Renormalization and Euler-Maclaurin Formula on Cones Li GUO (joint - PowerPoint PPT Presentation

Renormalization and Euler-Maclaurin Formula on Cones Li GUO (joint work with Sylvie Paycha and Bin Zhang) Rutgers University at Newark

Outline ◮ Conical zeta values and multiple zeta values; ◮ Double shuffle relations and double subdivision relations; ◮ Renormalization of conical zeta values; ◮ Euler-Maclaurin formula. 2

Cones ◮ A (closed polyhedral) cone in R k ≥ 0 is defined to be the convex set � v 1 , · · · , v n � := R ≥ 0 v 1 + · · · + R ≥ 0 v n , v i ∈ R k ≥ 0 , 1 ≤ i ≤ n . ◮ The interior of a cone � v 1 , · · · , v n � is an open (polyhedral) cone � v 1 , · · · , v n � o := R > 0 v 1 + · · · + R > 0 v n . ◮ The set { v 1 , · · · , v n } is called the generating set or the spanning set of the cone. The dimension of a cone is the dimension of linear subspace generated by it. ◮ Let C k (resp. OC k ) denote the set of closed (resp. open cones) in R k , k ≥ 1. For k = 0 we set C 0 = { 0 } (resp. OC 0 = { 0 } ) by convention. Through the natural inclusions C k → C k + 1 (resp. OC k → OC k + 1 ) from the natural inclusion R k → R k + 1 , we define C = lim → C k (resp. − OC = lim → OC k ). − 3

◮ A simplicial cone is defined to be a cone spanned by linearly independent vectors. ◮ A rational cone is a cone spanned by vectors in Z k ⊆ R k . ◮ A smooth cone is a rational cone with a spanning set that is a part of a basis of Z k ⊆ R k . In this case, the spanning set is unique and is called the primary set of the cone. ◮ A cone is called strongly convex or pointed if it does not contain any linear subspace. ◮ A subdivision of a closed cone C ∈ C k is a set { C 1 , · · · , C r } ⊆ C k such that C = ∪ r i = 1 C i , C 1 , · · · , C r have the same dimension C and intersect along their faces. The faces of the relative interior give an open subdivision of C o : � e 1 , e 2 � = � e 1 , e 1 + e 2 � ⊔ � e 1 + e 2 , e e � ⇒ � e 1 , e 2 � o = � e 1 , e 1 + e 2 � o ⊔ � e 1 + e 2 , e e � o ⊔ � e 1 + e 2 � o . y = ( y 1 , · · · , y k ) in R k , let ( � ◮ For � x = ( x 1 , · · · , x k ) and � x ,� y ) denote the inner product x 1 y 1 + · · · + x k y k . Through this inner product, R k is identified with its own dual space ( R k ) ∗ . 4

Conical zeta values ◮ Let C be a smooth cone. The conical zeta function of C is 1 � s ∈ C k , ζ ( C ; � ,� s ) := n s 1 1 · · · n s k k ( n 1 , ··· , n k ) ∈ C o ∩ Z k if the sum converges. When s i , 1 ≤ i ≤ k , are integers, ζ ( � s ) is called a conical zeta value (CZV). Convention: 0 s = 1 for any s . Hence ζ ( � s ) does not depend on the choice of k . ◮ If s i ≥ 2 , 1 ≤ i ≤ k , then ζ ( C ; � s ) converges. ◮ If { C i } i is an open cone subdivision of C , then � ζ ( C ; � ζ ( C i ; � s ) = s ) . i ◮ The cone subdivision � e 1 , e 2 � o = � e 1 , e 1 + e 2 � o ⊔ � e 1 + e 2 , e 2 � o ⊔ � e 1 + e 2 � o gives ζ ( � e 1 , e 2 � o ; ( s 1 , s 2 )) = ζ ( � e 1 , e 1 + e 2 � o ; ( s 1 , s 2 )) + ζ ( � e 1 + e 2 , e 2 � o ; ( s 1 , s 2 ) + ζ ( � e 1 + e 2 � o ; ( s 1 , s 2 ) . 5

Chen cones and multiple zeta values ◮ A Chen cone of dimension k is a cone C k ,σ := � e σ ( 1 ) , e σ ( 1 ) + e σ ( 2 ) , · · · , e σ ( 1 ) + · · · + e σ ( k ) � , where σ ∈ S k . Let C k denote the standard Chen cone spanned by { e 1 , · · · , e k } . ◮ Then ζ ( C k ,σ ; s 1 , · · · , s k ) = ζ ( s σ ( 1 ) , · · · , s σ ( k ) ) , ζ ( C k , id ; s 1 , · · · , s k ) = ζ ( s 1 , · · · , s k ) . ◮ The stuffle product of two MZVs ζ ( r 1 , · · · , r k ) and ζ ( s 1 , · · · , s ℓ ) is recovered by the subdivision of the cone C k × C ℓ (direct product) into Chen cones. ◮ For example, the open cone subdivision relation ζ ( � e 1 , e 2 � o ; ( s 1 , s 2 )) = ζ ( � e 1 , e 1 + e 2 � o ; ( s 1 , s 2 )) + ζ ( � e 1 + e 2 , e 2 � o ; ( s 1 , s 2 ) + ζ ( � e 1 + e 2 � o ; ( s 1 , s 2 ) gives the stuffle relation ζ ( s 1 ) ζ ( s 2 ) = ζ ( s 1 , s 2 ) + ζ ( s 2 , s 1 ) + ζ ( s 1 + s 2 ) . 6

Multiple zeta values ◮ The multiple zeta value algebra is MZV := Q { ζ ( s 1 , · · · , s k ) | s i ≥ 1 , s 1 ≥ 1 } . ◮ The quasi-shuffle algebra H ∗ has the underlying vector space Q � z s | s ≥ 1 � with the quasi-shuffle product. It contains the subalgebra   � H ∗  ⊆ H ∗ . 0 := Q . 1 ⊕ Q z s 1 · · · z s k s 1 ≥ 2 The stuffle relation of MZVs is encoded in the algebra homomorphism ζ ∗ : H ∗ 0 − → MZV , z s 1 · · · z s k �→ ζ ( s 1 , · · · , s k ) . 7

� � � Double shuffle relation ◮ The shuffle algebra H X has the underlying vector space Q � x 0 , x 1 � equipped with the shuffle product of words. It contains the subalgebra � H X x 0 H X x 1 . 0 := Q . 1 The shuffle relation of the MZVs is encoded in the algebra homomorphism ζ X : H X x s 1 − 1 x 1 · · · x s k − 1 0 → MZV , x 1 �→ ζ ( s 1 , · · · , s k ) . 0 0 ◮ There is a natural bijection of abelian groups (but not algebras) 1 ↔ 1 , x s 1 − 1 x 1 · · · x s k − 1 η : H X 0 → H ∗ x 1 ↔ z s 1 · · · z s k . 0 , 0 0 ◮ Then the fact that MZVs can be multiplied in two ways is reflected by η H ∗ H X 0 0 ζ ∗ ζ X MZV Double shuffle relation ζ ∗ � w 1 ∗ w 2 − η ( η − 1 ( w 1 ) X η − 1 ( w 2 )) � w 1 , w 2 ∈ H ∗ , 0 . 8

Linearly constrained zeta values (LCZ) ◮ Let � v 1 , · · · , v k � be a smooth close cone with ita (unique) primitive generating set. ◮ For s 1 , · · · , s k ≥ 1, called the formal expression [ v 1 ] s 1 · · · [ v k ] s k a decorated smooth cone. ◮ Define the linearly constrained zeta value (LZV) ζ c ([ v 1 ] s 1 · · · [ v k ] s k ) ∞ ∞ 1 � � := · · · ( a 11 m 1 + · · · + a 1 r m r ) s 1 · · · ( a k 1 m 1 + · · · + a kr m r ) s k m 1 = 1 m r = 1 if the sum is convergent, where v i = � r j = 1 a ij e j , 1 ≤ i ≤ k . When [ v 1 ] · · · [ v k ] is a Chen cone [ e 1 ] · · · [ e 1 + · · · + e k ] , then we have ζ c ([ v 1 ] s 1 · · · [ v k ] s k ) = ζ ( s 1 , · · · , s k ) . 9

Subdivision of decorated closed cones ◮ Let {� v i 1 , · · · , v ik �} i be a smooth subdivision of the smooth cone � v 1 , · · · , v k � . Call � i [ v i 1 ] · · · [ v ik ] an algebraic subdivision of [ v 1 ] · · · [ v k ] . ◮ Let [ v 1 ] s 1 · · · [ v k ] s k be a decorated smooth closed cone. j s j ( e i , v j )[ v 1 ] s 1 · · · [ v j ] s j + 1 · · · [ v k ] s k . For ◮ Define δ e i ([ v 1 ] s 1 · · · [ v k ] s k ) = � u = � i c i e i , define δ u = � i c i δ e i . Then [ v 1 ] s 1 · · · [ v k ] s k = ( s 1 − 1 )! ··· ( s k − 1 )! δ s 1 − 1 · · · δ s k − 1 1 ([ v i 1 ] · · · [ v ik ]) . v ∗ v ∗ 1 k ◮ Call 1 � ( s 1 − 1 )! · · · ( s k − 1 )! δ s 1 − 1 · · · δ s k − 1 ([ v i 1 ] · · · [ v ik ]) v ∗ v ∗ 1 k i an algebraic subdivision of [ v 1 ] s 1 · · · [ v k ] s k . Here v ∗ 1 , · · · , v ∗ k is a dual basis of v 1 , · · · , v k . ◮ Let D = � i a i D i be an algebraic subdivision of a decorated smooth cone D . Then � ζ c ( D ) = a i ζ c ( D i ) . i ◮ This generalizes the shuffle relation of MZVs. 10

Relating open and closed subdivisions ◮ Let GL r ( Z ) denote the set of r × r unimodular matrices. Let s := ( s 1 , . . . , s r ) ∈ Z r M ∈ GL r ( Z ) and � ≥ 0 . Let v 1 , · · · , v r and u 1 , · · · , u r be the row and column vectors of M . The (decorated) cone pair associated with M and � s is the pair ( C , D ) consisting of the s = ( � u 1 , · · · , u r � o ,� decorated open cone C := C M ,� s ) and the s = [ v 1 ] s 1 · · · [ v r ] s r . We call the pair decorated closed cone D := D M ,� convergent if the corresponding ζ -values ζ 0 ( C ) and ζ c ( D ) converge. ◮ Let DTP denote the set of cone pairs ( C M ,� s , D M ,� s ) where M ∈ O ( Z ) s ∈ Z r and � ≥ 0 . Let p o : Q DTP → Q DC and p c : Q DTP → Q DMC denote the natural projections. ◮ For any cone pair ( C , D ) ∈ DTP , we have ζ o ( C ) = ζ c ( D ) , if either side makes sense. 11

Double subdivision relation ◮ Let ( C , D ) be a convergent cone pair. Let { C i } i be an open subdivision of the decorated open cone C and let � j c j D j be a subdivision of the decorated closed cone D . Also let D T j ∈ DC be the transpose cone of D j , that is, ( D T j , D j ) is a cone pair. Then � � c j D T C i − (1) j i j lies in the kernel of ζ o . It is called a double subdivision relation. ◮ For any not necessarily convergent cone pair ( C , D ) , let { C i } be a j a j D T subdivision of C and � j a j D j a subdivision of D . If � i C i − � j is in Q DC , then it is called an extended double subdivision relation. ◮ Hunch. The kernel of ζ o is the subspace I EDS of Q DC generated by the extended double subdivision relations. 12

� � �� �� � � � � �� � � � � � � � � � � � � Double subdivision relation ◮ po pc � Q DMC c Q DOC 0 Q DTP 0 0 � � T Q DCH o Q DCH c 0 0 η H X H ∗ 0 0 ζ X ζ ∗ ζ o ζ c Q MZV Q OCMZV 13