The algebra of cell-zeta values

Abstract

In this paper, we introduce cell-forms on M 0,n , which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space M 0,n (R). We show that the cell-forms generate the top-dimensional cohomology group of 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 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.

Periods and Superstring Amplitudes

“…with σ(N) = N and compatible with the cyclic structure γ. The cell-form corresponding to γ is defined as [16]…”

“…As a consequence the cohomology group H N −3 (M 0,N ) is canonically isomorphic to the subspace of polygons having the edge 0 adjacent to edge 1 [16]. Figure 1: N-gon describing the 01 cyclic structure γ = (0, 1, ρ).…”

“…By changing variables the period integral (2.15) can be brought into an integral over the standard cell δ parameterized in (2.8). To obtain a convergent integral (2.15) in [16] certain linear combinations of 01 cell-forms (2.13) (called insertion forms) have been constructed with the properties of having no poles along the boundary of the standard cell δ and converging on the closure δ. E.g. in the case of M 0,5 the cell-form ω γ corresponding to the cell γ = (0, 1, t 1 , ∞, t 2 ) can be integrated over the compact standard cell δ defined in (2.8):…”

Symbolic integration and multiple polylogarithms

“…The integrals (3.15) play a role in pure mathematics, such as in reference [7,10], and in physics in the context of deformation quantization [17], superstring theory [31], Schnetz' model of graphical functions [32] and in perturbative quantum field theory. Here we focus on the latter and give a very brief outlook on how the use of B m Ω n can facilitate the computation of Feynman integrals.…”

Sequences, modular forms and cellular integrals

“…then I σ (n) converges if and only if σ is convergent. For n = 0, we then obtain the cellzeta values ζ σ (N − 3) = I σ (0) studied in [14], which are multiple zeta values of weight N − 3. More generally, by [13, corollary 8•2], I σ (n) is a Q-linear combination of multiple zeta values of weight less than or equal to N − 3.…”