Formal multiple Eisenstein series and their derivations

Technical Summary: Formal Multiple Eisenstein Series and Their Derivations

Problem Statement
The paper addresses the algebraic structure of multiple Eisenstein series and their relationship to multiple zeta values (MZVs) and (quasi)modular forms. While classical multiple Eisenstein series (introduced by Gangl–Kaneko–Zagier) and their regularized versions satisfy complex algebraic relations, a purely formal framework that captures these relations without relying on analytic convergence or specific $q$-series realizations has been lacking. Specifically, the authors seek to define a "formal" algebra that mirrors the algebraic properties of classical Eisenstein series, including their behavior under derivations, and to establish a rigorous link between this formal algebra and the algebra of formal multiple zeta values.

Methodology
The authors construct the algebra of formal multiple Eisenstein series, denoted $\mathcal{G}_f$, through a combinatorial and algebraic approach:

1. Algebraic Setup: They define an alphabet $\mathcal{A} = \{ \binom{k}{d} \mid k \ge 1, d \ge 0 \}$ and consider the free algebra $\mathbb{Q}\langle \mathcal{A} \rangle$. They equip this space with a "stuffle" product (a quasi-shuffle product) defined by the rule $\binom{k_1}{d_1} \ast \binom{k_2}{d_2} = \binom{k_1+k_2}{d_1+d_2}$, extending the standard stuffle product used for MZVs.
2. Swap Invariance: A crucial step is the introduction of a linear map $\sigma$ (the "swap") acting on generating series of words in $\mathcal{A}$. This map is motivated by the conjugation of partitions observed in the $q$-series expansions of multiple Eisenstein series. The algebra $\mathcal{G}_f$ is defined as the quotient of $(\mathbb{Q}\langle \mathcal{A} \rangle, \ast)$ by the ideal generated by elements of the form $\sigma(w) - w$. This imposes "swap invariance" as a fundamental relation.
3. Derivations: The authors introduce three specific derivations, $W$, $D$, and $\delta$, on the underlying free algebra. They prove these derivations are $\sigma$-equivariant (commute with the swap map), ensuring they descend to well-defined derivations on the quotient algebra $\mathcal{G}_f$.
   • $W$ acts as a weight operator.
   • $D$ acts as a derivation increasing the weight by 2 (analogous to $q \frac{d}{dq}$).
   • $\delta$ is a more complex derivation of weight $-2$, constructed as a sum of five specific derivations on quasi-shuffle algebras.
4. Quotient to MZVs: They define the algebra of formal multiple zeta values, $\mathcal{Z}_f$, as the quotient of $\mathcal{G}_f$ by an ideal $N$ generated by words not of the form $\binom{1, \dots, 1, k_1, \dots, k_r}{d_1, \dots, d_s, 0, \dots, 0}$. This corresponds to projecting onto the "constant term."

Key Contributions and Results

• Definition of $\mathcal{G}_f$: The paper formally defines $\mathcal{G}_f$ as the algebra of formal multiple Eisenstein series. It is shown that the single-indexed elements $G_f(k_1, \dots, k_r)$ (where $d_i=0$) satisfy the same algebraic relations as the classical Eisenstein series, including recursive formulas (Corollary 2.11) and generalized Euler relations (Corollary 2.12).
• $sl_2$-Algebra Structure: The primary result (Theorem 1.2, Theorem 4.12) establishes that $\mathcal{G}_f$ is an $sl_2$-algebra. The derivations $W, D, \delta$ satisfy the commutator relations:
  $$[W, D] = 2D, \quad [W, \delta] = -2\delta, \quad [\delta, D] = W.$$
  This generalizes the known $sl_2$-structure of the algebra of quasimodular forms to the setting of multiple Eisenstein series.
• Connection to Formal MZVs: Theorem 1.1 proves the existence of a surjective algebra homomorphism $\pi: \mathcal{G}_f \to \mathcal{Z}_f$ (formal multiple zeta values) that maps $G_f(k_1, \dots, k_r)$ to $\zeta_f(k_1, \dots, k_r)$. The kernel of this map is explicitly described. Theorem 2.17 demonstrates that the formal multiple zeta values defined this way satisfy the extended double shuffle relations, confirming the consistency of the formal approach with classical conjectures.
• Formal Quasimodular Forms: The subalgebra $\mathcal{M}_f \subset \mathcal{G}_f$ generated by $G_f(2), G_f(4), G_f(6)$ is shown to be isomorphic to the algebra of classical quasimodular forms $\tilde{M}$ as an $sl_2$-algebra (Theorem 5.4). This subalgebra satisfies the Ramanujan differential equations and the Chazy equation.
• Formal Modular and Cusp Forms: The authors define formal modular forms $\mathcal{M}_f$ as the kernel of $\delta$ within $\mathcal{M}_f$ and formal cusp forms $\mathcal{S}_f$ as the kernel of $\pi$ within $\mathcal{M}_f$. They prove these spaces are isomorphic to their classical counterparts (Proposition 5.7, Corollary 5.9).
• Balanced Setup: The paper includes an appendix (and Section 6) describing a "balanced setup" introduced by Nils Matthes (and Burmester), which provides an alternative isomorphism between $\mathcal{G}_f$ and a quotient of a free algebra on a different alphabet, offering a different perspective on the stuffle product and swap invariance.

Significance
The paper claims to provide a "novel and purely formal approach" to classical (quasi)modular forms. By constructing $\mathcal{G}_f$, the authors build a new link between formal multiple zeta values and modular forms that does not rely on the analytic properties of $q$-series or the convergence of Eisenstein series.

The significance lies in:

1. Unification: It unifies the algebraic structures of multiple zeta values and multiple Eisenstein series under a single formal framework where the "constant term" projection is an algebra homomorphism.
2. Derivations: It extends the $sl_2$-algebra structure of quasimodular forms to the realm of multiple Eisenstein series, providing explicit formulas for derivations that were previously only known in the classical analytic setting.
3. Formal Relations: It offers a new perspective on the extended double shuffle relations, showing they are equivalent to the combination of swap invariance, the stuffle product, and the specific quotient defining formal MZVs.

The authors note that while the formal algebra $\mathcal{G}_f$ is conjectured to be spanned by single-indexed elements (Conjecture 2.6) and isomorphic to the space of combinatorial multiple Eisenstein series (Conjecture 4.18), these remain open problems. The work establishes the algebraic foundations necessary to explore these connections further.