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The algebra of cell-zeta values

Part of: Computational aspects in algebraic geometry Computational number theory

Published online by Cambridge University Press:  10 March 2010

Francis Brown ,
Sarah Carr  and
Leila Schneps
Francis Brown
Affiliation:
CNRS and IMJ, 175 rue du Chevaleret, 75013 Paris, France (email: [email protected])
Sarah Carr
Affiliation:
Mathématiques Bât. 425, Université Paris-Sud, 91405 Orsay Cedex, France (email: [email protected])
Leila Schneps
Affiliation:
CNRS and IMJ, 175 rue du Chevaleret, 75013 Paris, France (email: [email protected])
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Abstract

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In this paper, we introduce cell-forms on 𝔐0,n, which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space 𝔐0,n(ℝ). We show that the cell-forms generate the top-dimensional cohomology group of 𝔐0,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 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.

Keywords

multiple zeta valuesLie algebrascohomologymoduli spacespolygons

MSC classification

Secondary: 14Q15: Higher-dimensional varieties 11Y40: Algebraic number theory computations
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 3 , May 2010 , pp. 731 - 771
Copyright
Copyright © Foundation Compositio Mathematica 2010

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