Cellular Multizeta Values. Sarah Carr

Transcription

1 Cellular Multizeta Values Sarah Carr August, 2007

2 Goals The following is joint work with Francis Brown and Leila Schneps based on a combinatorial theory of pairs of polygons. 1. Objects: Periods on M 0,n 2. Method: Pairs of Polygons = Periods 3. Theorem: H l (M δ 0,n ) = Insertion Polygons 4. Algebra: Polygon algebra 5. Theorem: Polygon algebra Multizeta algebra 1

3 1. Periods on Moduli Space, M 0,n Definition. M 0,n = M 0,n (C) is the space of Riemann spheres with n distinct, ordered marked points modulo isomorphism, i.e. modulo the action of PSL 2 (C). We will denote a point in M 0,n by (z 1, z 2,..., z n ) or by the representative in its equivalence class (0, t 1,..., t l,1, ), l = n 3. M 0,n (P 1 \ {0,1, }) n 3 \, where is the fat diagonal = {t i = t j }. 2

4 Stable Compactification of M 0,n : M 0,n The boundary components added to M 0,n to compactify correspond to loops around r points, 2 r n 2. These boundary components (divisors) are blowups of the regions {z i1 = z i2 = = z ir 2 r n 2 } (P1 ) n 3. z 1 z 2 z 3 z 4 z 5 z 6 z 1 z 2 z 3 z 4 z 5 z 6 = 3

5 M 0,n (R): {(0, t 1,..., t l,1, ) s.t. t i R}. The connected components (cells) of M 0,n (R) are given by fixed orderings of the t i. Example t 2 0<t 1 <1<t 2 < 1 < t 2 < t 1 < 0<t 1 <t 2 <1 t 1 M 0,5 (R) Standard Cell := δ = 0 < t 1 <... < t l < 1 Boundary of δ in M 0,n \ M 0,n = loops around consecutive points 4

6 Definition. We will define a period on M 0,n as a convergent integral, γ ω, over a cell γ in M 0,n (R), of a form ω which is holomorphic on M 0,n and has at most simple poles on M 0,n \ M 0,n. Up to a variable change corresponding to permuting {0,1, t 1,..., t l }, all periods can be written as integrals over the standard cell, 0 < t 1 <... < t l < 1. Proposition. H l (M 0,n ) Vector space of differential l-forms convergent on M 0,n with at most simple poles on M 0,n \ M 0,n. 5

7 2. Polygons We can associate an n-polygon decorated by the marked points, {0, t 1,..., t l,1, }, to the cell in M 0,n (R) defined by the clockwise order of the marked points around the polygon. polygon, γ 0 t 1 cell, γ (0, t 1, t 2, t 3,1, ) = 0 < t 1 < t 2 < t 3 < 1 < 1 t 3 t 2 6

8 Definition. Let γ be a polygon decorated by the marked points, {0, t 1,..., t l,1, }. The cell form associated to γ, ω γ, is the volume form, dt 1 dt l, divided by the product of successive differences of marked points around the polygon. Example. polygon, γ cell form, ω γ 0 1 t 2 [t 2,0,1, t 1, t 3, ] = dt 1 dt 2 dt 3 ( t 2 )(t 3 t 1 )(t 1 1) t 3 t 1 Boundary = Singularity divisor So we have a map from pairs of polygons to {periods} { }: (γ, ω) γ ω. 7

9 Theorem. (BCS) The cell forms [0,1,...], which we call 01-cell forms, form a basis for H l (M 0,n ). Proof. The proof of this theorem is based on results by Arnol d who displayed (n 2)! explicit linearly independent differential forms. These forms form a basis because dim(h l (M 0,n )) = (n 2)! by induction, since M 0,n M 0,n 1 is a fibration of fiber P 1 \ n 1 points. It is easy to express Arnol d s forms in terms of 01-cell forms. 8

10 Recall that the shuffle product of sequences, a = (a 1,..., a i ) and b = (b 1,..., b j ) is defined as a X b = σ S i+j σ(a 1,..., a i, b 1,...b j ) over all σ that preserve the ordering of both a and b. Example. (1,2) X(3) = (1,2,3) + (1,3,2) + (3,1,2). Shuffle product of polygons γ 1 = [A 1, z i1,..., A r, z ir ] γ 2 = [B 1, z i1,..., B r, z ir ]. We define the shuffle product of polygons with respect to the common points as γ 1 X γ 2 = [A 1 X B 1, z i1,..., A r X B r, z ir ]. 9

11 Let P n be the Q-vector space generated by oriented n-gons with the sides indexed by 0, t 1,..., t l,1,. So we have a natural map, φ : P n H l (M 0,n ). Definition. Let I n P n be the vector subspace generated by shuffle products with respect to one point [A X B, d], with d {z 1,..., z n } and A B = {z 1,..., z n } \ {d}. Lemma. I n ker(φ). Theorem. (BCS) P n /I n H l (M 0,n ). Proof. P n H l (M 0,n ) I n kernel dim(p n /I n ) = (n 2)! by Lyndon basis argument dim(h l (M 0,n )) = (n 2)! 10

12 3. Insertion basis for H l (M δ 0,n ) Definition. M δ 0,n := M 0,n {the boundary divisors of δ}. To study H l (M δ 0,n ) we use the polygon structure to calculate the residues of linear combinations of 01-cell forms. Example. The residue of a polygon (or cell form) along a divisor, d is a chord is given by Res = d t 3 d t 1 t 3 d d t 1 t 2 t 2 11

13 Example. Convergence of ω = [0,1, t 1, t 2,, t 3 ] + [0,1, t 2, t 1,, t 3 ] Identify bad divisors as chords, d := t 1 = t 2. Res d (ω) = [0,1, d,, t 3 ] [t 1 X t 2, d]. Image in forms is 0, since the right hand factor is a shuffle w.r.t. one element t 3 t 1 d + d = ω t 3 t 2 t 2 t 1 12

14 We write the form, ω = [0,1, t 1, t 2,, t 3 ] + [0,1, t 2, t 1,, t 3 ] = [0,1, t 1 X t 2,, t 3 ]. It is obtained by inserting the shuffle t 1 X t 2 into the smaller convergent cell form, [0,1, t 1,, t 2 ]. Insertion polygons are all obtained in this way and they map to insertion forms. Theorem. (BCS) The insertion forms form a basis for H l (M δ 0,n ) and the dimension can be counted using a recursion formula based on their construction. 13

15 4. The algebra of periods, C We define a product map, f : M 0,n M 0,r M 0,s by specifying subsets T 1, T 2 of T = {z 1,..., z n } such that if T 1 = {z i1,..., z ir }, T 2 = {z j1,..., z js }, then T 1 T 2 = 3 and T 1 T 2 = T. We define the product map as the product of two forgetful maps: f : (z 1,..., z n ) (z i1,..., z ir ) (z j1,..., z js ) such that the order of the original sequence is preserved (i k 1 <i k, j k 1 <j k ). 14

16 Theorem. Given two periods, on M 0,r and M 0,s and a product map f, we have ω 1 γ 1 γ 2 ω 2 = = f 1 (γ 1 γ 2 ) f (ω 1 ω 2 ) γ 1 X γ 2 ω 1 X ω 2. - (BCS) Product of cellforms equals their polygon shuffle product. - Preimage of product domain is a shuffle. Example. f : (0, t 1, t 2, t 3, t 4,1, ) (0, t 1, t 2,1, ) (0, t 3, t 4,1, ) (0,(t 1, t 2 ) X(t 3, t 4 ),1, ) = (0, t 1, t 2, t 3, t 4,1, ) (0, t 1, t 3, t 2, t 4,1, ) (0, t 3, t 1, t 4, t 2,1, )... (0, t 1, t 2,1, ) (0, t 3, t 4,1, ) 15

17 Definition. We denote by C the algebra of periods on M 0,n with multiplication given by the product maps. Conjecture. All of the relations in C are given by variable changes and relations coming from the different product maps. Definition. We denote by FC the formal algebra of pairs of polygons, (δ, ω), decorated by marked points where the ω is an insertion polygon modulo the following relations : 1. (δ, ω) = (σ(δ), σ(ω)), for all σ S n (variable changes) 2. (γ,[a X B, z i ]) = 0 (I n 0) 3. For each product map f, a corresponding shuffle product of polygons, (γ 1, ω 1 )(γ 2, ω 2 ) = (γ 1 X γ 2, ω 1 X ω 2 ). 16

18 5. FC C = Z Definition. Let n 1,..., n r N \ {0} such that n 1 2. Multizeta values are defined by nested sums 1 ζ(n 1,..., n r ) = k 1 >...>k r 1 k n kn r r where the weight is l = n n r. R Definition. We denote by Z the algebra of multizeta values. Theorem. (Brown) Z = C Proof. (Kontsevich, Zagier) Any multizeta can be expressed as a period on M o,n (explicitly) Any convergent period on M o,n can be written as a linear combination of multizeta values (not explicitly) 17

19 Since periods satisfy the three defining relations of FC, we have FC C = Z. Let Z n be the Q-vector space generated by weight n multizeta values and products of multizeta values of total weight n. Conjecture.(Zagier) Let d n = dim Q Z n. Then d n = d n 2 + d n 3, where d 0 = 1, d 1 = 0, d 2 = 1. This conjecture is true for FC n+3, n = 0,1,2,3, 4,5,6. We hope that the combinatorial structure will make this conjecture accessible for FC. 18