Motives in Quantum Field Theory

Transcription

1 String-Math conference, 2011

2 Motives of algebraic varieties (Grothendieck) Universal cohomology theory for algebraic varieties (with realizations) Pure motives: smooth projective varieties with correspondences Hom((X, p, m), (Y, q, n)) = qcorr m n /,Q (X, Y ) p Algebraic cycles mod equivalence (rational, homological, numerical), composition Corr(X, Y ) Corr(Y, Z) Corr(X, Z) (π X,Z ) (π X,Y (α) π Y,Z (β)) intersection product in X Y Z; with projectors p 2 = p and q 2 = q and Tate twists Q(m) with Q(1) = L 1 Numerical pure motives: M num,q (k) semi-simple abelian category (Jannsen)

3 Mixed motives: varieties that are possibly singular or not projective (much more complicated theory!) Triangulated category DM (Voevodsky, Levine, Hanamura) m(y ) m(x ) m(x Y ) m(y )[1] m(x A 1 ) = m(x )( 1)[2] Mixed Tate motives DMT DM generated by the Q(m) Over a number field: t-structure, abelian category of mixed Tate motives (vanishing result, M.Levine)

4 Quantum Field Theory perturbative (massless) scalar field theory S(φ) = L(φ)d D x = S 0 (φ) + S int (φ) in D dimensions, with Lagrangian density L(φ) = 1 2 ( φ)2 m2 2 φ2 L int (φ) Perturbative expansion: Feynman rules and Feynman diagrams S eff (φ) = S 0 (φ) + Γ(φ) (1PI graphs) #Aut(Γ) Γ Γ(φ) = 1 N! ˆφ(p 1 ) ˆφ(p N )U(Γ(p 1,..., p N ))dp 1 dp N i p i =0 U(Γ(p 1,..., p N )) = I Γ (k 1,..., k l, p 1,..., p N )d D k 1 d D k l l = b 1 (Γ) loops

5 Feynman rules for 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 U(Γ) = δ( n i=1 ɛ v,ik i + N j=1 ɛ v,jp j ) q 1 q n n = #E int (Γ), N = #E ext (Γ) +1 t(e) = v ɛ e,v = 1 s(e) = v 0 otherwise, d D k 1 d D k n

6 Formal properties reduce combinatorics to 1PI graphs: Connected graphs: Γ = v T Γ v 1PI graphs: U(Γ 1 Γ 2, p) = U(Γ 1, p 1 )U(Γ 2, p 2 ) U(Γ, p) = v T U(Γ v, p v ) δ((p v ) e (p v ) e ) q e ((p v ) e ) Note: formal properties can be used to construct abstract algebro-geometric Feynman rules (Chern classes; Grothendieck ring) P. Aluffi, M.M. Algebro-geometric Feynman rules, arxiv:

7 Parametric Feynman integrals Schwinger parameters q k1 1 q kn 1 Γ(k 1 ) Γ(k n ) 0 Feynman trick 1 = (n 1)! q 1 q n 0 n = e (s1q1+ +snqn) s k1 1 1 s kn 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 ± ei loop l k η ik = 0 otherwise U(Γ) = Γ(n Dl/2) (4π) ld/2 σ n = {t R n + i t i = 1}, vol form ω n σ n ω n Ψ Γ (t) D/2 V Γ (t, p) n Dl/2

8 Graph polynomials Ψ Γ (t) = det M Γ (t) = T Massless case m = 0: V Γ (t, p) = P Γ(t, p) Ψ Γ (t) t e with (M Γ ) kr (t) = e / T and P Γ (p, t) = C Γ s C e C n t i η ik η ir cut-sets C (complement of spanning tree plus one edge) s C = ( v V (Γ P 1) v ) 2 with P v = e E p ext(γ),t(e)=v e for e E p ext(γ) e = 0 with deg Ψ Γ = b 1 (Γ) = deg P Γ 1 U(Γ) = Γ(n Dl/2) (4π) ld/2 σ n P Γ (t, p) n+dl/2 ω n Ψ Γ (t) n+d(l+1)/2 stable range n + Dl/2 0; log divergent n = Dl/2: ω n Ψ Γ (t) D/2 σ n i=0 t e

9 Graph hypersurfaces Residue of U(Γ) (up to divergent Gamma factor) P Γ (t, p) n+dl/2 ω n σ n Ψ Γ (t) n+d(l+1)/2 Graph hypersurfaces ˆX Γ = {t A n Ψ Γ (t) = 0} X Γ = {t P n 1 Ψ Γ (t) = 0} deg = b 1 (Γ) Relative cohomology: (range n + Dl/2 0) H n 1 (P n 1 X Γ, Σ n (Σ n X Γ )) with Σ n = { i t i = 0} σ n Periods: σ ω integrals of algebraic differential forms ω on a cycle σ defined by algebraic equations in an algebraic variety

10 Feynman integrals and periods Parametric Feynman integral: algebraic differential form on cycle in algebraic variety But... divergent: where X Γ σ n, inside divisor Σ n σ n of coordinate hyperplanes Blowups of coordinate linear spaces defined by edges of 1PI subgraphs (toric variety P(Γ)) Iterated blowup P(Γ) separates strict transform of X Γ from non-negative real points Deform integration chain: monodromy problem; lift to P(Γ) Subtraction of divergences: Poincaré residuces and limiting mixed Hodge structure S. Bloch, E. Esnault, D. Kreimer, On motives associated to graph polynomials, arxiv:math/ S. Bloch, D. Kreimer, Mixed Hodge Structures and Renormalization in Physics, arxiv:

11 Periods and motives: Constraints on numbers obtained as periods from the motive of the variety! Periods of mixed Tate motives are Multiple Zeta Values ζ(k 1, k 2,..., k r ) = n k 1 1 n k 2 2 nr kr n 1 >n 2 > >n r 1 Conjecture proved recently: Francis Brown, Mixed Tate motives over Z, arxiv: Feynman integrals and periods: MZVs as typical outcome: D. Broadhurst, D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, arxiv:hep-th/ Conjecture (Kontsevich 1997): Motives of graph hypersurfaces are mixed Tate (or counting points over finite fields behavior)

12 Conjecture was first verified for all graphs up to 12 edges: J. Stembridge, Counting points on varieties over finite fields related to a conjecture of Kontsevich, 1998 But... Conjecture is false! P. Belkale, P. Brosnan, Matroids, motives, and a conjecture of Kontsevich, arxiv:math/ Dzmitry Doryn, On one example and one counterexample in counting rational points on graph hypersurfaces, arxiv: Francis Brown, Oliver Schnetz, A K3 in phi4, arxiv: Belkale Brosnan: general argument shows motives of graph hypersurfaces can be arbitrarily complicated Doryn and Brown Schnetz: explicit counterexamples with small enough graphs (14 edges)

13 Motives and the Grothendieck ring of varieties Difficult to determine explicitly the motive of X Γ (singular variety!) in the triangulated category of mixed motives Simpler invariant (universal Euler characteristic for motives): class [X Γ ] in the Grothendieck ring of varieties K 0 (V) generators [X ] isomorphism classes [X ] = [X Y ] + [Y ] for Y X closed [X ] [Y ] = [X Y ] Tate motives: Z[L, L 1 ] K 0 (M) (K 0 group of category of pure motives: virtual motives)

14 Universal Euler characteristics: Any additive invariant of varieties: χ(x ) = χ(y ) if X = Y χ(x ) = χ(y ) + χ(x Y ), Y X χ(x Y ) = χ(x )χ(y ) values in a commutative ring R is same thing as a ring homomorphism χ : K 0 (V) R Examples: Topological Euler characteristic Couting points over finite fields Gillet Soulé motivic χ mot (X ): χ mot : K 0 (V)[L 1 ] K 0 (M), χ mot (X ) = [(X, id, 0)] for X smooth projective; complex χ mot (X ) = W (X )

15 Universality: a dichotomy After localization (Belkale-Brosnan): the graph hypersurfaces X Γ generate the Grothendieck ring localized at L n L, n > 1 Stable birational equivalence: the graph hypersurfaces span Z inside Z[SB] = K 0 (V) L=0 P. Aluffi, M.M. Graph hypersurfaces and a dichotomy in the Grothendieck ring, arxiv:

16 Graph hypersurfaces: computing in the Grothendieck ring Example: banana graphs Ψ Γ (t) = t 1 t n ( 1 t t n ) [X Γn ] = Ln 1 L 1 (L 1)n ( 1) n n (L 1) n 2 L where L = [A 1 ] Lefschetz motive and T = [G m ] = [A 1 ] [A 0 ] X Γ = L hyperplane in P n 1 (Γ = polygon) [L Σ n ] = [L] [L Σ n ] = Tn 1 ( 1) n 1 T + 1 X Γn Σ n = S n with [S n ] = [Σ n ] nt n 2 [X Γn ] = [X Γn Σ n ] + [X Γn Σ n ] Using Cremona transformation: [X Γn ] = [S n ] + [L Σ n ] P. Aluffi, M.M. Feynman motives of banana graphs, arxiv:

17 Dual graph and Cremona transformation C : (t 1 : : t n ) ( 1 t 1 : : 1 t n ) outside S n singularities locus of Σ n = { i t i = 0}, ideal I Sn = (t 1 t n 1, t 1 t n 2 t n,, t 1 t 3 t n ) Ψ Γ (t 1,..., t n ) = ( e t e )Ψ Γ (t 1 1,..., t 1 n ) C(X Γ (P n 1 Σ n )) = X Γ (P n 1 Σ n ) isomorphism of X Γ and X Γ outside of Σ n

18 Sum over graphs Even when non-planar: can transform by Cremona (new hypersurface, not of dual graph) graphs by removing edges from complete graph: fixed vertices S N = #V (Γ)=N N! [X Γ ] #Aut(Γ) Z[L], Tate motive (though [X Γ ] individually need not be) Spencer Bloch, Motives associated to sums of graphs, arxiv: Suggests that although individual graphs need not give mixed Tate contribution, the sum over graphs in Feynman amplitudes (fixed loops, not vertices) may be mixed Tate

19 Deletion contraction relation In general cannot compute explicitly [X Γ ]: would like relations that simplify the graph... but cannot have true deletion-contraction relation, else always mixed Tate... What kind of deletion-contraction? P. Aluffi, M.M. Feynman motives and deletion-contraction relations, arxiv: Graph polynomials: Γ with n 2 edges, deg Ψ Γ = l > 0 Ψ Γ = t e Ψ Γ e + Ψ Γ/e Ψ Γ e = Ψ Γ t n and Ψ Γ/e = Ψ Γ tn=0 General fact: X = {ψ = 0} P n 1, Y = {F = 0} P n 2 ψ(t 1,..., t n ) = t n F (t 1,..., t n 1 ) + G(t 1,..., t n 1 ) Y = cone of Y in P n 1 : Projection from (0 : : 0 : 1) isomorphism X (X Y ) P n 2 Y

20 Then deletion-contraction: for X Γ A n [A n X Γ ] = L [A n 1 ( X Γ e X Γ/e )] [A n 1 X Γ e ] if e not a bridge or a looping edge if e bridge [A n X Γ ] = L [A n 1 X Γ e ] = L [A n 1 X Γ/e ] if e looping edge [A n X Γ ] = (L 1) [A n 1 X Γ e ] = (L 1) [A n 1 X Γ/e ] Note: intersection X Γ e X Γ/e difficult to control motivically: first place where non-tate contributions will appear

21 Example of application: Multiplying edges Γ me obtained from Γ by replacing edge e by m parallel edges (Γ 0e = Γ e, Γ e = Γ) Generating function: T = [G m ] K 0 (V) U(Γ me ) sm m! = ets e s T + 1 U(Γ) m 0 + ets + Te s U(Γ e) T (s e Ts ets e s ) U(Γ/e). T + 1 e not bridge nor looping edge: similar for other cases For doubling: inclusion-exclusion then cancellation U(Γ 2e ) = L [A n ( ˆX Γ ˆX Γo )] U(Γ) [ ˆX Γ ˆX Γo ] = [ ˆX Γ/e ] + (L 1) [ ˆX Γ e ˆX Γ/e ] U(Γ 2e ) = (L 2) U(Γ) + (L 1) U(Γ e) + L U(Γ/e)

22 Example of application: Lemon graphs and chains of polygons Λ m = lemon graph m wedges; Γ Λ m = replacing edge e of Γ with Λ m Generating function: m 0 U(ΓΛ m)s m = (1 (T + 1)s) U(Γ) + (T + 1)Ts U(Γ e) + (T + 1) 2 s U(Γ/e) 1 T(T + 1)s T(T + 1) 2 s 2 e not bridge or looping edge; similar otherwise Recursive relation: U(Λ m+1 ) = T(T + 1)U(Λ m ) + T(T + 1) 2 U(Λ m 1 ) a m = U(Λ m ) is a divisibility sequence: U(Λ m 1 ) divides U(Λ n 1 ) if m divides n

23 Determinant hypersurfaces and Schubert cells Mixed Tate question reformulated in terms of determinant hypersurfaces and intersections of unions of Schubert cells in flag varieties P. Aluffi, M.M. Parametric Feynman integrals and determinant hypersurfaces, arxiv: Υ : A n A l2, Υ(t) kr = i t i η ik η ir, ˆX Γ = Υ 1 ( ˆD l ) determinant hypersurface ˆD l = {det(x ij ) = 0} [A l2 ˆD l ] = L (l 2) l (L i 1) mixed Tate When Υ embedding U(Γ) = Υ(σ n) i=1 If ˆΣ Γ normal crossings divisor in A l2 m(a l2 ˆD l, ˆΣ Γ (ˆΣ Γ ˆD l )) P Γ (x, p) n+dl/2 ω Γ (x) det(x) n+(l+1)d/2 with Υ( σ n ) ˆΣ Γ mixed Tate motive?

24 Combinatorial conditions for embedding Υ : A n ˆX Γ A l2 ˆD l Closed 2-cell embedded graph ι : Γ S g with S g Γ union of open disks (faces); closure of each is a disk. Two faces have at most one edge in common Every edge in the boundary of two faces Sufficient: Γ 3-edge-connected with closed 2-cell embedding of face width 3. Face width: largest k N, every non-contractible simple closed curve in S g intersects Γ at least k times ( for planar). Note: 2-edge-connected =1PI; 2-vertex-connected conjecturally implies face width 2

25 Identifying the motive m(x, Y ). Set ˆΣ Γ ˆΣ l,g (f = l 2g + 1) { ˆΣ l,g = L 1 L ( f 2) x ij = 0 1 i < j f 1 x i1 + + x i,f 1 = 0 1 i f 1 m(a l2 ˆD l, ˆΣ l,g (ˆΣ l,g ˆD l )) ˆΣ l,g = normal crossings divisor Υ Γ ( σ n ) ˆΣ l,g depends only on l = b 1 (Γ) and g = min genus of S g Sufficient condition: Varieties of frames mixed Tate? F(V 1,..., V l ) := {(v 1,..., v l ) A l2 ˆD l v k V k }

26 Varieties of frames Two subspaces: (d 12 = dim(v 1 V 2 )) [F(V 1, V 2 )] = L d1+d2 L d1 L d2 L d L d12 + L Three subspaces (D = dim(v 1 + V 2 + V 3 )) [F(V 1, V 2, V 3 )] = (L d1 1)(L d2 1)(L d3 1) (L 1)((L d1 L)(L d23 1) + (L d2 L)(L d13 1) + (L d3 L)(L d12 1) +(L 1) 2 (L d1+d2+d3 D L d123+1 ) + (L 1) 3 Higher: difficult to find suitable induction Other formulation: Flag l,{di,e i }({V i }) locus of complete flags 0 E 1 E 2 E l = E, with dim E i V i = d i and dim E i V i+1 = e i : are these mixed Tate? (for all choices of d i, e i ) F(V 1,..., V l ) fibration over Flag l,{di,e i }({V i }): class [F(V 1,..., V l )] = [Flag l,{di,e i }({V i })](L d1 1)(L d2 L e1 )(L d3 L e2 ) (L dr L er 1 ) Flag l,{di,e i }({V i }) intersection of unions of Schubert cells in flag varieties Kazhdan Lusztig?

27 Other approach: Feynman integrals in configuration space Özgür Ceyhan, M.M. Feynman integrals and motives of configuration spaces, arxiv: Singularities of Feynman amplitude along diagonals e = {(x v ) v VΓ x v1 = x v2 for Γ (e) = {v 1, v 2 }} Conf Γ (X ) = X V Γ e E Γ e = X VΓ γ GΓ γ, with G Γ subgraphs induced (all edges of Γ between subset of vertices) and 2-vertex-connected Conf Γ (X ) γ G Γ Bl γ X VΓ iterated blowup description (wonderful compactifications : generalize Fulton-MacPherson) Conf Γ (X ) = Conf Γ (X ) N G nests X N stratification by G-nests of subgraphs (based on work of Li Li)

28 Voevodsky motive (quasi-projective smooth X ) m(conf Γ (X )) = m(x V Γ ) m(x VΓ/δN (Γ) )( µ )[2 µ ] N G Γ -nests,µ M N where M N := {(µ γ ) γ G Γ : 1 µ γ r γ 1, µ γ Z} with r γ = r γ,n := dim( γ N :γ γ γ ) dim γ and µ := γ G Γ µ γ Class in the Grothendieck ring [Conf Γ (X )] = [X ] VΓ + [X ] V Γ/δ N (Γ) L µ N G Γ -nests µ M N Key ingredient: Blowup formulae For mixed motives: codim m(bl V (Y )) Y (V ) 1 = m(y ) m(v )(k)[2k] k=1 Bittner relation in K 0 (V): exceptional divisor E [Bl V (Y )] = [Y ] [V ] + [E] = [Y ] + [V ]([P codim Y (V ) 1 ] 1)

29 Conf Γ (X ) are mixed Tate motives if X is To regularize Feynman integrals: lift to blowup Conf Γ (X ) Ambiguities by monodromies along exceptional divisors of the iterated blowups Residues of Feynman integrals and periods on hypersurface complement in Conf Γ (X ) Poincaré residues: periods on intersections of divisors of the stratification

30 Some other recent results: All the original Broadhurst Kreimer cases now proved Mapping to moduli space M 0,n and using results on multiple zeta values as periods of M 0,n (Goncharov-Manin, Brown) Francis Brown, On the periods of some Feynman integrals, arxiv: Chern classes of graph hypersurfaces Mixed Tate cases possible thanks to X Γ being singular (in low codimension): Chern Schwartz MacPherson classes measure singularities and can be assembled into an algebro-geometric Feynman rule: deletion-contraction and recursions Paolo Aluffi, Chern classes of graph hypersurfaces and deletion-contraction relations, arxiv:

31 More questions: Piece of cohomology supporting period always mixed Tate? Why think yes? Computations of middle cohomology D. Doryn, Cohomology of graph hypersurfaces associated to certain Feynman graphs, arxiv: Why think no? New invariant [X Γ ] c 2 (Γ)L 2 mod L 3 should be framing of motive (smallest piece carrying period) F. Brown, O. Schnetz, A K3 in phi4, arxiv: Action of Motivic Galois group of mixed Tate motives? Non-canonical isomorphism to Galois group of category of differential systems parameterizing divergences A.Connes, M.M. Renormalization and motivic Galois theory, math.nt/ )...likely related to the sum over graphs mixed Tate property