_1,k_2}[\tau] - c_{k_1,k_2} - \frac{1}{2} \left\{ \int_{\tau}^{i\infty} \mathfrak{f}_{k_1,k_2}[\tau_1] + \int_{\bar{\tau}}^{-i\infty} \overline{\mathfrak{g}_{k_1,k_2}[\tau_1]} \right\} \quad (5.18) \\ = \frac{1}{16}(k_1-1)!(k_2-1)! \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \frac{\binom{k_1-2}{j_1} \binom{k_2-2}{j_2}}{(-4y)^{j_1+j_2}} \beta^{\text{eqv}} \begin{bmatrix} j_2 & j_1 \\ k_2 & k_1 \end{bmatrix} \\ \times (X_1 - \tau Y_1)^{j_1} (X_1 - \bar{\tau} Y_1)^{k_1-2-j_1} (X_2 - \tau Y_2)^{j_2} (X_2 - \bar{\tau} Y_2)^{k_2-2-j_2}, $$

such that

$$ \frac{(i\pi)^\ell}{(\ell!)^2} \delta^\ell(M_{k_1,k_2}) = M_{k_1,k_2}^{(\ell)}, \qquad \frac{(i\pi)^\ell}{(\ell!)^2} \delta^\ell(c_{k_1,k_2}) = c_{k_1,k_2}^{(\ell)}, \quad (5.19) $$

$$ \frac{(i\pi)^\ell}{(\ell!)^2} \delta^\ell(\mathfrak{f}_{k_1,k_2}) = \mathfrak{f}_{k_1,k_2}^{(\ell)}, \qquad \frac{(i\pi)^\ell}{(\ell!)^2} \delta^\ell(\overline{\mathfrak{g}_{k_1,k_2}}) = \overline{\mathfrak{g}_{k_1,k_2}^{(\ell)}}. $$

For fixed values of $(k_1, k_2)$, the $\beta^{\text{eqv}} \begin{bmatrix} j_2 & j_1 \\ k_2 & k_1 \end{bmatrix}$ in (5.18) enter the generating series $\mathcal{J}^{\text{eqv}}(e_k; \tau)$ with words $e_{k_1}^{(j_1)} e_{k_2}^{(j_2)}$ in the $\mathfrak{sl}_2$ tensor product $V(e_{k_1}) \otimes V(e_{k_2})$ as coefficients. The highest-dimensional module $V(e_{k_1+k_2-2})$ in this tensor product is isolated by $M_{k_1,k_2}^{(0)}$ in (5.19) resulting from the projector (5.15) at $\ell = 0$. The operator $\delta^0$ preserves the combined homogeneity degree in $X_i, Y_i$ which is $k_1+k_2-4$ in both $M_{k_1,k_2}^{(0)}$ and the unprojected $M_{k_1,k_2}$. In the same way as the differential operator entering $\delta^\ell$ in (5.15) reduces this homogeneity degree by $2\ell$, the projected equivariant integrals in $M_{k_1,k_2}^{(\ell)}$ at generic $\ell = 0, 1, \dots, \min(k_1, k_2)-2$ fall into the $\mathfrak{sl}_2$ irreducible $V(e_{k_1+k_2-2-2\ell})$.

By isolating the contributions to (5.18) at modular depth zero, the $\tau$-independent quantities $c_{k_1,k_2}$ can be assembled from the expansion coefficients $c^{\text{sv}}$ in (2.87) or (3.15) [28]

$$ c_{k_1,k_2} = -\frac{1}{16}(k_1-1)!(k_2-1)! \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} (-1)^{j_1+j_2} \binom{k_1-2}{j_1} \binom{k_2-2}{j_2} \\ \times \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} c^{\text{sv}} \begin{bmatrix} j_2 & j_1 \\ k_2 & k_1 \end{bmatrix}. \quad (5.20) $$

The $c^{\text{sv}}$ on the right-hand side are rational multiples of $\zeta_{2n_1+1}$ or $\zeta_{2n_1+1}\zeta_{2n_2+1}$ with $n_i \in \mathbb{N}$ that can be explicitly determined in all degrees from (2.83) and (3.47).

The Eisenstein contributions to $\overline{\mathfrak{g}_{k_1,k_2}}$ in (5.18) are known in closed form from (4.49) of [28]. In the setting of this work, the expressions $\overline{\mathfrak{g}_{k_1,k_2}} \sim \zeta_{\min(k_1,k_2)-1} \overline{\mathfrak{G}_{|k_1-k_2|+2}}$ of the reference can be derived from the terms $\sim [e_{k_1}^{(j_1)}, e_{k_2}^{(j_2)}]$ in $\mathcal{M}^{\text{sv}}(\hat{z}_i, \hat{z}_\varpi)^{-1} \overline{\mathbb{I}}_-(e_k; \tau) \mathcal{M}^{\text{sv}}(\hat{z}_i, \hat{z}_\varpi)$ and their contributions to the $\beta^{\text{eqv}} \begin{bmatrix} j_2 & j_1 \\ k_2 & k_1 \end{bmatrix}$ in (5.18). Similarly, the cusp forms entering $\mathfrak{f}_{k_1,k_2}$ and $\overline{\mathfrak{g}_{k_1,k_2}}$ have rational multiples of $\frac{\Lambda(\Delta_{2s}, 2s+2n)}{\Lambda(\Delta_{2s}, 2s-2)}$ or $\frac{\Lambda(\Delta_{2s}, 2s+2n+1)}{\Lambda(\Delta_{2s}, 2s-1)}$ as coefficients ($n \in \mathbb{N}_0$), see section 4.3.4 of [28] for explicit examples. The exact form of these expressions can be reproduced from the terms $\sim [e_{k_1}^{(j_1)}, e_{k_2}^{(j_2)}]$ in the letters $e_{\Delta_{2s} \pm}$ of section 3.3 and their contributions to the $\beta^{\text{eqv}} \begin{bmatrix} j_2 & j_1 \\ k_2 & k_1 \end{bmatrix}$ in (5.18).

From expression (5.18) along with (5.9) and (5.7) it can be verified that

$$ 2 \partial_{\tau} M_{k_1,k_2}[\tau] d\tau = \underline{\mathfrak{G}}_{k_1}[\tau] M_{k_2}[\tau] + \mathfrak{f}_{k_1,k_2}[\tau], \quad 2 \partial_{\bar{\tau}} M_{k_1,k_2}[\tau] d\bar{\tau} = M_{k_1}[\tau] \underline{\mathfrak{G}}_{k_2}[\tau] + \overline{\mathfrak{g}}_{k_1,k_2}[\tau]. \quad (5.21) $$
With the expansions (5.5) and (5.18) of $M_k[\tau]$ and $M_{k_1,k_2}[\tau]$ in terms of the modular forms $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}]$, the differential equations (5.7) and (5.21) are an alternative way of projecting the differential equation (3.30) of the equivariant series $\mathbb{I}^{\text{eqv}}$ to modular depth one and two. The projectors (5.15) to irreducible representations of $\mathfrak{sl}_2$ then isolate those combinations of $\beta^{\text{eqv}}$ from (5.18) which form an $\mathfrak{sl}_2$-multiplet under the raising and lowering operators $\nabla$ and $\bar{\nabla}$.

## 5.2 Modular completion at modular depth three

We will now repeat at modular depth three the analysis of the preceding section. The first step is to find an equivariant closed form involving triple integrals and the second step is to find a completion that is modular.

### 5.2.1 Towards equivariant integrals at modular depth three

As an analogue of $D_{k_1,k_2}[\tau]$ in (5.9) at modular depth three, we aim to construct a closed and equivariant one-form that comprises differentials of triple Eisenstein integrals. A first guess compatible with equivariance and the desired triple integrals is $\underline{G}_{k_1}[\tau]M_{k_2,k_3}[\tau] + M_{k_1,k_2}[\tau]\underline{G}_{k_3}[\tau]$, where we recall the convention that $M_{k_i,k_j}[\tau] := M_{k_i,k_j}[X_i, Y_i, X_j, Y_j; \tau]$. However, starting from this initial candidate one-form we see that closure requires the addition of two further terms, leading us to the definition

$$ D_{k_1,k_2,k_3}[\tau] := -\frac{1}{2} \left( \underline{G}_{k_1}[\tau] M_{k_2,k_3}[\tau] + M_{k_1,k_2}[\tau] \underline{G}_{k_3}[\tau] + \underline{f}_{k_1,k_2}[\tau] M_{k_3}[\tau] + M_{k_1}[\tau] \overline{G_{k_2,k_3}[\tau]} \right). \quad (5.22) $$

Closure of $D_{k_1,k_2,k_3}$ in the sense of (5.4) can be checked using (5.7) and (5.21).

Since all of $M_{k_i,k_j}[\tau]$, $\underline{f}_{k_i,k_j}[\tau]$ and $\overline{G_{k_i,k_j}[\tau]}$ depend on $X_i, Y_i, X_j, Y_j$, the closed and equivariant one-form $D_{k_1,k_2,k_3}[\tau]$ depends on three pairs $X_i, Y_i$ at $i = 1, 2, 3$. Considering its integral then gives the following lengthy expression

$$ \begin{aligned} K_{k_1,k_2,k_3}[\tau] = & -\frac{1}{8} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \int_{\tau_1}^{i\infty} \underline{G}_{k_2}[\tau_2] \int_{\tau_2}^{i\infty} \underline{G}_{k_3}[\tau_3] - \frac{1}{8} \int_{\tau}^{-i\infty} \overline{G_{k_3}[\tau_3]} \int_{\tau_3}^{-i\infty} \overline{G_{k_2}[\tau_2]} \int_{\tau_2}^{-i\infty} \overline{G_{k_1}[\tau_1]} \\ & -\frac{1}{8} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \times \int_{\tau}^{-i\infty} \overline{G_{k_3}[\tau_3]} \int_{\tau_3}^{-i\infty} \overline{G_{k_2}[\tau_2]} - \frac{1}{8} \int_{\tau}^{-i\infty} \overline{G_{k_3}[\tau_3]} \times \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \int_{\tau_1}^{i\infty} \underline{G}_{k_2}[\tau_2] \\ & -\frac{1}{4} c_{k_1} \int_{\tau}^{-i\infty} \overline{G_{k_3}[\tau_3]} \int_{\tau_3}^{-i\infty} \overline{G_{k_2}[\tau_2]} - \frac{1}{4} c_{k_2} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \times \int_{\tau}^{-i\infty} \overline{G_{k_3}[\tau_3]} - \frac{1}{4} c_{k_3} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \int_{\tau_1}^{i\infty} \underline{G}_{k_2}[\tau_2] \\ & +\frac{1}{4} \int_{\tau}^{-i\infty} \overline{G_{k_3}[\tau_3]} \times \int_{\tau}^{i\infty} \underline{f}_{k_1,k_2}[\tau_1] + \frac{1}{4} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \int_{\tau_1}^{i\infty} \underline{f}_{k_2,k_3}[\tau_2] + \frac{1}{4} \int_{\tau}^{i\infty} \underline{f}_{k_1,k_2}[\tau_1] \int_{\tau_1}^{i\infty} \underline{G}_{k_3}[\tau_3] \\ & +\frac{1}{4} \int_{\tau}^{-i\infty} \overline{G_{k_2,k_3}[\tau_2]} \int_{\tau_2}^{-i\infty} \overline{G_{k_1}[\tau_1]} + \frac{1}{4} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] \times \int_{\tau}^{-i\infty} \overline{G_{k_2,k_3}[\tau_2]} + \frac{1}{4} \int_{\tau}^{-i\infty} \overline{G_{k_3}[\tau_3]} \int_{\tau_3}^{-i\infty} \overline{G_{k_1,k_2}[\tau_1]} \\ & +\frac{1}{2} c_{k_3} \int_{\tau}^{i\infty} \underline{f}_{k_1,k_2}[\tau_1] + \frac{1}{2} c_{k_1} \int_{\tau}^{-i\infty} \overline{G_{k_2,k_3}[\tau_2]} + \frac{1}{2} c_{k_2,k_3} \int_{\tau}^{i\infty} \underline{G}_{k_1}[\tau_1] + \frac{1}{2} c_{k_1,k_2} \int_{\tau}^{-i\infty} \overline{G_{k_3}[\tau_3]}. \quad (5.23) \end{aligned} $$
The simple monomials $c_{k_i} \sim \zeta_{k_i-1} Y_i^{k_i-2}$ are given by (5.11), and the explicit form of $c_{k_i,k_j}$, $\underline{f}_{k_i,k_j}$ and $\underline{g}_{k_i,k_j}$ defined by (5.18) is discussed below (5.19).

### 5.2.2 Equivariant completion at modular depth three

Similar to the case of modular depth two, the combinations $K_{k_1,k_2,k_3}$ in (5.23) are not equivariant in spite of their construction by integrating an equivariant one-form (5.22). Still, it is possible to find an equivariant completion of $K_{k_1,k_2,k_3}$ by solely adding terms of modular depth zero and one. The existence of the equivariant completion once more follows from the Eichler-Shimura theorem whose application relies on a generalisation of the projector $\delta^j$ in (5.15) to triple integrals. More specifically, this projector needs to be designed to handle polynomials in three pairs of bookkeeping variables $X_i, Y_i$ (i.e. for $i = 1, 2, 3$, and not just for $i = 1, 2$). With this in mind, we first introduce the following generalisation of $\delta^\ell$ defined in (5.15) (for $\ell \ge 0$),

$$ \delta_{a,b}^{\ell} := m_{a,b} \circ \left( \frac{\partial}{\partial X_a} \otimes \frac{\partial}{\partial Y_b} - \frac{\partial}{\partial Y_a} \otimes \frac{\partial}{\partial X_b} \right)^{\ell}, \qquad (5.24) $$

where $m_{a,b}$ is the multiplication map $\mathbb{Q}[X_a, Y_a] \otimes \mathbb{Q}[X_b, Y_b] \to \mathbb{Q}[X_a, Y_a]$ setting $X_b = X_a$ and $Y_b = Y_a$ after evaluating the derivatives in (5.24). Using a pair of these projectors we can construct an equivariant projector $\delta^{\ell_1, \ell_2}$ via

$$ \delta^{\ell_1, \ell_2} := \delta_{1,2}^{\ell_1} \circ \delta_{2,3}^{\ell_2}, \qquad (5.25) $$

where we note that $\delta_{1,2}^\ell$ is simply the original $\delta^\ell$ in (5.15). For example we have

$$ \begin{aligned} \delta^{0,0}(X_1 X_2 X_3 Y_1 Y_2 Y_3) &= \delta_{1,2}^0(\delta_{2,3}^0(X_1 X_2 X_3 Y_1 Y_2 Y_3)) = \delta_{1,2}^0(m_{2,3}(X_1 X_2 X_3 Y_1 Y_2 Y_3)) \\ &= \delta_{1,2}^0(X_1 X_2^2 Y_1 Y_2^2) = m_{1,2}(X_1 X_2^2 Y_1 Y_2^2) = X_1^3 Y_1^3, \end{aligned} $$

$$ \delta^{0,1}(X_1^{a_1} X_2^{a_2} X_3^{a_3} Y_1^{b_1} Y_2^{b_2} Y_3^{b_3}) = (a_2 b_3 - a_3 b_2) X_1^{a_1+a_2+a_3-1} Y_1^{b_1+b_2+b_3-1}, \qquad (5.26) $$

$$ \delta^{1,0}(X_1^{a_1} X_2^{a_2} X_3^{a_3} Y_1^{b_1} Y_2^{b_2} Y_3^{b_3}) = (a_1(b_2+b_3)-(a_2+a_3)b_1)X_1^{a_1+a_2+a_3-1}Y_1^{b_1+b_2+b_3-1} $$

and the last two cases here presented illustrate that generically $\delta^{\ell_1, \ell_2} \neq \delta^{\ell_2, \ell_1}$. The homogeneity degrees with respect to the $X_i, Y_i$ translate into degrees $k_1+k_2+k_3 = 6 + \sum_{i=1}^3 (a_i+b_i)$ of the accompanying iterated Eisenstein integrals in (5.23). The order of writing the projectors as in (5.25) corresponds to a specific choice of putting parentheses when evaluating the triple tensor product (3.86) of $V(e_{k_i})$. We will comment more on the group-theoretic interpretation of $\delta^{\ell_1, \ell_2}$ in section 5.2.3.

Using the projector in (5.24) and (5.25), we can define the new function

$$ K_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau] := \frac{(i\pi)^{\ell_1+\ell_2}}{(\ell_1!)^2(\ell_2!)^2} \delta^{\ell_1,\ell_2} (K_{k_1,k_2,k_3}[\tau]), \qquad (5.27) $$

which, as before, is equivariant up to a cocycle depending on one pair $X_1, Y_1$ of bookkeeping variables. An application of the Eichler-Shimura theorem then gives a systematic completion $M_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}$ of $K_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}$ which results in an equivariant function:

$$ \begin{aligned} M_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau] ={}& K_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau] - c_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1] \\ & - \frac{1}{2} \left\{ \int_{\tau}^{i\infty} \underline{f}_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau_1] + \int_{\bar{\tau}}^{-i\infty} \underline{g}_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau_1] \right\}. \end{aligned} \qquad (5.28) $$
In perfect analogy with the modular-depth-two case discussed below (5.17), the $\tau$-independent $c_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1]$ are polynomials in the bookkeeping variables $X_1, Y_1$ of homogeneity degree $w-2 = k_1+k_2+k_3-2\ell_1-2\ell_2-6$. Moreover, $f_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau_1]$ is an equivariant (1,0)-form composed of holomorphic cusp forms of weight $w$ multiplied by $(X_1-\tau_1 Y_1)^{w-2}$, and $g_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau_1]$ is an equivariant (0,1)-form composed of antiholomorphic modular forms of weight $w$ multiplied by $(X_1-\bar{\tau}_1 Y_1)^{w-2}$.

Similar to our reasoning at modular depth two in section 5.1.3, we have projected the functions $K_{k_1,k_2,k_3}[X_1, Y_1, X_2, Y_2, X_3, Y_3; \tau]$ in (5.23) to the $K_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau]$ in (5.27) in order to infer the existence of the equivariant completion (5.28) from the Eichler-Shimura theorem. The union of the projected $M_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau]$ at all $\ell_1, \ell_2 \ge 0$ with $\ell_1+\ell_2 \le \frac{1}{2}(k_1+k_2+k_3)-3$ can be used to reconstruct an equivariant function

$$M_{k_1,k_2,k_3}[\tau] = K_{k_1,k_2,k_3}[\tau] - c_{k_1,k_2,k_3} - \frac{1}{2} \left\{ \int_{\tau}^{i\infty} \mathfrak{f}_{k_1,k_2,k_3}[\tau_1] + \int_{\bar{\tau}}^{-i\infty} \overline{g_{k_1,k_2,k_3}[\tau_1]} \right\}, \quad (5.29)$$

of three pairs $X_i, Y_i$ of bookkeeping variables subject to

$$M_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau] = \frac{(i\pi)^{\ell_1+\ell_2}}{(\ell_1!)^2(\ell_2!)^2} \delta^{\ell_1,\ell_2} (M_{k_1,k_2,k_3}[\tau]). \quad (5.30)$$

The same conventions apply to the extraction of the projected $c_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}$, $f_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}$ and $g_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}$ from the more general quantities $c_{k_1,k_2,k_3}$, $f_{k_1,k_2,k_3}$ and $g_{k_1,k_2,k_3}$ in (5.29).

In perfect analogy with (5.18) at modular depth two, the $M_{k_1,k_2,k_3}[\tau]$ in (5.29) admit an alternative description in terms of the equivariant functions $\beta^{\text{eqv}}[\begin{smallmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{smallmatrix}]$,

$$M_{k_1,k_2,k_3}[\tau] = \frac{-(k_1-1)!(k_2-1)!(k_3-1)!}{64} \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \sum_{j_3=0}^{k_3-2} \frac{\binom{k_1-2}{j_1} \binom{k_2-2}{j_2} \binom{k_3-2}{j_3}}{(-4y)^{j_1+j_2+j_3}} \beta^{\text{eqv}}[\begin{smallmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{smallmatrix}] \quad (5.31) \\ \times (X_1-\tau Y_1)^{j_1} (X_1-\bar{\tau} Y_1)^{k_1-2-j_1} (X_2-\tau Y_2)^{j_2} (X_2-\bar{\tau} Y_2)^{k_2-2-j_2} (X_3-\tau Y_3)^{j_3} (X_3-\bar{\tau} Y_3)^{k_3-2-j_3},$$

see appendix C.4 for a detailed proof of equivalence to (5.29).

In summary, the Eichler-Shimura theorem implies the existence of equivariant integrals $M_{k_1,k_2,k_3}^{(\ell_1,\ell_2)}[X_1, Y_1; \tau]$ at arbitrary degree $k_1+k_2+k_3$. Their multivariate combinations $M_{k_1,k_2,k_3}[\tau]$ obtained from assembling the contributions from all projectors $\delta^{\ell_1,\ell_2}$ at $0 \le \ell_1+\ell_2 \le \frac{1}{2}(k_1+k_2+k_3)-3$ determine the $\beta^{\text{eqv}}$ at modular depth three via (5.31). As a consequence, the generating-series construction of modular forms $\beta^{\text{eqv}}[\begin{smallmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{smallmatrix}]$ as in section 3 is bound to succeed at arbitrary degree, i.e. beyond the explicit checks at $k_1+k_2+k_3 \le 20$ we have performed.

### 5.2.3 $\delta^{\ell_1,\ell_2}$ versus $\mathfrak{sl}_2$ representations

Similar to the decomposition (5.18) of $M_{k_1,k_2}[\tau]$ at modular depth two, the projector $\delta^{\ell_1,\ell_2}$ in (5.25) isolates irreducible $\mathfrak{sl}_2$ representations of the tensor product $V(e_{k_1}) \otimes V(e_{k_2}) \otimes V(e_{k_3})$ comprising the coefficients $e_{k_1}^{(j_1)} e_{k_2}^{(j_2)} e_{k_3}^{(j_3)}$ of $\beta^{\text{eqv}}$ in $\mathfrak{J}^{\text{eqv}}(e_k; \tau)$. The particular choice of order in (5.25) is parallel to the way the triple tensor product was written in (3.86): The inner projector $\delta_{2,3}^{\ell_2}$ decomposes the product $V(e_{k_2}) \otimes V(e_{k_3})$ into irreducibles and the outer operator
$\delta_{1,2}^{\ell_1}$ then projects onto irreducibles in the product of these with $V(e_{k_1})$. All possible ways of putting parentheses when evaluating the triple tensor correspond to one of the three projectors $\delta_{1,2}^{\ell_1} \circ \delta_{2,3}^{\ell_2}$, $\delta_{1,2}^{\ell_1} \circ \delta_{1,3}^{\ell_2}$ and $\delta_{1,3}^{\ell_1} \circ \delta_{1,2}^{\ell_2}$. These projectors can be checked to have the correct linear dependencies expected from associativity of the tensor product. Therefore, it is sufficient to simply choose one ordering — as we have done in (5.25)—and to consider all possible values of $\ell_1$ and $\ell_2$.

At fixed $(k_1, k_2, k_3)$, the highest-dimensional representation $V(e_{k_1+k_2+k_3-4})$ in (3.86) occurs with multiplicity one and it is isolated by selecting the $\ell_1 = \ell_2 = 0$ instance of $\delta^{\ell_1, \ell_2}$ which preserves the homogeneity degree $k_1+k_2+k_3-6$ of (5.31) in $X_i, Y_i$. The $M_{k_1, k_2, k_3}^{(\ell_1, \ell_2)}[X_1, Y_1; \tau]$ in (5.30) obtained from $\delta^{\ell_1, \ell_2}$ at non-zero $\ell_1$ or $\ell_2$ have reduced homogeneity degree given by $k_1+k_2+k_3-2\ell_1-2\ell_2-6$. Accordingly, these $M_{k_1, k_2, k_3}^{(\ell_1, \ell_2)}[X_1, Y_1; \tau]$ correspond to sl2 modules $V(e_{k_1+k_2+k_3-2\ell_1-2\ell_2-4})$ whose reduction in dimensions by $2\ell_1+2\ell_2$ matches the shift of homogeneity degree by $\delta^{\ell_1, \ell_2}$.

In contrast to the sl2 modules in the tensor products $V(e_{k_1}) \otimes V(e_{k_2})$ at modular depth two, the $V(e_{k_1+k_2+k_3-2\ell_1-2\ell_2-4})$ isolated from $M_{k_1, k_2, k_3}^{(\ell_1, \ell_2)}[X_1, Y_1; \tau]$ at modular depth three have non-trivial multiplicities for generic $\ell_1, \ell_2 \ge 0$. For instance, the sl2 modules $V(e_{k_1+k_2+k_3-6})$ and $V(e_{k_1+k_2+k_3-8})$ in $V(e_{k_1}) \otimes V(e_{k_2}) \otimes V(e_{k_3})$ of subleading and subsubleading dimensions in (3.87) and (3.88) have multiplicities two and three, respectively. The multiplicities 2 and 3 in

$$V(e_{k_1}) \otimes V(e_{k_2}) \otimes V(e_{k_3}) = V(e_{k_1+k_2+k_3-4}) \oplus 2 \times V(e_{k_1+k_2+k_3-6}) \oplus 3 \times V(e_{k_1+k_2+k_3-8}) \oplus \dots \quad (5.32)$$

(with $V(e_k)$ at $k \le k_1+k_2+k_3-10$ in the ellipsis) are in one-to-one correspondence with the counting of independent projectors $\delta^{\ell_1, \ell_2}$ at a given sum $\ell_1+\ell_2 = 1, 2$. More specifically, the distinct equivariant functions $M_{k_1, k_2, k_3}^{(0,1)}[X_1, Y_1; \tau]$ and $M_{k_1, k_2, k_3}^{(1,0)}[X_1, Y_1; \tau]$ in (5.28) are expressible in terms of those $\beta^{\text{eqv}}[j_3 j_2 j_1]$ which occur in the projection of $\mathcal{J}^{\text{eqv}}(e_k; \tau)$ to the modules $2 \times V(e_{k_1+k_2+k_3-6})$ in (5.32). Similarly, the three distinct $M_{k_1, k_2, k_3}^{(\ell_1, \ell_2)}[X_1, Y_1; \tau]$ resulting from $\delta^{0,2}, \delta^{1,1}$ and $\delta^{2,0}$ comprise those combinations of $\beta^{\text{eqv}}[j_3 j_2 j_1]$ in the projection of $\mathcal{J}^{\text{eqv}}(e_k; \tau)$ to $3 \times V(e_{k_1+k_2+k_3-8})$.

For the projectors $\delta^{\ell_1, \ell_2}$ at higher $N = \ell_1+\ell_2 \ge 3$, however, not all of the $N+1$ partitions $\delta^{0,N}, \delta^{1,N-1}, \dots, \delta^{N-1,1}, \delta^{N,0}$ lead to linearly independent $M_{k_1, k_2, k_3}^{(\ell_1, \ell_2)}[X_1, Y_1; \tau]$ via (5.30). This corresponds to the fact that the largest representation $V(e_{k_1+k_2+k_3-10})$ in the ellipsis of (5.32) often appears with multiplicity $\le 3$, see for instance (3.87) and (3.88). In other words, relations among the $\delta^{0,3}, \delta^{1,2}, \delta^{2,1}, \delta^{3,0}$ actions on $M_{k_1, k_2, k_3}[\tau]$ explain the deviation from multiplicity $4 \times V(e_{k_1+k_2+k_3-10})$ as one may naively expect. The same effect is responsible for analogous multiplicity drops for $V(e_k)$ in (5.32) at $k \le k_1+k_2+k_3-12$.

Nevertheless, by exhausting the $\delta^{\ell_1, \ell_2}$ with all $\ell_1, \ell_2 \ge 0$ subject to the bound $\ell_1+\ell_2 \le \frac{1}{2}(k_1+k_2+k_3)-3$, one can be sure to capture each sl2 module in the triple tensor product (3.86) at least once. This is why the reconstruction of $M_{k_1, k_2, k_3}[\tau]$ from equivariant functions (5.28) will always be possible. On these grounds, the existence of equivariant $M_{k_1, k_2, k_3}^{(\ell_1, \ell_2)}[X_1, Y_1; \tau]$ resulting from the Eichler-Shimura theorem propagates to that of the $M_{k_1, k_2, k_3}[\tau]$ where the complete tensor product (3.86) is unified via three pairs $X_i, Y_i$ of bookkeeping variables.

Note that the combinations of $\beta^{\text{eqv}}$ in a given sl2 irreducible $M_{k_1, k_2, k_3}^{(\ell_1, \ell_2)}[X_1, Y_1; \tau]$ obtained from the projection (5.30) of (5.31) form a multiplet under the raising and lowering operators $\nabla$ and $\bar{\nabla}$.
### 5.2.4 Properties and examples of the equivariant completions

We shall next exploit the representation (5.31) of the equivariant triple integrals $M_{k_1,k_2,k_3}[\tau]$ to further specify the ingredients $c_{k_1,k_2,k_3}$, $\mathfrak{f}_{k_1,k_2,k_3}[\tau_1]$ and $\overline{\mathfrak{g}_{k_1,k_2,k_3}[\tau_1]}$ of the equivariant completion in (5.29). Similar to (5.20) at modular depth two, the $\tau$-independent $c_{k_1,k_2,k_3}$ are easily determined by truncating (5.31) to terms of modular depth zero,

$$c_{k_1,k_2,k_3} = \frac{1}{64}(k_1-1)!(k_2-1)!(k_3-1)! \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \sum_{j_3=0}^{k_3-2} (-1)^{j_1+j_2+j_3} \binom{k_1-2}{j_1} \binom{k_2-2}{j_2} \binom{k_3-2}{j_3} \\ \times \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \left(\frac{Y_2}{2\pi i}\right)^{j_2} X_2^{k_2-2-j_2} \left(\frac{Y_3}{2\pi i}\right)^{j_3} X_3^{k_3-2-j_3} c^{\text{sv}}\left[\begin{smallmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{smallmatrix}\right]. \quad (5.33)$$

As a consequence, the coefficients of $X_i$ and $\frac{Y_i}{2\pi i}$ in $c_{k_1,k_2,k_3}$ of degree $k_1+k_2+k_3 \le 16$ are single-valued MZVs, for instance

$$c_{4,4,6}^{(0,0)} = \frac{3Y_1^8}{2560\pi^8} \left( \zeta_{3,3,5}^{\text{sv}} - \frac{14573}{96}\zeta_{11} \right) - \frac{iX_1^3 Y_1^5 \zeta_3 \zeta_5}{7680\pi^5} + \frac{iX_1^5 Y_1^3 \zeta_3^2}{80640\pi^3} + \frac{X_1^6 Y_1^2 \zeta_5}{1382400\pi^2} - \frac{X_1^8 \zeta_3}{19353600}, \quad (5.34)$$

where the projection $c_{k_1,k_2,k_3}^{(\ell_1, \ell_2)}$ of $c_{k_1,k_2,k_3}$ is defined in analogy with (5.30). Starting from degree $k_1+k_2+k_3 = 18$, however, one additionally encounters the new periods $\overline{\omega}_{k,w}^{d_1, d_2}$ of section 3.3.3 among the coefficients of $c_{k_1,k_2,k_3}$. From the examples (3.78) of $c^{\text{sv}}$ involving new periods, one can straightforwardly generate instances of $\overline{\omega}_{k,w}^{d_1, d_2}$ in $c_{k_1,k_2,k_3}$ via (5.33).

Note that the $c_{k_1,k_2,k_3}^{(\ell_1, \ell_2)}$ in singlet representations of $\mathfrak{sl}_2$ (i.e. at $\ell_1+\ell_2 = \frac{1}{2}(k_1+k_2+k_3)-3$) are independent on $X_1, Y_1$ and therefore unconstrained by equivariance. By (5.33), this reflects the freedom to redefine combinations of $c^{\text{sv}}\begin{bmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{bmatrix}$ in $\mathfrak{sl}_2$ singlets as described in section 3.2.3 and at the end of section 3.3.4. In the context of the generating series $\mathcal{J}^{\text{eqv}}$, these ambiguities in $c^{\text{sv}}\begin{bmatrix} j_3 & j_2 & j_1 \\ k_3 & k_2 & k_1 \end{bmatrix}$ were fixed by imposing certain conditions on the arithmetic parts $\hat{z}_i, \hat{z}_\omega$ of the generators $\hat{\sigma}_i, \hat{\sigma}_\omega$. By the relations (5.31) and (5.33), our conventions for $\hat{z}_i, \hat{z}_\omega$ single out a preferred choice for $c_{k_1,k_2,k_3}$ and thereby for $M_{k_1,k_2,k_3}$.

The Eisenstein part of $\overline{\mathfrak{g}_{k_1,k_2,k_3}[\tau_1]}$ is determined by those terms in $M^{\text{sv}}(\hat{z}_i, \hat{z}_\omega)^{-1}\tilde{\mathbb{I}}_-(e_k; \tau)$ $M^{\text{sv}}(\hat{z}_i, \hat{z}_\omega)$ where $\tilde{\mathbb{I}}_-$ contributes with modular depth one and the change of alphabet maps the respective letter $e_k$ to $[e_{k_1}^{(j_1)}, [e_{k_2}^{(j_2)}, e_{k_3}^{(j_3)}]]$. These nested brackets can arise from two types of contributions from the change of alphabet via $M^{\text{sv}}$ — either from $[\hat{z}_{2n+1}, e_k]$ with coefficients $\zeta_{2n+1}$ or from $[\hat{z}_{2n+1}, [\hat{z}_{2n+2}, e_k]]$ with coefficients $\zeta_{2n+1}\zeta_{2n+2}$. Hence, the Eisenstein part of $\overline{\mathfrak{g}_{k_1,k_2,k_3}[\tau_1]}$ is given by (0, 1)-forms $\overline{\mathfrak{G}_k[\tau_1]}$ multiplied by $\mathcal{Q}[(2\pi i)^{\pm 1}]$-linear combinations of $\zeta_{2n+1}$ and $\zeta_{2n+1}\zeta_{2n+2}$.

Finally, the cusp forms entering $\mathfrak{f}_{k_1,k_2,k_3}[\tau_1]$ and $\overline{\mathfrak{g}_{k_1,k_2,k_3}[\tau_1]}$ are obtained from $\mathbb{I}_+(e_k; \tau)$ and $\tilde{\mathbb{I}}_-(e_k; \tau)$ by isolating the contributions $\sim [e_{k_1}^{(j_1)}, [e_{k_2}^{(j_2)}, e_{k_3}^{(j_3)}]]$ to the expansion of $e_{\Delta^\pm} = e_{\Delta^{\text{even}}} \mp e_{\Delta^{\text{odd}}}$.
periods $\Lambda_{k,w}^{d_1,d_2}$ of section 3.3.2. This is most conveniently exemplified via

$$ \left( \mathfrak{f}_{8,6,6}^{(0,0)}[X_1, Y_1; \tau], \overline{\mathfrak{g}_{8,6,6}^{(0,0)}}[X_1, Y_1; \tau] \right) = i\pi \left( \frac{2861}{748632192} \frac{\Lambda(\Delta_{16}, 17)}{\Lambda(\Delta_{16}, 15)} - \frac{9}{385} \frac{\Lambda_{6,14}^{2,2}}{\Lambda(\Delta_{16}, 15)} \right) \times \left( (X_1 - \tau Y_1)^{14} \Delta_{16}(\tau) \, \mathrm{d}\tau, -(X_1 - \bar{\tau} Y_1)^{14} \overline{\Delta_{16}(\tau)} \, \mathrm{d}\bar{\tau} \right). \quad (5.35) $$

In summary, the new periods in $\beta^{\text{eqv}}$ at modular depth three enter their alternative construction via $M_{k_1,k_2,k_3}$ through the equivariant completion of $K_{k_1,k_2,k_3}$ in (5.29). The two classes $\varpi_{k,w}^{d_1,d_2}$ and $\Lambda_{k,w}^{d_1,d_2}$ of new periods originate from $c_{k_1,k_2,k_3}$ and $\mathfrak{f}_{k_1,k_2,k_3}[\tau_1]$, $\overline{\mathfrak{g}_{k_1,k_2,k_3}[\tau_1]}$, respectively. In both cases, the need for new periods can be traced back to the multiple modular values $\mathfrak{m}[\begin{smallmatrix} j_1 & j_2 \\ k_1 & k_2 \end{smallmatrix}]$, $\mathfrak{m}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ and $\mathfrak{m}[\begin{smallmatrix} j_1 & j_2 \\ k_1 & \Delta_{k_2} \end{smallmatrix}]$ in the modular S-transformation of the iterated integrals in the expression (5.23) for $K_{k_1,k_2,k_3}$.

## 5.3 Equivariant integrals at arbitrary modular depth

The examples at modular depth up to three suggest a natural generalisation to equivariant iterated Eisenstein integrals at arbitrary modular depth. In these examples, the construction of equivariant integrals $M_{k_1,k_2}[\tau]$ and $M_{k_1,k_2,k_3}[\tau]$ was initiated through the identification of closed and equivariant one-forms $D_{k_1,k_2}[\tau]$ and $D_{k_1,k_2,k_3}[\tau]$ explicitly given in (5.9) and (5.22), respectively. One aim of this section is to generalise these formulae to closed and equivariant one-forms $D[\tau] = D_{k_1,k_2,...,k_n}[\tau]$ at arbitrary modular depth $n$.

Closure is a key feature of these forms, since it is this property that ensures homotopy invariance of their integrals $K[\tau] = K_{k_1,k_2,...,k_n}[\tau]$. Equivariance of the one-forms $D[\tau]$ in turn implies that the associated $K[\tau]$ function is equivariant up to a cocycle. We then used this equivariance to give a modular completion $M[\tau]$ of $K[\tau]$. The second aim of this section is to generalise this modular completion to any modular depth $n$. We will achieve both of the above goals via an inductive proof.

### 5.3.1 Setting up the inductive hypothesis and the base case

We will take the modular-depth-two example to be our base case and show how this lines up with a recursive formula we introduce below. Although it is not strictly necessary, we will often discuss the modular-depth-three case too, in order to familiarise the reader with the recursive definitions.

We now proceed to construct a closed one-form $D_{k_1,...,k_n}[\tau]$ at arbitrary modular depth $n$. For $n=1$, we recall the following functions from section 5.1:

$$ \begin{gathered} D_k[\tau] = \underline{\underline{G}}_k[\tau] + \overline{\overline{G}}_k[\tau], \qquad \mathfrak{f}_k[\tau] = 0, \quad (5.36) \\ M_k[\tau] = -\frac{1}{2} \left\{ \int_{\tau}^{i\infty} \underline{\underline{G}}_k[\tau_1] + \int_{\bar{\tau}}^{-i\infty} \overline{\overline{G}}_k[\tau_1] \right\} + c_k, \qquad \underline{\underline{g}}_k[\tau] = 0, \end{gathered} $$

where the $\tau$-independent quantities $c_k \sim \zeta_{k-1}Y^{k-2}$ are given by (5.11).

We then recursively define, for $n \ge 2$, the following one-forms:

$$ D_{k_1,k_2,...,k_n}[\tau] := -\frac{1}{2} \left( D_{k_1,k_2,...,k_n}^{+}[\tau] + D_{k_1,k_2,...,k_n}^{-}[\tau] \right), \quad (5.37) $$
with

$$D_{k_1, k_2, \dots, k_n}^{+}[\tau] := \underline{\mathcal{G}}_{k_1}[\tau] M_{k_2, k_3, \dots, k_n}[\tau] + \sum_{r=2}^{n-1} \underline{f}_{k_1, \dots, k_r}[\tau] M_{k_{r+1}, \dots, k_n}[\tau], \quad (5.38)$$

$$D_{k_1, k_2, \dots, k_n}^{-}[\tau] := M_{k_1, k_2, \dots, k_{n-1}}[\tau] \overline{\underline{\mathcal{G}}}_{k_n}[\tau] + \sum_{r=1}^{n-2} M_{k_1, \dots, k_r}[\tau] \overline{\underline{\mathcal{G}}}_{k_{r+1}, \dots, k_n}[\tau].$$

where, for $m < n$, all of $M_{k_1, \dots, k_m}[\tau]$, $\underline{f}_{k_1, \dots, k_m}[\tau]$ and $\overline{\underline{\mathcal{G}}}_{k_1, \dots, k_m}[\tau]$ are some equivariant functions satisfying the following differential equations:

$$2 \partial_{\tau} M_{k_1, \dots, k_m}[\tau] d\tau = D_{k_1, \dots, k_m}^{+}[\tau] + \underline{f}_{k_1, \dots, k_m}[\tau], \quad (5.39)$$

$$2 \partial_{\tau} M_{k_1, \dots, k_m}[\tau] d\bar{\tau} = D_{k_1, \dots, k_m}^{-}[\tau] + \overline{\underline{\mathcal{G}}}_{k_1, \dots, k_m}[\tau],$$

which are generalisations of (5.7) and (5.21). We wish to show that, for every $n$, the one-forms $D_{k_1, k_2, \dots, k_n}[\tau]$ are closed and equivariant, and that equivariant functions $M_{k_1, \dots, k_n}[\tau]$ satisfying (5.39) exist.

The instances of (5.37) at $n=2$ and $n=3$ are easily seen to reproduce the expressions (5.9) and (5.22) for $D_{k_1, k_2}[\tau]$ and $D_{k_1, k_2, k_3}[\tau]$, respectively. Similarly, for these $n$, equivariant $M[\tau]$, $\underline{f}[\tau]$ and $\overline{\underline{\mathcal{G}}}[\tau]$ satisfying (5.39) have already been shown to exist via the modular completions (5.18) and (5.29).

To summarise, at the base case $n=2$, we have the closure and equivariance of (5.37) and the existence of equivariant functions satisfying (5.39). For $n \ge 3$, we will now assume, as our inductive hypothesis, the closure and equivariance of $D_{k_1, k_2, \dots, k_m}[\tau]$ (5.37) and the existence of equivariant $M[\tau]$, $\underline{f}[\tau]$ and $\overline{\underline{\mathcal{G}}}[\tau]$ satisfying (5.39) at modular depth $m < n$. In the next section, we will prove, under this inductive hypothesis, that $D_{k_1, \dots, k_n}$ at modular depth $n$ is closed. In the following section 5.3.3, we will then show the existence of equivariant functions $M_{k_1, \dots, k_n}[\tau]$ satisfying (5.39) at modular depth $n$, completing our proof by induction.

### 5.3.2 Proof of equivariance and closure at modular depth $n$

We recall that $D_{k_1, k_2, \dots, k_n}$ is defined via $D_{k_1, k_2, \dots, k_n}^{+}$ and $\overline{\underline{\mathcal{G}}}_{k_1, k_2, \dots, k_n}$ according to (5.37) and these definitions, in turn, contain various $M[\tau]$, $\underline{f}[\tau]$ and $\overline{\underline{\mathcal{G}}}[\tau]$ at modular depth $n-1$ or lower. All of these objects are well defined and equivariant by our induction hypothesis. This observation, coupled with the fact that the $\underline{\mathcal{G}}_{k_1}[\tau]$ and $\overline{\underline{\mathcal{G}}}_{k_n}[\tau]$ are also equivariant, proves the equivariance of $D_{k_1, k_2, \dots, k_n}[\tau]$ at modular depth $n$.

The closure condition (5.4) that we need to prove at modular depth $n$ is equivalent to proving

$$\partial_{\tau} D_{k_1, \dots, k_n}^{+}[\tau] \wedge 2d\bar{\tau} = 2d\tau \wedge \partial_{\tau} D_{k_1, \dots, k_n}^{-}[\tau]. \quad (5.40)$$

Using (5.38) and (5.39), the left-hand side can be written as

$$\underline{\mathcal{G}}_{k_1}[\tau] \wedge (D_{k_2, \dots, k_n}^{-}[\tau] + \overline{\underline{\mathcal{G}}}_{k_2, \dots, k_n}[\tau]) + \sum_{r=2}^{n-2} \underline{f}_{k_1, \dots, k_r}[\tau] \wedge (D_{k_{r+1}, \dots, k_n}^{-}[\tau] + \overline{\underline{\mathcal{G}}}_{k_{r+1}, \dots, k_n}[\tau]) \\ + \underline{f}_{k_1, \dots, k_{n-1}}[\tau] \wedge \overline{\underline{\mathcal{G}}}_{k_n}[\tau], \quad (5.41)$$
where we have used the fact that $D_{k_n}^-[\tau] = \overline{\mathcal{G}}_{k_n}[\tau]$ and $\overline{\mathcal{G}}_{k_n}[\tau] = 0$. Similarly, we can write the right-hand side of (5.40) as

$$ (D_{k_1, \dots, k_{n-1}}^{+}[\tau] + \overline{f}_{k_1, \dots, k_{n-1}}[\tau]) \wedge \overline{\mathcal{G}}_{k_n}[\tau] + \sum_{r=2}^{n-2} (D_{k_1, \dots, k_r}^{+}[\tau] + \overline{f}_{k_1, \dots, k_r}[\tau]) \wedge \overline{\mathcal{G}}_{k_{r+1}, \dots, k_n}[\tau] \\ + \overline{\mathcal{G}}_{k_1}[\tau] \wedge \overline{\mathcal{G}}_{k_2, \dots, k_n}[\tau]. \qquad (5.42) $$

The terms in (5.41) and (5.42) not containing any $D^\pm$ match by inspection. The remaining terms in (5.41) and (5.42) can be shown to agree by inserting (5.38) for the $D^\pm$ which, as before, are of lower modular depth and well defined. In this setting, the terms involving $D^\pm$ in both (5.41) and (5.42) simplify to

$$ \overline{\mathcal{G}}_{k_1}[\tau] M_{k_2, \dots, k_{n-1}}[\tau] \wedge \overline{\mathcal{G}}_{k_n}[\tau] + \overline{\mathcal{G}}_{k_1}[\tau] \wedge \sum_{r=2}^{n-2} M_{k_2, \dots, k_r}[\tau] \overline{\mathcal{G}}_{k_{r+1}, \dots, k_n}[\tau] \\ + \sum_{r=2}^{n-2} \overline{f}_{k_1, \dots, k_r}[\tau] M_{k_{r+1}, \dots, k_{n-1}}[\tau] \wedge \overline{\mathcal{G}}_{k_n}[\tau] + \sum_{2 \le r < s \le n-2} \overline{f}_{k_1, \dots, k_r}[\tau] \wedge M_{k_{r+1}, \dots, k_s}[\tau] \overline{\mathcal{G}}_{k_{s+1}, \dots, k_n}[\tau]. \qquad (5.43) $$

Therefore, equation (5.40) holds and $D_{k_1, k_2, \dots, k_n}$ is closed.

### 5.3.3 Proving the existence of a modular completion at depth n

We have demonstrated the closure and equivariance of $D_{k_1, k_2, \dots, k_n}[\tau]$ in the previous section and, therefore, the following integral is equivariant up to a cocycle depending on n pairs $X_i, Y_i$ of commutative bookkeeping variables, $i = 1, 2, \dots, n$:

$$ K_{k_1, k_2, \dots, k_n}[\tau] = -\frac{1}{2} \left\{ \int_{\tau}^{i\infty} D_{k_1, k_2, \dots, k_n}^{+}[\tau_1] + \int_{\bar{\tau}}^{-i\infty} D_{k_1, k_2, \dots, k_n}^{-}[\tau_1] \right\}. \qquad (5.44) $$

The cancellation of its cocycle in multiple pairs $X_i, Y_i$ follows the logic of sections 5.1.3 and 5.2.2. We first introduce projectors analogous to (5.25) which reduce the cocycle from the integral over $D_{k_1, k_2, \dots, k_n}^\pm$ to a function of only one pair $X_1, Y_1$. One possible choice for the desired family of projectors is

$$ \delta^{\ell_1, \ell_2, \dots, \ell_{n-1}} := \delta_{1,2}^{\ell_1} \circ \delta_{2,3}^{\ell_2} \circ \dots \circ \delta_{n-1, n}^{\ell_{n-1}} \qquad (5.45) $$

with $\delta_{a,b}^\ell$ given by (5.24). This choice of projector corresponds to a particular way of putting parentheses when decomposing the n-fold tensor product $V(e_{k_1}) \otimes V(e_{k_2}) \otimes \dots \otimes V(e_{k_n})$ into irreducible representations of $\mathfrak{sl}_2$. This definition is a natural generalisation of (5.25). We formally allow for arbitrary non-negative values of the $\ell_i$, but in practice only a finite number of projectors will be non-zero when acting on a polynomial in the $(X_i, Y_i)$.

For a given choice of the integers $\ell_i$ characterising the projector (5.45), we reduce the function $K_{k_1, k_2, \dots, k_n}[\tau]$ of 2n bookkeeping variables $X_i, Y_i$ to a function of only $X_1, Y_1$:

$$ K_{k_1, k_2, \dots, k_n}^{(\ell_1, \dots, \ell_{n-1})}[X_1, Y_1; \tau] = \frac{(i\pi)^{\ell_1 + \dots + \ell_{n-1}} \delta^{\ell_1, \ell_2, \dots, \ell_{n-1}}}{(\ell_1!)^2 \dots (\ell_{n-1}!)^2} K_{k_1, k_2, \dots, k_n}[\tau], \qquad (5.46) $$
which is equivariant up to a cocycle now only depending on one pair $X_1, Y_1$ of bookkeeping variables. An application of the Eichler-Shimura theorem then gives a systematic completion which results in the equivariant function at fixed $\ell_1, \dots, \ell_{n-1}$:

$$ M_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}[X_1, Y_1; \tau] = K_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}[X_1, Y_1; \tau] - c_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}[X_1, Y_1] \tag{5.47} \\ - \frac{1}{2} \left\{ \int_{\tau}^{i\infty} \mathfrak{f}_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}[X_1, Y_1; \tau_1] + \int_{\bar{\tau}}^{-i\infty} \overline{\mathfrak{g}_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}[X_1, Y_1; \tau_1]} \right\}, $$

where $c_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}[X_1, Y_1]$ is a polynomial and

* $\mathfrak{f}_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}[X_1, Y_1; \tau_1]$ is a unique equivariant (1,0)-form composed of holomorphic cusp forms of weight $w = k_1 + \dots + k_n - 2(\ell_1+1) - \dots - 2(\ell_{n-1}+1)$ multiplying $(X_1 - \tau_1 Y_1)^{w-2}$,

* $\overline{\mathfrak{g}_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}[X_1, Y_1; \tau_1]}$ is a unique equivariant (0,1)-form composed of antiholomorphic modular forms of the same weight $w$ multiplying $(X_1 - \bar{\tau}_1 Y_1)^{w-2}$.

By constructing these equivariant completions $M_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}[X_1, Y_1; \tau]$ for all integers $\ell_i \ge 0$ in the finite range $\sum_{i=1}^{n-1} \ell_i \le \frac{1}{2}(k_1+k_2+\dots+k_n)-n$, one captures all irreducible representation of $\mathfrak{sl}_2$ in the tensor product $V(e_{k_1}) \otimes V(e_{k_2}) \otimes \dots \otimes V(e_{k_n})$ relevant for $M_{k_1,k_2,\dots,k_n}$ at least once. Hence, it is possible to construct an unprojected equivariant function

$$ M_{k_1,k_2,\dots,k_n}[\tau] = K_{k_1,k_2,\dots,k_n}[\tau] - c_{k_1,k_2,\dots,k_n} \tag{5.48} \\ - \frac{1}{2} \left\{ \int_{\tau}^{i\infty} \mathfrak{f}_{k_1,k_2,\dots,k_n}[\tau_1] + \int_{\bar{\tau}}^{-i\infty} \overline{\mathfrak{g}_{k_1,k_2,\dots,k_n}[\tau_1]} \right\}, $$

of $n$ pairs of bookkeeping variables subject to

$$ M_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}[X_1, Y_1; \tau] = \frac{(i\pi)^{\ell_1+\dots+\ell_{n-1}} \delta^{\ell_1,\ell_2,\dots,\ell_{n-1}}}{(\ell_1!)^2 \dots (\ell_{n-1}!)^2} M_{k_1,k_2,\dots,k_n}[\tau], \tag{5.49} $$

and analogous conditions for $c_{k_1,k_2,\dots,k_n}, \mathfrak{f}_{k_1,k_2,\dots,k_n}[\tau_1]$ and $\overline{\mathfrak{g}_{k_1,k_2,\dots,k_n}[\tau_1]}$. The functions defined in (5.48) can be seen to satisfy the differential equation (5.39).

To summarise, under our induction hypothesis, we have proven the closure and equivariance of $D_{k_1,k_2,\dots,k_n}$ and shown the existence of equivariant $M_{k_1,k_2,\dots,k_n}[\tau], \mathfrak{f}_{k_1,k_2,\dots,k_n}[\tau]$ and $\overline{\mathfrak{g}_{k_1,k_2,\dots,k_n}[\tau]}$ satisfying (5.39). Therefore, by induction, $D_{k_1,k_2,\dots,k_n}$ is equivariant and closed and equivariant $M_{k_1,k_2,\dots,k_n}, \mathfrak{f}_{k_1,k_2,\dots,k_n}$ and $\overline{\mathfrak{g}_{k_1,k_2,\dots,k_n}}$ satisfying (5.39) exist, for all $n \ge 2$. Furthermore, we can always assume that $M_{k_1,k_2,\dots,k_n}$ is the modular completion of $K_{k_1,k_2,\dots,k_n}$ (5.44) and takes the form given by (5.48).

### 5.3.4 Consequences for $\beta^{\text{eqv}}$ at modular depth $n$

The equivariant integrals $M_{k_1,k_2}$ and $M_{k_1,k_2,k_3}$ offer an equivalent description of the modular forms $\beta^{\text{eqv}}$ at modular depths two and three via (5.18) and (5.31), respectively. The construction of equivariant $M_{k_1,k_2,\dots,k_n}$ at arbitrary modular depth should similarly capture the $\beta^{\text{eqv}}$ at all orders in the generating series $\mathfrak{J}^{\text{eqv}}$ for suitable choices of the $\tau$-independent
$c_{k_1, k_2, \dots, k_n}$ in the form

$$
M_{k_1, k_2, \dots, k_n}[\tau] = \left(-\frac{1}{4}\right)^n (k_1-1)!(k_2-1)! \dots (k_n-1)! \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \dots \sum_{j_n=0}^{k_n-2} \frac{\binom{k_1-2}{j_1} \binom{k_2-2}{j_2} \dots \binom{k_n-2}{j_n}}{(-4y)^{j_1+j_2+\dots+j_n}} \\
\times \beta^{\text{eqv}} \begin{bmatrix} j_n & \dots & j_2 & j_1 \\ k_n & \dots & k_2 & k_1 \end{bmatrix} (X_1-\tau Y_1)^{j_1} (X_1-\bar{\tau} Y_1)^{k_1-2-j_1} \dots (X_n-\tau Y_n)^{j_n} (X_n-\bar{\tau} Y_n)^{k_n-2-j_n}. \tag{5.50}
$$

The sl₂ irreducible representations (5.49) obtained from the $\delta^{\ell_1, \ell_2, \dots, \ell_{n-1}}$ projections of (5.50) then feature combinations of $\beta^{\text{eqv}}$ where $\nabla$ and $\bar{\nabla}$ act as raising and lowering operators.

In the same way as the $c_{k_1,k_2,\dots,k_n}^{(\ell_1,\dots,\ell_{n-1})}$ in sl₂-singlet representations are unconstrained by equivariance, there is an analogous freedom of redefining the modular-depth-zero contributions $c^{\text{sv}}\begin{bmatrix} j_1 & \cdots & j_\ell \\ k_1 & \cdots & k_\ell \end{bmatrix}$ to $\beta^{\text{eqv}}\begin{bmatrix} j_1 & \cdots & j_\ell \\ k_1 & \cdots & k_\ell \end{bmatrix}$ in an sl₂-invariant way. As detailed in section 3.2.3 and at the end of section 3.3.4, the ambiguities in the constants $c^{\text{sv}}$ entering the generating series $\mathcal{J}^{\text{eqv}}$ are fixed by imposing certain conditions on its arithmetic generators $\hat{z}_i, \hat{z}_\omega$. This choice can be propagated to fix ambiguities in $M_{k_1,k_2,\dots,k_n}$ by imposing the following corollary of (5.50) iteratively in $n$:

$$
c_{k_1, k_2, \dots, k_n} = - \left(-\frac{1}{4}\right)^n (k_1-1)!(k_2-1)! \dots (k_n-1)! \sum_{j_1=0}^{k_1-2} \sum_{j_2=0}^{k_2-2} \dots \sum_{j_n=0}^{k_n-2} \frac{\binom{k_1-2}{j_1} \binom{k_2-2}{j_2} \dots \binom{k_n-2}{j_n}}{(-4y)^{j_1+j_2+\dots+j_n}} \\
\times c^{\text{sv}} \begin{bmatrix} j_n & \dots & j_2 & j_1 \\ k_n & \dots & k_2 & k_1 \end{bmatrix} \left(\frac{Y_1}{2\pi i}\right)^{j_1} X_1^{k_1-2-j_1} \dots \left(\frac{Y_n}{2\pi i}\right)^{j_n} X_n^{k_n-2-j_n}. \quad (5.51)
$$

We should highlight another advantage of the generating-series approach to $\beta^{\text{eqv}}$ over the alternative construction via (5.50): the structure of the generating series $\mathcal{J}^{\text{eqv}}$ — in particular the conjectural organisation of MZVs through the group-like series $M^{\text{sv}}$ in (3.74) and (3.75) — fixes the appearance of arbitrary (products of) MZVs from that of odd Riemann zeta values. This has implications on the MZVs entering all of $c_{k_1, \dots, k_n}$, $\tilde{f}_{k_1, \dots, k_n}[\tau_1]$ and $\overline{g}_{k_1, \dots, k_n}[\tau_1]$.

Still, both approaches to the construction of $\beta^{\text{eqv}}$ have in common that the new periods at modular depth $n$ (i.e. generalisations of the $\Lambda_{k,w}^{d_1,d_2}$ and $\bar{\omega}_{k,w}^{d_1,d_2}$ of section 3.3) have to be determined from modularity. By imposing equivariance of (5.48), the constants $c_{k_1, \dots, k_n}$ in non-singlet representations of $\mathfrak{sl}_2$ as well as the real coefficients of the modular forms in $\tilde{f}_{k_1, \dots, k_n}[\tau_1]$ and $\overline{g}_{k_1, \dots, k_n}[\tau_1]$ at modular depth $n$ are determined by the MMVs (2.14) and (2.17) from the $S$-transformation of $K_{k_1, \dots, k_n}[\tau]$.

In the formulation via $\mathcal{J}^{\text{eqv}}$, the new periods enter via $c^{\text{sv}}$, $\hat{\psi}^{\text{sv}}$ as well as higher-order terms in the expansion of the letters $e_{\Delta \pm}$ in (3.73). Their detailed expressions in terms of MMVs are generated by the $S$-cocycle condition (4.44). Within the discussion of section 4.2, the existence of a series $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\omega)$ subject to (4.44) was conjectural and relied on order-by-order checks at increasing modular depth and degree. The inductive proof of this section together with the relation (5.50) in turn establishes the existence of a solution to (4.44) at arbitrary modular depth and degree.

In order to determine the new periods entering $c_{k_1, \dots, k_n}$, $\tilde{f}_{k_1, \dots, k_n}[\tau_1]$ and $\overline{g}_{k_1, \dots, k_n}[\tau_1]$ at a given modular depth and degree, one would still need additional information on the MMVs entering the $S$-transformation of $K_{k_1, \dots, k_n}[\tau]$. From the methods of this work, one would have to test order by order whether the $S$-cocycle of $K_{k_1, \dots, k_n}[\tau]$ can be expressed in terms of
(sums of products of) known periods and add a minimal set of new ones whenever this is
not the case. This kind of case-by-case study for instance leads to the new periods $\Lambda_{k,w}^{d_1,d_2}$
and $\overline{\varpi}_{k,w}^{d_1,d_2}$ at modular depth three and degree $\le 20$ described in section 3.3 which were
initially derived from $\mathcal{J}^{\text{eqv}}$.

As another open problem for the future, the procedure of this section to determine $c_{k_1, \dots, k_n}$,
$f_{k_1, \dots, k_n}[\tau_1]$ and $g_{k_1, \dots, k_n}[\overline{\tau_1}]$ from MMVs does not manifest any correlations among the MZVs.
More specifically, beyond our explicit checks at modular depth three and degree $\le 20$ it
is still conjectural that the appearances of $\zeta_{i_k}$ and $\rho^{-1}(\text{sv}(f_{i_1} f_{i_2} \dots f_{i_\ell}))$ in $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\varpi)$ are
interlocked through the expansion (2.90) of the generating series $M^{\text{sv}}(\hat{\sigma}_i)$ of its MZV sector.

# 6 Conclusion and outlook

In this work, we have connected different perspectives on non-holomorphic modular forms built from iterated integrals of holomorphic modular forms. The explicit construction of such modular forms was advanced in several directions, and elegant number-theoretic structures have been identified in the coefficients of the iterated integrals.

A first line of results concerns modular graph forms (MGFs) known from the low-energy
expansion of genus-one string amplitudes [5, 6]. Following Brown's generating-series approach
to assemble MGFs from iterated Eisenstein integrals [14], we give the first explicit description
of all contributions from single-valued multiple zeta values (MZVs). The key ingredients in
our construction are so-called zeta generators (see section 2.4) that conspire with the non-
commutative variables $\epsilon_k$ in the generating series. As a main result of this work, we find simple
composition rules for zeta generators which interlock products and higher-depth instances of
MZVs with the appearance of odd Riemann zeta values $\zeta_{2k+1}$ as explained in section 3.2.

A more general class of non-holomorphic modular forms is obtained by augmenting
iterated Eisenstein integrals by holomorphic cusp forms as additional integration kernels.
The modular completions of double Eisenstein integrals via cusp forms are known to feature
odd Riemann zeta values and L-values of holomorphic cusp forms [13, 37]. Another main
result of this work is the systematic construction of the analogous modular completions of
triple Eisenstein integrals. Apart from double integrals that mix holomorphic Eisenstein
series and cusp forms, we find irreducible MZVs of depth $\ge 3$ and two classes of new periods.
This is explained in detail in section 3 with general arguments related to the modular
properties of the generating series in section 4. The existence of similar modular completions
is established for iterated integrals over an arbitrary number of Eisenstein series by the more
direct construction methods in section 5.

Our results motivate and support a variety of directions of follow-up research:

* The use of zeta generators in this work manifests striking parallels between the constructions of single-valued genus-zero polylogarithms in one variable and modular graph forms at genus one, see section 3.2.4. It would be rewarding to pinpoint echoes of these parallels in the multi-variable case and at higher genus. For instance, the description of single-valued polylogarithms in $n$ variables via zeta generators [50] may shed light on the appearance of MZVs in elliptic modular graph forms [97–99] depending on $n-1$ marked points on a torus. At higher genus, it remains to find realisations of zeta generators
that introduce MZVs into the respective modular graph forms [100] and tensors [101],
starting with the appearance of $\zeta_3$ in the expansion of the Zhang-Kawazumi invariant
in [102].

* In this work, zeta generators are applied to modular completions of iterated Eisenstein integrals where the integration variables are modular parameters. The tight interplay of zeta generators with the fundamental group of the once punctured torus [34] suggests to reformulate and extend our results at the level of configuration-space integrals over marked points on genus-one surfaces. More specifically, one could try to (i) organize the MZVs in the $\tau \to i\infty$ asymptotics of elliptic associators [103] via zeta generators, (ii) describe the low-energy expansions of the open- and closed-string integrals in [29, 104, 105] via matrix representations of zeta generators, (iii) connect the proposal for a single-valued map between open- and closed-string integrals at genus one [10] with the more recent double-copy formulae for integrals over marked points on a torus [106–109].

* The new periods encountered in the modular completion of triple Eisenstein integrals call for an organizing principle akin to the *f*-alphabet description of MZVs [32, 33]. It is tempting to aim for an extension of the group-like series $M^{\text{sv}}$ in MZVs to accommodate L-values of holomorphic cusp forms and the new periods that start appearing at the level of triple integrals. Such extensions of $M^{\text{sv}}$ should involve the counterparts of zeta generators for non-critical L-values discussed in [38], see section 3.3 for preliminary suggestions. Moreover, a group-like series featuring L-values and beyond may offer a natural interpretation of the composite letters $e_{\Delta\pm}$ accompanying the holomorphic cusp forms in the generating series of this work: The expansions of $e_{\Delta\pm}$ in terms of brackets of Eisenstein letters $e_k$ could be traced back to conjugations of elementary letters for cusp forms [82] with suitable extensions or cuspidal analogues of $M^{\text{sv}}$.

* The Laplace equations discussed in section 3.4 simplify the integration of modular-invariant MGFs over the fundamental domain of SL(2, $\mathbb{Z}$) as demonstrated in [1, 110-112]. Our results on equivariant iterated Eisenstein integrals of arbitrary modular depth in section 5.3 are a first step to extend the integration techniques of the references to MGFs of modular depth four and higher. Since having the zero-mode of MGFs with respect to $\text{Re}(\tau)$ is enough to integrate them over the fundamental domain [112-115], another possible investigation is to take advantage of the contributions $\sim (q\bar{q})^n$ to the modular invariants in this work exposed by their iterated-integral representations. With this approach, we can gain valuable insights into the number-theoretic properties of the integrated MGFs and evaluate previously inaccessible coefficients of low-energy interactions $D^w R^4$ of type-II superstrings with $w \ge 14$, going beyond the state of the art on $D^{\le 12}R^4$ [2, 63, 116] summarised with a view to transcendentality in [7].

* Modular forms involving double Eisenstein integrals were obtained from Poincaré series of integrals over a single Eisenstein series [37, 74]. It would be interesting to generalize these results to new representations of modular triple integrals in terms of Poincaré series of suitable double integrals and to make contact with the cuspidal Poincaré series
of [117]. These connections between triple, double and single integrals should contain key information on the systematics of how Poincaré series relate iterated integrals over different numbers of holomorphic modular forms.

Acknowledgments

We are grateful to Francis Brown, Emiel Claasen, Eric D'Hoker, Benjamin Enriquez, Hadleigh Frost, Michael Green, Deepak Kamlesh, Carlos Rodriguez and Congkao Wen for combinations of inspiring discussions, collaboration on related topics and helpful comments on a draft version of this work. In particular, we are indebted to Francis Brown for bringing the zeta generators $\sigma_w$ to our attention. The authors would like to thank the organisers of the workshops “Geometries and Special Functions for Physics and Mathematics” at the BCTP Bonn and “New connections between physics and number theory” at the Pollica Physics Centre for creating a stimulating atmosphere. AK would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Black holes: bridges between number theory and holographic quantum information” where work on this paper was undertaken.

This research was supported in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP). This work was supported by EPSRC grant no EP/R014604/1. The research of MD was supported by the IMPRS for Mathematical and Physical Aspects of Gravitation, Cosmology and Quantum Field Theory. JD is supported by the Royal Society (Spectral theory of automorphic forms: trace formulas and more). MH, OS and BV are partially supported by the European Research Council under ERC-STG-804286 UNISCAMP. MH and BV are furthermore supported by the Knut and Alice Wallenberg Foundation under grants KAW 2018.0116 and KAW 2018.0162, respectively. MH is also partially supported by the Swiss National Science Foundation through the NCCR SwissMAP. The research of OS is partially supported by the strength area “Universe and mathematical physics” which is funded by the Faculty of Science and Technology at Uppsala University.

A Multiple zeta values and f-alphabet

This appendix gathers basics of multiple zeta values (MZVs), their f-alphabet description and their single-valued map. We follow conventions where MZVs, $\zeta_{n_1,n_2,...,n_r}$, of depth $r$ are defined by nested sums over integers $k_i$

$$ \zeta_{n_1,n_2,...,n_r} := \sum_{1 \le k_1 < k_2 < ... < k_r}^{\infty} k_1^{-n_1} k_2^{-n_2} ... k_r^{-n_r}, \qquad (A.1) $$

indexed by integers $n_i$ in the range $n_1, ..., n_{r-1} \ge 1$ and $n_r \ge 2$. MZVs admit an alternative description in terms of multiple polylogarithms (3.60) at unit argument $z = 1$,

$$ \zeta_{n_1,n_2,...,n_r} = (-1)^r G(\underbrace{0, ..., 0}_{n_r-1}, 1, ..., \underbrace{0, ..., 0}_{n_2-1}, 1, ..., \underbrace{0, ..., 0}_{n_1-1}, 1; 1). \qquad (A.2) $$

By combining the properties of their representations via nested sums and iterated integrals, one can generate infinite families of relations among MZVs over $\mathbb{Q}$. All known $\mathbb{Q}$-relations
among MZVs preserve the weight $w = n_1+n_2+...+n_r$ of $\zeta_{n_1,n_2,...,n_r}$ and can therefore be solved weight by weight. From the joint effort of all currently known Q-relations up to weight $w = 18$, for instance, one can express all MZVs in this range via products of Riemann zeta values $\zeta_n$ (i.e. MZVs at depth one) and the following list of conjecturally indecomposable higher-depth MZVs [118]:

<table><thead><tr><th>w</th><th>basis MZVs</th><th>w</th><th>basis MZVs</th></tr></thead><tbody><tr><td>8</td><td>ζ<sub>3,5</sub></td><td>14</td><td>ζ<sub>3,3,3,5</sub>, ζ<sub>3,11</sub>, ζ<sub>5,9</sub></td></tr><tr><td>10</td><td>ζ<sub>3,7</sub></td><td>15</td><td>ζ<sub>5,3,7</sub>, ζ<sub>3,3,9</sub>, ζ<sub>1,1,3,4,6</sub></td></tr><tr><td>11</td><td>ζ<sub>3,3,5</sub></td><td>16</td><td>ζ<sub>3,3,3,7</sub>, ζ<sub>3,3,5,5</sub>, ζ<sub>3,13</sub>, ζ<sub>5,11</sub>, ζ<sub>1,1,6,8</sub></td></tr><tr><td>12</td><td>ζ<sub>3,9</sub>, ζ<sub>1,1,4,6</sub></td><td>17</td><td>ζ<sub>3,3,3,3,5</sub>, ζ<sub>1,1,3,6,6</sub>, ζ<sub>5,5,7</sub>, ζ<sub>3,3,11</sub>, ζ<sub>5,3,9</sub>, ζ<sub>3,5,9</sub></td></tr><tr><td>13</td><td>ζ<sub>3,3,7</sub>, ζ<sub>3,5,5</sub></td><td>18</td><td>ζ<sub>3,15</sub>, ζ<sub>5,13</sub>, ζ<sub>1,1,6,10</sub>, ζ<sub>3,5,5,5</sub>, ζ<sub>5,3,3,7</sub>, ζ<sub>3,3,3,9</sub>, ζ<sub>3,5,3,7</sub>, ζ<sub>1,1,3,3,4,6</sub></td></tr></tbody></table>

## A.1 The f-alphabet

The systematics of conjecturally indecomposable higher-depth MZVs in (A.3) can be conveniently captured by mapping them to certain non-commutative variables $f_k$ known as the f-alphabet [32, 33]. However, in view of the conjectural status of Q-linear independence results for MZVs and their products, one performs the mapping into the f-alphabet at the level of so-called motivic MZVs $\zeta_{n_1,n_2,...,n_r}^m$. By their definition in the algebraic-geometry literature [31, 32, 119, 120], motivic MZVs obey all currently known Q-relations of their real counterparts in (A.1) but are otherwise independent over Q.

The key idea behind the f-alphabet is to map motivic MZVs to a Hopf-algebra comodule

$$ \mathcal{F} := \mathbb{Q}[f_2] \otimes_{\mathbb{Q}} \mathbb{Q}\langle f_3, f_5, f_7, \dots \rangle, \quad (A.4) $$

spanned by monomials in non-commutative generators $f_3, f_5, f_7, \dots$ of odd degree and a single commutative generator $f_2$ of even degree. Given an isomorphism $\rho$ mapping $\zeta_{n_1,n_2,...,n_r}^m$ to non-commutative polynomials in $\mathcal{F}$, all Q-relations among motivic MZVs are exposed. Such isomorphisms $\rho$ are taken to obey the normalisation condition on Riemann zeta values

$$ \rho(\zeta_n^m) = f_n, \quad (A.5) $$

(with $f_{2k} := \frac{\zeta_{2k}}{(\zeta_2)^k} f_2^k \in \mathbb{Q} f_2^k$ in case of even degree) and to translate products of motivic MZVs into shuffles$^{28}$ (with $\vec{n}, \vec{m} \in \mathbb{N}^\times$),

$$ \rho(\zeta_n^m \cdot \zeta_m^m) = \rho(\zeta_n^m) \sqcup \rho(\zeta_m^m). \quad (A.6) $$

Most importantly, $\rho$-isomorphisms are required to translate the Goncharov-Brown coaction $\Delta_G$ of motivic MZVs [31–33, 121] into the simple deconcatenation coaction $\Delta_{\text{dec}}$ in $\mathcal{F}^{29}$

$$ \rho(\Delta_G(\zeta_{n_1,n_2,...,n_r}^m)) = \Delta_{\text{dec}}(\rho(\zeta_{n_1,n_2,...,n_r}^m)), \quad (A.7) $$

$$ \Delta_{\text{dec}}(f_2^n f_{i_1} f_{i_2} \cdots f_{i_r}) = \sum_{j=0}^{r} (f_2^n f_{i_1} f_{i_2} \cdots f_{i_j}) \otimes f_{i_{j+1}} \cdots f_{i_r}. $$

$$ f_2^a f_{i_1} \cdots f_{i_r} \sqcup f_2^b f_{i_{r+1}} \cdots f_{i_s} = f_2^{a+b} (f_{i_1} \cdots f_{i_r} \sqcup f_{i_{r+1}} \cdots f_{i_s}), \quad i_j \in 2\mathbb{N}+1. $$

<footnote>$^{28}$The shuffle product in $\mathcal{F}$ is understood to act trivially on the commutative generator $f_2$,</footnote>

<footnote>$^{29}$We note that $f_2$ here is only mapped to the left factor in the tensor product under the coaction. This choice is not uniform in the literature.</footnote>
The conditions (A.5) to (A.7) determine the $\rho$-image of motivic MZVs $\zeta_{n_1, n_2, \dots, n_r}^m$ of depth $r \ge 2$ up to the coefficient of $f_{n_1+n_2+\dots+n_r}$.

A canonical choice of these leftover coefficients, i.e. a canonical choice of isomorphism $\rho$, is described in a companion paper [34] and, from now on and everywhere in this work, $\rho$ will denote this preferred isomorphism. As discussed in [34], this leads to the following $f$-alphabet images of the simplest indecomposable motivic MZVs in (A.3)

$$ 
\begin{align} 
\rho(\zeta_{3,5}^{\mathbf{m}}) &= -5f_3f_5 + \frac{100471}{35568}f_8, \tag{A.8} \\ 
\rho(\zeta_{3,7}^{\mathbf{m}}) &= -14f_3f_7 - 6f_5f_5 + \frac{408872741707}{40214998720}f_{10}, \\ 
\rho(\zeta_{3,3,5}^{\mathbf{m}}) &= -5f_3f_3f_5 - 45f_2f_9 - \frac{6}{5}f_2^2f_7 + \frac{4}{7}f_2^3f_5 + \frac{1119631493}{14735232}f_{11}. \nonumber 
\end{align} 
$$

The large integers appearing in the denominators of the coefficients of $f_w$ in (A.8) are purely artifacts of the choice of $\zeta_{3,5}^m, \zeta_{3,7}^m$ etc. as irreducibles (similar artifacts occur in (A.9) and (A.13) below). In the companion paper [34] we give a natural method to choose weight-$w$ irreducibles such that the coefficients of the $f_w$ in their image under the canonical map $\rho$ is zero.

Being an isomorphism, $\rho$ is invertible, and one can retrieve arbitrary motivic MZVs from $\mathbb{Q}$-linear combinations of $\rho^{-1}(f_2^n f_{i_1} f_{i_2} \dots f_{i_r})$ for suitable choices of $n$ and $i_j$. One can readily check at fixed weight that the indecomposable motivic MZVs obtained from (A.3) are in one-to-one correspondence with compositions $f_{i_1} f_{i_2} \dots \ne f_{i_2} f_{i_1} \dots$ with $i_j \in 2\mathbb{N}+1$ that cannot be written as shuffles. For instance, (A.8) relates the existence of indecomposable $\zeta_{3,5}^m$ and $\zeta_{3,3,5}^m$ to $\rho^{-1}(f_3 f_5) \ne \rho^{-1}(f_5 f_3)$ and $\rho^{-1}(f_3 f_3 f_5)$ being different from $\rho^{-1}(f_3 f_5 f_3)$ and $\rho^{-1}(f_5 f_3 f_3)$. We note that, starting from weights $w = 12, 15, \dots$, the minimal depth of some of the motivic MZVs in a $\mathbb{Q}$-basis necessarily exceeds the number of odd letters in their $f$-alphabet images. For instance, the two shuffle independent combinations $\rho^{-1}(f_3 f_9)$ and $\rho^{-1}(f_5 f_7)$ at $w = 12$ necessitate at least one motivic MZV of depth $\ge 4$ such as $\zeta_{1,1,4,6}$ in (A.3) [118, 122].

By a slight abuse of notation, we will drop the motivic superscript when stating explicit $\rho^{-1}$-images in the remainder of this appendix and in the main body of the paper, for instance

$$ 
\begin{align} 
\rho^{-1}(f_3 f_5) &= -\frac{1}{5} \zeta_{3,5} + \frac{100471}{177840} \zeta_8, \tag{A.9} \\ 
\rho^{-1}(f_3 f_7) &= -\frac{1}{14} \zeta_{3,7} - \frac{3}{14} \zeta_5^2 + \frac{408872741707}{563009982080} \zeta_{10}, \\ 
\rho^{-1}(f_3 f_3 f_5) &= \frac{4}{35} \zeta_2^3 \zeta_5 - \frac{6}{25} \zeta_2^2 \zeta_7 - 9 \zeta_2 \zeta_9 + \frac{1}{5} \zeta_{3,3,5} + \frac{1119631493}{73676160} \zeta_{11}. \nonumber 
\end{align} 
$$

In other words, $\rho^{-1}$ in (A.9) and below is understood to automatically comprise the period map $\zeta_{n_1, n_2, \dots, n_r}^m \mapsto \zeta_{n_1, n_2, \dots, n_r}$ which is well-defined in spite of the currently unproven statements for MZVs (see e.g. the textbooks [123, 124] for an overview). Note that (A.9) determines $\rho^{-1}$ images of permutations in $f_{i_j}$ via shuffle relations (A.6), e.g. $\rho^{-1}(f_{i_1} \sqcup f_{i_2}) = \zeta_{i_1} \zeta_{i_2}$.

Before spelling out additional $\rho^{-1}$-images relevant to the group-like series $M^{\text{sv}}$ in (2.90), we shall briefly review the single-valued map $\text{sv}$.
## A.2 Single-valued MZVs

In the same way as MZVs are multiple polylogarithms at unit argument, see (A.2), one arrives at so-called single-valued MZVs by evaluating single-valued polylogarithms at $z = 1$ [20, 21]

$$ \zeta_{n_1, n_2, \dots, n_r}^{\text{sv}} = (-1)^r G^{\text{sv}}(\underbrace{0, \dots, 0}_{n_r-1}, 1, \dots, \underbrace{0, \dots, 0}_{n_2-1}, 1, \underbrace{0, \dots, 0}_{n_1-1}, 1; 1). \quad (A.10) $$

The single-valued polylogarithms, $G^{\text{sv}}$, in one variable were firstly constructed in [51] and generalised to multiple variables in [125, 126]. With the ingredients reviewed in this work, single-valued polylogarithms $G^{\text{sv}}(a_1, \dots, a_r; z)$ at $a_i \in \{0, 1\}$ can also be obtained from the coefficient of $x_{a_r} \dots x_{a_1}$ in the series $G_{\{0,1\}}^{\text{sv}}(x_i; z)$ given by (3.58) [50]. These constructions of $G^{\text{sv}}$ manifest that single-valued MZVs (A.10) are again expressible as $\mathbb{Q}$-linear combinations of products of MZVs.

At the level of motivic MZVs, one can define a single-valued map via

$$ \text{sv} : \zeta_{n_1, n_2, \dots, n_r}^{\mathbf{m}} \mapsto \zeta_{n_1, n_2, \dots, n_r}^{\text{sv}}, \quad (A.11) $$

which takes a particularly convenient form in the $f$-alphabet [20, 21]$^{30}$

$$ \text{sv}(f_2^n f_{i_1} f_{i_2} \dots f_{i_r}) = \delta_{n,0} \sum_{j=0}^{r} (f_{i_j} \dots f_{i_2} f_{i_1}) \sqcup (f_{i_{j+1}} \dots f_{i_{r-1}} f_{i_r}), \quad (A.12) $$

and preserves the (shuffle) multiplication in the sense that $\text{sv}(\zeta_n^m \cdot \zeta_m^m) = \text{sv}(\zeta_n^m)\text{sv}(\zeta_m^m)$. For Riemann zeta values, one readily finds $\zeta_{2n}^{\text{sv}} = 0$ and $\zeta_{2n+1}^{\text{sv}} = 2\zeta_{2n+1}$ from (A.12) at $r = 0$ and $r = 1$, respectively. For words in $r = 2$ and $r = 3$ odd-degree generators $f_{i_j}$ with $i_j \in 2\mathbb{N}+1$, (A.12) specialises to

$$ \begin{aligned} \text{sv}(f_{i_1} f_{i_2}) &= 2f_{i_1} f_{i_2} + 2f_{i_2} f_{i_1} = 2f_{i_1} \sqcup f_{i_2}, && (A.13) \\ \text{sv}(f_{i_1} f_{i_2} f_{i_3}) &= 2(f_{i_1} f_{i_2} f_{i_3} + f_{i_2} f_{i_1} f_{i_3} + f_{i_3} f_{i_2} f_{i_1} + f_{i_2} f_{i_3} f_{i_1}) = \text{sv}(f_{i_3} f_{i_2} f_{i_1}). \end{aligned} $$

By the first line, the expansion of the group-like $M^{\text{sv}}$ in (2.90) up to and including the second order in $f_i$ is expressible via Riemann zeta values since $\rho^{-1}(\text{sv}(f_{i_1})) = 2\zeta_{i_1}$ and $\rho^{-1}(\text{sv}(f_{i_1}f_{i_2})) = 2\zeta_{i_1}\zeta_{i_2}$. Starting from the third order, the $\rho^{-1}$ images of $\text{sv}(f_{i_1}f_{i_2}f_{i_3})$ in the second line of (A.13) generically involve (conjecturally indecomposable) single-valued MZVs beyond depth one, for instance

$$ \zeta_{3,3,5}^{\text{sv}} = 2\zeta_{3,3,5} - 5\zeta_3^2\zeta_5 + 90\zeta_2\zeta_9 + \frac{12}{5}\zeta_2^2\zeta_7 - \frac{8}{7}\zeta_2^3\zeta_5. \quad (A.14) $$

The construction (3.74) of the generating series $\mathbb{J}^{\text{eqv}}$ from the series $M^{\text{sv}}(\hat{\sigma}_i, \hat{\sigma}_\varpi) = M^{\text{sv}}(\hat{\sigma}_i) + \dots$ in zeta generators implies that the modular forms $\beta^{\text{eqv}}[\begin{smallmatrix} j_1 & j_2 & j_3 \\ k_1 & k_2 & k_3 \end{smallmatrix}]$ at modular depth three involve the single-valued MZV $\rho^{-1}(\text{sv}(f_{k_1-1}f_{k_2-1}f_{k_3-1}))$. Hence, our investigations up to degree $k_1+k_2+k_3 = 20$ lead to all single-valued MZVs $\rho^{-1}(\text{sv}(f_{i_1}f_{i_2}f_{i_3}))$ up to and including weight

<footnote>$^{30}$By slight abuse of notation, we do not distinguish the map $\text{sv}$ in (A.11) from its composition $\rho \circ \text{sv} \circ \rho^{-1}$ seen in (A.12).</footnote>
$i_1+i_2+i_3 = 17$ and three non-commutative letters in the $f$-alphabet. Their representations in terms of conventional MZVs are given by

$$ \rho^{-1}(\mathrm{sv}(f_3 f_3 f_5)) = -\frac{1}{5}\zeta_{3,3,5}^{\mathrm{sv}} + \frac{1119631493}{36838080}\zeta_{11}, \qquad (\mathrm{A.15}) $$

$$ \rho^{-1}(\mathrm{sv}(f_3 f_5 f_3)) = \frac{2}{5}\zeta_{3,3,5}^{\mathrm{sv}} + 4\zeta_3^2\zeta_5 - \frac{1119631493}{18419040}\zeta_{11}, $$

$$ \rho^{-1}(\mathrm{sv}(f_3 f_3 f_7)) = -\frac{1}{14}\zeta_{3,3,7}^{\mathrm{sv}} + \frac{3}{35}\zeta_{3,5,5}^{\mathrm{sv}} + \frac{8607216661079268929}{999844297536624120}\zeta_{13}, $$

$$ \rho^{-1}(\mathrm{sv}(f_3 f_7 f_3)) = \frac{1}{7}\zeta_{3,3,7}^{\mathrm{sv}} - \frac{6}{35}\zeta_{3,5,5}^{\mathrm{sv}} + 4\zeta_3^2\zeta_7 - \frac{8607216661079268929}{499922148768312060}\zeta_{13}, $$

$$ \rho^{-1}(\mathrm{sv}(f_3 f_5 f_5)) = \frac{1}{25}\zeta_{3,5,5}^{\mathrm{sv}} - \frac{839332307937696179}{39676361013358100}\zeta_{13}, $$

$$ \rho^{-1}(\mathrm{sv}(f_5 f_3 f_5)) = -\frac{2}{25}\zeta_{3,5,5}^{\mathrm{sv}} + 4\zeta_3\zeta_5^2 + \frac{839332307937696179}{19838180506679050}\zeta_{13}, $$

$$ \rho^{-1}(\mathrm{sv}(f_5 f_3 f_7)) = \frac{1}{70}\zeta_{5,3,7}^{\mathrm{sv}} + \frac{4}{5}\zeta_5^3 + \frac{24}{5}\zeta_3\zeta_5\zeta_7 - \frac{1156747681600394679684642590233}{52156108470099943251903139200}\zeta_{15}, $$

$$ \rho^{-1}(\mathrm{sv}(f_3 f_5 f_7)) = -\frac{1706}{68409}\zeta_{3,3,9}^{\mathrm{sv}} - \frac{58001}{1596210}\zeta_{5,3,7}^{\mathrm{sv}} + \frac{144}{7601}\zeta_{1,1,3,4,6}^{\mathrm{sv}} + \frac{759436}{114015}\zeta_3\zeta_5\zeta_7 + \frac{384}{38005}\zeta_3^5 \\ + \frac{557516}{114015}\zeta_5^3 + \frac{5024}{691}\zeta_3^2\zeta_9 + \frac{70011715120065369545804422936104641}{396438580481229668657715761059200}\zeta_{15}, $$

$$ \rho^{-1}(\mathrm{sv}(f_3 f_3 f_9)) = -\frac{655}{22803}\zeta_{3,3,9}^{\mathrm{sv}} - \frac{17203}{532070}\zeta_{5,3,7}^{\mathrm{sv}} - \frac{48}{7601}\zeta_{1,1,3,4,6}^{\mathrm{sv}} - \frac{178972}{38005}\zeta_3\zeta_5\zeta_7 - \frac{128}{38005}\zeta_3^5 \\ - \frac{88972}{38005}\zeta_5^3 - \frac{5024}{2073}\zeta_3^2\zeta_9 + \frac{26869796704014139979194459442197511}{264292386987486445771810507372800}\zeta_{15}, $$

and a comprehensive list of all cases up to weight 17 in the ancillary file. Note that instances of $\rho^{-1}(\mathrm{sv}(f_{i_1}f_{i_2}f_{i_3}))$ with $i_1 > i_3$ can be inferred from the symmetry under $i_1 \leftrightarrow i_3$ noted in (A.13), and we reiterate that we have implicitly applied the period map $\zeta_{n_1,n_2,\dots,n_r}^m \mapsto \zeta_{n_1,n_2,\dots,n_r}$ to the images of $\rho^{-1}$ in this section.

In order to find the numerical values of the above $\rho^{-1}(\mathrm{sv}(f_a f_b f_c))$, we have rewritten the single-valued MZVs on the right-hand sides of (A.15) via (A.14) and

$$ \zeta_{3,3,7}^{\mathrm{sv}} = 2\zeta_{3,3,7} + 12\zeta_{3,5}\zeta_5 + 60\zeta_3\zeta_5^2 - \frac{64}{35}\zeta_2^3\zeta_7 - 14\zeta_3^2\zeta_7 + \frac{112}{5}\zeta_2^2\zeta_9 + 407\zeta_2\zeta_{11}, \qquad (\mathrm{A.16}) $$

$$ \zeta_{3,5,5}^{\mathrm{sv}} = 2\zeta_{3,5,5} + 10\zeta_{3,5}\zeta_5 + 50\zeta_3\zeta_5^2 + 20\zeta_2^2\zeta_9 + 275\zeta_2\zeta_{11}, $$

$$ \zeta_{5,3,7}^{\mathrm{sv}} = 2\zeta_{5,3,7} - \frac{192}{385}\zeta_5^5\zeta_5 - 12\zeta_{3,7}\zeta_5 - 78\zeta_5^3 - \frac{96}{25}\zeta_2^4\zeta_7 - 28\zeta_{3,5}\zeta_7 - 336\zeta_3\zeta_5\zeta_7 \\ - \frac{272}{35}\zeta_2^3\zeta_9 + 44\zeta_2^2\zeta_{11} + 1001\zeta_2\zeta_{13}, $$

$$ \zeta_{3,3,9}^{\mathrm{sv}} = 2\zeta_{3,3,9} + 12\zeta_{3,7}\zeta_5 + 72\zeta_5^3 + \frac{144}{175}\zeta_2^4\zeta_7 + 30\zeta_{3,5}\zeta_7 + 318\zeta_3\zeta_5\zeta_7 + \frac{232}{35}\zeta_2^3\zeta_9 \\ - 27\zeta_3^2\zeta_9 + \frac{504}{5}\zeta_2^2\zeta_{11} + 1209\zeta_2\zeta_{13}, $$
$$
\begin{aligned}
\zeta_{1,1,3,4,6}^{\text{sv}} = & 2\zeta_{1,1,3,4,6} + 4\zeta_2\zeta_{3,3,7} - \frac{28}{5}\zeta_2\zeta_{3,5,5} + 2\zeta_{1,1,4,6}\zeta_3 + 8\zeta_2^2\zeta_{3,3,5} - \frac{16}{5}\zeta_2^2\zeta_{3,5}\zeta_3 \\
& - 12\zeta_2\zeta_{3,7}\zeta_3 + \frac{58}{9}\zeta_{3,9}\zeta_3 + 21\zeta_2\zeta_{3,5}\zeta_5 - \frac{145}{56}\zeta_{3,7}\zeta_5 - \frac{481}{10}\zeta_{3,5}\zeta_7 + \frac{2903944\zeta_2^6\zeta_3}{716625} \\
& + \frac{24}{35}\zeta_2^3\zeta_3^3 - \frac{2\zeta_3^5}{3} - \frac{47270\zeta_2^5\zeta_5}{1617} - \frac{23}{5}\zeta_2^2\zeta_3^2\zeta_5 - 16\zeta_2\zeta_3\zeta_5^2 - \frac{13133\zeta_5^3}{56} + \frac{2792\zeta_2^4\zeta_7}{2625} \\
& - 52\zeta_2\zeta_3^2\zeta_7 - \frac{12509}{24}\zeta_3\zeta_5\zeta_7 + \frac{16837}{63}\zeta_2^3\zeta_9 - \frac{2927}{9}\zeta_3^2\zeta_9 - \frac{27199}{30}\zeta_2^2\zeta_{11} - \frac{56717}{60}\zeta_2\zeta_{13},
\end{aligned}
$$

see the ancillary files for their weight-17 analogues.

The examples of (A.15) illustrate that the modular depth of $\beta^{\text{eqv}}$ does not bound the depth $r$ of the MZVs $\zeta_{n_1, \dots, n_r}$ in its Fourier expansion. Instead, the modular depth of $\beta^{\text{eqv}}$ sets an upper bound on the number of non-commutative letters in the $f$-alphabet representation of the associated MZVs, see [86] for the analogous statement for MMVs. The MZV representation of a given $\rho^{-1}(f_{i_1} f_{i_2} \dots f_{i_l})$ will in general necessitate representatives of depth $> \ell$ as exemplified by $\zeta_{1,1,3,4,6}^{\text{sv}}$ in the last four lines of (A.15).

## B Changing from holomorphic to modular frame

In this section we will prove the validity of (3.10) relating the two types of integration kernels $\nu[j_k; \tau_1]$ and $\omega_+[j_k; \tau, \tau_1]$ through the $\text{SL}_2$ transformation $U_{\text{SL}_2}(\tau)$ in (3.1). To this end we will actually prove a stronger statement. Let us define the auxiliary operator

$$ \widetilde{U}_{\text{SL}_2}(a, b) := \exp(a e_0^\vee) \exp(b e_0). \qquad (\text{B.1}) $$

**Lemma 3** *From Lemma 1 we can derive the identity:*

$$ \widetilde{U}_{\text{SL}_2}(a, b) \left[ \sum_{j=0}^{k-2} \frac{(-1)^j}{j!} (2\pi i \tau_1)^j e_k^{(j)} \right] \widetilde{U}_{\text{SL}_2}(a, b)^{-1} = \sum_{\ell=0}^{k-2} \frac{e_k^{(\ell)}}{\ell!} [1 + a(b - 2\pi i \tau_1)]^{k-2-\ell} (b - 2\pi i \tau_1)^{\ell}. \qquad (\text{B.2}) $$

Note that given our definition (3.1), we simply have

$$ U_{\text{SL}_2}(\tau) = \widetilde{U}_{\text{SL}_2}(-\frac{1}{4y}, 2\pi i \bar{\tau}), \quad a = -\frac{1}{4y}, \quad b = 2\pi i \bar{\tau}, \qquad (\text{B.3}) $$

such that (3.10) follows as a simple corollary of this more general lemma.

*Proof:* We start by expanding both exponential factors in $\widetilde{U}_{\text{SL}_2}(a, b)$ on the left-hand side of (B.2) as a power series:

$$
\begin{aligned}
& \sum_{\ell_1, \ell_2=0}^{\infty} \frac{a^{\ell_1} b^{\ell_2}}{\ell_1! \ell_2!} (\text{ad}_{e_0^\vee})^{\ell_1} [(\text{ad}_{e_0})^{\ell_2} \left[ \sum_{j=0}^{k-2} \frac{(-1)^j}{j!} (2\pi i \tau_1)^j e_k^{(j)} \right]] \qquad (\text{B.4}) \\
& = \sum_{j=0}^{k-2} \sum_{\ell_2=0}^{k-2-j} \sum_{\ell_1=0}^{j+\ell_2} \frac{a^{\ell_1} b^{\ell_2}}{\ell_1! \ell_2!} \frac{(-1)^j}{j!} (2\pi i \tau_1)^j \frac{(j+\ell_2)! (k+\ell_1-2-j-\ell_2)!}{(j+\ell_2-\ell_1)! (k-2-j-\ell_2)!} e_k^{(j+\ell_2-\ell_1)},
\end{aligned}
$$

where we used crucially (2.55).
We then perform the change in summation variables $j \to \ell := j+\ell_2-\ell_1$ to arrive at

$$ \begin{aligned} & \widetilde{U}_{\text{SL}_2}(a, b) \left[ \sum_{j=0}^{k-2} \frac{(-1)^j}{j!} (2\pi i \tau_1)^j e_k^{(j)} \right] \widetilde{U}_{\text{SL}_2}(a, b)^{-1} \tag{B.5} \\ &= \sum_{\ell=0}^{k-2} \frac{e_k^{(\ell)}}{\ell!} \sum_{\ell_1=0}^{k-2-\ell} \binom{k-2-\ell}{\ell_1} a^{\ell_1} \sum_{\ell_2=0}^{\ell+\ell_2} \binom{\ell+\ell_1}{\ell_2} b^{\ell_2} (-2\pi i \tau_1)^{\ell+\ell_1-\ell_2} \\ &= \sum_{\ell=0}^{k-2} \frac{e_k^{(\ell)}}{\ell!} \sum_{\ell_1=0}^{k-2-\ell} \binom{k-2-\ell}{\ell_1} a^{\ell_1} (b - 2\pi i \tau_1)^{\ell+\ell_1} \\ &= \sum_{\ell=0}^{k-2} \frac{e_k^{(\ell)}}{\ell!} [1 + a(b - 2\pi i \tau_1)]^{k-2-\ell} (b - 2\pi i \tau_1)^{\ell}, \end{aligned} $$

thus concluding our proof of (B.2).

Note that this lemma only relies on the relations (2.55) coming from the sl₂ action on the module spanned by $e_k^{(j)}$ with $0 \le j \le k-2$. In particular, it can be also reformulated in terms of the formal variables $e_k^{(j)}$ rather than $e_k^{(j)}$.

If we specialise $a, b$ according to (B.3) we obtain

$$ U_{\text{SL}_2}(\tau) \left[ \sum_{j=0}^{k-2} \frac{(-1)^j}{j!} (2\pi i \tau_1)^j e_k^{(j)} \right] U_{\text{SL}_2}(\tau)^{-1} = (2\pi i)^{k-2} \sum_{\ell=0}^{k-2} \frac{(-1)^\ell e_k^{(\ell)}}{\ell!} \left(\frac{\tau - \tau_1}{4y}\right)^{k-2-\ell} (\bar{\tau} - \tau_1)^\ell, \quad (B.6) $$

and upon adding some factors of $(2\pi i)$ we arrive at the kernels $\omega_\pm$ defined in (2.20),

$$ \begin{aligned} & U_{\text{SL}_2}(\tau) \left[ \sum_{j=0}^{k-2} \frac{(-1)^j}{j!} e_k^{(j)} \nu \left[ \begin{smallmatrix} j \\ k \\ \tau_1 \end{smallmatrix} \right] \right] U_{\text{SL}_2}(\tau)^{-1} = \sum_{\ell=0}^{k-2} \frac{(-1)^\ell}{\ell!} e_k^{(\ell)} \omega_+ \left[ \begin{smallmatrix} \ell \\ k \\ \tau, \tau_1 \end{smallmatrix} \right], \quad (B.7) \\ & U_{\text{SL}_2}(\tau) \left[ \sum_{j=0}^{k-2} \frac{1}{j!} e_k^{(j)} \overline{\nu \left[ \begin{smallmatrix} j \\ k \\ \tau_1 \end{smallmatrix} \right]} \right] U_{\text{SL}_2}(\tau)^{-1} = \sum_{\ell=0}^{k-2} \frac{(-1)^\ell}{\ell!} e_k^{(\ell)} \omega_- \left[ \begin{smallmatrix} \ell \\ k \\ \tau, \tau_1 \end{smallmatrix} \right]. \end{aligned} $$

This dictionary between the differential forms $\nu$ and $\omega_\pm$ also holds for holomorphic cusp forms instead of Eisenstein series, i.e. for kernels $\nu[\Delta_k^j; \tau]$ defined in (2.9) such that

$$ \begin{aligned} & U_{\text{SL}_2}(\tau) \left[ \sum_{j=0}^{k-2} \frac{(-1)^j}{j!} e_{\Delta_k^+}^{(j)} \nu \left[ \begin{smallmatrix} j \\ \Delta_k \\ \tau_1 \end{smallmatrix} \right] \right] U_{\text{SL}_2}(\tau)^{-1} = \sum_{\ell=0}^{k-2} \frac{(-1)^\ell}{\ell!} e_{\Delta_k^+}^{(\ell)} \omega_+ \left[ \begin{smallmatrix} \ell \\ \Delta_k \\ \tau, \tau_1 \end{smallmatrix} \right], \quad (B.8) \\ & U_{\text{SL}_2}(\tau) \left[ \sum_{j=0}^{k-2} \frac{1}{j!} e_{\Delta_k^-}^{(j)} \overline{\nu \left[ \begin{smallmatrix} j \\ \Delta_k \\ \tau_1 \end{smallmatrix} \right]} \right] U_{\text{SL}_2}(\tau)^{-1} = \sum_{\ell=0}^{k-2} \frac{(-1)^\ell}{\ell!} e_{\Delta_k^-}^{(\ell)} \omega_- \left[ \begin{smallmatrix} \ell \\ \Delta_k \\ \tau, \tau_1 \end{smallmatrix} \right], \end{aligned} $$

where the only property used is that $e_{\Delta_k^\pm}^{(j)}$ span an sl₂ multiplet.

### C Comparing $M_{k_1,k_2,k_3}$ with combinations of $\mathcal{E}$ and $c^{\text{sv}}$

A first goal of this appendix is to derive more explicit formulae for the expansion coefficients $\mathcal{E}^{\text{eqv}}[j_1 \dots j_\ell]$ of equivariant iterated Eisenstein integrals from the generating series $\mathbb{I}^{\text{eqv}}(e_k; \tau)$