slides (pdf, 0.36 MB) - Frontiers of Fundamental Physics (FFP14)

Gravitational Scattering via Twistor Theory
Tim Adamo
DAMTP, University of Cambridge
Frontiers of Fundamental Physics 14
14 July 2014
Work with L. Mason
T Adamo (FFP14)
[arXiv:1307.5043, 1308.2820]
Twistor Gravity Amplitudes
14 July 2014
1 / 15
Motivation
Why gravity scattering amplitudes?
Provide important constraints on any theory of quantum gravity
Theoretical ‘data’
May point to novel formulations of underlying theory or new ways to
compute observables
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
2 / 15
Motivation
Why gravity scattering amplitudes?
Provide important constraints on any theory of quantum gravity
Theoretical ‘data’
May point to novel formulations of underlying theory or new ways to
compute observables
c.f., ongoing progress in (planar) gauge theory
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
2 / 15
Today: tree-level (semi-classical) scattering amplitudes.
Defined by the classical action of a theory:
Definition (Tree-level amplitudes)
Given an action functional S[φ], non-linear solution to the FEs
(‘background’) φcl , and n solutions {φi } to the linearized FEs (‘scattering
states’), then the tree-level scattering amplitude for the {φi } on φcl is:
P
n S φcl −
∂
φ
i
i
i
M0n (φ1 , . . . , φn ) =
∂1 · · · ∂n
1 =···n =0
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
3 / 15
Today: tree-level (semi-classical) scattering amplitudes.
Defined by the classical action of a theory:
Definition (Tree-level amplitudes)
Given an action functional S[φ], non-linear solution to the FEs
(‘background’) φcl , and n solutions {φi } to the linearized FEs (‘scattering
states’), then the tree-level scattering amplitude for the {φi } on φcl is:
P
n S φcl −
∂
φ
i
i
i
M0n (φ1 , . . . , φn ) =
∂1 · · · ∂n
1 =···n =0
Usually compute by summing Feynman diagrams...hard!
Resulting formulae (drastically) simpler than expected! [DeWitt,
Berends-Giele-Kuijf, Mason-Skinner, Nguyen-Spradlin-Volovich-Wen, Hodges, Cachazo-Skinner,
Cachazo-He-Yuan, ...]
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
3 / 15
Today: tree-level (semi-classical) scattering amplitudes.
Defined by the classical action of a theory:
Definition (Tree-level amplitudes)
Given an action functional S[φ], non-linear solution to the FEs
(‘background’) φcl , and n solutions {φi } to the linearized FEs (‘scattering
states’), then the tree-level scattering amplitude for the {φi } on φcl is:
P
n S φcl −
∂
φ
i
i
i
M0n (φ1 , . . . , φn ) =
∂1 · · · ∂n
1 =···n =0
Usually compute by summing Feynman diagrams...hard!
Resulting formulae (drastically) simpler than expected! [DeWitt,
Berends-Giele-Kuijf, Mason-Skinner, Nguyen-Spradlin-Volovich-Wen, Hodges, Cachazo-Skinner,
Cachazo-He-Yuan, ...]
Many of these simplifications are related to expressing amplitudes in
twistor theory.
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
3 / 15
Questions to Answer:
Is there some classical action principle giving rise to these
simplifications?
Can we learn anything about the associated twistor geometry?
Are there new expressions for ‘amplitudes’ in backgrounds that aren’t
asymptotically flat (e.g., de Sitter space)?
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
4 / 15
Twistor Toolbox
Basic idea of twistor theory:
Physical info on M
⇔
Geometric data on PT
PT a (deformation of a) 3-dimensional complex projective manifold
Points x ∈ M ↔ holomorphic rational curves X ⊂ PT
x, y ∈ M null separated iff X , Y ⊂ PT intersect.
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
5 / 15
Twistor Toolbox
Basic idea of twistor theory:
Physical info on M
⇔
Geometric data on PT
PT a (deformation of a) 3-dimensional complex projective manifold
Points x ∈ M ↔ holomorphic rational curves X ⊂ PT
x, y ∈ M null separated iff X , Y ⊂ PT intersect.
Conformal structure on M ↔ C-structure on PT .
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
5 / 15
Does a twistor space exist for every M?
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
6 / 15
Does a twistor space exist for every M?
No!
Basic result is
[Penrose, Ward]
:
Theorem (Non-linear Graviton)
∃ a 1:1 correspondence between:
M with self-dual holomorphic conformal structure, and
PT with integrable almost complex structure.
Thing to remember: ∂¯2 = 0 on PT ⇔ ΨABCD = 0 on M.
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
6 / 15
Twistor theory good for describing self-duality
T Adamo (FFP14)
Twistor Gravity Amplitudes
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7 / 15
Twistor theory good for describing self-duality
Is there a way to formulate GR as an expansion around the SD sector?
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
7 / 15
Conformal Gravity
Start with an un-physical theory:
Z
1
S[g ] = 2
dµ C µνρσ Cµνρσ
ε M
Z
2
= 2
dµ ΨABCD ΨABCD + top. terms
ε M
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
8 / 15
Conformal Gravity
Start with an un-physical theory:
Z
1
S[g ] = 2
dµ C µνρσ Cµνρσ
ε M
Z
2
= 2
dµ ΨABCD ΨABCD + top. terms
ε M
Conformally invariant, with 4th -order equations of motion (non-unitary):
B
AB
A0 B 0
A0 B 0 e
(∇A
A0 ∇B 0 + ΦA0 B 0 )ΨABCD = (∇A ∇B + ΦAB )ΨA0 B 0 C 0 D 0 = 0
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
8 / 15
Conformal Gravity
Start with an un-physical theory:
Z
1
S[g ] = 2
dµ C µνρσ Cµνρσ
ε M
Z
2
= 2
dµ ΨABCD ΨABCD + top. terms
ε M
Conformally invariant, with 4th -order equations of motion (non-unitary):
B
AB
A0 B 0
A0 B 0 e
(∇A
A0 ∇B 0 + ΦA0 B 0 )ΨABCD = (∇A ∇B + ΦAB )ΨA0 B 0 C 0 D 0 = 0
0
AA Ψ
Note: Einstein (ΦAB
ABCD ) and SD/ASD are subsectors of
A0 B 0 = 0 = ∇
solutions.
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
8 / 15
Upshot: Perturbative expansion around SD sector.
[Berkovits-Witten]
Introduce Lagrange multiplier GABCD :
Z
Z
ε2
ABCD
S[g ] → S[g , G ] =
dµ G
ΨABCD −
dµ G ABCD GABCD
2 M
M
Field Equations:
ΨABCD = ε2 GABCD ,
T Adamo (FFP14)
B
AB
(∇A
A0 ∇B 0 + ΦA0 B 0 )GABCD = 0
Twistor Gravity Amplitudes
14 July 2014
9 / 15
Upshot: Perturbative expansion around SD sector.
[Berkovits-Witten]
Introduce Lagrange multiplier GABCD :
Z
Z
ε2
ABCD
S[g ] → S[g , G ] =
dµ G
ΨABCD −
dµ G ABCD GABCD
2 M
M
Field Equations:
ΨABCD = ε2 GABCD ,
B
AB
(∇A
A0 ∇B 0 + ΦA0 B 0 )GABCD = 0
ε2 an expansion parameter around the SD sector.
But we can formulate this in twistor space!
T Adamo (FFP14)
[Mason]
Twistor Gravity Amplitudes
14 July 2014
9 / 15
Conformal Gravity in Twistor Space
Translation:
ΨABCD ↔ N[J] ∈ Ω0,2 (PT , TPT ),
T Adamo (FFP14)
GABCD ↔ b ∈ Ω1,1 (PT , O(−4))
Twistor Gravity Amplitudes
14 July 2014
10 / 15
Conformal Gravity in Twistor Space
Translation:
ΨABCD ↔ N[J] ∈ Ω0,2 (PT , TPT ),
GABCD ↔ b ∈ Ω1,1 (PT , O(−4))
Action functional:
Z
Z
ε2
3
dµ ∧ b1 ∧ b2 (σ1 σ2 )4
S[b, J] =
D Z ∧ Nyb −
2
PT
PT ×M PT
Using standard results
[Penrose, Atiyah-Hitchin-Singer]
N[J] = ΨABCD σ A ΣBC
∂
,
∂σD
:
Z
GABCD =
σA σB σC σD b|X
X
Implies FEs on twistor space equivalent to those on space-time.
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
10 / 15
Why do we care?
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
11 / 15
Why do we care?
Theorem (Anderson, Maldacena)
For M asymptotically de Sitter,
S CG [M] = −
2 Λ2
Λ κ2 EH
V
(M)
=
−
S [M] ,
ren
3ε2
3ε2 ren
and asymptotic Einstein states can be singled out in the conformal theory.
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
11 / 15
Why do we care?
Theorem (Anderson, Maldacena)
For M asymptotically de Sitter,
S CG [M] = −
2 Λ2
Λ κ2 EH
V
(M)
=
−
S [M] ,
ren
3ε2
3ε2 ren
and asymptotic Einstein states can be singled out in the conformal theory.
⇒ for tree-level amplitudes,
MEin =
T Adamo (FFP14)
1 CG
M |Ein
Λ
Twistor Gravity Amplitudes
14 July 2014
11 / 15
Einstein Gravity in Twistor Space
Einstein degrees of freedom ⇒ break conformal invariance
Infinity twistor:
I
αβ
=
ΛAB
0
0
0
A B
0
,
Iαβ =
AB
0
0
ΛA0 B 0
Obey I αβ Iβγ = Λδγα , induce geometric structures:
τ = Iαβ Z α dZ β ∈ Ω1,0 (PT , O(2)),
T Adamo (FFP14)
Twistor Gravity Amplitudes
{·, ·} = I αβ ∂α ∂β
14 July 2014
12 / 15
Einstein Gravity in Twistor Space
Einstein degrees of freedom ⇒ break conformal invariance
Infinity twistor:
I
αβ
=
ΛAB
0
0
0
A B
0
,
Iαβ =
AB
0
0
ΛA0 B 0
Obey I αβ Iβγ = Λδγα , induce geometric structures:
τ = Iαβ Z α dZ β ∈ Ω1,0 (PT , O(2)),
{·, ·} = I αβ ∂α ∂β
(Like fixing the conformal factor for metric in Klein representation:
ds 2 =
αβγδ dX αβ dX γδ
)
(Iαβ X αβ )2
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
12 / 15
Write complex structure on PT as finite deformation of ‘flat’ structure.
Compatability with I αβ , Iαβ ⇒
∂¯ = ∂¯0 + I αβ ∂α h ∂β ,
b → Iαβ Z α dZ β h˜ = τ h˜
T Adamo (FFP14)
h ∈ Ω0,1 (PT , O(2))
h˜ ∈ Ω0,1 (PT , O(−6))
Twistor Gravity Amplitudes
14 July 2014
13 / 15
Write complex structure on PT as finite deformation of ‘flat’ structure.
Compatability with I αβ , Iαβ ⇒
∂¯ = ∂¯0 + I αβ ∂α h ∂β ,
b → Iαβ Z α dZ β h˜ = τ h˜
h ∈ Ω0,1 (PT , O(2))
h˜ ∈ Ω0,1 (PT , O(−6))
Action becomes:
˜ h] = Λ
S[b, J] → S[h,
Z
PT
1
D Z ∧ h˜ ∧ ∂¯0 h + {h, h}
2
Z
2
ε
−
dµ τ1 ∧ τ2 ∧ h˜1 ∧ h˜2 (σ1 σ2 )4
2
3
PT ×M PT
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
13 / 15
˜ h] should compute tree-level Einstein gravity
By earlier Theorem, Λ−1 S[h,
amplitudes. Is this actually true?
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
14 / 15
˜ h] should compute tree-level Einstein gravity
By earlier Theorem, Λ−1 S[h,
amplitudes. Is this actually true?
Yes! Lots of technical detail, but upshots are
[Adamo-Mason]
:
Second term is generating functional for MHV amplitudes
Flat space limit = Hodges formulae
New Λ 6= 0 formulae
Apparent MHV formalism induced on twistor space
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
14 / 15
Further Directions
Twistor action for Einstein gravity itself?
More general (i.e., Nk MHV) amplitudes?
What do the Λ 6= 0 formulae mean?
Momentum space prescription for MHV formalism?
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
15 / 15
Further Directions
Twistor action for Einstein gravity itself?
More general (i.e., Nk MHV) amplitudes?
What do the Λ 6= 0 formulae mean?
Momentum space prescription for MHV formalism?
Thanks!
T Adamo (FFP14)
Twistor Gravity Amplitudes
14 July 2014
15 / 15

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