Multiplicative convolution and double shuffle relations

Abstract: We develop a geometric approach to the regularized double shuffle relations for multiple zeta values, based on convolution of perverse sheaves on $\mathbb{C}^*$ and inspired by the approach of Deligne and Terasoma. We introduce semi-holonomy isomorphisms associated with pro-unipotent paths and show that their compatibility with multiplicative convolution is equivalent to a condition on the pro-unipotent fundamental group, the homological pentagon equation. We prove that this condition is equivalent to the regularized double shuffle relations, yielding a geometric proof that the pentagon equation implies these relations. The approach is purely topological and avoids Hodge-theoretic and Tannakian methods.

• The paper presents a geometric-topological framework establishing an explicit equivalence between the pentagon equation and the regularized double shuffle relations for MZVs.
• It introduces novel tools such as perverse sheaf convolution, semi-holonomy isomorphisms, and a transport algebra to analyze multiple zeta values.
• The work bypasses traditional Hodge-theoretic methods, offering concrete topological constructions that unify period, motivic, and associator theories.

Multiplicative Convolution and Double Shuffle Relations: A Geometric and Topological Approach

Overview

The paper "Multiplicative convolution and double shuffle relations" (2604.26357) develops a geometric and topological framework for understanding the regularized double shuffle relations for multiple zeta values (MZVs). It establishes the equivalence between these relations and compatibility conditions arising from the multiplicative convolution of perverse sheaves on $\mathbb{C}^*$. Importantly, this work provides a geometric proof that the pentagon equation implies the double shuffle relations without reliance on Hodge-theoretic or Tannakian methods, leveraging pro-unipotent path spaces, semi-holonomy isomorphisms, and the transport algebra framework.

Geometric Formalism for Double Shuffle Relations

The study begins by recognizing multiple zeta values as periods of iterated integrals, which consequently satisfy the relations dictated by the Drinfeld associator via the pentagon and hexagon equations. Another family of relations, the regularized double shuffle relations, arises from algebraic manipulations within the defining series of MZVs. The connection between these relations and geometric objects is articulated using Mellin transforms and convolution of polylogarithms, leading to perverse sheaf-theoretic interpretations.

Inspired by previous frameworks (notably by Deligne-Terasoma and Furusho), the author constructs a purely topological regime:

• Sheaf Convolution: The multiplicative convolution of perverse sheaves is formulated as a geometric analog of the product in the MZV context.
• Semi-holonomy: Introduced as isomorphisms between vanishing cycles at $1$ and certain cohomological realizations on projective space, depending parametrically on pro-unipotent paths. The standard interval $[0,1]$ yields the "canonical" semi-holonomy.
• Transport Algebra: The algebra $W$, generated by the action of the pro-unipotent completion of the fundamental group, encapsulates the structure needed for the analysis of vanishing cycles and convolution-induced maps.
• Harmonic Coproduct: The symmetric monoidal structure (arising from convolution) induces a canonical "harmonic" coproduct on $W$, closely related to the structures in the theory of mixed Tate motives.

Main Results

Compatibility of Semi-holonomy with Convolution

A central technical achievement is the explicit demonstration that the standard semi-holonomy is compatible with the tensor structure induced by convolution (Theorem in Section 4). This compatibility is encoded in an equivalence between the regularized double shuffle relations for MZVs and a geometric "homological pentagon equation" imposed on pro-unipotent paths in the moduli space $M_4$.

• The approach avoids both Hodge-theoretic machinery and quotient category/Tannakian arguments used in prior work, providing explicit geometric representatives for all relevant isomorphisms.

Geometric Construction of the Harmonic Coproduct

A geometric realization of the harmonic coproduct is constructed directly, bypassing categorical quotient constructions and instead working in the subcategory of perverse sheaves without sections supported at $1$. The action of the transport algebra $W$ on vanishing cycles of convolutions is shown to be governed by this coproduct, and its explicit formula in terms of free generators is presented.

Semi-holonomy and Fox Derivatives

The paper introduces the explicit formula for general semi-holonomy isomorphisms along arbitrary pro-unipotent paths using Fox derivatives in both augmentation and logarithmic coordinates. This provides detailed coherence data relating different choices of paths and recovers classical relations in the group algebra setting.

Equivalence of Homological Pentagon and Regularized Double Shuffle Relations

A key conceptual outcome is the proof that the homological pentagon equation for symmetric elements in the completed fundamental group is equivalent to the regularized double shuffle relations. The homological pentagon equation is characterized via commutators in certain intersection subgroups and the (co)homology of associated local systems in the fibers of the relevant moduli space projections.

• The equivalence (summarized in the final theorem) unifies several approaches in the motivic and period domains, linking the combinatorics of MZVs' defining relations to geometric operations in sheaf categories.

Implications and Future Directions

This work advances the understanding of the deep correspondence between geometric/topological operations (convolution, vanishing cycles, holonomy) and the algebraic structures of MZVs (double shuffle relations, associator equations). The proofs eschew the use of mixed Hodge modules or motivic Galois groups, favoring explicit and topologically motivated constructions.

Theoretical implications include:

• A unified framework linking the motivic, topological, and period-theoretic perspectives on MZV relations.
• New tools and perspectives for analyzing the Turaev cobracket, the Kashiwara–Vergne problem, and the interplay between geometric representation theory and period/cohomology phenomena.

Practical implications may emerge in explicit computations within the cohomology of moduli spaces, and in the algorithmic generation or verification of relations in the motivic Lie algebra of MZVs.

Future developments may include:

• Extensions of the semi-holonomy formalism to other moduli spaces or configuration spaces, or its application to other families of periods.
• Clarification of the connections between Hodge-theoretic realizations and the geometric-topological constructions detailed here, possibly leading to more conceptual and computationally tractable presentations of categories of mixed Tate motives and their period maps.
• Further articulation of the connections with the Kashiwara–Vergne problem, possibly yielding new progress on the quantization of Lie bialgebras.

Conclusion

The paper provides a topologically grounded, explicit, and highly structured framework for understanding the equivalence between the pentagon equation and the regularized double shuffle relations for MZVs. It leverages perverse sheaf convolution, pro-unipotent path spaces, semi-holonomy isomorphisms, and transport algebra structures, bypassing the need for motive or Hodge-theoretic formalism. This geometric and algebraic synthesis yields new connections and provides a versatile platform for further investigation into the structure of periods, associators, and related problems in the theory of motives and algebraic geometry.