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 γ1 ◮ 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 γ1 ◮ 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 γ1 ◮ 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 = Q1 ⊕ ∞

j=1 Hj

◮ The coproduct

∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ +

∆′(Γ)

• γ=∪iγi,ω4(γi)≥0

γ ⊗ Γ/γ (1)

Hopf algebra of graphs H = Q1 ⊕ ∞

j=1 Hj

◮ The coproduct

∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ +

∆′(Γ)

• γ=∪iγi,ω4(γi)≥0

γ ⊗ Γ/γ (1)

◮ The antipode

S(Γ) = −Γ −

• S(γ)Γ/γ = −m(S ⊗ P)∆

(2)

Hopf algebra of graphs H = Q1 ⊕ ∞

j=1 Hj

◮ The coproduct

∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ +

∆′(Γ)

• γ=∪iγi,ω4(γi)≥0

γ ⊗ Γ/γ (1)

◮ The antipode

S(Γ) = −Γ −

• S(γ)Γ/γ = −m(S ⊗ P)∆

(2)

◮ The character group

G H

V ∋ Φ ⇔ Φ : H → V , Φ(h1 ∪ h2) = Φ(h1)Φ(h2)

(3)

Hopf algebra of graphs H = Q1 ⊕ ∞

j=1 Hj

◮ The coproduct

∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ +

∆′(Γ)

• γ=∪iγi,ω4(γi)≥0

γ ⊗ Γ/γ (1)

◮ The antipode

S(Γ) = −Γ −

• S(γ)Γ/γ = −m(S ⊗ P)∆

(2)

◮ The character group

G H

V ∋ Φ ⇔ Φ : H → V , Φ(h1 ∪ h2) = Φ(h1)Φ(h2)

(3)

◮ The counterterm

SΦ

R (Γ)

= −R

• Φ(h) −
• SΦ

R (γ)Φ(Γ/γ)

• =

−R Φ

• m(SΦ

R ⊗ Φ P)∆(Γ)

• (4)

Hopf algebra of graphs H = Q1 ⊕ ∞

j=1 Hj

◮ The coproduct

∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ +

∆′(Γ)

• γ=∪iγi,ω4(γi)≥0

γ ⊗ Γ/γ (1)

◮ The antipode

S(Γ) = −Γ −

• S(γ)Γ/γ = −m(S ⊗ P)∆

(2)

◮ The character group

G H

V ∋ Φ ⇔ Φ : H → V , Φ(h1 ∪ h2) = Φ(h1)Φ(h2)

(3)

◮ The counterterm

SΦ

R (Γ)

= −R

• Φ(h) −
• SΦ

R (γ)Φ(Γ/γ)

• =

−R Φ

• m(SΦ

R ⊗ Φ P)∆(Γ)

• (4)

◮ The renormalized Feynman rules

ΦR = m(SΦ

R ⊗ Φ)∆

(5)

An Example

◮ The co-product

∆′

+ + + + + + +

• =

3 ⊗ +2 ⊗ + ⊗ .

An Example

◮ The co-product

∆′

+ + + + + + +

• =

3 ⊗ +2 ⊗ + ⊗ .

◮ The counterterm

SΦ

R (

+ + + + + + +

) = −Rm

• SΦ

R ⊗ ΦP

• ×

×∆

• +

+ + + + + +

• = −R
• Φ
• +

+ + + + + +

• +

+R [Φ (3 + 2 + )] Φ ( )}

An Example

◮ The co-product

∆′

+ + + + + + +

• =

3 ⊗ +2 ⊗ + ⊗ .

◮ The counterterm

SΦ

R (

+ + + + + + +

) = −Rm

• SΦ

R ⊗ ΦP

• ×

×∆

• +

+ + + + + +

• = −R
• Φ
• +

+ + + + + +

• +

+R [Φ (3 + 2 + )] Φ ( )}

◮ The renormalized result

ΦR = (id − R)m(SΦ

R ⊗ ΦP)∆

• +

+ + + + + +

• = (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

ZΓ, δΓ′ = δKronecker

Γ,Γ′

(6)

Lie algebra of graphs

◮ The Milnor Moore Theorem

H = U⋆(L)

◮ The pairing

ZΓ, δΓ′ = δKronecker

Γ,Γ′

(6)

◮ the Lie algebra

[ZΓ, ZΓ′] = ZΓ′⋆Γ−Γ⋆Γ′ (7) ⋆ = ⋆ = 2

Lie algebra of graphs

◮ The Milnor Moore Theorem

H = U⋆(L)

◮ The pairing

ZΓ, δΓ′ = δKronecker

Γ,Γ′

(6)

◮ the Lie algebra

[ZΓ, ZΓ′] = ZΓ′⋆Γ−Γ⋆Γ′ (7) ⋆ = ⋆ = 2

◮ Leads to an identification of β-functions and anomalous

dimenions, and lifts the Birkhoff decomposition ΦR = SΦ

R ⋆ Φ

to diffeomorphisms of physical parameters.

sub-Hopf algebras

◮ summing order by order

cr

k =

• |Γ|=k,res(Γ)=r

1 |Aut(Γ)|Γ, (8) then ∆(cr

k) =

• j

Polj(cs

m) ⊗ cr k−j.

(9)

sub-Hopf algebras

◮ summing order by order

cr

k =

• |Γ|=k,res(Γ)=r

1 |Aut(Γ)|Γ, (8) then ∆(cr

k) =

• j

Polj(cs

m) ⊗ cr k−j.

(9)

◮ Hochschild closedness

X r = 1 ±

• j

cr

j αj = 1 ±

• j

αjBr;j

+ (X rQj(α)),

(10) Qj =

X v

√Q

edges e at v X e . Evaluates to invariant charge.

sub-Hopf algebras

◮ summing order by order

cr

k =

• |Γ|=k,res(Γ)=r

1 |Aut(Γ)|Γ, (8) then ∆(cr

k) =

• j

Polj(cs

m) ⊗ cr k−j.

(9)

◮ Hochschild closedness

X r = 1 ±

• j

cr

j αj = 1 ±

• j

αjBr;j

+ (X rQj(α)),

(10) Qj =

X v

√Q

edges e at v X e . Evaluates to invariant charge.

◮ bBr;j + = 0.

∆Br;j

+ (X) = Br;j + (X) ⊗ 1 + (id ⊗ Br;j + )∆(X).

(11) Implies locality of counterterms upon application of Feynman rules.

Symmetry

◮ Ward and Slavnov–Taylor ids

ik := c

¯ ψψ k

+ c

¯ ψA /ψ k

(12) span Hopf (co-)ideal I: ∆(I) ⊆ H ⊗ I + I ⊗ H. (13) ∆(i2) = i2 ⊗ 1 + 1 ⊗ i2 + (c

1 4F 2

1

+ c

¯ ψA /ψ 1

+ i1) ⊗ i1 + i1 ⊗ c

¯ ψA /ψ 1

.

Symmetry

◮ Ward and Slavnov–Taylor ids

ik := c

¯ ψψ k

+ c

¯ ψA /ψ k

(12) span Hopf (co-)ideal I: ∆(I) ⊆ H ⊗ I + I ⊗ H. (13) ∆(i2) = i2 ⊗ 1 + 1 ⊗ i2 + (c

1 4F 2

1

+ c

¯ ψA /ψ 1

+ i1) ⊗ i1 + i1 ⊗ c

¯ ψA /ψ 1

.

◮ Feynman rules vanish on I ⇔ Feynman rules respect

quantized symmetry: ΦR : H/I → V .

Symmetry

◮ Ward and Slavnov–Taylor ids

ik := c

¯ ψψ k

+ c

¯ ψA /ψ k

(12) span Hopf (co-)ideal I: ∆(I) ⊆ H ⊗ I + I ⊗ H. (13) ∆(i2) = i2 ⊗ 1 + 1 ⊗ i2 + (c

1 4F 2

1

+ c

¯ ψA /ψ 1

+ i1) ⊗ i1 + i1 ⊗ c

¯ ψA /ψ 1

.

◮ Feynman rules vanish on I ⇔ Feynman rules respect

quantized symmetry: ΦR : 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

ΦR(Γ) =

corad(Γ)

• j

cj(Γ) lnj s (15) ⇒ cj = 1 j! σ ⊗ · · · ⊗ σ

• j times

∆j−1, j ≥ 1 (16) where σ = ΦR ◦ S ⋆ Y ↔ γk ≡ γk(γ1).

Kinematics and Cohomology

◮ Exact co-cycles

[Br,j

+ ] = Br;j + + bφr;j

(17) with φr;j : H → C

Kinematics and Cohomology

◮ Exact co-cycles

[Br,j

+ ] = Br;j + + bφr;j

(17) with φr;j : H → C

◮ Variation of momenta

G R({g}, ln s, {Θ}) = 1 ± ΦR

ln s,{Θ}(X r({g}))

(18) with X r = 1 ±

j gjBr;j + (X rQj(g)), bBr;j + = 0. Also,

G r =  

∞

• j=1

γj({g}, {Θ}) lnj s   +

abelian factor

• G r

(19) Then, for MOM and similar schemes (not MS!): {Θ} → {Θ′} ⇔ Br;j

+ → Br,j + + bφr,j.

Leading log expansions and the RGE

◮ The invariant charge Qv

For each vertex v, a charge Qv: Qv(g) = X v(g)

• e

√ X e , (20) e adjacent to v.

Leading log expansions and the RGE

◮ The invariant charge Qv

For each vertex v, a charge Qv: Qv(g) = X v(g)

• e

√ X e , (20) e adjacent to v.

◮

 ∂L + β(g)∂g −

• e adj r

γe

1

  G r(g, L) = 0 (21) rewrites in terms of the Dynkin operator (γr

1(g) = S ⋆ Y (X r(g))):

γr

k(g) = 1

k  γr

1(g) −

• j∈R

sjγj

1g∂g

  γr

k−1(g)

(22)

Ordinary differential equations vs DSE

◮ RGE+DSE

the iterated integral structure ΦR(Br;j

+ (X)) =

• ΦR(X)dµr;j

(23) allows to combine X r = 1 ±

j B+(X rQj) with RGE to

γr

1 = P(g) − [γr 1(g)]2 +

• j∈R

sjγj

1g∂gγr 1(g).

(24)

Ordinary differential equations vs DSE

◮ RGE+DSE

the iterated integral structure ΦR(Br;j

+ (X)) =

• ΦR(X)dµr;j

(23) allows to combine X r = 1 ±

j B+(X rQj) with RGE to

γr

1 = P(g) − [γr 1(g)]2 +

• j∈R

sjγj

1g∂gγr 1(g).

(24)

◮ massless gauge theories

β(g) = gγ1(g)/2 for γ1 anomalous dim of gauge propagator γ1(g) =

existence assumed

P(g) −γ1(g)(1 − g∂g)γ1(g) (25) (Ward Id QED, background field gauge (Abbott) QCD)

QED

◮ sub Hopf algebra for vacuum polarization suffices

QED

◮ sub Hopf algebra for vacuum polarization suffices ◮ γ1(x) = P(x) − γ1(x)2 + γ1(x)x∂xγ1(x) with P(x) > 0

QED

◮ sub Hopf algebra for vacuum polarization suffices ◮ γ1(x) = P(x) − γ1(x)2 + γ1(x)x∂xγ1(x) with P(x) > 0

P(x) twice differentiable γ1(x0) = γ0 > 0 different solutions distinguished by e− 1

x

behaviour

dγ1 dx = γ1 − γ2 1 − P, dx dL = xγ1

L = x(L)

x0 dz zγ1(z)

−1 x γ1(x)

◮ separatrix exists and might have no Landau pole:

D(P) = ∞

x0 P(z)dz z3

< ∞, ∞

x0 2dz z√ 1+4P(z)−1 < ∞

QCD

◮ sub Hopf algebra for gluon polarization suffices in background

field gauge

QCD

◮ sub Hopf algebra for gluon polarization suffices in background

field gauge

◮ γ1(g) = P(g) − γ1(g)2 + γ1(g)g∂gγ1(g) with P(g) < 0

QCD

◮ sub Hopf algebra for gluon polarization suffices in background

field gauge

◮ γ1(g) = P(g) − γ1(g)2 + γ1(g)g∂gγ1(g) with P(g) < 0

P(g) twice differentiable and concave near 0 unique solution which flows into (0, 0) at large Q2 L = g(L)

g0 dz zγ1(z) →

LΛ = − ∞

g(LΛ) dz zγ1(z),

LΛ = ln Q2/ΛQCD fdisp(Q2) = ∞

ℑ(f (σ))dσ σ+Q2−iη

and ODE

−1 γ1(g) g

◮ separatrix exists and gives asymptotic free solution, with finite mass

gap for inverse propagator iff γ1(x) < −1 for some x > 0. |D(P)| < ∞ → γ1(x) ∼ sx, x → ∞. That allows for dispersive methods as introduced by Shirkov et.al. in field theory.

Limiting mixed Hodge structures

◮ Hopf algebra from flags

f := γ1 ⊂ γ2 ⊂ . . . ⊂ Γ, ∆′(γi+1/γi) = 0 (26) The set of all such flags FΓ ∋ f determines Hopf algebra structure, |FΓ| is the length of the flag.

Limiting mixed Hodge structures

◮ Hopf algebra from flags

f := γ1 ⊂ γ2 ⊂ . . . ⊂ Γ, ∆′(γi+1/γi) = 0 (26) The set of all such flags FΓ ∋ f determines Hopf algebra structure, |FΓ| is the length of the flag.

◮ It also determines a column vector v = v(FΓ) and a nilpotent

matrix (N) = (N(|FΓ|)), (N)k+1 = 0, k = corad(Γ) such that

lim

t→0 (e− ln t(N))ΦR(v(FΓ)) = (cΓ 1 (Θ) ln s, cΓ 2 (Θ), cΓ k (Θ) lnk s)T

(27)

where k is determined from the co-radical filtration and t is a regulator say for the lower boundary in the parametric representation.

P(x) and Witt algebras

◮ A graded commutative Hopf algebra H can be regarded as the dual

• f the universal enveloping algebra U(L) of a Lie algebra L. We need

zr

m ⊗ zs n − zs n ⊗ zr m, ∆ct j = [zs n, zr m], ∆ct j ,

(28) ∀j > 0, t ∈ R.

P(x) and Witt algebras

◮ A graded commutative Hopf algebra H can be regarded as the dual

• f the universal enveloping algebra U(L) of a Lie algebra L. We need

zr

m ⊗ zs n − zs n ⊗ zr m, ∆ct j = [zs n, zr m], ∆ct j ,

(28) ∀j > 0, t ∈ R.

◮

[zs

k, zt l ] = −Q(s)kzs k+l + Q(t)lzt k+l.

(29) In QED one finds Q( ¯ ψA /ψ)) = Q( ¯ ψψ) = 2, Q( 1

4F 2) = 1.

P(x) and Witt algebras

◮ A graded commutative Hopf algebra H can be regarded as the dual

• f the universal enveloping algebra U(L) of a Lie algebra L. We need

zr

m ⊗ zs n − zs n ⊗ zr m, ∆ct j = [zs n, zr m], ∆ct j ,

(28) ∀j > 0, t ∈ R.

◮

[zs

k, zt l ] = −Q(s)kzs k+l + Q(t)lzt k+l.

(29) In QED one finds Q( ¯ ψA /ψ)) = Q( ¯ ψψ) = 2, Q( 1

4F 2) = 1.

◮ We identify this Lie algeb as a subalgebra of the generalized Witt

algebra W . For integers Q(t) as above, set zs

m :=

• t∈R

xQ(t

t

m xs∂xs. (30) This puts Lgrad ⊂ W +. We can now augment the algebra W + by an R-matrix: [Y , zq

1 ] = zq 1 , → r := Y ⊗ zq 1 − zq 1 ⊗ Y .

P(x) and Witt algebras

◮ A graded commutative Hopf algebra H can be regarded as the dual

• f the universal enveloping algebra U(L) of a Lie algebra L. We need

zr

m ⊗ zs n − zs n ⊗ zr m, ∆ct j = [zs n, zr m], ∆ct j ,

(28) ∀j > 0, t ∈ R.

◮

[zs

k, zt l ] = −Q(s)kzs k+l + Q(t)lzt k+l.

(29) In QED one finds Q( ¯ ψA /ψ)) = Q( ¯ ψψ) = 2, Q( 1

4F 2) = 1.

◮ We identify this Lie algeb as a subalgebra of the generalized Witt

algebra W . For integers Q(t) as above, set zs

m :=

• t∈R

xQ(t

t

m xs∂xs. (30) This puts Lgrad ⊂ W +. We can now augment the algebra W + by an R-matrix: [Y , zq

1 ] = zq 1 , → r := Y ⊗ zq 1 − zq 1 ⊗ Y .

◮ P(x) comes from S ⋆ Y on flags, and from dualizing Lie brackets in

• Lgrad. Bounds from counting in U(Lgrad) and constructive

estimates a possibility.

Periods and functions

◮ Wanted: ρ : Graphs → Periods

(ρ ⊗ ρ)∆Graphs = ∆periodsρ. (31) What is ρ? Which ∆Graphs? Is ∆MZV enough??? (waiting for Steph Belcher...)

Periods and functions

◮ Wanted: ρ : Graphs → Periods

(ρ ⊗ ρ)∆Graphs = ∆periodsρ. (31) What is ρ? Which ∆Graphs? Is ∆MZV enough??? (waiting for Steph Belcher...)

◮ What is the role of shuffle/stuffle algebras on graphs?

They are there for flags. Is there a free Lie algebra structure on graphs?

Periods and functions

◮ Wanted: ρ : Graphs → Periods

(ρ ⊗ ρ)∆Graphs = ∆periodsρ. (31) What is ρ? Which ∆Graphs? Is ∆MZV enough??? (waiting for Steph Belcher...)

◮ What is the role of shuffle/stuffle algebras on graphs?

They are there for flags. Is there a free Lie algebra structure on graphs?

◮ What is the number-theoretic meaning of all the graph Hopf

algebras? Not all of this is hopeless. See Francis Brown, Oliver Schnetz,... In general, we need a better algebro-geometric understanding. See identification of zig-zag graphs by Dzmitri Doryn. But still no understanding of rational coefficients.

core Hopf algebra structures: unitarity, gravity, BCFW

◮ The core Hopf algebra

∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ +

• γ=∪iγi

γ ⊗ Γ/γ (32) Only primitive graphs are one-loop graphs. Appears as the endpoint in tower H0 ⊂ H2 ⊂ H4 ⊂ H6 ⊂ · · · ⊂ H∞ = Hcore (33)

core Hopf algebra structures: unitarity, gravity, BCFW

◮ The core Hopf algebra

∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ +

• γ=∪iγi

γ ⊗ Γ/γ (32) Only primitive graphs are one-loop graphs. Appears as the endpoint in tower H0 ⊂ H2 ⊂ H4 ⊂ H6 ⊂ · · · ⊂ H∞ = Hcore (33)

◮ Gravity

ω4(γ) = 2|γ| + 2 Hren = Hcore

(34) All skeletons are one-loop.

core Hopf algebra structures: unitarity, gravity, BCFW

◮ The core Hopf algebra

∆(Γ) = Γ ⊗ 1 + 1 ⊗ Γ +

• γ=∪iγi

γ ⊗ Γ/γ (32) Only primitive graphs are one-loop graphs. Appears as the endpoint in tower H0 ⊂ H2 ⊂ H4 ⊂ H6 ⊂ · · · ⊂ H∞ = Hcore (33)

◮ Gravity

ω4(γ) = 2|γ| + 2 Hren = Hcore

(34) All skeletons are one-loop.

◮ Britto-Cachazo-Feng-Witten recursion holds →

Maximal Co-ideals of Hcore respected by Feynman rules. Gravity possibly renormalizable iff full cut-reconstrucbility holds (∞-ly many Ward ids suggested).

Integral kernels in gravity

◮

∆′(Γ) = 0 ⇔ |Γ| = 1 (35) Holds after taking derivatives for projective integral kernels

Integral kernels in gravity

◮

∆′(Γ) = 0 ⇔ |Γ| = 1 (35) Holds after taking derivatives for projective integral kernels

◮ Situation is dual between renormalizable theory and gravity:

• ne one-cocycle per loop number for the gauge boson

determines DSE in massless gauge theories

• ne one-cocycle per n-point one-loop graphs determines DSE

in gravity loop-leg duality

Integral kernels in gravity

◮

∆′(Γ) = 0 ⇔ |Γ| = 1 (35) Holds after taking derivatives for projective integral kernels

◮ Situation is dual between renormalizable theory and gravity:

• ne one-cocycle per loop number for the gauge boson

determines DSE in massless gauge theories

• ne one-cocycle per n-point one-loop graphs determines DSE

in gravity loop-leg duality

◮

X n+1 X n = X n X n−1 (36) core Hopf ideal = renormalization ideal

Integral kernels in gravity

◮

∆′(Γ) = 0 ⇔ |Γ| = 1 (35) Holds after taking derivatives for projective integral kernels

◮ Situation is dual between renormalizable theory and gravity:

• ne one-cocycle per loop number for the gauge boson

determines DSE in massless gauge theories

• ne one-cocycle per n-point one-loop graphs determines DSE

in gravity loop-leg duality

◮

X n+1 X n = X n X n−1 (36) core Hopf ideal = renormalization ideal

◮ same powercounting holds for field diffeomorphisms of free

theory same ideal I: Φ(I) = 0 (37) delivers the equivalence theorem

Conclusions

◮ Hopf algebras are the natural habitat of renormalization

Conclusions

◮ Hopf algebras are the natural habitat of renormalization ◮ Locality reflected in Hochschild cohomology

Conclusions

◮ Hopf algebras are the natural habitat of renormalization ◮ Locality reflected in Hochschild cohomology ◮ perturbative structures suggest non-perturbative approaches

Conclusions

◮ Hopf algebras are the natural habitat of renormalization ◮ Locality reflected in Hochschild cohomology ◮ perturbative structures suggest non-perturbative approaches ◮ reflected nicely in the periods and special functions known by

practitioners

Conclusions

◮ Hopf algebras are the natural habitat of renormalization ◮ Locality reflected in Hochschild cohomology ◮ perturbative structures suggest non-perturbative approaches ◮ reflected nicely in the periods and special functions known by

practitioners

◮ Unitarity, internal symmetry, gravity, multi-leg-recursions vs

co-ideals...

Conclusions

◮ Hopf algebras are the natural habitat of renormalization ◮ Locality reflected in Hochschild cohomology ◮ perturbative structures suggest non-perturbative approaches ◮ reflected nicely in the periods and special functions known by

practitioners

◮ Unitarity, internal symmetry, gravity, multi-leg-recursions vs

co-ideals...

◮ Don’t loose trust in local point-particle quantum fields!