Modular Iterated Integrals

• Modular iterated integrals are constructed from modular forms or related automorphic kernels and exhibit defining properties such as the shuffle algebra and recursive differential relations.
• They unify holomorphic, real-analytic, and Eichler-type formulations, connecting algebraic, cohomological, and analytical frameworks while encoding multiple modular L-functions.
• Their applications span perturbative quantum field theory, string theory, and gauge theory computations, showcasing their practical impact in advanced mathematical physics.

Searching arXiv for recent and foundational papers on modular iterated integrals. Modular iterated integrals are iterated integrals whose kernels are modular forms or closely related automorphic objects, typically integrated along paths in the upper half–plane from the cusp $i\infty$ to a point $\tau$ or $z$. In the literature they occur in several closely related forms: as holomorphic iterated integrals of modular forms on congruence subgroups, as real-analytic or single-valued completions built from holomorphic and antiholomorphic pieces, as Eichler-type iterated integrals attached to cusp forms, and as equivariant constructions valued in polynomial or representation-theoretic modules. Their defining features are the shuffle algebra, recursive differential relations, compatibility with modular transformations up to cocycles or lower-depth corrections, and a rich interaction with cohomology, multiple modular $L$-functions, modular graph forms, and elliptic Feynman integrals (Brown, 2017).

1. Definitions and basic forms

A standard holomorphic definition starts with a congruence subgroup $\Gamma\subset SL_2(\mathbb Z)$ and modular forms $f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$. For a base point $\tau_0$, often $i\infty$, the $n$-fold iterated integral is defined by

$$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$

or equivalently in the $\tau$0-variable by repeated integration against $\tau$1 (Adams et al., 2017). In the formulation used in physics, the last kernel is often required to vanish at the cusp in order to ensure convergence for $\tau$2 (Adams et al., 2018).

For real-analytic modular forms one works on the upper half-plane $\tau$3 with bi-weights $\tau$4. If $\tau$5 is a Fuchsian group of the first kind and

$\tau$6

then the space $\tau$7 consists of smooth functions satisfying

$\tau$8

together with polynomial growth at the cusp; the cusp-type subspace $\tau$9 consists of those whose Fourier expansion at each cusp has no constant term (Diamantis, 2020). This bi-graded setting supports a length filtration of modular iterated integrals defined recursively via the Maass operators

$z$0

with $z$1 and, for $z$2,

$z$3

(Diamantis, 2020).

A different but related family is given by Eichler-type iterated integrals. For cusp forms $z$4, one considers

$z$5

which transforms under slash operators of total weight $z$6 and is designed to encode noncommutative modular symbols and iterated $z$7-data (Bringmann et al., 2024).

These variants are not competing definitions so much as different realizations of the same general phenomenon. A plausible implication is that “modular iterated integral” functions as an umbrella term for a family of constructions sharing Chen-type iteration, modular covariance, and depth filtrations, but differing in regularization, coefficient modules, and whether one emphasizes holomorphic, real-analytic, or cohomological structures.

2. Algebraic structure and filtrations

The basic algebraic property is the shuffle product. If $z$8 are words in modular one-forms, then

$z$9

and similarly for the standard holomorphic iterated integrals $L$0 (Matthes et al., 2022). This identifies modular iterated integrals with a commutative shuffle algebra generated by modular kernels, subject to the regularization conventions at the cusp.

In the quasimodular and modular setting for $L$1, one can make this completely explicit. Let $L$2 and $L$3. The algebra $L$4 of iterated integrals of quasimodular forms is the smallest extension of $L$5 closed under integration, and the corresponding modular subalgebra is $L$6. The main structure theorem states that

$L$7

is an isomorphism of filtered $L$8-algebras, where $L$9, and analogously

$\Gamma\subset SL_2(\mathbb Z)$0

for genuine modular forms (Matthes, 2017). Via Radford’s theorem, this yields polynomial presentations on generators indexed by Lyndon words: $\Gamma\subset SL_2(\mathbb Z)$1

In Brown’s real-analytic level-one theory, the corresponding algebra is the ring of equivariant iterated Eisenstein integrals. It is constructed from a generating series $\Gamma\subset SL_2(\mathbb Z)$2 of iterated Eisenstein integrals, corrected by single-valued data so that the resulting $\Gamma\subset SL_2(\mathbb Z)$3 is genuinely $\Gamma\subset SL_2(\mathbb Z)$4-invariant. Its coefficients define real-analytic modular forms of bi-weight $\Gamma\subset SL_2(\mathbb Z)$5, and the resulting ring admits expansions in $\Gamma\subset SL_2(\mathbb Z)$6, $\Gamma\subset SL_2(\mathbb Z)$7, and $\Gamma\subset SL_2(\mathbb Z)$8 with coefficients in rational numbers and single-valued multiple zeta values (Brown, 2017).

Length and weight filtrations coexist throughout the subject. In the physics-oriented definitions, an iterated integral $\Gamma\subset SL_2(\mathbb Z)$9 has length $f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$0 and total transcendental weight $f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$1 when the kernels have weights $f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$2 (Abreu et al., 2019). In Brown-type real-analytic theories, one instead works with length filtrations $f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$3 inside the space of real-analytic modular forms (Drewitt, 2021). This suggests that “depth,” “length,” and “weight” are not interchangeable: different subfields fix different filtrations according to analytic or arithmetic priorities.

3. Cusp forms, Eichler integrals, and invariant completions

A central issue is how cusp forms enter the theory. Brown’s initial real-analytic length-two constructions were largely Eisenstein-origin. Diamantis isolates the complementary cuspidal part by introducing an extended space $f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$4, characterized by

$f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$5

where $f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$6 replaces $f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$7 on the right-hand side and the restriction $f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$8 is removed (Diamantis, 2020).

For a fixed cusp form $f_i(\tau)\in \mathcal M_{k_i}(\Gamma)$9 with Eichler integrals

$\tau_0$0

Diamantis defines Poincaré-type series

$\tau_0$1

and similarly $\tau_0$2, for $\tau_0$3 (Diamantis, 2020). These converge absolutely, are invariant under the representation of bi-weight $\tau_0$4 on $\tau_0$5, and their coefficients in

$\tau_0$6

lie in $\tau_0$7. Moreover,

$\tau_0$8

$\tau_0$9

up to an explicit sign in the “$i\infty$0” case, so each coefficient $i\infty$1 lies in the extended space $i\infty$2 (Diamantis, 2020).

The family $i\infty$3 spans a subspace fitting into the short exact sequence

$i\infty$4

The surjection is induced by the Eichler–Shimura map sending each $i\infty$5 to the underlying cusp form $i\infty$6 and its conjugate (Diamantis, 2020). In particular, every pair $i\infty$7 arises from an explicit invariant real-analytic iterated integral.

A parallel phenomenon appears in modular graph theory. Dorigoni, Kleinschmidt, and Schlotterer show that single-valued iterated Eisenstein integrals are insufficient for all depth-two modular-invariant functions defined by inhomogeneous Laplace equations. When cusp forms $i\infty$8 exist, one must adjoin single-valued lifts

$i\infty$9

satisfying

$n$0

but transforming under $n$1 with polynomial cocycles governed by completed $n$2-values (Dorigoni et al., 2021). The coefficients needed to restore modularity can be chosen proportional to ratios such as

$n$3

depending on the parity sector (Dorigoni et al., 2021).

These constructions clarify a common misconception: modular iterated integrals are not exhausted by Eisenstein series. Several theories require explicit cusp-form contributions in order to capture invariant or single-valued objects of higher depth.

4. Cohomology, modular symbols, and $n$4-functions

Iterated Eichler integrals admit a precise cohomological interpretation through extended or higher-depth cohomology. Bringmann and Diamantis introduce depth-$n$5 cochain spaces in which a depth-$n$6 $n$7-cochain fails to satisfy the usual cocycle relation by sums of lower-depth factorizations. In this language, if $n$8 and $n$9, then

$$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$0

satisfies the depth-two cocycle relation

$$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$1

hence defines a class in

$$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$2

(Bringmann et al., 2024). For general depth $$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$3, the failure of the cocycle identity is controlled by lower-depth cocycles $$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$4, and the classes $$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$5 are non-trivial in $$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$6 (Bringmann et al., 2024).

This cohomological perspective ties directly to Manin’s noncommutative modular symbols. For weight-two cusp forms $$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$7 on $$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$8, the iterated integrals

$$I(f_1,\dots,f_n;\tau) =(2\pi i)^n\int_{\tau_0}^{\tau} d\tau_1\, f_1(\tau_1) \int_{\tau_0}^{\tau_1} d\tau_2\, f_2(\tau_2)\cdots \int_{\tau_0}^{\tau_{n-1}} d\tau_n\, f_n(\tau_n),$$9

depend only on the endpoints by Chen’s theory and satisfy both shuffle and path-composition formulas (Matthes et al., 2022). Matthes and Risager analyze the asymptotic distribution of the values $\tau$00 as the cusp $\tau$01 varies. For length $\tau$02, the renormalized symbols converge in distribution to the standard complex normal law; for length $\tau$03, they converge to a radially symmetric law depending only on the Gram matrix of the forms; for lengths at least $\tau$04, all asymptotic moments exist but in general do not determine a unique distribution (Matthes et al., 2022).

The connection with $\tau$05-functions is equally direct. Yokomizo studies modular iterated integrals

$\tau$06

allowing general modular forms, including those with nonzero constant terms (Yokomizo, 7 May 2026). These are related to Manin’s multiple modular $\tau$07-functions

$\tau$08

and the paper generalizes the Choie–Ihara correspondence beyond the cusp-form case (Yokomizo, 7 May 2026). It also proves a functional equation

$\tau$09

where $\tau$10 and

$\tau$11

(Yokomizo, 7 May 2026).

The arithmetic significance is therefore twofold: iterated integrals encode multiple modular $\tau$12-values, and their transformation failures or completions are naturally expressed in cohomological terms.

5. Real-analytic, equivariant, and higher-length theories

Brown’s equivariant iterated Eisenstein integrals provide a level-one prototype for single-valued or modularly completed real-analytic objects. The coefficients of the corrected generating series $\tau$13 define real-analytic modular forms of bi-weight $\tau$14, and the first non-trivial examples are the real-analytic Eisenstein series (Brown, 2017). This construction strongly influenced later work on modular graph forms and string perturbation theory.

Drewitt develops a length-three theory inside the space $\tau$15 of real-analytic modular forms for $\tau$16. Writing

$\tau$17

with

$\tau$18

he constructs closed, $\tau$19-equivariant $\tau$20-forms $\tau$21 from length-two primitives $\tau$22 and Eisenstein forms, integrates them to obtain a primitive $\tau$23, and then corrects by an Eichler–Shimura cocycle to produce $\tau$24-equivariant functions whose components define length-three iterated integrals $\tau$25 (Drewitt, 2021). For $\tau$26, where $\tau$27, these satisfy recursive $\tau$28-equations and associated Laplace-eigenvalue equations (Drewitt, 2021).

A closely related line studies modular graph functions and their depth filtration. Doroudiani introduces a depth-dependent basis up to depth three built from completed Eisenstein series $\tau$29, shuffle products, and new functions $\tau$30, $\tau$31, $\tau$32, $\tau$33, and $\tau$34, each defined by inhomogeneous Laplace equations of the form

$\tau$35

with $\tau$36 of lower depth (Doroudiani, 2023). The basis is then integrated over the truncated fundamental domain using Stokes’ theorem and Rankin–Selberg–Zagier methods, yielding closed-form expressions in completed zeta values and their derivatives (Doroudiani, 2023).

This suggests a useful distinction. Real-analytic modular iterated integrals in the Brown–Drewitt–Diamantis sense are built as modular objects from the outset or after equivariant correction; modular graph function bases are often defined by Laplace equations and then identified with iterated-integral expressions. The two approaches are technically different but converge on the same function space in many examples.

6. Applications in physics and geometry

One major application is perturbative quantum field theory. Adams and Weinzierl show that the equal-mass sunrise and kite Feynman integrals can be expressed to all orders in the dimensional regularization parameter $\tau$37 as iterated integrals of modular forms (Adams et al., 2017). In the sunrise case, all kernels can be chosen as modular forms for $\tau$38, with three basic letters

$\tau$39

and the all-orders $\tau$40-expansion takes the form of a generating series in iterated integrals of these kernels (Adams et al., 2017). Related work shows how to put the differential equations of elliptic Feynman integrals into $\tau$41-form after changing variables to the modular parameter $\tau$42, so that the solution is manifestly a path-ordered exponential of modular one-forms (Adams et al., 2018, Broedel et al., 2018).

At three loops, Hidding, Moriello, and collaborators compute analytic expressions for the Standard Model $\tau$43-parameter involving precisely elliptic polylogarithms and iterated integrals of modular forms (Abreu et al., 2019). They analytically continue the relevant iterated Eisenstein integrals to all kinematic regions by patching local period solutions and mapping the period ratio $\tau$44 to the standard fundamental domain. The resulting $\tau$45-series converge rapidly in every region, giving manifestly real and fast-converging expansions (Abreu et al., 2019).

A second major application is string perturbation theory. Generating series of modular graph forms can be expressed through real-analytic combinations $\tau$46 built from holomorphic iterated Eisenstein integrals and antiholomorphic integration constants (Gerken et al., 2020). For one-variable elliptic modular graph forms, Broedel, Matthes, Schlotterer, and Tourkine translate lattice-sum realizations into iterated $\tau$47-integrals involving both Eisenstein series $\tau$48 and Kronecker–Eisenstein coefficients $\tau$49; this produces concrete realizations of single-valued elliptic polylogarithms at arbitrary depth and a basis-counting formalism based on an extension of Tsunogai’s derivation algebra (Hidding et al., 2022). More recently, equivariant generating series have been used to solve differential equations for elliptic modular graph forms and to construct single-valued elliptic multiple polylogarithms in one variable (Schlotterer et al., 19 Nov 2025), while a separate algorithm converts lattice-sum modular graph forms into equivariant iterated Eisenstein integrals and implements all topologies up to four vertices in a \textsc{Mathematica} package (Claasen et al., 8 Feb 2025).

Modular iterated integrals also appear in supersymmetric gauge theory. In Vafa–Witten theory on $\tau$50, the modular anomaly of the $\tau$51 partition function involves a double Eichler–Shimura integral: $\tau$52 or equivalently $\tau$53. Since the shadow of $\tau$54 is a depth-one mock modular form times $\tau$55, the $\tau$56 are pure mock modular forms of depth two (Manschot, 2017).

These applications show that modular iterated integrals are not merely analogues of multiple polylogarithms. They form a computational language for elliptic and modular phenomena in arithmetic geometry, quantum field theory, and string theory.

7. Numerical, asymptotic, and conceptual directions

Effective computation relies on $\tau$57-expansions, regularization at cusps, and analytic continuation. Walden and Weinzierl implement numerical evaluation of iterated integrals of modular forms and of Kronecker coefficient functions $\tau$58 in GiNaC, using the canonical one-form

$\tau$59

and absolutely convergent multiple sums derived from the local Laurent expansions of the kernels (Walden et al., 2020). This framework also accommodates iterated $\tau$60- and $\tau$61-integrals built from the Kronecker function, including elliptic multiple polylogarithms (Walden et al., 2020).

In applications to Feynman integrals, one frequently improves convergence by modular transformations that move $\tau$62 into the standard fundamental domain. For the three-loop $\tau$63-parameter, one introduces $\tau$64 with $\tau$65 chosen so that $\tau$66, and since $\tau$67, the $\tau$68-expansions converge very rapidly (Abreu et al., 2019).

Conceptually, several open directions emerge from the current literature. Extended higher-order modular forms over general Fuchsian groups of the first kind provide a representation-theoretic framework that contains extended second-order real-analytic forms and classical iterated Eichler integrals as special cases (Diamantis, 2020). Higher-depth cohomology gives a systematic way to measure the failure of strict cocycle relations by lower-depth factorizations and suggests modular analogues of double-shuffle structures (Bringmann et al., 2024). Arithmetic applications to period polynomials and $\tau$69-values are explicitly indicated in the cusp-form literature (Diamantis, 2020), while the multiple modular $\tau$70-function formalism now extends beyond the cusp-form case to general modular forms with nonzero constant terms (Yokomizo, 7 May 2026).

A final recurring theme is modular completion. Holomorphic iterated integrals alone rarely transform modularly; real-analytic, equivariant, or single-valued corrections are typically necessary. This is true for Brown’s level-one equivariant iterated Eisenstein integrals (Brown, 2017), for cusp-form completions in modular graph theory (Dorigoni et al., 2021), for the cohomological interpretation of Manin’s symbols (Bringmann et al., 2024), and for depth-two mock modularity in Vafa–Witten theory (Manschot, 2017). The persistent need for such completions suggests that modular iterated integrals are best understood not as isolated special functions but as objects living naturally at the interface of differential equations, automorphic representation theory, and periods.