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The algebra of cell-zeta values. (English) Zbl 1195.14078

Summary: We introduce cell-forms on \(\mathfrak M_{0,n}\), which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space \(\mathfrak M_{0,n}(\mathbb R)\). We show that the cell-forms generate the top-dimensional cohomology group of \(\mathfrak M_{0,n}\), 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.

MSC:

14Q15 Computational aspects of higher-dimensional varieties
11Y40 Algebraic number theory computations
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