The algebra of cell-zeta values

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1 The algebra of cell-zeta values Sarah Carr May 0, 200 A multizeta value or MZV is a real number, ζ(k i,..., k d ) = X n > >n d >0 Abstract n k nk d d, k i Z, k 2. () We are interested in studying the Q algebra generated by these numbers. In this talk we would like to present a candidate for this algebra, by giving its generators and relations. The motivation for this work is the recent theorem by F. Brown: Theorem. All periods on M 0,n+ are Q linear combinations of MZV s. Background. M 0,n+ Definition 2. M 0,n+ is the moduli space of genus 0 complex projective curves with n + distinct ordered marked points (punctures) modulo isomorphism. M 0,n+ (P C \ {0,, })n \ where := {t i = t j }. We denote a point in this space by an ordered n + -tuple: (0, t i,..., t n,, ), where the t i denote the punctures on the sphere, run through C \ {0, }, are distinct and are different from 0,, and..2 Compactification and Boundary Divisors We denote by M 0,n+, the stable compactification of Deligne and Mumford of M 0,n+. The irreducible boundary divisors M 0,n+ of codimension can be indexed by partitions of {0, t,..., t n,, } = S n. In this talk the important spaces are M 0,n+ and M 0,n+ \ M 0,n+.. Periods We denote by M 0,n+ (R) the set of points (0, t,..., t n,, ) where t i R. The space is not connected, but is partitioned into cells. We call the cell 0 < t < < t n < n, the standard cell. The boundary of n intersects the irreducible boundary divisors indexed by subsets of consecutive numbered marked points, {t i, t i+,..., t ij }. Let the collection of of boundary divisors sharing a boundary with n be denoted by n. Example. Definition 4. A period on M 0,n+ is an integral, n ω, where ω has rational function coefficient which is holomorphic on M 0,n+ and has at most simple poles on all the boundary divisors and of course has no poles on n.

2 t 2 0<t <<t 2 < < t 2 < t < 0<t <t 2 < t polygon, γ cell form, ω γ 0 t 2 t t [t 2, 0,, t, t, ] = dt dt 2 dt ( t 2 )(t t )(t ) Boundary = Singularity divisor Figure : A cell-form in M 0,6 2 Generators of the period algebra, cell-forms I will start this section with an example from M 0,6. Example 5. Consider the 6-gon with sides decorated by S. I may obtain a differential form by taking the - volume form and dividing by the product of successive differences of the sides, leaving any side containing out of the product as in figure. Definition 6. A cell-form, ω γ, on M 0,n+ is a differential n form associated to a cyclic order (or polygon) on S n, γ = [, t i0,..., t in+ ]: ω γ := dt dt n n+ k= (t i k t ik ). We will refer to a block of a cyclic order as a sublist of that order. Note: Let I = {t i, t i+,..., t i+j } denote a set of marked points with consecutive indices (as in the definition of n ). The integral, ω γ converges if and only if γ contains no consecutive blocks which are orderings of such I. Definition 7. ω γ is a 0-cell-form if γ contains the block (0, ). Lemma 8. The 0-cell-forms form a basis for H n dr (M 0,n+). Definition 9. The shuffle product on lists, A = (a,..., a j ) and B = (b,..., b k ), is defined as AxB = σ σ(a B), where denotes concatenation and the sum runs over all permutations σ S j+k where σ preserves the orders of both A and B. 2

3 Example 0. (a, b)x(c, d) = (a, b, c, d) + (a, c, b, d) + (a, c, d, b) + (c, a, b, d) + (c, a, d, b) + (c, d, a, b) Definition. A degree k shuffle product, A x xa k, is called a convergent shuffle in I S n if. Aj = I where I is a set with consecutive indices, I = {t i, t i+,..., t i+j }, 2. no factor, A j, contains a block of consecutive indices. Example 2. t 2 xt is a convergent shuffle. t t t 4 t 2 xt 5 is not a convergent shuffle because the first factor is a block whose associated set is a set of consecutive indices, {t, t 2, t, t 4 }. Definition. An insertion form is a linear combination of cell forms gotten from inserting a convergent shuffle on I = {t i,...} into the place of t i in a convergent cell-form and renumbering the indices so that there are no repeated marked points. Example 4. t xt 2 is a convergent shuffle and [0,, t,, t 2 ] is a convergent cell-form. From this we get the insertion form: [0,, (t xt 2 ),, t ]. In the form we renamed t 2 as t in order to make the variables in the form distinct. Example 5. All forms naturally associated to multizeta values are insertion forms gotten from inserting into a form on M 0,5. ζ() = [0,, t,, t 2 xt ] ζ(2, ) = [0,, t xt 2,, t ] ζ(2,, ) = [0,, t xt 2 xt,, t 4 ] 4 ζ(2, 2) = [0,, t xt,, t 2 xt 4 ] 4 ζ(4) = [0,, t,, t 2 xt xt 4 ] 4 Theorem 6. The Q vector space of periods on M 0,n+ is generated by integrals over n of insertion forms. Method of proof: Let M 0,n+ = M 0,n+ n, and let Res di : HdR(M n 0,n+ ) H n dr (M 0,n+2) be the map which calculates the residue of a form along the irreducible boundary divisor, d I. Then HdR(M n 0,n+) Ker(Res di ) ω : ω is an insertion form. d I n The formal cell number algebra Lemma 7. Let ω γ = [, σ (0,, t,..., t j )] = [, A, 0, A 2,, A ] ω γ2 = [, σ 2 (0,, t j+,..., t j+k )] = [, B, 0, B 2,, B ].

4 Then where ω γ ω γ2 = ω γ xγ 2, j k j+k γ xγ 2 = [, A xb, 0, A 2 xb 2, A xb ] and x 2 = (i,...i j+k )0 < t i < t ik <, where the lists, (i,..., i j+k ) run over all terms in the shuffle product, (t,..., t j )x(t j+,..., t j+k ). Lemma 8. Let P n denote the Q vector space of n + -gons decorated by S n. Let π : P n H n dr(m 0,n+ ) Then, Ker(π) = I = [e, AxB] : e S n, A B = S n \ {e}. γ ω γ. (2) We call I the space of shuffles with respect to one point. This lemma is easy to prove using lemma 8. Now, we will think of n as an element of P n by associating it with the order, n = [0, t,..., t n,, ]. This is natural if one just pictures this list replacing the commas by the symbol <. Then we have the natural map, where σ S n+ such that σ(α) =. π 2 : P n P n periods (α, γ) σ(ω γ ), Definition 9. The formal cell number algebra, FC. FC is a Q vector subspace of n=0 P n P n generated by the pairs, ( σ( n ), σ(ω) ) : σ S n+, ω is an insertion form, with the following relations, σ(α). ( σ(n ), σ(ω) ) = ( n, ω ) (variable changes), 2. (α, [e, AxB]) = 0 (I = Ker(π)),. according to the rules given in lemma 7. (α, ω )(α 2, ω 2 ) = (α xα 2, ω xω 2 ), There is a possibility that FC is isomorphic to the multizeta value algebra, Z. This conjecture seems out of reach at the moment because of questions of transcendence of MZV s. However, up to weight 9, FC verifies Zagier s dimension conjecture on the formal multizeta value algebra, FZ. Therefore, we believe that the following conjecture is reasonable. Conjecture 20. FC FZ There is much work to be done. We are for the moment unable to find a possible candidate for such an isomorphism, because we don t know how to explicitly dig out the stuffle relation from the three relations on FC, though we know that it can be done in low weights. 4

5 References [BCS] F. Brown, S. Carr, L. Schneps, The algebra of cell-zeta values, to appear Compositio Math. (200) [DM] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, IHES Sci. Publ. Math., Vol. 6 (969), pp [GM] A. Goncharov and Y. Manin, Multiple ζ-motives and moduli spaces M 0,n, Compositio Math. Vol. 40 no. (2004), pp. -4 5