Multiple zeta values in deformation quantization Brent Pym w/ Peter - PowerPoint PPT Presentation

Multiple zeta values in deformation quantization Brent Pym w/ Peter Banks and Erik Panzer 1 / 26

Hamiltonian mechanics Particle in 1d: position momentum energy H ( x , p ) = p 2 x p = m · ˙ x 2 m + V ( x ) Equations of motion x = p m = ∂ H p = force = − ∂ V ∂ x = − ∂ H ˙ ˙ ∂ p ∂ x In general, f ( x , p ) evolves according to f = ∂ f ∂ H ∂ p − ∂ H ∂ f ˙ ∂ p =: { f , H } ∂ x ∂ x using Poisson bracket {− , −} = ∂ x ∧ ∂ p { x , p } = 1 2 / 26

Quantization Promote to operators on C [ x ]: p = − i � ∂ x x � � x · p � � Canonical commutation relations: x = i � [ � x , � p ] = � x � p − � p � { x , p } = 1 Weyl: given f ∈ C [ x , p ], define ˆ f by symmetrization: � ( ax + bp ) n = ( a ˆ p ) n x + b ˆ Defines new associative product ⋆ on C [ x , p ]: C � � x , � p � f ⋆ g = � � ( C [ x , p ] , ⋆ ) ∼ f � g = x = i � � x � p − � p � 3 / 26

The star product “Explicit” formula: � f ( v ) g ( w ) e iS ( u , v , w ) / � dv dw ( f ⋆ g )( u ) = v , w ∈ R 2 x , p   u   S ( u , v , w ) = 4 · Area v w Groenewold 1946, Moyal 1949: � ∞ � n ( i � ) n ∂ n f ∂ n g ( − 1) i f ⋆ g ∼ 2 n n ! ∂ x i ∂ p n − i ∂ x n − i ∂ p i n =0 i =0 Key point: f ⋆ g − g ⋆ f = i � { f , g } + O ( � 2 ) 4 / 26

Abstraction “Phase space” = manifold/variety X equipped with a Poisson bracket {− , −} : O X × O X → O X Axioms: 1 { f , g } = −{ g , f } 2 { f , gh } = { f , g } h + g { f , h } 3 { f , { g , h }} + { g , { h , f }} + { h , { f , g }} = 0 5 / 26

Examples of Poisson manifolds Darboux: X = R 2 n with { f , g } = � ∂ f ∂ p i − ∂ g ∂ g ∂ x i ∂ f i ∂ x i ∂ p i ( X , ω ) symplectic, e.g. X = T ∗ Q . Angular momentum: X = R 3 x , y , z with { x , y } = z { y , z } = x { z , x } = y Linear brackets on X = R n � c k Lie algebra g = ( R n ) ∗ { x i , x j } = ⇐ ⇒ ij x k k moduli spaces in gauge theory ... 6 / 26

The “deformation quantization” problem Formulated by Bayen–Flato–Fronsdal–Lichnerowicz–Sternheimer (1978) A deformation quantization of ( X , {− , −} ) is a family of associative products ⋆ � such that when � = 0 f ⋆ � g = fg f ⋆ � g − g ⋆ � f = � { f , g } + O ( � 2 ) Today: only formal deformations ⋆ : O X × O X → O X [[ � ]] f ⋆ g = fg + � B 1 ( f , g ) + � 2 B 2 ( f , g ) + · · · Basic question: Does a quantization always exist? Answer in symplectic case: yes – Berezin, Deligne, Fedosov, Kirillov, Kostant, Schlichenmaier, Souriau, Toeplitz, ... 7 / 26

Kontsevich formality theorem Theorem (Kontsevich 1997) Every smooth Poisson manifold X has a canonical quantization. In fact there is an equivalence { Poisson brackets on X } = { noncommutative deformations of O X } ∼ ∼ ∼ Precise statement is stronger: CC ∗ ( O X ) ∼ = ∧ • T X as dg Lie algebras Explicit Feynman expansion when X = R n :   � �   + � 2 f ⋆ g = fg + � + +  + · · ·  � complicated � � derivatives of � = · integral f , g and {− , −} 8 / 26

Kontsevich formula: differential operator Given {− , −} in coordinates x 1 , . . . , x n on R n , want to compute f ⋆ g { x i , x j } { x k , x l } ∂ x k ∂ x j ∂ x i ∂ x l g f � derivatives of � � � � := ( ∂ x i f ) · ∂ x j ∂ x l g · ( ∂ x k { x i , x j } ) · { x k , x l } f , g and {− , −} i , j , k , l 9 / 26

Kontsevich formula: Feynman integrals   ∞ z 1 z 2     �     z 2 C n , m = holomorphic iso. z 1 �       y 1 y 2   y 1 y 2    ∞  = H n \ { z i = z j } i � = j e.g. C n , 2 ∼ ∼ =   0 1 a 2 α e := d log( a , a ; b , ∞ ) + d log( a , a ; b , ∞ ) e � 2 i π 2 i π b � d log f � � � ω := α e 1 ∧ · · · ∧ α e N ∈ 2 − N Z � ⊂ Ω • ( C n , m ) � f a cross ratio 2 i π � complicated � � ω ∈ R := integral C n , m 10 / 26

Recovering Groenewold–Moyal � { x , x } � � 0 � { f , g } = ∂ f ∂ g ∂ p − ∂ g ∂ f { x , p } 1 = ∂ x ∂ x ∂ p { p , x } { p , p } − 1 0   � �   + � 2 f ⋆ g = fg + � + +  + · · ·  � � � � � � + � 2 + � 3 = fg + � + · · · ∞ n � � ( i � ) n ∂ n f ∂ n g ( − 1) i = 2 n n ! ∂ x i ∂ p n − i ∂ x n − i ∂ p i n =0 i =0 11 / 26

Linear case � c k { x i , x j } = ij x k ↔ Lie algebra g Similar analysis: Series truncates for f , g ∈ C [ x i ] Can compute � c k x i ⋆ x j − x j ⋆ x i = � ij x k Conclude C � x i � ( C [ x i ] , ⋆ � ) ∼ =: U ( g , � ) x i x j − x j x i = � � c k = ij x k 12 / 26

Quadratic case { X , P } = XP X ⋆ P = q ( � ) XP P ⋆ X = q ( − � ) XP Our software: � 251 ζ (3) 2 � 2 + � 2 24 − � 3 1440 + � 5 � 4 q ( � ) = 1 + � 17 � 6 + · · · 48 − 480 + 2048 π 6 − 184320 Nevertheless: algebra determined by X ⋆ P = q ( � ) q ( − � ) P ⋆ X = e � P ⋆ X Morally: X = e x and P = e p where { x , p } = 1. 13 / 26

Special values of Riemann zeta � 1 ζ ( s ) = k s k ≥ 1 Theorem (Euler 1735): ζ (2 m ) = ( − 1) m +1 B 2 m (2 π ) 2 m ∈ Q π 2 m 2(2 m )! Open Question: Is ζ (2 m + 1) ∈ Q ( π )? Conjecture: π, ζ (3) , ζ (5) , ζ (7) , . . . are algebraically independent over Q . ∈ Q Theorem (Ap´ ery 1978): ζ (3) / ∈ Q Theorem ((Ball–)Rivoal 2000): Infinitely many ζ (3) , ζ (5) , ζ (7) , . . . / ∈ Q Theorem (Zudilin 2000): At least one of ζ (5) , ζ (7) , ζ (9) , ζ (11) / 14 / 26

MZVs (Euler, ´ Ecalle, Zagier, ...) Definition A normalized multiple zeta value (MZV) of weight n is a number of the form � � R 1 1 n even � ζζ ( n 1 , . . . , n d ) = ∈ k n 1 1 k n 2 2 · · · k n d (2 i π ) n i R n odd d k 1 > k 2 > ··· > k d ≥ 1 where n 1 ≥ 2 and n 1 + · · · + n d = n . Additional “honourary” normalized MZVs: 1 ∈ R has weight 0 1 2 = i π 2 i π ∈ R has weight 1 15 / 26

Algebra of MZVs � Z := Z · { normalized MZVs } ⊂ C Weight filtration: � � � � Z 0 ⊂ Z 1 ⊂ Z 2 ⊂ · · · ⊂ Z = = = Z · ζ (2) Z · 1 Z ⊂ ⊂ ⊂ · · · 2 (2 i π ) 2 � �� � = − 1 24 Shuffle product: Z m � � Z n ⊂ � Z m + n e.g. ζ ( m ) � � ζ ( n ) = � ζ ( m , n ) + � ζ ( n , m ) + � ζ ( n + m ) 16 / 26

How many MZVs are there? For unnormalized MZVs: Q -dimension of weight spaces conjectured by Zagier ◮ Proven to be an upper bound (Terasoma, Deligne–Goncharov) Q -basis conjectured by Hoffman: ζ (2 s and 3 s ) ◮ Proven to generate (Brown) For normalized MZVs: Z -module generators of � Z n 0 1 2 3 4 5 6 n 1 1 1 1 1 1 real 1 2 24 48 5760 11520 2903040 ζ (3) 2 128 π 6 i ζ (3) i ζ (3) i ζ (3) i ζ (3) imaginary 8 π 3 16 π 3 192 π 3 384 π 3 i ζ (5) i ζ (5) 64 π 5 128 π 5 17 / 26

Ubiquity of MZVs Quantum groups: coefficients of Drinfel’d associator Knot theory: coefficients of Kontsevich integral Homotopical algebra: formality of the operad E 2 Algebraic geometry: periods integrals on moduli space M 0 , N (Brown 2006, conj. by Goncharov–Manin) Physics: values of certain Feynman integrals ... Theorem (Brown 2011, building on Deligne–Goncharov, Levine, Voevodsky, Zagier, . . . ) All periods of unramified mixed Tate motives lie in Q � Z [ 1 2 i π ] . 18 / 26

Main result   �   C n , m = holomorphic iso.   � � d log f � � � A • ( C n , m ) := Z ⊂ Ω • ( C n , m ) � f a cross ratio 2 i π Theorem (Banks–Panzer–P.) Suppose that ω ∈ A • ( C n , m ) is absolutely integrable. Then � � � Z n + m − 2 m > 0 ω ∈ � Z n − 1 m = 0 C n , m Corollary (case m = 2) Coefficients at order � n in Kontsevich’s star product lie in 4 − n � Z n ∩ R 19 / 26

Alternate definitions of MZVs � 1 1 � ζ ( n 1 , . . . , n d ) = = L n 1 ,..., n d (1) k n 1 1 k n 2 2 · · · k n d (2 i π ) n d k 1 > k 2 > ··· > k d ≥ 1 in terms of multiple polylogarithm � z k 1 1 L n 1 ,..., n d ( z ) := k n 1 1 k n 2 2 · · · k n d (2 i π ) n d k 1 > k 2 > ··· > k d ≥ 1 e.g. � z k k = log(1 − z ) L 1 ( z ) = L 2 ( z ) = dilogarithm 2 i π k ≥ 1 Alternate notation: n 1 , . . . , n d ↔ s 1 · · · s n = 00 · · · 01 00 · · · 01 · · · 00 · · · 01 � �� � � �� � � �� � n 1 n 2 n d Check: dL s 1 ··· s n = ( − 1) s 1 L s 2 ··· s n dz 2 i π ( z − s 1 ) 20 / 26

Alternate definitions of MZVs, II Rewrite dL s 1 ··· s n = ( − 1) s 1 L s 2 ··· s n dz � ζ ( n 1 , . . . , n d ) = L s 1 ··· s n (1) 2 i π ( z − s 1 ) and therefore (Kontsevich, Le–Murakami) � 1 � t 1 � t n − 1 � ζ ( n 1 , . . . , n d ) = ( − 1) d dt 1 dt 2 dt n 2 i π ( t 2 − s 2 ) · · · 2 i π ( t 1 − s 1 ) 2 i π ( t n − s n ) 0 0 0 � �� � � 1 s 1 · · · s n 0 Chen iterated integral NB: diverges if s 1 = 1 or s n = 0, so “regularize”: log( ǫ ) = 0 21 / 26