Gravity from the viewpoint of local fields Dirk Kreimer, IHES - PowerPoint PPT Presentation

Gravity from the viewpoint of local fields Dirk Kreimer, IHES February 2010

Acknowledgments and Literature ◮ Thanks to people involved: Christoph Bergbauer, Spencer Bloch, David Broadhurst, Francis Brown, Alain Connes, Dzimitri Doryn, H´ el` ene Esnault, Kurusch Ebrahimi-Fard, Loic Foissy, Herbert Gangl, Dominique Manchon, Oliver Schnetz, Walter van Suijlekom, Matt Szczesny, Andrea Velenich, Karen Yeats

Acknowledgments and Literature ◮ Thanks to people involved: Christoph Bergbauer, Spencer Bloch, David Broadhurst, Francis Brown, Alain Connes, Dzimitri Doryn, H´ el` ene Esnault, Kurusch Ebrahimi-Fard, Loic Foissy, Herbert Gangl, Dominique Manchon, Oliver Schnetz, Walter van Suijlekom, Matt Szczesny, Andrea Velenich, Karen Yeats ◮ Literature: D. Kreimer, Algebra for quantum fields, arXiv:0906.1851 [hep-th], Clay Math. Inst. Proc. and references there.

Overview of talk ◮ Feynman graphs and their algebraic properties ◮ Hopf algebras ◮ Lie algebras ◮ sub-Hopf algebras ◮ Dynkin operators S ⋆ Y

Overview of talk ◮ Feynman graphs and their algebraic properties ◮ Hopf algebras ◮ Lie algebras ◮ sub-Hopf algebras ◮ Dynkin operators S ⋆ Y ◮ The structure of a Green function ◮ Kinematics as cohomology ◮ Leading-log expansions - the RGE from S ⋆ Y ◮ Reductions to γ 1 ◮ ODEs for β -functions

Overview of talk ◮ Feynman graphs and their algebraic properties ◮ Hopf algebras ◮ Lie algebras ◮ sub-Hopf algebras ◮ Dynkin operators S ⋆ Y ◮ The structure of a Green function ◮ Kinematics as cohomology ◮ Leading-log expansions - the RGE from S ⋆ Y ◮ Reductions to γ 1 ◮ ODEs for β -functions ◮ Nonperturbative aspects of QED and QCD ◮ QED ◮ QCD

Overview of talk ◮ Feynman graphs and their algebraic properties ◮ Hopf algebras ◮ Lie algebras ◮ sub-Hopf algebras ◮ Dynkin operators S ⋆ Y ◮ The structure of a Green function ◮ Kinematics as cohomology ◮ Leading-log expansions - the RGE from S ⋆ Y ◮ Reductions to γ 1 ◮ ODEs for β -functions ◮ Nonperturbative aspects of QED and QCD ◮ QED ◮ QCD ◮ Hodge structures and Feynman graphs ◮ renormalization as a limiting mixed Hodge structure ◮ Core Hopf algebras, gravity, BCFW

Hopf algebra of graphs H = Q 1 ⊕ � ∞ j =1 H j ◮ The coproduct ∆ ′ (Γ) � �� � � ∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ + γ ⊗ Γ /γ (1) γ = ∪ i γ i ,ω 4 ( γ i ) ≥ 0

Hopf algebra of graphs H = Q 1 ⊕ � ∞ j =1 H j ◮ The coproduct ∆ ′ (Γ) � �� � � ∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ + γ ⊗ Γ /γ (1) γ = ∪ i γ i ,ω 4 ( γ i ) ≥ 0 ◮ The antipode � S (Γ) = − Γ − S ( γ )Γ /γ = − m ( S ⊗ P )∆ (2)

Hopf algebra of graphs H = Q 1 ⊕ � ∞ j =1 H j ◮ The coproduct ∆ ′ (Γ) � �� � � ∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ + γ ⊗ Γ /γ (1) γ = ∪ i γ i ,ω 4 ( γ i ) ≥ 0 ◮ The antipode � S (Γ) = − Γ − S ( γ )Γ /γ = − m ( S ⊗ P )∆ (2) ◮ The character group G H V ∋ Φ ⇔ Φ : H → V , Φ( h 1 ∪ h 2 ) = Φ( h 1 )Φ( h 2 ) (3)

Hopf algebra of graphs H = Q 1 ⊕ � ∞ j =1 H j ◮ The coproduct ∆ ′ (Γ) � �� � � ∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ + γ ⊗ Γ /γ (1) γ = ∪ i γ i ,ω 4 ( γ i ) ≥ 0 ◮ The antipode � S (Γ) = − Γ − S ( γ )Γ /γ = − m ( S ⊗ P )∆ (2) ◮ The character group G H V ∋ Φ ⇔ Φ : H → V , Φ( h 1 ∪ h 2 ) = Φ( h 1 )Φ( h 2 ) (3) ◮ The counterterm � � � S Φ S Φ R (Γ) = − R Φ( h ) − R ( γ )Φ(Γ /γ ) � � m ( S Φ = − R Φ R ⊗ Φ P )∆(Γ) (4)

Hopf algebra of graphs H = Q 1 ⊕ � ∞ j =1 H j ◮ The coproduct ∆ ′ (Γ) � �� � � ∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ + γ ⊗ Γ /γ (1) γ = ∪ i γ i ,ω 4 ( γ i ) ≥ 0 ◮ The antipode � S (Γ) = − Γ − S ( γ )Γ /γ = − m ( S ⊗ P )∆ (2) ◮ The character group G H V ∋ Φ ⇔ Φ : H → V , Φ( h 1 ∪ h 2 ) = Φ( h 1 )Φ( h 2 ) (3) ◮ The counterterm � � � S Φ S Φ R (Γ) = − R Φ( h ) − R ( γ )Φ(Γ /γ ) � � m ( S Φ = − R Φ R ⊗ Φ P )∆(Γ) (4) ◮ The renormalized Feynman rules Φ R = m ( S Φ R ⊗ Φ)∆ (5)

An Example ◮ The co-product ∆ ′ � � = 3 ⊗ + + + + + + + +2 ⊗ + ⊗ .

An Example ◮ The co-product ∆ ′ � � = 3 ⊗ + + + + + + + +2 ⊗ + ⊗ . ◮ The counterterm � � S Φ S Φ R ( ) = − Rm R ⊗ Φ P × + + + + + + + � � × ∆ + + + + + + + � � � = − R Φ + + + + + + + + + R [Φ (3 + 2 + )] Φ ( ) }

An Example ◮ The co-product ∆ ′ � � = 3 ⊗ + + + + + + + +2 ⊗ + ⊗ . ◮ The counterterm � � S Φ S Φ R ( ) = − Rm R ⊗ Φ P × + + + + + + + � � × ∆ + + + + + + + � � � = − R Φ + + + + + + + + + R [Φ (3 + 2 + )] Φ ( ) } ◮ The renormalized result � � Φ R = ( id − R ) m ( S Φ R ⊗ Φ P )∆ + + + + + + + � � � = ( id − R ) Φ + + + + + + + + R [Φ (3 + 2 + )] Φ ( ) }

Lie algebra of graphs ◮ The Milnor Moore Theorem H = U ⋆ ( L )

Lie algebra of graphs ◮ The Milnor Moore Theorem H = U ⋆ ( L ) ◮ The pairing � Z Γ , δ Γ ′ � = δ Kronecker (6) Γ , Γ ′

Lie algebra of graphs ◮ The Milnor Moore Theorem H = U ⋆ ( L ) ◮ The pairing � Z Γ , δ Γ ′ � = δ Kronecker (6) Γ , Γ ′ ◮ the Lie algebra [ Z Γ , Z Γ ′ ] = Z Γ ′ ⋆ Γ − Γ ⋆ Γ ′ (7) ⋆ = ⋆ = 2

Lie algebra of graphs ◮ The Milnor Moore Theorem H = U ⋆ ( L ) ◮ The pairing � Z Γ , δ Γ ′ � = δ Kronecker (6) Γ , Γ ′ ◮ the Lie algebra [ Z Γ , Z Γ ′ ] = Z Γ ′ ⋆ Γ − Γ ⋆ Γ ′ (7) ⋆ = ⋆ = 2 ◮ Leads to an identification of β -functions and anomalous dimenions, and lifts the Birkhoff decomposition Φ R = S Φ R ⋆ Φ to diffeomorphisms of physical parameters.

sub-Hopf algebras ◮ summing order by order 1 � c r k = | Aut (Γ) | Γ , (8) | Γ | = k , res (Γ)= r then � ∆( c r Pol j ( c s m ) ⊗ c r k ) = k − j . (9) j

sub-Hopf algebras ◮ summing order by order 1 � c r k = | Aut (Γ) | Γ , (8) | Γ | = k , res (Γ)= r then � ∆( c r Pol j ( c s m ) ⊗ c r k ) = k − j . (9) j ◮ Hochschild closedness � � X r = 1 ± j α j = 1 ± α j B r ; j c r + ( X r Q j ( α )) , (10) j j Q j = X v √ Q edges e at v X e . Evaluates to invariant charge.

sub-Hopf algebras ◮ summing order by order 1 � c r k = | Aut (Γ) | Γ , (8) | Γ | = k , res (Γ)= r then � ∆( c r Pol j ( c s m ) ⊗ c r k ) = k − j . (9) j ◮ Hochschild closedness � � X r = 1 ± j α j = 1 ± α j B r ; j c r + ( X r Q j ( α )) , (10) j j Q j = X v √ Q edges e at v X e . Evaluates to invariant charge. ◮ bB r ; j + = 0. ∆ B r ; j + ( X ) = B r ; j + ( X ) ⊗ 1 + ( id ⊗ B r ; j + )∆( X ) . (11) Implies locality of counterterms upon application of Feynman rules.

Symmetry ◮ Ward and Slavnov–Taylor ids ¯ ¯ ψψ ψ A /ψ i k := c + c (12) k k span Hopf (co-)ideal I : ∆( I ) ⊆ H ⊗ I + I ⊗ H . (13) 1 4 F 2 ¯ ¯ ψ A /ψ ψ A /ψ ∆( i 2 ) = i 2 ⊗ 1 + 1 ⊗ i 2 + ( c + c + i 1 ) ⊗ i 1 + i 1 ⊗ c . 1 1 1

Symmetry ◮ Ward and Slavnov–Taylor ids ¯ ¯ ψψ ψ A /ψ i k := c + c (12) k k span Hopf (co-)ideal I : ∆( I ) ⊆ H ⊗ I + I ⊗ H . (13) 4 F 2 1 ¯ ¯ ψ A /ψ ψ A /ψ ∆( i 2 ) = i 2 ⊗ 1 + 1 ⊗ i 2 + ( c + c + i 1 ) ⊗ i 1 + i 1 ⊗ c . 1 1 1 ◮ Feynman rules vanish on I ⇔ Feynman rules respect quantized symmetry: Φ R : H / I → V .

Symmetry ◮ Ward and Slavnov–Taylor ids ¯ ¯ ψψ ψ A /ψ i k := c + c (12) k k span Hopf (co-)ideal I : ∆( I ) ⊆ H ⊗ I + I ⊗ H . (13) 1 4 F 2 ¯ ¯ ψ A /ψ ψ A /ψ ∆( i 2 ) = i 2 ⊗ 1 + 1 ⊗ i 2 + ( c + c + i 1 ) ⊗ i 1 + i 1 ⊗ c . 1 1 1 ◮ Feynman rules vanish on I ⇔ Feynman rules respect quantized symmetry: Φ R : H / I → V . ◮ Ideals for Slavnov–Taylor ids generated by equality of renormalized charges, also for the master equation in Batalin-Vilkovisky (see Walter van Suijlekom’s work)

Dynkin operators ◮ S ⋆ Y Y (Γ) = | Γ | Γ the grading operator S ⋆ Y (Γ) = m ( S ⊗ Y )∆(Γ) . (14) Vanishes on products.

Dynkin operators ◮ S ⋆ Y Y (Γ) = | Γ | Γ the grading operator S ⋆ Y (Γ) = m ( S ⊗ Y )∆(Γ) . (14) Vanishes on products. ◮ The leading log expansion corad (Γ) � c j (Γ) ln j s Φ R (Γ) = (15) j ⇒ c j = 1 ∆ j − 1 , j ≥ 1 j ! σ ⊗ · · · ⊗ σ (16) � �� � j times where σ = Φ R ◦ S ⋆ Y ↔ γ k ≡ γ k ( γ 1 ).

Kinematics and Cohomology ◮ Exact co-cycles [ B r , j + ] = B r ; j + + b φ r ; j (17) with φ r ; j : H → C

Kinematics and Cohomology ◮ Exact co-cycles [ B r , j + ] = B r ; j + + b φ r ; j (17) with φ r ; j : H → C ◮ Variation of momenta G R ( { g } , ln s , { Θ } ) = 1 ± Φ R ln s , { Θ } ( X r ( { g } )) (18) with X r = 1 ± � j g j B r ; j + ( X r Q j ( g )), bB r ; j + = 0. Also,   abelian factor ∞ ���� � G r = γ j ( { g } , { Θ } ) ln j s  + G r (19)  0 j =1 Then, for MOM and similar schemes (not MS!): { Θ } → { Θ ′ } ⇔ B r ; j + → B r , j + + b φ r , j .