The Scattering Equations in Curved Spacepeople.maths.ox.ac.uk/lmason/NGSA14/Slides/Tim-Adamo.pdf ·...

The Scattering Equations in Curved Space

Tim AdamoDAMTP, University of Cambridge

New Geometric Structures in Scattering Amplitudes

22 September 2014

Work with E. Casali & D. Skinner [arXiv:1409.????]

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 1 / 35

Motivation

We’ve learned a lot about perturbative classical GR in recent years:

Simpler on-shell than Einstein-Hilbert action makes it seem

Increasingly simple/compact/general formulae for tree-level S-matrix[deWitt, Hodges, Cachazo-Geyer, Cachazo-Skinner, Cachazo-He-Yuan, ...]


What are these simple amplitude formulae telling us?

There should be some simpler formulation of GR

as a non-linear theory of gravity!


What are these simple amplitude formulae telling us?

There should be some simpler formulation of GR

as a non-linear theory of gravity!


An analogy...

The Veneziano amplitude:

Remarkably compact

Lots of nice properties

Can be generalized to higher-points

But the real upshot is string theory!


An analogy...

The Veneziano amplitude:

Remarkably compact

Lots of nice properties

Can be generalized to higher-points

But the real upshot is string theory!


We have a similar situation with gravity amplitudes:

Remarkably compact/general formulae, but where are they coming from?

Partial answer:

Worldsheet theories which produce these formulae [Skinner, Mason-Skinner,

Berkovits, Geyer-Lipstein-Mason]

Know about linearized Einstein equations around flat space

Give a formulation of perturbative gravity, linearized around flat space

We want to learn something about the non-linear theory!




Partial answer:









Partial answer:







Back to analogy...

In (closed) string theory, tree-level (sphere) amps:

Arise from the flat target sigma model

Give tree-level S-matrix of gravity in α′ → 0 limit [Scherk, Yoneya,

Scherk-Schwarz]

How to get non-linear field equations?

Formulate non-linear sigma model on curved target space

Demand worldsheet conformal invariance → compute β-functions

Conformal anomaly vanishes as α′ → 0 ⇔ non-linear field eqns.satisfied

[Callan-Martinec-Perry-Friedan, Banks-Nemeschansky-Sen]


Back to analogy...

In (closed) string theory, tree-level (sphere) amps:

Arise from the flat target sigma model

Give tree-level S-matrix of gravity in α′ → 0 limit [Scherk, Yoneya,

Scherk-Schwarz]

How to get non-linear field equations?

Formulate non-linear sigma model on curved target space

Demand worldsheet conformal invariance → compute β-functions

Conformal anomaly vanishes as α′ → 0 ⇔ non-linear field eqns.satisfied

[Callan-Martinec-Perry-Friedan, Banks-Nemeschansky-Sen]


Since non-linear sigma model is an interacting CFT on the worldsheet,

Must work perturbatively in α′

Higher powers of α′ ↔ higher-curvature corrections to field equations[Gross-Witten, Grisaru-van de Ven-Zanon]

Evident in S-matrix and β-function approaches

But we have a worldsheet theory giving the tree-level S-matrix EXACTLY

No higher-derivative corrections


Since non-linear sigma model is an interacting CFT on the worldsheet,

Must work perturbatively in α′

Higher powers of α′ ↔ higher-curvature corrections to field equations[Gross-Witten, Grisaru-van de Ven-Zanon]

Evident in S-matrix and β-function approaches

But we have a worldsheet theory giving the tree-level S-matrix EXACTLY

No higher-derivative corrections


Basic idea

So we want to:

Formulate the worldsheet theory on a curved target space

Do it so that the theory is solveable (no backgroundfield/perturbative expansion required)

See non-linear field equations as some sort of anomaly cancellationcondition


Starting Point

One particular representation of the tree-level S-matrix [Cachazo-He-Yuan] :

Mn,0 =

∫1

vol SL(2,C)

|z1z2z3|dz1 dz2 dz3

n∏i=4

δ

∑j 6=i

ki · kjzi − zj

Pf ′(M)Pf ′(M)

zi ⊂ Σ ∼= CP1, ki null momenta,

M =

(A −CT

C B

), Pf ′(M) = (−1)i+j

√dzi dzj

zi − zjPf(M ij

ij ) ,

Aij = ki · kj√dzi dzj

zi − zj, Bij = εi · εj

√dzi dzj

zi − zj, Cij = εi · kj

√dzi dzj

zi − zj

Aii = Bii = 0, Cii = −dzi∑

j 6=iCij√dzi dzj


Starting Point

One particular representation of the tree-level S-matrix [Cachazo-He-Yuan] :

Mn,0 =

∫1

vol SL(2,C)

|z1z2z3|dz1 dz2 dz3

n∏i=4

δ

∑j 6=i

ki · kjzi − zj

Pf ′(M)Pf ′(M)

zi ⊂ Σ ∼= CP1, ki null momenta,

M =

(A −CT

C B

), Pf ′(M) = (−1)i+j

√dzi dzj

zi − zjPf(M ij

ij ) ,

Aij = ki · kj√

dzi dzj

zi − zj, Bij = εi · εj

√dzi dzj

zi − zj, Cij = εi · kj

√dzi dzj

zi − zj

Aii = Bii = 0, Cii = −dzi∑

j 6=iCij√dzi dzj


This representation of Mn,0 manifests (gauge)2=(gravity), and related toBCJ duality

All integrals over M0,n fixed by delta functions, imposing the scatteringequations [Fairlie-Roberts, Gross-Mende, Witten] :

i ∈ 4, . . . , n ,∑j 6=i

ki · kjzi − zj

= 0

So the locations zi ⊂ Σ are fixed by the scattering equations.

Structure of Mn,0 hints at natural origin...


This representation of Mn,0 manifests (gauge)2=(gravity), and related toBCJ duality

All integrals over M0,n fixed by delta functions, imposing the scatteringequations [Fairlie-Roberts, Gross-Mende, Witten] :

i ∈ 4, . . . , n ,∑j 6=i

ki · kjzi − zj

= 0

So the locations zi ⊂ Σ are fixed by the scattering equations.

Structure of Mn,0 hints at natural origin...


Worldsheet theory, I

Consider worldsheet action [Mason-Skinner] :

S =1

2π

∫Σ

Pµ ∂Xµ + Ψµ∂Ψµ − χPµΨµ + Ψµ∂Ψµ − χPµΨµ − e

2P2

Pµ ∈ Ω0(Σ,K ) and Ψµ, Ψµ ∈ ΠΩ0(Σ,K 1/2)

gauge-fixing−−−−−−−−−→ 1

2π

∫Σ

Pµ ∂Xµ + Ψµ∂Ψµ + Ψµ∂Ψµ + Sgh

where fixing e = 0 enforces the constraint

P2 = 0 .


Worldsheet theory, I

Consider worldsheet action [Mason-Skinner] :

S =1

2π

∫Σ

Pµ ∂Xµ + Ψµ∂Ψµ − χPµΨµ + Ψµ∂Ψµ − χPµΨµ − e

2P2

Pµ ∈ Ω0(Σ,K ) and Ψµ, Ψµ ∈ ΠΩ0(Σ,K 1/2)

gauge-fixing−−−−−−−−−→ 1

2π

∫Σ

Pµ ∂Xµ + Ψµ∂Ψµ + Ψµ∂Ψµ + Sgh

where fixing e = 0 enforces the constraint

P2 = 0 .


Scattering equations from the worldsheet

In the presence of vertex operator insertions, Pµ becomes meromorphic:

∂Pµ(z) = 2πi dz ∧ dzn∑

i=1

ki µ δ2(z − zi ).

Likewise, quadratic differential P2 becomes meromorphic, with residues:

Resz=zi P2(z) = ki · P(zi ) = dzi

∑j 6=i

ki · kjzi − zj

Setting Resz=zi P2(z) = 0 for i = 4, . . . , n is sufficient to set P2(z) = 0

globally on Σ.


Scattering equations from the worldsheet

In the presence of vertex operator insertions, Pµ becomes meromorphic:

∂Pµ(z) = 2πi dz ∧ dzn∑

i=1

ki µ δ2(z − zi ).

Likewise, quadratic differential P2 becomes meromorphic, with residues:

Resz=zi P2(z) = ki · P(zi ) = dzi

∑j 6=i

ki · kjzi − zj

Setting Resz=zi P2(z) = 0 for i = 4, . . . , n is sufficient to set P2(z) = 0

globally on Σ.


But these are the scattering equations!

P2(z) = 0 ↔ Resz=zi P2(z) = 0 =

∑j 6=i

ki · kjzi − zj

i ∈ 4, . . . , n

The condition P2(z) = 0 globally on Σ defines the scattering equations forany genus worldsheet [TA-Casali-Skinner]

g = 0 (n − 3) × Resz=zi P2(z) = 0

g = 1 (n − 1) × Resz=zi P2(z) = 0 , P2(z1) = 0

g ≥ 2 n × Resz=zi P2(z) = 0 , (3g − 3)× P2(zr ) = 0


But these are the scattering equations!

P2(z) = 0 ↔ Resz=zi P2(z) = 0 =

∑j 6=i

ki · kjzi − zj

i ∈ 4, . . . , n

The condition P2(z) = 0 globally on Σ defines the scattering equations forany genus worldsheet [TA-Casali-Skinner]

g = 0 (n − 3) × Resz=zi P2(z) = 0

g = 1 (n − 1) × Resz=zi P2(z) = 0 , P2(z1) = 0

g ≥ 2 n × Resz=zi P2(z) = 0 , (3g − 3)× P2(zr ) = 0


This theory has a BRST-charge

Q =

∮c Tm+ : bc∂c : +

c

2P2 + γPµΨµ + γPµΨµ ,

which is nilpotent Q2 = 0 provided the space-time has d = 10.

Fixed and integrated vertex operators:

ccδ(γ)δ(γ) U ,

∫Σδ(ReszP2

)V

for U ∈ Ω0(Σ,K ), V ∈ Ω0(Σ,K 2).

Anomalies in BRST-closure ↔ double contractions between currents

P2 , PµΨµ , PµΨµ ,

and U ,V .

For momentum eigenstates, this constrains:

QU = QV = 0 ⇔ k2 = 0 = ε · k = ε · k

i.e., obey the linearized Einstein equations around flat space



Q =

∮c Tm+ : bc∂c : +

c




ccδ(γ)δ(γ) U ,

∫Σδ(ReszP2

)V

for U ∈ Ω0(Σ,K ), V ∈ Ω0(Σ,K 2).



and U ,V .


QU = QV = 0 ⇔ k2 = 0 = ε · k = ε · k




Q =

∮c Tm+ : bc∂c : +

c




ccδ(γ)δ(γ) U ,

∫Σδ(ReszP2

)V

for U ∈ Ω0(Σ,K ), V ∈ Ω0(Σ,K 2).



and U ,V .


QU = QV = 0 ⇔ k2 = 0 = ε · k = ε · k



The g = 0 correlators in this model reproduce the CHY formulae[Mason-Skinner]

Other vertex operators for dilatons, B-fields, gravitini, R-R form fields

Explicit amplitude candidates at higher genus passing non-trivial checks[TA-Casali-Skinner] :

Modular invariance

Factorization onto rational functions

Explicit loop momenta (zero modes of Pµ(z))


Upshot

So, we have a worldsheet theory that:

Knows about the entire tree-level S-matrix of type II SUGRA ind = 10 exactly

Gives scattering equations in the form P2 = 0

Enforces the linearized Einstein equations about flat space on vertexoperators via BRST-closure

Question: can this theory be extended to an arbitrary curved manifold,with the non-linear Einstein equations emerging as an anomalycancellation condition?


Upshot

So, we have a worldsheet theory that:

Knows about the entire tree-level S-matrix of type II SUGRA ind = 10 exactly

Gives scattering equations in the form P2 = 0

Enforces the linearized Einstein equations about flat space on vertexoperators via BRST-closure

Question: can this theory be extended to an arbitrary curved manifold,with the non-linear Einstein equations emerging as an anomalycancellation condition?


Once more, analogy with strings:

String theory

Tree-level S-matrixα′→0−−−→ supergravity

linearized EFEs ↔ anomalous conformal weights

Worldsheet theory

Exact supergravity tree-level S-matrix

linearized EFEs ↔ anomalies w/ currents

⇒ Look for solvable worldsheet theory with curved target space


Once more, analogy with strings:

String theory

Tree-level S-matrixα′→0−−−→ supergravity

linearized EFEs ↔ anomalous conformal weights

Worldsheet theory

Exact supergravity tree-level S-matrix

linearized EFEs ↔ anomalies w/ currents

⇒ Look for solvable worldsheet theory with curved target space


Worldsheet theory, II

Naive generalization to curved target, M:

S =1

2π

∫Σ

Pµ∂Xµ + ψµDψµ + Sgh

=1

2π

∫Σ

Pµ∂Xµ + ψµ(δµν ∂ + Γµνρ∂X ρ

)ψν + Sgh

with complex fermion ψµ = Ψµ + iΨµ to make life easier.

Why this way?


Field redefinition

Make the redefinitionΠµ ≡ Pµ + Γρµνψρψ

ν

so action becomes:

S =1

2π

∫Σ

Πµ ∂Xµ + ψµ ∂ψµ .

Free action and OPEs:

Xµ(z) Πν(w) ∼ δµνz − w

, ψµ(z) ψν(w) ∼ δµνz − w

.

Covariance non-manifest, due to transformation:

Πµ =∂X ν

∂XµΠν +

∂2Xκ

∂Xµ∂X ν

∂X ν

∂X σψκψ

σ


Field redefinition


ν

so action becomes:

S =1

2π

∫Σ





.


Πµ =∂X ν

∂XµΠν +

∂2Xκ

∂Xµ∂X ν

∂X ν

∂X σψκψ

σ


Field redefinition


ν

so action becomes:

S =1

2π

∫Σ





.


Πµ =∂X ν

∂XµΠν +

∂2Xκ

∂Xµ∂X ν

∂X ν

∂X σψκψ

σ


Classical currents

Action has fermionic symmetries generated by:

G = ψµΠµ , G = gµνψµ(Πν − Γρνσψρψ

σ).

Classically, obey the algebra G,G = G, G = 0 , G , G = Hwith

H = gµν(Πµ − Γρµσψρψ

σ) (

Πν − Γκνλψκψλ)− 1

2ψµψνψρψσRρσ

µν

These are analogues of the flat space currents:

ψµPµ → G , gµνψµPν → G , P2 → H


Classical currents



σ).



σ) (



µν




Classical currents



σ).



σ) (



µν




BRST charge

Gauge these currents ⇒

Q =

∮c Tm+ : bc∂c : +

c

2H + γ G + γ G

Does this agree with what we’re expecting?


At the naive level, yes:

Free OPEs

Only conformal anomaly condition remains d = 10

So where are potential anomalies?

BRST-charge is nilpotent iff

G(z)G(w) ∼ 0 ∼ G(z) G(w) , G(z) G(w) ∼ H

z − w.

But we only know this classically; need to extend to quantum level


At the naive level, yes:

Free OPEs

Only conformal anomaly condition remains d = 10

So where are potential anomalies?

BRST-charge is nilpotent iff

G(z)G(w) ∼ 0 ∼ G(z) G(w) , G(z) G(w) ∼ H

z − w.

But we only know this classically; need to extend to quantum level


Quantum issues

Before we can look at these anomalies, we still have lots to worry about atthe quantum level:

Diffeomorphism covariance of the fields

Diffeomorphism covariance of the currents

In other words, do the currents even make sense quantum mechanically?


Infinitesimal diffeomorphism on M generated by vector field V = V µ∂µ.

At quantum level, look for an operator OV obeying:

OV (z) OW (w) ∼O[V ,W ](w)

z − w

and acting on fields as:

OV (z) Xµ(w) ∼ V µ

z − w, OV (z)ψµ(w) ∼ ∂νV µ ψν

z − w,

OV (z) ψµ(w) ∼ −∂µV ν ψνz − w

,

Ov (z) Πµ(w) ∼ −∂µV ν Πν + ∂µ∂νV ρ ψρψν

z − w

Implemented by:OV = −

(V µΠµ + ∂νV µψµψ

ν)


Infinitesimal diffeomorphism on M generated by vector field V = V µ∂µ.

At quantum level, look for an operator OV obeying:

OV (z) OW (w) ∼O[V ,W ](w)

z − w

and acting on fields as:

OV (z) Xµ(w) ∼ V µ

z − w, OV (z)ψµ(w) ∼ ∂νV µ ψν

z − w,

OV (z) ψµ(w) ∼ −∂µV ν ψνz − w

,

Ov (z) Πµ(w) ∼ −∂µV ν Πν + ∂µ∂νV ρ ψρψν

z − w

Implemented by:OV = −

(V µΠµ + ∂νV µψµψ

ν)


Quantum currents

How does OV act on composite operators like G, G?

On any J(F(X )), infinitesimal diffeos should act geometrically:

OV (z) J(F(X ))(w) ∼ · · · +J(LVF)

z − w+ · · ·

But our currents G, G don’t obey this. (double contractions!)

Solution: add quantum corrections


Quantum currents



OV (z) J(F(X ))(w) ∼ · · · +J(LVF)

z − w+ · · ·




Quantum currents



OV (z) J(F(X ))(w) ∼ · · · +J(LVF)

z − w+ · · ·




To fix OPE with OV , take

G = G + ∂(ψµΓνµν

)G = G − gνσ∂

(ψµΓµνσ

)Great, but now G, G no longer worldsheet primaries.


Resolution ⇒ quantum correction to stress tensor:

T = −Πµ ∂Xµ − 1

2ψµ ∂ψ

µ − 1

2ψµ ∂ψµ −

1

2∂2 log(

√g)

Note: doesn’t alter central charge!

Action now invariant under quantum charges, and free OPEs unaffected


Some observations:

Similar methods for removing anomalous OPEs in study of curvedβγ-systems [Nekrasov, Witten]

See also math literature, sheaves of chiral algebras, chiral de Rhamcomplex [Malikov-Schechtman-Vaintrob, Gorbounov-Malikov-Schechtman, Ben-Zvi-Heluani-Szczesny,

Frenkel-Losev-Nekrasov, Ekstrand-Heluani-Kallen-Zabzine]

Related constructions in 1st-order formalism for string theory[Schwarz-Tseytlin, Losev-Marshakov-Zeitlin]


Quantum model

We now have a well-defined worldsheet theory, and a BRST operator builtfrom ghosts and the currents:

Quantum Currents

G = ψµΠµ + ∂(ψµΓνµν

)G = gµνψµ

(Πν − Γρνσψρψ

σ)− gνσ∂

(ψµΓµνσ

)T = −Πµ ∂Xµ − 1

2ψµ ∂ψ

µ − 1

2ψµ ∂ψµ −

1

2∂2 log(

√g)

Only potential anomalies to Q2 = 0 from algebra of currents

G(z)G(w) ∼ 0 ∼ G(z) G(w) , G(z) G(w) ∼ Hz − w


Anomaly calculation

Do the OPEs (lots of fun!) and find:

G(z) G(w) ∼ 0 ,

G(z) G(w) ∼ 1

2

ψµψνψρψσ

z − wR µνρσ +

∂(ψµψνRµν

)z − w

+ 2ψµψν∂X σ

z − w

[ΓναβRβαµ

σ + Γασβ(Rµβνα + Rνβµ

α)]

G(z) G(w) ∼ 2

(z − w)3R + 2

(Γµσν∂X σ + ψµψν)

(z − w)2Rµν +

Hz − w


The only anomaly cancellation conditions are:

Rµν = 0 = R ,

the vacuum Einstein equations!

Note:

Free OPEs, so anomalies are exact

No background field expansion

No perturbative (α′) expansion on worldsheet



Rµν = 0 = R ,

the vacuum Einstein equations!

Note:

Free OPEs, so anomalies are exact

No background field expansion

No perturbative (α′) expansion on worldsheet


Other fields

Can also add dilaton and B-field:

G = ψµΠµ +1

6Hµνρψ

µψνψρ + ∂(ψµΓνµν

)− 2∂ (ψµ∂µΦ)

G = gµνψµ(Πν − Γρνσψρψ

σ)

+1

6Hµνρψµψνψρ

−gνσ∂(ψµΓµνσ

)− 2∂

(ψµgµν∂νΦ

)T = −Πµ ∂Xµ − 1

2ψµ ∂ψ

µ − 1

2ψµ ∂ψµ −

1

2∂2 log

(√ge−2Φ

)and do the same sort of calculations...



Field Equations

Rµν −1

4Hµρσ H ρσ

ν + 2∇µ∇νΦ = 0 ,

∇ρHρµν − 2Hρ

µν∇ρΦ = 0 ,

R + 4∇µ∇µΦ− 4∇µΦ∇µΦ− H2

12= 0 .


Back to scattering equations

In flat space, the scattering equations were P2 = 0.

On M, they becomeG(z) G(w) ∼ 0.

This has a quasi-classical piece, H = 0, and quantum pieces.

The quantum pieces of the scattering equations in curved space are thefield equations!


Back to scattering equations

In flat space, the scattering equations were P2 = 0.

On M, they becomeG(z) G(w) ∼ 0.

This has a quasi-classical piece, H = 0, and quantum pieces.

The quantum pieces of the scattering equations in curved space are thefield equations!


Summary

Worldsheet CFT which is

Solvable (basically free)

Background independent

Encodes scattering equations and field equations

Reduces to flat space model (linearize H around flat space to get V )


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