The algebra of cell-zeta values

TL;DR

This paper introduces cell-forms on the moduli space of genus zero curves, establishing their relation to multizeta values and defining a new algebraic structure called the cell-zeta algebra.

Contribution

It constructs an explicit basis of differential forms on ,n, linking geometric cell structures to algebraic multizeta values and proposing a conjectural isomorphism with the formal multizeta algebra.

Findings

Cell-forms generate the top cohomology of ,n.

Cell-zeta values form a -algebra equal to multizeta values.

Quadratic relations arise from moduli space geometry.

Abstract

In this paper, we introduce cell-forms on $\mathcal{M}_{0,n}$, which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space $\mathcal{M}_{0,n}(\mathbb{R})$. We show that the cell-forms generate the top-dimensional cohomology group of $\mathcal{M}_{0,n}$, so that there is a natural duality between cells and cell-forms. In the heart of the paper, we determine an explicit basis for the subspace of differential forms which converge along a given cell $X$. The elements of this basis are called insertion forms, their integrals over $X$ are real numbers, called cell-zeta values, which generate a $\mathbb{Q}$-algebra called the cell-zeta algebra. By a result of F. Brown, the cell-zeta algebra is equal to the algebra of multizeta values. The cell-zeta values satisfy a family of simple quadratic relations coming from the geometry of moduli spaces, which leads to a natural definition of a formal version of the cell-zeta algebra, conjecturally isomorphic to the formal multizeta algebra defined by the much-studied double shuffle relations.