Renormalization and Euler-Maclaurin Formula on Cones

Transcription

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

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

3 Cones A (closed polyhedral) cone in R k 0 is defined to be the convex set v 1,, v n := R 0 v 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 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

4 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. For x = (x 1,, x k ) and y = (y 1,, y k ) in R k, let ( x, y) denote the inner product x 1 y x k y k. Through this inner product, R k is identified with its own dual space (R k ). 4

5 Conical zeta values Let C be a smooth cone. The conical zeta function of C is 1 ζ(c; s) := n s 1 (n 1,,n k ) C o Z k 1, s C k, ns k 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; s) = i ζ(c i ; s). 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

6 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 )) gives the stuffle relation +ζ( e 1 + e 2, e 2 o ; (s 1, s 2 ) + ζ( e 1 + e 2 o ; (s 1, s 2 ) ζ(s 1 )ζ(s 2 ) = ζ(s 1, s 2 ) + ζ(s 2, s 1 ) + ζ(s 1 + s 2 ). 6

7 Multiple zeta values 7 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 0 := Q.1 Qz s1 z sk H. s1 2 The stuffle relation of MZVs is encoded in the algebra homomorphism ζ : H 0 MZV, z s 1 z sk ζ(s 1,, s k ).

8 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 0 := Q.1 x 0 H X x 1. The shuffle relation of the MZVs is encoded in the algebra homomorphism ζ X : H0 X MZV, x s x 1 x s k 1 0 x 1 ζ(s 1,, s k ). There is a natural bijection of abelian groups (but not algebras) η : H X 0 H 0, 1 1, x s x 1 x s k 1 0 x 1 z s1 z sk. Then the fact that MZVs can be multiplied in two ways is reflected by H 0 η H X 0 ζ 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 ζ X

9 Linearly constrained zeta values (LCZ) 9 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 ] s1 [v k ] s k ) 1 := (a 11 m a 1r m r ) s 1 (ak1 m a kr m r ) s k m 1 =1 m r =1 if the sum is convergent, where v i = r j=1 a ije j, 1 i k. When [v 1 ] [v k ] is a Chen cone [e 1 ] [e e k ], then we have ζ c ([v 1 ] s1 [v k ] s k ) = ζ(s 1,, s k ).

10 This generalizes the shuffle relation of MZVs. 10 Subdivision of decorated closed cones Let { v i1,, v ik } i be a smooth subdivision of the smooth cone v 1,, v k. Call i [v i1] [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. Define δ ei ([v 1 ] s 1 [v k ] s k ) = j s j(e i, v j )[v 1 ] s 1 [v j ] s j +1 [v k ] s k. For u = i c ie i, define δ u = i c iδ ei. Then [v 1 ] s 1 [v k ] s k 1 = (s 1 1)! (s k 1)! δs 1 1 v1 δ s k 1 vk ([v i1 ] [v ik ]). Call 1 (s 1 1)! (s k 1)! δs 1 1 v1 δ s k 1 vk ([v i1 ] [v ik ]) i an algebraic subdivision of [v 1 ] s 1 [v k ] s k. Here v1,, v k is a dual basis of v 1,, v k. Let D = i a id i be an algebraic subdivision of a decorated smooth cone D. Then ζ c (D) = a i ζ c (D i ). i

11 Relating open and closed subdivisions Let GL r (Z) denote the set of r r unimodular matrices. Let M GL r (Z) and s := (s 1,..., s r ) Z r 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 decorated open cone C := C M, s = ( u 1,, u r o, s) and the decorated closed cone D := D M, s = [v 1 ] s 1 [v r ] sr. We call the pair 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) and s Z r 0. Let p o : QDTP QDC and p c : QDTP QDMC 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

12 Double subdivision relation 12 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 jd j be a subdivision of the decorated closed cone D. Also let Dj T DC be the transpose cone of D j, that is, (Dj T, D j ) is a cone pair. Then C i c j Dj T (1) 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 subdivision of C and j a jd j a subdivision of D. If i C i j a jd T j is in QDC, then it is called an extended double subdivision relation. Hunch. The kernel of ζ o is the subspace I EDS of QDC generated by the extended double subdivision relations.

13 Double subdivision relation 13 QDOC 0 p o QDTP 0 p c QDMC c 0 QDCH o 0 T QDCH c 0 H 0 η H X 0 ζ o ζ ζ X ζ c Q MZV Q OCMZV

14 Algebraic Birkhoff Decomposition Algebraic Birkhoff Decomposition. Let H be a connected filtered Hopf algebra, R = P(R) (id P)(R) a commutative Rota-Baxter algebra with an idempotent Rota-Baxter operator P. Any algebra homomorphism φ : H R has a unique decomposition into algebra homomorphisms φ = φ 1 φ +, { φ : H C + P(R) (counter term) φ + : H C + (id P)(R) (renormalization) H formal rules formal expressions(=!) φ R φ + 14 renormalized values ε 0 C + (id P)(R)

15 In QFT renormalization (Dim-Reg scheme), we take the triple (H FG, R FG, φ FG ) with Hopf algebra H FG of Feynman graphs; R FG = C[ε 1, ε]] of Laurent series, with the pole part projection P; φ FG : H FG R FG from dimensional regularized Feynman rule. Then Algebraic Birkhoff Decomposition gives φ FG = φ FG, φ FG,+ H FG Feynman rules Feynman integrals(=!) φ FG φ FG,+ renormalized values ε 0 C[ε 1, ε]] 15 C[[ε]]

16 Generalized Algebraic Birkhoff Decomposition Let C = n 0 C(n) be a (co)differential connected coalgebra (so C (0) = kj) with counit ε : C k and coderivations δ σ, σ Σ. Let A be a differential algebra with derivations σ, σ Σ. Let A = A 1 A 2 be a linear decomposition such that 1 A A 1 and σ (A i ) A i, i = 1, 2, σ Σ. Let P be the projection of A to A 1 along A 2. Denote G(C, A) := {φ : C A φ(j) = 1 A, σ φ = φδ σ, σ Σ}. Then any φ G(C, A) has a unique decomposition ϕ = ϕ ( 1) 1 ϕ 2, where ϕ i G(C, A), i = 1, 2, satisfy (ker ε) A i (hence ϕ i : C k1 A + A i ). If moreover A 1 is a subalgebra of A then φ ( 1) 1 lies in G(C, A 1 ). 16

17 Transverse cones 17 Identify V k := R k with its dual through a fixed inner product (, ). For a cone C, let lin(c) denote the subspace spanned by C. For any closed cone C and its face F, define the transverse cone (Berline and Vergne) t(c, F) along F to be the projection of C to F, where F = lin C (F) is the orthogonal completion of lin(f) in lin(c). For example, the transverse cone of e 1, e 1 + e 2 along e 1 + e 2 is e 1 e 2.

18 Coproduct of cones 18 We equip the linear space QCC of close cones with a coproduct and a counit : QCC QCC QCC, C := F C ε : QCC Q, ε(c) = t(c, F) F { 1, C = {0}, 0, C {0}. With CC (n) := {C CC dim C = n}, n 0, we have a connected coalgebra CC = n 0 CC (n).

19 Decorated closed cones 19 Let QDC denote the space of decorated cones (C; s) for s Z 0. Extend on QCC to QDC by derivation: (C; s) = ( δ i )(C; s + e i ) = (D i )(C; s + e i ). Then QDC is a connected coalgebra with derivations.

20 Regularized CZVs A meromorphic function f ( z) on C k is said to have linear poles at zero if there are linear forms L i ( z) = j a ijz j, such that ( i L i)f is homomorphic at zero. Let M(C k ) be the algebra of such functions and let M(C ) = k M(C k ). We also have the summation map S : QCC M(C ), S(C)( z) := e ( z, n). n C o Z k By taking derivations, S extends to S : QDCC M(C ), S(C; s) := ζ(c; s; z) := n C o Z k e ( z, n) n s 1 1 ns k k This can be regarded as a regularization of ζ(c; s; 0) = 1 n s 1 n C o Z k 1. ns k k 20.

21 Algebraic Birkhoff Decomposition There is a linear decomposition M(C ) = M + (C ) M (C ) = M 1 (C ) M 2 (C ), where M + (C ) = Hol(C ) is the space of functions holomorphic at 0 and M (C ) is spanned by h(l1,, l m ) L r, 1 1 L rn n where h M + (C ), l 1,, l m, L 1,, L n independent linear forms such that (l i, L j ) = 0, i, j. Together with the coproduct on QDC, we obtain a (Birkhoff) decomposition S = S ( 1) 1 S 2, where S i : QDC M i (C ). The value ζ(c; s) := S ( 1) 1 (S; s)(0) is called the renormalized conical zeta value of (C; s). 21

22 Classical Euler-Maclaurin Formula 22 The (classical) Euler-Maclaurin formula relates the discrete sum S(ε) := k=0 e εk = 1 for positive ε to the integral 1 e ε I(ε) := by means of the interpolator 0 e εx dx = 1 ε µ(ε) := S(ε) I(ε) = S(ε) 1 ε = K k=1 B 2k (2k)! ε2k 1 +o(ε 2K ) for all K N which is holomorphic at ε = 0. This formula becomes a special case of the Euler-Maclaurin formula for cone, of Berline and Vergne, when the cone is taken to be [0, ).

23 Euler-Maclaurin Formula for Cones 23 For a smooth cone C, define I(C)( z) := This gives rise to a map C e ( x, z) d x. I : QCC M(C ). Euler-Maclaurin Formula (Berline-Vergne) There is a map (interpolator) µ : QCC Hol(C ), such that S(C) = µ(t(c, F))I(F). F face of C

24 Birkhoff Factorization and Euler-Maclaurin Note that S + and S are unique such that S + (ker ε) M + and S (ker ε) M where ε : QDCC Q is the counit. Thus comparing with S = µ I and I : QDCC M, we obtain µ = S ( 1) +, I = S. Further, µ = π + S. 24

25 References N. Berline and M. Vergne, Euler-Maclaurin formula for polytopes, Mosc. Math. J. 7 (2007) N. Berline and M. Vergne, Local asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron, arxiv: v1. L. G., S. Paycha and B. Zhang, Conical zeta values and their double subdivision relations, Adv. Math. 252 (2014) L. G., S. Paycha and B. Zhang, Counting an infinite number of points: a testing ground for renormalization methods, In: Geometric, algebraic and topological methods for quantum field theory L. G., S. Paycha and B. Zhang, Decompositions and residue of meromorphic functions with linear poles in the light of the geometry of cones, arxiv: L. G., S. Paycha and B. Zhang, Algebraic Birkhoff Factorization and the Euler-Maclaurin Formula on cones, arxiv: (revised December 2015). L. G., S. Paycha and B. Zhang, Renormalized conical zeta values, preprint,

26 26 Thank You!