Combinatorial multiple Eisenstein series

This paper constructs combinatorial multiple Eisenstein series, a family of $q$-series that interpolate between a given rational solution of the extended double shuffle equations and multiple zeta values, serving as $q$-analogues closely related to modular forms and defined via symmetril and swap invariant bimoulds.

The Gist

Imagine you are trying to understand a massive, complex city called Mathematics. In this city, there are two very famous, very different neighborhoods that seem to have nothing to do with each other:

1. The Neighborhood of Numbers (Multiple Zeta Values): This is a place of pure, static numbers. Think of it like a library of infinite sums that describe deep, fundamental truths about numbers. They are "frozen" in time.
2. The Neighborhood of Shapes (Modular Forms/Eisenstein Series): This is a place of flowing, dynamic patterns. Think of it like a kaleidoscope or a spinning top. These objects change and shift beautifully, but they follow strict rules of symmetry. They are "alive" and moving.

For a long time, mathematicians knew these two neighborhoods were related, but they couldn't build a bridge between them. They knew the "frozen" numbers appeared as the starting points (the constant terms) of the "moving" shapes, but they didn't have a way to walk from one to the other while keeping the rules of the city intact.

Enter the "Combinatorial Multiple Eisenstein Series."

This paper by Henrik Bachmann and Annika Burmester is essentially the construction of a magical bridge (or a high-speed train) that connects these two neighborhoods.

The Core Idea: A Shape-Shifting Object

Imagine you have a special object, let's call it a "q-Shape."

• When you turn a dial to 0 ($q \to 0$): The object freezes and turns into a rational number (a simple fraction). This is the "frozen" side, related to the rational solutions of the city's laws.
• When you turn the dial to 1 ($q \to 1$): The object melts and becomes a famous, complex number called a "Multiple Zeta Value." This is the "number" side.
• In between (anywhere else): The object is a q-series. It's a flowing, dynamic formula that acts as a perfect interpolation. It is neither just a number nor just a shape; it is a hybrid that carries the DNA of both.

The authors call these objects Combinatorial Multiple Eisenstein Series.

The "Double Shuffle" Puzzle

To understand why this bridge is so important, you need to understand the "Double Shuffle" rules.

Imagine you have a deck of cards. You can shuffle them in two different ways:

1. The "Stuffle" Shuffle: You can merge cards if they have the same value.
2. The "Shuffle" Shuffle: You can interleave cards without merging them.

In the world of numbers, these two shuffling methods usually give you different results, but they are related by a complex set of equations (the "Extended Double Shuffle Equations").

• The frozen numbers satisfy these rules perfectly.
• The moving shapes also satisfy these rules perfectly.

The problem was: Does the bridge (the q-series) also satisfy these rules?

The authors say YES. They constructed these q-series specifically so that they obey the "Double Shuffle" laws. This means the bridge isn't just a random path; it's a structurally sound highway that respects the fundamental laws of the city.

How They Built It: The "Bimould" Blueprint

How did they build this bridge? They didn't just guess. They used a blueprint called a "Bimould."

Think of a Mould as a mold for making cookies. You put dough in, and it comes out in a specific shape.

• A Bimould is a super-mold that has two handles. It takes two sets of instructions (variables) and creates a complex, multi-layered cookie.

The authors took a "rational solution" (a simple, known set of rules) and used this Bimould mold to bake a new family of cookies.

• They proved that these new cookies are Symmetril: They look the same no matter how you rearrange the ingredients (like a perfect snowflake).
• They proved they are Swap Invariant: If you flip the mold upside down and swap the handles, the cookie still looks the same (like a reflection in a mirror).

These two properties are the "secret sauce" that ensures the bridge connects the two neighborhoods correctly.

Why Should You Care? (The "So What?")

1. It Unifies Math: It shows that the static world of numbers and the dynamic world of shapes are actually two sides of the same coin. You can now translate problems from one side to the other.
2. It Solves a Mystery: It answers a question raised by mathematician Alexander Okounkov: "Can we find a version of these numbers that behaves like a modular form?" The answer is yes, and this paper provides the recipe.
3. It's a New Tool: Just as a bridge allows people to travel between cities, this mathematical tool allows researchers to travel between different areas of math, potentially solving problems that were previously stuck on one side of the river.

The Analogy Summary

• Multiple Zeta Values: The "Ancestors" (frozen, historical numbers).
• Eisenstein Series: The "Descendants" (living, breathing shapes).
• The Bridge (Combinatorial Multiple Eisenstein Series): A time-traveling vehicle that is an ancestor at the start of the trip and a descendant at the end, but is a unique, hybrid creature in the middle.
• The Rules (Double Shuffle): The traffic laws that the vehicle must obey to stay on the road.
• The Bimould: The engineering blueprint that guarantees the vehicle is built correctly.

In short, this paper builds a mathematical time machine that smoothly transitions between the rigid world of numbers and the fluid world of shapes, proving that they are deeply connected by a hidden, symmetrical structure.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the relationship between Multiple Zeta Values (MZVs) and Modular Forms, specifically focusing on the algebraic structures governing them.

• Multiple Zeta Values (MZVs): Defined as $\zeta(k_1, \dots, k_r) = \sum_{m_1 > \dots > m_r > 0} \frac{1}{m_1^{k_1} \dots m_r^{k_r}}$. They satisfy the Extended Double Shuffle (EDS) relations, which arise from two different product structures: the stuffle (harmonic) product and the shuffle product.
• Eisenstein Series: Classical Eisenstein series $G_k(\tau)$ are modular forms whose constant terms (Fourier coefficients at $q=0$) are related to MZVs (specifically $\zeta(k)$).
• The Gap: While depth-1 Eisenstein series interpolate between MZVs and rational numbers (related to Bernoulli numbers), a unified combinatorial framework for arbitrary depth that satisfies the EDS relations and interpolates between MZVs and a rational solution was lacking. Previous attempts (e.g., by Gangl-Kaneko-Zagier) defined "Multiple Eisenstein Series" but their Fourier expansions involved complex dependencies on MZVs and did not form a purely combinatorial $q$-series algebra independent of the specific values of MZVs.

Goal: Construct a family of $q$-series with rational coefficients, called Combinatorial Multiple Eisenstein Series (CMES), that:

1. Lift a given rational solution to the EDS equations.
2. Interpolate between this rational solution (as $q \to 0$) and MZVs (as $q \to 1$).
3. Satisfy a variant of the EDS relations.
4. Are defined purely combinatorially via generating series, avoiding convergence issues inherent in lattice sums.

2. Methodology

The authors employ the theory of Moulds and Bimoulds (introduced by Jean Ecalle) to construct these series. The methodology involves several key steps:

A. Algebraic Framework

• Bimoulds: Instead of standard generating series, the authors work with bimoulds $G(X_1, \dots, X_r; Y_1, \dots, Y_r)$, which depend on two sets of variables. The coefficients of these bimoulds are the combinatorial bi-multiple Eisenstein series $G\binom{k_1, \dots, k_r}{d_1, \dots, d_r}$.
• Symmetrility: A property analogous to the stuffle product. A bimould is symmetril if its coefficient map is an algebra homomorphism from a quasi-shuffle algebra (based on the alphabet $L_z^{\text{bi}} = \{z_k^d\}$) to the ring of $q$-series.
• Swap Invariance: A functional equation reflecting the conjugation of integer partitions. A bimould is swap invariant if it satisfies a specific transformation rule swapping $X$ and $Y$ variables with cumulative sums.

B. Construction of Building Blocks

The construction relies on four specific bimoulds inspired by the Fourier expansion of classical Multiple Eisenstein Series:

1. $b$ (The Rational Solution): A bimould derived from a fixed rational solution $\beta$ to the EDS equations (depth 1 given by Bernoulli numbers). It is constructed to be both symmetril and swap invariant.
2. $g$ (The $q$-analogue): A bimould defined by sums over partitions, generalizing divisor sums. It is swap invariant but not symmetril under the standard stuffle product; it requires a modified product $\hat{\ast}$.
3. $L_m$ (Generating Series): A family of bimoulds acting as generating functions for "combinatorial monotangent functions." They are shown to be symmetril.
4. $g^\ast$ (The Harmonic Lift): Constructed from $L_m$ via a specific summation over indices, mimicking the structure of the classical $g^\ast$ in the Fourier expansion of Multiple Eisenstein Series. It is proven to be symmetril.

C. The Main Definition

The Combinatorial Multiple Eisenstein Series are defined as the coefficients of the product bimould:
$$G = g^\ast \hat{\times} b$$
where $\hat{\times}$ is the standard product of bimoulds. The series $G(k_1, \dots, k_r)$ corresponds to the case where all derivative indices $d_i = 0$.

3. Key Contributions and Results

A. Existence and Properties of CMES

The authors prove the existence of a bimould $G$ that is:

• Symmetril: Satisfies the stuffle product relations.
• Swap Invariant: Satisfies the shuffle-like relations derived from the swap involution.
• Interpolation:
  • $\lim_{q \to 0} G(k_1, \dots, k_r) = \beta(k_1, \dots, k_r)$ (The rational solution).
  • $\lim_{q \to 1} (1-q)^{w} G(k_1, \dots, k_r) = \zeta(k_1, \dots, k_r)$ (The Multiple Zeta Value), where $w$ is the weight.

B. Generalized Double Shuffle Relations

The paper establishes that CMES satisfy a variant of the EDS equations.

• Stuffle Relation: Direct consequence of symmetrility.
• Shuffle Relation: Derived from the combination of symmetrility and swap invariance.
• Derivative Obstacle: Unlike MZVs, the product of CMES does not close perfectly under the standard stuffle product due to derivatives. The authors show:
  $$q \frac{d}{dq} G(w) = G(z_2 \ast w - z_2 \shuffle w)$$
  This non-vanishing term is expressed using bi-multiple Eisenstein series (where $d_i > 0$), effectively treating derivatives as "partial" indices in the bimould framework.

C. Connection to Modular Forms

• The space spanned by CMES contains the algebra of quasi-modular forms with rational coefficients.
• Specific linear combinations of CMES yield modular forms.
• The authors provide a criterion: If a linear combination of CMES is a modular form of weight $k$, the corresponding linear combination of MZVs equals $(2\pi i)^k$ times the corresponding linear combination of the rational solution $\beta$.

D. Algebraic Structure

• The space of CMES, denoted $\mathcal{G}^{bi}$, is conjectured to be isomorphic to the algebra of formal multiple Eisenstein series introduced in previous work.
• The paper suggests that $\mathcal{G}^{bi}$ carries an $sl_2$-action, extending the action on quasi-modular forms.

4. Significance

1. Unified Framework: This work provides the first purely combinatorial model of $q$-analogues of MZVs that are simultaneously compatible with the weight grading of quasi-modular forms and satisfy the full set of EDS relations (stuffle and shuffle) in arbitrary depth.
2. Resolution of the "Derivative" Problem: By introducing the bi- indices ($d_i$), the authors successfully incorporate derivatives into the algebraic structure. This allows the EDS relations to hold exactly, with the "error terms" (derivatives) being absorbed into the larger space of bi-multiple Eisenstein series.
3. Interpolation: It rigorously realizes the idea of interpolating between the "arithmetic" world (MZVs) and the "modular" world (Eisenstein series) via a single family of $q$-series.
4. New Tools for MZV Theory: The use of bimoulds and the specific "swap" involution offers a new perspective on the algebraic relations of MZVs, potentially leading to new proofs of conjectures regarding the dimension of spaces of MZVs (e.g., the Broadhurst-Kreimer conjecture).
5. Connection to Okounkov's Question: The results suggest a positive answer to a question by Okounkov regarding the space of $q$-analogues of MZVs, as the CMES span the space $Z_q$ defined in prior literature.

In summary, Bachmann and Burmester successfully construct a robust algebraic object (Combinatorial Multiple Eisenstein Series) that bridges the gap between number theory (MZVs) and the theory of modular forms, resolving technical difficulties regarding derivatives and providing a concrete realization of the extended double shuffle relations in the $q$-series setting.