Renormalization, Galois symmetries, and motives. Matilde Marcolli

Transcription

1 Renormalization, Galois symmetries, and motives Matilde Marcolli July 2008

2 Quantum Fields and Motives (an unlikely match) Feynman diagrams: graphs and integrals (2π) 2D 1 k 4 1 (k p) 2 1 (k + l) 2 1 l 2 dd k d D l p k k p Divergences Renormalization Γ k p Algebraic varieties and motives V K smooth proj alg varieties over K category of pure motives M K Hom((X, p, m),(y, q, n)) = qcorr m n / (X, Y ) p p 2 = p, q 2 = q, Q(m) = Tate motives Universal cohomology theory for algebraic varieties What do they have in common? 1

3 Supporting evidence Multiple zeta values from Feynman integral calculations (Broadhurst Kreimer) Parametric Feynman integrals as periods (Bloch Esnault Kreimer) Graph hypersurfaces and their motives (Belkale Brosnan) Hopf algebras of renormalization (Connes Kreimer) Flat equisingular connections and Galois symmetries (Connes-M.) Feynman integrals and Hodge structures (Bloch Kreimer; M.) 2

4 Perturbative QFT in a nutshell T = scalar field theory in spacetime dimension D S(φ) = L(φ)d D x = S 0 (φ) + S int (φ) with Lagrangian density L(φ) = 1 2 ( φ)2 m2 2 φ2 L int (φ) Effective action and perturbative expansion (1PI graphs) Γ(φ) = 1 N! S eff (φ) = S 0 (φ) + Γ i p i=0 U(Γ(p 1,..., p N )) = l = b 1 (Γ) loops Γ(φ) #Aut(Γ) ˆφ(p 1 ) ˆφ(p N )U z µ (Γ(p 1,..., p N ))dp 1 dp N I Γ (k 1,..., k l, p 1,..., p N )d D k 1 d D k l Dimensional Regularization: Uµ(Γ(p z 1,..., p N )) = µ zl d D z k 1 d D z k l I Γ (k 1,..., k l, p 1,..., p N ) (Laurent series in z C ) 3

5 Two aspects: Individual Feynman graphs (Feynman rules, parametric representations) Construction of I Γ (k 1,..., k l, p 1,..., p N ): - Internal lines propagator = quadratic form q i 1 q 1 q n, q i (k i ) = k 2 i + m 2 - Vertices: conservation (valences = monomials in L) k i = 0 e i E(Γ):s(e i )=v - Integration over k i, internal edges - Parameterizations of the integral - Graph hypersurfaces and periods All Feynman graphs (subdivergences and renormalization) - Regularization and BPHZ renormalization - Connes Kreimer Hopf algebra - Birkhoff factorization - Beta function - Equisingular connections - Ward identities/gauge theories 4

6 Regularization (Dim Reg) p + k p k p φ 3 -theory D = 4 divergent Schwinger parameters 1 1 k 2 + m 2 (p + k) 2 + m = 2 1 k 2 + m 2 1 ((p + k) 2 + m 2 ) dd k s>0, t>0 diagonalize quadratic form in exp e s(k2 +m 2 ) t((p+k) 2 +m 2) ds dt Q(k) = λ((k + xp) 2 + ((x x 2 )p 2 + m 2 )) with s = (1 x)λ and t = x λ Gaussian q = k + xp e λ q2 d D q = π D/2 λ D/ = π D/2 1 e (λ(x x2 )p 2 +λ m 2 ) 0 = π D/2 Γ(2 D/2) 0 e λq2 d D q λ dλ dx e (λ(x x2 )p 2 +λ m 2) λ D/2 λ dλ dx 1 0 ((x x 2 )p 2 + m 2 ) D/2 2 dx 5

7 Adjust bare constants to cancel polar part L E = 1 ( m 2 δm 2 ) 2 ( φ)2 (1 δz)+ φ 2 g + δg φ But with subdivergences: p k Γ k p k p (2π) 2D 1 k 4 1 (k p) 2 1 (k + l) 2 1 l 2 dd k d D l (4π) D Γ(2 D 2 )Γ(D 2 1)3 Γ(5 D)Γ(D 4) Γ(D 2)Γ(4 D 2 )Γ(3D 2 5) (p 2 ) D 5 (p 2 /µ 2 ) z = ( z) n n! log n (p 2 /µ 2 ) (4π) p2 (log(p 2 /µ 2 ) + constant) non-polynomial term coefficient of 1 z 6

8 Renormalization (BPHZ) Preparation: R(Γ) = U(Γ) + C(γ)U(Γ/γ) γ V(Γ) coeff of pole is local Counterterm: C(Γ) = T( R(Γ)) Projection onto polar part of Laurent series Renormalized value: R(Γ) = R(Γ) + C(Γ) = U(Γ) + C(Γ) + C(γ)U(Γ/γ) γ V(Γ) 7

9 Connes Kreimer Hopf algebra H = H(T ) (depend on theory L(φ)) Free commutative algebra in generators Γ 1PI Feynman graphs Grading: loop number (or internal lines) deg(γ 1 Γ n ) = i deg(γ i ), deg(1) = 0 Coproduct: (Γ) = Γ Γ + γ Γ/γ γ V(Γ) Antipode: inductively S(X) = X S(X )X for (X) = X X + X X Extended to gauge theories (van Suijlekom): Ward identities as Hopf ideals 8

10 Connes Kreimer theory H dual to affine group scheme G (diffeographisms) G(C) pro-unipotent Lie group γ(z) = γ (z) 1 γ + (z) Birkhoff factorization of loops exists Recursive formula for Birkhoff = BPHZ loop = φ Hom(H, C({z})) (germs of meromorphic functions) Feynman integral U(Γ) = φ(γ) counterterms C(Γ) = φ (Γ) renormalized value R(Γ) = φ + (Γ) z=0 9

11 Bottom-up method: Feynman parameters U(Γ) = δ( n i=1 ǫ v,ik i + N j=1 ǫ v,jp j ) q 1 q n n = #E int (Γ), N = #E ext (Γ) +1 t(e) = v ǫ e,v = 1 s(e) = v 0 otherwise, Schwinger parameters q k 1 1 Γ(k 1 ) Γ(k n ) 0 Feynman trick 1 q 1 q n = (n 1)! 0 1 q k n d D k 1 d D k n n = e (s 1q 1 + +s n q n ) s k s k n 1 n ds 1 ds n. δ(1 n i=1 t i) (t 1 q t n q n ) n dt 1 dt n then change of variables k i = u i + l k=1 η ikx k +1 edge e i loop l k, same orientation η ik = 1 edge e i loop l k, reverse orientation 0 otherwise. 10

12 Parametric form U(Γ) = Γ(n Dl/2) (4π) ld/2 σ n ω n Ψ Γ (t) D/2 V Γ (t, p) n Dl/2 σ n = {t R n + i t i = 1}, vol form ω n Graph polynomials Ψ Γ (t) = det M Γ (t) = T e/ T t e V Γ (t, p) = P Γ(t, p) Ψ Γ (t) (M Γ ) kr (t) = and n t i η ik η ir i=0 P Γ (p, t) = C Γ s C e C cut-sets C (compl of spanning tree plus one edge) s C = ( v V (Γ 1 ) P v) 2 with P v = e E ext (Γ),t(e)=v p e for e E ext (Γ) p e = 0 degψ Γ = b 1 (Γ) = deg P Γ 1 t e Dim Reg (D+z)l U µ (Γ)(z) = µ zlγ(n (4π) l(d+z) 2 2 ) σ n ω n Ψ Γ (t) (D+z) 2 V Γ (t, p) n (D+z)l 2 11

13 Divergent case Renormalization (BPHZ) Convergent: period X Γ = {t = (t 1 : : t n ) P n 1 Ψ Γ (t) = 0} On complement P n 1 X Γ P Γ (p, t) σn n+dl/2 Ψ Γ (t) n+(l+1)d/2 ω n alg diff form on semi-alg set (period) Σ n = {t = (t 1 : : t n ) P n 1 i t i = 0} (coordinate simplex) then σ n H n 1 (P n 1,Σ n, Z) Motive: H n 1 (P n 1 X Γ,Σ n X Γ Σ n ) How complex? Classes [X Γ ] generate K 0 (V) Grothendieck ring of varieties (Belkale-Brosnan) but is the part of the motive involved in the period simpler? a mixed Tate motive? (Note: Tate part of K 0 (M) is just Z[L, L 1 ]) 12

14 A simple example: Banana graphs Class in the Grothendieck group [X Γn ] = Ln 1 L 1 (L 1)n ( 1) n n(l 1) n 2 L where L = [A 1 ] K 0 (V) Lefschetz motive (also characteristic classes: Aluffi-M.) σn (t 1 t n ) (D 2 1)(n 1) 1 ω n Ψ Γ (t) (D 2 1)n Ψ Γ (t) = t 1 t n ( 1 t t n ) More information on X Γ from other invariants e.g. Characteristic classes of singular varieties More complicated examples: wheels with n-spokes (Bloch Esnault Kreimer) 13

15 Observations on the bottom-up method P n 1 X Γ can be very complicated motivically (by Belkale-Brosnan) but for l loops, when X Γ not a cone: P n 1 X Γ P l2 1 D l D l = determinant variety P n 1 mapped linearly Υ : P n 1 P l2 1, Υ(t)kr = i t i η ki η ir The motive of P l2 1 D l can be computed mixed Tate P Γ (t, p) extends (non-uniquely) to P l2 1 D l Ψ Γ (t) becomes det(x) σ n and Σ n mapped linearly to P l2 1 difficulty: explicit control of Σ D l for H (P l2 1 D l,σ (Σ D l )) (from Aluffi-M. work in progress) 14

16 Top-down method: Galois theory (Connes M.) Compare renormalization and motives by comparing Tannakian categories Tannakian formalism C neutral Tannakian category G affine group scheme C Rep G abelian category (homological algebra) - Hom k-vector spaces - products and coproducts - kernels and cokernels (decomp of morphisms) rigid tensor category - : C C C, 1 Obj(C) - : C C op - ǫ : X X 1, δ : 1 X X - (ǫ 1)(1 δ) = 1 X, (1 ǫ)(δ 1) = 1 X Tannakian - fiber functor ω : C V ect K (faithful exact tensor) - K=k neutral 15

17 Main results (Connes M.) Counterterms as iterated integrals ( t Hooft Gross relations) Solutions of irregular singular differential equations (flat equisingular connections) Flat equisingular vector bundles form a neutral Tannakian category E Free graded Lie algebra L = F(e n ; n N) E Rep U, U = U G m U = Hom(H U, ), with H U = U(L) Motivic Galois group (Deligne Goncharov) U Gal(M S ) M S mixed Tate motives on S = Spec(Z[i][1/2]) 16

18 Counterterms as iterated integrals γ µ (z) = γ (z) 1 γ µ,+ (z) Time ordered exponential γ (z) = Te 1 z d n (β) = s 1 s 2 s n 0 0 θ t(β)dt = 1 + with β Lie(G) beta function n=1 d n (β) z n θ s1 (β) θ sn (β)ds 1 ds n (infinitesimal generator of renormalization group) θ u (X) = u n X, u G m, X H,deg(X) = n 1-param action generated by grading Y (X) = nx γ e t µ (z) = θ tz(γ µ (z)), Birkhoff factorization: z log µ µ γ (z) = 0 γ µ,+ (z) = Te 1 z so γ µ (z) specified by β and γ reg (z) up to equivalence (same γ ) just β 0 θ t (β)dt θz log µ (γ reg (z)) 17

19 Iterated integrals to differential systems g(t) = Te b a α(t)dt solution of dg(t) = g(t)α(t)dt Differential field (K = C({z}), δ) affine group scheme G G(K) f D(f) = f 1 δ(f) LieG(K) Differential equations: D(f) = ω flat LieG(C)-valued ω (singular at z = 0 ) Existence of solutions: trivial monodromy on M(ω)(l) = Te 1 0 l ω = 1, l π1 ( ) Gauge equivalence D(fh) = Dh + h 1 Df h (regular h C{z}) ω = Dh + h 1 ωh f ω = f ω for D(f ω ) = ω and D(f ω ) = ω solutions 18

20 Flat equisingular connections fibration G b B, principal bundle P = B G u(b, g) = (u(b), u Y (g)) u G m Flat connection on P equisingular u G m action (z, u(v)) = u Y ( (z, u)) Dγ = solution restrictions σ 1 (γ) σ 2 (γ) σ 1, σ 2 sections of B with σ 1 (0) = σ 2 (0) and f 1 f 2 iff f 1 1 f 2 G(O) Restrictions to different sections same type of singularity Up to gauge equivalence 19

21 Flat equisingular vector bundles category E Obj(E) pairs Θ = (V,[ ]) - V = fin dim Z-graded vector space - E = B V filtered W n (V ) = m n V m - G m action from grading - compatible with filtration, trivial on Gr n W (V ) - W-equivalent: T 1 = 2 T for a T Aut(E) compatible with filtration, trivial on Gr n W (V ) - equisingular: G m -invariant and restrictions to sections with same σ(0) are W-equivalent Hom E (Θ,Θ ) linear maps T : V V - compatible with grading - on E E connections ( ) 0 W equiv 0 ( T T 0 ) No longer depends on particular G (particular physical theory) 20

22 The Riemann Hilbert correspondence E Rep U, U = U G m U(A) = Hom(H U, A), Hopf algebra L = F(e 1, e 2, e 3, ), H U = U(L) free graded Lie algebra Renormalization group rg : G a U generator e = n=1 e n Universal singular frame (source of counterterms) γ U (z, v) = Te 1 z v 0 uy (e) du u For Θ = (V,[ ]) in E exists unique ρ Rep U Fiber functor Dρ(γ U ) W equiv ω n (Θ) = Hom(Q(n), Gr W n (Θ)) Q(n) = 1-dim in degree n, trivial connection Deligne Goncharov: U is the motivic Galois group of a category of mixed Tate motives M S Rep U 21

23 Remark: Bottom-up approach with periods works well in convergent (or log-divergent) case Top-down approach with Tannakian categories works well in divergent case Do they meet? Dim Reg of parametric integrals (divergent case): seen as oscillatory integrals and mixed Hodge structures (M. 2008) 22

24 Parametric Feynman integrals = σ P Γ (t, p) n+dl/2 Ψ Γ (t) n+(l+1)d/2ω n σ π (η) + σ df π (η) f forms on hypersurface complements f = Ψ Γ π (η) = (ω) f m (ω) = contraction with Euler vect field E = t i i π : A n {0} P n 1 (ω) = P n+dl/2 Γ Ω n (ω n ) = Ω n = i ( 1) i+1 t i dt 1 ˆ dt i dt n ω mdeg(f) Σ f = (ω) m Σ f + df (ω) m Σ f m+1 ω = closed k-form homogeneous of deg mdeg(f) 23

25 Nonisolated singularities (example: Banana graphs: codim 2) Slicing the Feynman integral Π ξ = {t A n ξ i, t = 0, i = 1,..., n k} Ω n = ξ 1, dt ξ n k, dt Ω ξ U(Γ) ξ F Σ,k (h Γ )(ξ) ξ, dt h Γ = V k+dl/2 Γ /Ψ D/2 Γ F Σ,k (h Γ )(ξ) = = σ F Γ (t, p) n k i=1 σ Π ξ h Γ (t, p)ω ξ (t) δ( ξ i, t )Ω ξ (t) for ξ, dt ω ξ = ω n and ξ, dt = ξ 1, dt ξ n k, dt forms h (ω ξ )/f m span subspace of H r (F ξ ) of Milnor fiber, with r = dimπ ξ 1 24

26 DimReg, Mellin transforms, Gelfand-Leray F Γ,ξ (z) = Ψ z Γ χ ξp l Γ Ω ξ from DimReg of (sliced) parametric Feynman integrals J Γ,ξ (s) = X s α ξ df Gelfand Leray function (f = Ψ Γ Πξ, α ξ = χ ξ P l Γ Ω ξ) F Γ,ξ (z) = 0 sz J Γ,ξ (s)ds Mellin transform (Note: Ψ Γ > 0 and P Γ R on σ) Regular singular Picard Fuchs equation J (l) Γ,ξ (s)+p 1(s)J (l 1) Γ,ξ (s)+ +p l (s)j Γ,ξ (s) = 0 25

27 Regular to irregular singular connections S Γ = manifold of planes Π ξ of dim codim Sing X Γ Distributions σ Γ C (S Γ ) induce σ γ and σ Γ/γ Hopf algebra: (Γ, σ) with coproduct (Γ, σ Γ ) = (Γ, σ Γ ) 1+1 (Γ, σ Γ )+ (γ, σ γ ) (Γ/γ, σ Γ/γ ) γ V(Γ) Dual affine group scheme G with g = LieG A solution J Γ,ξ of Picard Fuchs equation flat g(k)-valued equisingular connection φ µ (Γ, σ)(ǫ) = µ ǫb 1(Γ) σ(fγ,ξ (z)) z= (D+ǫ)/2 a µ (ǫ) = (φ µ S) d dǫ φ µ b µ (ǫ) = (φ µ S) Y (φ µ ) ω(ǫ, u) = u Y (a µ (ǫ))dǫ + u Y (b µ (ǫ)) du u flat db dǫ Y (a) + [a, b] = 0 26

28 The geometry of Dim Reg Dim Reg basic rule: Gaussian integral in dim z C e λt2 d z t := π z/2 λ z/2 What is a space of dim z C? Two possible models: Noncommutative Geometry (Connes-M. 2008) Motives (M. 2008) In both cases: product of an ordinary geometry by something else - NCG: of spacetime by an NC space X z - Motives: of individual graph motives by Log motive 27

29 Noncommutative geometry of DimReg Spectral triples X = (A, H, D) (metric NC spaces) Dim = {s C ζ a (s) = Tr(a D s ) have poles } Exists (type II) spectral triple X z with: Dim = {z} Tr(e λd2 z) = π z/2 λ z/2 D z = ρ(z)f Z 1/z Z = F Z self-adj affiliated to a type II N ρ(z) = π 1/2 (Γ(1 + z/2)) 1/z and spectral measure Tr(χ [a,b] (Z)) = 1 dt 2 (Tr = type II) [a,b] 28

30 Ordinary spacetime = commutative (A, H, D) = (C (X), L 2 (X, S), /D X ) Product X X z = cup product (A, H, D) (A z, H z, D z ) = (A A z, H H z, D 1 + γ D z ) (adapted to type II case) Breitenlohner Maison prescription γ 5 problem in DimReg D 1 + γ D z Example of X z : adèle class space N = L (Ẑ R ) GL 1 (Q) (partially defined action) Tr(f) = Z(1, ρ, λ) = λ, f(1, a) da Ẑ R Z(r, ρ, λ) = 0, r 1 Q 29

31 Motivic interpretation of DimReg Kummer motive M = [u : Z G m ] Ext 1 DM(K) (Q(0), Q(1)) with u(1) = q K and period matrix ( 1 0 ) log q 2πi Kummer extension of Tate sheaves K Ext 1 DM(G m )(Q G m (0),Q Gm (1)) Q Gm (1) K Q Gm (0) Q Gm (1)[1] Logarithmic motives Log n = Sym n (K) Log = lim n Log n log(s) (2πi) 0 0 log 2 (s) 2! (2πi)log(s) (2πi) 2 0. log n (s) (2πi) logn 1 (s) n! (n 1)!..... (2πi) 2 log n 2 (s) (2πi) n 1 (n 2)!.. 30

32 Graph polynomials and motivic sheaves M Γ (Ψ Γ : P n 1 X Γ G m,σ n X Γ Σ n, n 1, n 1) Arapura s category of motivic sheaves (f : X S, Y, i, w) DimReg = product M Γ Log fibered product (X 1 S X 2 S, Y 1 S X 2 X 1 S Y 2, i 1 + i 2, w 1 + w 2 ) π X 1 (ω) π X 2 (η) = ω f 1 (f 2) (η) on Σ 1 S Σ 2 for Σ i X i, Σ i Y i π X1 X 1 f 1 X 1 S X 2 π X2 X 2 f 2 S DimReg integral σ Ψz Γ α period on M Γ Log 31