The algebra of cell-zeta values

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.

References in corpus (1)

• Mixed Tate Motives and Multiple Zeta Values

Cited by in corpus (4)

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• Single-valued integration and superstring amplitudes in genus zero
• Sequences, modular forms and cellular integrals
• Periods and Superstring Amplitudes