▼✉❧t✐♣❧❡ ③❡t❛ ✈❛❧✉❡s ❛♥❞ ❛ss♦❝✐❛t♦rs ✐♥ ❣❡♥✉s ③❡r♦ ❛♥❞ ♦♥❡ Leila Schneps HIM Trimester Program January 12, 2018 1
Part I: Multizeta values and associators in genus zero For each sequence ( k 1 , . . . , k r ) of strictly positive integers, k 1 ≥ 2, the multiple zeta value is defined by the convergent series 1 � ζ ( k 1 , . . . , k r ) = . n k 1 1 · · · n k r r n 1 > ··· >n r > 0 These real numbers have been studied since Euler (1775). They form a Q -algebra, the multizeta algebra Z . 2
Two multiplications of multizeta values 1. Shuffle multiplication Straightforward integration shows that � 1 � t 1 � t n − 1 dt n dt 2 dt 1 ζ ( k 1 , . . . , k r ) = ( − 1) r · · · · · · t n − ǫ n t 2 − ǫ 2 t 1 − ǫ 1 0 0 0 where ( ǫ 1 , . . . , ǫ n ) = (0 , . . . , 0 , 1 , 0 , . . . , 0 , 1 , . . . , 0 , . . . , 0 , 1) . � �� � � �� � � �� � k 1 − 1 k 2 − 1 k r − 1 The product of two simplices is a union of simplices, giving an expres- sion for the product of two multizeta values as a sum of multizeta values. This is the shuffle product . Example. We have � 1 � t 1 dt 2 dt 1 ζ (2) = 1 − t 2 t 1 0 0 � 1 � t 1 � t 2 � t 3 dt 4 dt 3 dt 2 dt 1 ζ (2 , 2) = 1 − t 4 t 3 1 − t 2 t 1 0 0 0 0 � 1 � t 1 � t 2 � t 3 dt 4 dt 3 dt 2 dt 1 ζ (3 , 1) = 1 − t 4 1 − t 3 t 2 t 1 0 0 0 0 and ζ (2) 2 = 2 ζ (2 , 2) + 4 ζ (3 , 1) . 3
2. Stuffle multiplication The product of two series over ordered indices can be expressed as a sum of series over ordered indices. This is the stuffle product of multizeta values. Examples. We have �� 1 ��� 1 � ζ (2) 2 = n 2 m 2 n> 0 m> 0 1 1 1 � � � = n 2 m 2 + n 2 m 2 + n 4 n>m> 0 m>n> 0 n = m> 0 = 2 ζ (2 , 2) + ζ (4) . Using either multiplication shows that the multiple zeta values form a Q -algebra. We use the shuffle multiplication and call this algebra Z . Transcendence conjecture. The Q -algebra Z is graded by the weight. Zagier Dimension conjecture. The dimension of the n -th graded part Z n is equal to d n where d n = d n − 2 + d n − 3 with d 0 = 1 , d 1 = 0 , d 2 = 1 . Remark. The algebra of motivic multizetas MZ surjects onto Z . The dimension conjecture is known for MZ (Goncharov+Brown) so the d n pro- vide an upper bound for dim ( Z n ). 4
Convergent and non-convergent words A convergent word w ∈ Q � x, y � is a word w = xvy . The reason for this notation is that it gives a bijection { tuples with k 1 ≥ 2 } ↔ { convergent words } ( k 1 , . . . , k r ) ↔ x k 1 − 1 y · · · x k r − 1 y. As a notation, we use this to write ζ ( k 1 , . . . , k r ) = ζ ( x k 1 − 1 y · · · x k r − 1 y ) . We extend the definition to ζ ( w ) for any word w = y a ux b with u convergent: a b � � � � ( − 1) r + s ζ sh ( y r , y a − r ux b − s , x s ) ζ ( w ) = . r =0 s =0 The depth of a word w the number of y ’s and the weight is the degree; correspondingly, the depth of ζ ( k 1 , . . . , k r ) is r and the weight is k 1 + · · · + k r . 5
The Drinfel’d associator Definition. The Drinfel’d associator is the power series given by � ( − 1) d w ζ ( w ) w Φ KZ ( x, y ) = 1 + w ∈ Q � x,y � where d w is the number of y ’s in the word w . It is a generating series for multizeta values. Let Φ r KZ be the depth r part of Φ KZ . Then Φ r KZ can also be obtained as the iterated integral � x � x � � � y y Φ r KZ ( x, y ) = + · · · + dv r · · · dv 1 . v 1 1 − v 1 v r 1 − v r 0 <v r < ··· <v 1 < 1 It is obtained as monodromy of the KZB equation d � x y � dz G ( z ) = v + G ( z ); 1 − v more specifically Φ KZ ( x, y ) = G 1 ( z ) − 1 G 0 ( z ), where G 0 (resp. G 1 ) is the solution to the KZ equation that tends to z x as z → 0 (resp. to (1 − z ) y as z → 1). 6
Associator relations Theorem. [Drinfel’d] The Drinfeld associator also satisfies the associator relations ( I ) Φ KZ ( x, y )Φ KZ ( y, x ) = 1 e πix Φ KZ ( y, x ) e πiy Φ KZ ( z, y ) e iπz Φ KZ ( x, z ) = 1 ( II ) with x + y + z = 0 , and ( III ) The 5-cycle relation Φ KZ ( x 12 , x 23 )Φ KZ ( x 34 , x 45 )Φ KZ ( x 51 , x 12 )Φ KZ ( x 23 , x 34 )Φ KZ ( x 45 , x 51 ) = 1 , where the x ij generate the pure braid group on 5 strands. All these properties of Φ KZ ( x, y ) yield algebraic relations between mul- tizeta values. Conjecture. The double shuffle relations generate all algebraic relations between multizeta values. Much weaker conjecture! The double shuffle relations generate all alge- braic relations between motivic multizeta values. Definition. We say that a power series Φ( x, y ) ∈ C �� x, y �� is an associator if it is group-like with no linear term and satisfies the associator relations (I), (II), (III) with iπ replaced by µ/ 2 for some µ ∈ C . Let GRT µ be the set of associators Φ( x, y ) with a fixed µ . In particular Φ KZ ∈ GRT 2 πi . 7
Application of associators Formality: Every associator with µ � = 0 gives rise to a formality isomor- phism between the pro-unipotent group � P 1 − { 0 , 1 , ∞} � ≃ � e X , e Y , e Z | e X e Y e Z = e CH ( X,Y,Z ) = 1 � π pro − uni 1 and the graded version exp Lie( π 1 ( P 1 − { 0 , 1 , ∞} ) given by � � ≃ � e x , e y , e z | e x + y + z = 1 � . exp Lie[ x, y, z | x + y + z = 0] For Φ KZ , the isomorphism is given by e X �→ e 2 iπx e Y �→ Φ KZ ( y, x ) e 2 iπy Φ KZ ( x, y ) e Z �→ e iπx Φ KZ ( z, x ) e 2 iπz Φ KZ ( x, z ) e − iπx . The product of the three terms is equal to 1 thanks to relation (II). If instead if Φ ∈ GRT 0 , i.e. µ = 0, it gives an automorphism of exp Lie( π 1 ( P 1 − { 0 , 1 , ∞} ) given by e x �→ e x , e y �→ Φ( y, x ) e y Φ( x, y ) . This gives rise to a group law on GRT 0 coming from the composition: Φ( x, y ) ⊙ Ψ( x, y ) = Φ( x, y )Ψ( x, Φ − 1 y Φ) . Each GRT µ is a torsor under GRT 0 by composition. Braid extension: Following Grothendieck, we can identify P 1 − { 0 , 1 , ∞} with the moduli space of genus zero curves with 4 marked points, and the free π 1 is then identified with the pure sphere braid on 4 strands. Relation (III) ensures that the formality isomorphism of the 4-strand (free) group given by any associator above extends to one of the pro- unipotent pure sphere braid group on 5 strands, and elements of GRT 0 extend to automorphisms of the pro-unipotent pure 5-strand braid group. 8
Double shuffle relations satisfied by Φ KZ The two multiplications on mzvs can be expressed economically by two conditions on the power series. • The first condition is ∆(Φ KZ ) = Φ KZ ⊗ Φ KZ , where ∆( x ) = x ⊗ 1 + 1 ⊗ x and ∆( y ) = y ⊗ 1 + 1 ⊗ y . This means that Φ KZ is group-like , i.e. it lies in exp(Lie[ x, y ]). Viewed as a family of algebraic relations on the coefficients of Φ KZ , it is exactly the shuffle relations on the multizetas. • The second condition is obtained by considering the modified series ( − 1) n − 1 �� � ζ ( y n ) y n Φ corr = exp π y (Φ KZ ) 1 n n ≥ 1 where π y (Φ KZ ) is the projection of Φ KZ onto the words ending in y . We rewrite the series Φ corr in the variables y i = x i − 1 y , and the condition is ∆ ∗ (Φ corr ) = Φ corr ⊗ Φ corr , where � ∆ ∗ ( y i ) = y k ⊗ y l . k + l = i These relations are called the double shuffle relations on Φ KZ . Definition. Let DS λ denote the set of power series in x, y with • constant term 1 • no linear term • coefficient λ for the monomial xy • that satisfy the double shuffle relations. 9
Theorem. [Racinet, 2000] The set DS 0 of double shuffle power series forms a group under the multiplication f ( x, y ) ⊙ g ( x, y ) = f ( x, y ) g ( x, f − 1 yf ) . Theorem. [Furusho] The group GRT 0 injects into DS 0 by the map Φ( x, y ) �→ Φ( x, − y ) . In other words, even if we don’t know whether double shuffle relations imply all algebraic relations on multizeta values, we do know that they imply the associator relations. 10
Part II. A quick look at ´ Ecalle’s mould theory Very generally, a mould is a family ( P r ) r ≥ 0 where P r is a function of r commutative variables u 1 , . . . , u r . We will restrict attention to rational functions over Q , so P 0 ∈ Q and P r ∈ Q ( u 1 , . . . , u r ) for r ≥ 1. Let ARI denote the space of moulds with constant term 0 and GARI the set of moulds with constant term 1. The swap is an operation on moulds that is given by the following change of variables: swap ( A )( u 1 , . . . , u r ) = A ( u r , u r − 1 − u r , . . . , u 1 − u 2 ) . We can make GARI into a non-commutative group by the multiplica- tion law r � ( A × B )( u 1 , . . . , u r ) = A ( u 1 , . . . , u i ) B ( u i +1 , . . . , u r ) . i =0 Theorem. [easy] Let G be the set of power series in x, y with constant term 1 in the kernel of the derivation ∂ x ( x ) = 1 , ∂ x ( y ) = 0 . There is an injective homomorphism G ֒ → GARI given by rewriting elements in the non-commutative variables where c i = ad ( x ) i − 1 ( y ) and linearly extending c i 1 · · · c i r �→ u i 1 − 1 · · · u i r − 1 . 1 r Note in particular that ´ Ecalle extended the usual notion of multiplication of non-commutative power series to all moulds . 11
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