Iterated Eisenstein Integrals in Modular Theory

• Iterated integrals over holomorphic Eisenstein series are multi-variable, modular-invariant constructs that generalize classical modular forms.
• They encode period structures linking multiple zeta values, modular graph forms, and low-energy string amplitudes through systematic differential techniques.
• Computational methods employ modular-covariant differentiation and Poincaré series to manage integration constants and reveal transcendental contributions.

Iterated integrals over holomorphic Eisenstein series form a class of functions and constructions central to the interface of modern number theory, algebraic geometry, and theoretical physics. They generalize the concept of classical modular forms by defining multi-variable, often nonholomorphic, modular-invariant analytic objects whose algebraic, differential, and period structures encode rich connections between multiple zeta values, modular forms, and low-energy string amplitudes. In particular, they appear in the explicit analysis of modular graph forms, the algebraic structure of modular invariants, the paper of differential equations for Feynman amplitudes, and in the construction of higher-depth period polynomials. The development of this theory has led to new representations, computational tools, and structural understanding of both mathematical and physical phenomena.

1. Foundations: Iterated Integrals over Eisenstein Series

An iterated integral over holomorphic Eisenstein series is generally defined as a repeated integration along a chain in the upper half-plane, where each integrand is (up to normalization) a holomorphic Eisenstein series or a related modular form. These iterated integrals naturally generalize classical Eichler integrals. For Eisenstein series $G_{2k}(\tau)$, the depth-$r$ iterated integral takes the typical form: $$\mathcal{I} = \int_\tau^{i\infty} G_{2k_1}(\tau_1) d\tau_1 \int_{\tau_1}^{i\infty} G_{2k_2}(\tau_2) d\tau_2 \cdots \int_{\tau_{r-1}}^{i\infty} G_{2k_r}(\tau_r) d\tau_r.$$ These objects admit regularization schemes (e.g., by subtraction of constant terms) to ensure convergence near the cusp $\tau \to i\infty$ (Matthes, 2017, Duhr et al., 2019). The iterated Eisenstein integrals generate a shuffle algebra and play the role of transcendental period generators for the moduli space of elliptic curves.

2. Modular Graph Forms and Poincaré Series Representations

Modular graph forms (MGFs) are nonholomorphic modular invariants arising in the low-energy expansion of genus-one closed-string amplitudes (Gerken et al., 2020, Dorigoni et al., 21 Mar 2024, Claasen et al., 8 Feb 2025). The work of Gerken, Dorigoni, Kleinschmidt, Schlotterer, and collaborators provides systematic algorithms to rewrite lattice-sum representations of MGFs in terms of iterated integrals of holomorphic Eisenstein series (Dorigoni et al., 2021, Claasen et al., 8 Feb 2025). The fundamental procedure involves:

• Application of a modular-covariant derivative $\pi\nabla_0$ repeatedly to a given MGF, which lowers antiholomorphic weight and produces new terms interpreted via the action on graph edges.
• Reduction via momentum conservation relations and holomorphic subgraph reduction (HSR), employing Fay identities and partial-fraction expansions to isolate pure holomorphic contributions.
• Encoding the iterative differentiation in a tree representation. Each path terminates in a branch labeled by either a holomorphic Eisenstein series or a rational constant, which may include transcendental constants, notably single-valued multiple zeta values (svMZVs), determined by matching the Laurent expansion at the cusp (Claasen et al., 8 Feb 2025).

The resulting expressions for MGFs are modular-invariant sums over equivariant iterated Eisenstein integrals, with explicit coefficients and boundary (integration constant) contributions controlled by modular and differential constraints (Gerken et al., 2020, Dorigoni et al., 21 Mar 2024).

3. Laplace Equations, Seed Functions, and Depth Structure

MGFs and higher-depth modular invariants are solutions to specific inhomogeneous Laplace eigenvalue equations, whose source terms are products of nonholomorphic Eisenstein series (Dorigoni et al., 2021, Dorigoni et al., 2021). Poincaré series constructions enable one to express MGFs as sums over modular group images of “seed” functions built from lower-depth Eisenstein iterated integrals. Notably,

• For depth one (e.g., nonholomorphic Eisenstein series), the MGF is given by a sum over $(\mathrm{Im}\,\tau)^s$,
• For depth two, the seed involves a depth-one iterated Eisenstein integral, and the Poincaré sum increases the depth by one. The correct modular invariance and vanishing of constant terms are guaranteed by adding lower-depth corrections dictated by the Laplacian and Cauchy–Riemann equations (Dorigoni et al., 2021).

Decorating integrals with odd $\zeta$ values (e.g., in boundary terms and integration constants) ensures uniform transcendentality and resolves modular invariance at each step (Claasen et al., 8 Feb 2025). However, it is shown that for general depth-two nonholomorphic modular functions, the use of Eisenstein series alone does not suffice—additional iterated integrals of holomorphic cusp forms are required to fully restore modular invariance and capture the complete space of solutions to the Laplace systems (Dorigoni et al., 2021).

4. Extension to Cusp Forms and the Structure of Modular Invariants

When the Laplace eigenvalue coincides with half the weight of a cusp form, iterated Eisenstein integrals alone become insufficient for modular invariance. In such cases, one must include real-analytic single-valued iterated integrals of holomorphic cusp forms: $$I_\Delta(\tau) = y^{1-s} \int_\tau^{i\infty} (\tau-\tau_1)^{s-1} (\overline{\tau}-\tau_1)^{s-1} \Delta(\tau_1)\, d\tau_1 \pm \text{c.c.}$$ where $\Delta$ is a cusp form of even weight $2s$ (Dorigoni et al., 2021). Coefficients appearing in the expansion are ratios of completed $L$-values of cusp forms, reflecting (via period polynomials and multiple modular values) the algebraic obstructions to generating the full solution space from Eisenstein series alone.

This necessity derives from the analysis of modular transformation properties, in particular, the presence of nonvanishing cocycles (S-transformations) in the Eisenstein sector, which are canceled exactly by contributions from the cusp-form integrals. The Fourier coefficients of these extended modular invariants thus contain both odd zeta values and rational multiples of ratios of completed $L$-values, often in the nonzero modes and edge terms.

5. Algorithmic and Computational Aspects

The practical computation of iterated Eisenstein integral representations for MGFs and related modular invariants proceeds by:

• Systematic differentiation (lowering antiholomorphic weight) and reduction using algebraic identities (momentum conservation, HSR, Fay).
• Tree-encoding of branches, forming a blueprint for the structure of the final iterated integral representation (Claasen et al., 8 Feb 2025).
• Integration “up the tree” by solving the associated differential equations for equivariant iterated Eisenstein integrals. Integration constants at modular-invariant vertices are determined by matching Laurent expansions and imposing cusp or modular constraints.
• Implementation in symbolic computation packages (notably “MGFtoBeqv” in Mathematica), automating the expansion and conversion of MGFs of arbitrary topology up to a fixed number of vertices.

Explicit formulas for modular graph forms at order $\alpha'^8$ in string amplitudes, for example, demonstrate the algorithm’s capacity to yield new analytic and transcendental information, including terms involving $\zeta_3\zeta_5$ in the analytic expansion (Claasen et al., 8 Feb 2025).

6. Applications in String Theory and the Arithmetic of Modular Forms

Iterated integrals over Eisenstein series are deeply entwined with the low-energy expansion of both open- and closed-string genus-one amplitudes (Gerken et al., 2020, Dorigoni et al., 21 Mar 2024). They facilitate:

• The reformulation of string amplitude integrands in a basis manifesting transcendental weight and modular properties.
• Direct computation of analytic parts of amplitudes, including explicit transcendental contributions from multiple zeta values, which are crucial for matching with effective field theory and extracting nonperturbative modular properties relevant to S-duality and arithmetic geometry.

Moreover, such representations reveal structural analogies with genus-zero amplitudes and provide concrete realizations of the generating series framework of equivariant iterated Eisenstein integrals developed by Brown, including the role of zeta generators and derivation algebras (Dorigoni et al., 21 Mar 2024).

7. Impact, Limitations, and Future Directions

The analytic control over modular graph forms via iterated Eisenstein integrals unifies algebraic, automorphic, and transcendental facets of modular invariants. The necessity to include cusp-form integrals signals a deep link with the theory of period polynomials, modular $L$-values, and the structure of multiple modular values (Dorigoni et al., 2021).

Open directions and limitations include:

• The explicit determination of all integration constants and Laurent expansions for a general class of modular invariants.
• The extension from depth-two to arbitrary depth, requiring systematic incorporation of higher-order cusp-form iterated integrals and further analysis of the associated differential-algebraic structures.
• Application to higher-genus string amplitudes and the exploration of corresponding period structures in the arithmetic of moduli spaces.

The theory thus not only consolidates the analytic backbone of string perturbation theory but also stimulates new developments in the arithmetic of modular and automorphic forms.