Post on 27-Jun-2020
transcript
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, ...]
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 2 / 35
What are these simple amplitude formulae telling us?
There should be some simpler formulation of GR
as a non-linear theory of gravity!
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 3 / 35
What are these simple amplitude formulae telling us?
There should be some simpler formulation of GR
as a non-linear theory of gravity!
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 3 / 35
An analogy...
The Veneziano amplitude:
Remarkably compact
Lots of nice properties
Can be generalized to higher-points
But the real upshot is string theory!
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 4 / 35
An analogy...
The Veneziano amplitude:
Remarkably compact
Lots of nice properties
Can be generalized to higher-points
But the real upshot is string theory!
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 4 / 35
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!
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 5 / 35
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!
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 5 / 35
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!
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 5 / 35
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]
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 6 / 35
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]
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 6 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 7 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 7 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 8 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 9 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 9 / 35
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...
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 10 / 35
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...
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 10 / 35
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 .
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 11 / 35
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 .
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 11 / 35
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.
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 12 / 35
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.
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 12 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 13 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 13 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 14 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 14 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 14 / 35
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))
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 15 / 35
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?
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 16 / 35
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?
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 16 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 17 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 17 / 35
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?
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 18 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 19 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 19 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 19 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 20 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 20 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 20 / 35
BRST charge
Gauge these currents ⇒
Q =
∮c Tm+ : bc∂c : +
c
2H + γ G + γ G
Does this agree with what we’re expecting?
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 21 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 22 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 22 / 35
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?
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 23 / 35
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)
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 24 / 35
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)
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 24 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 25 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 25 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 25 / 35
To fix OPE with OV , take
G = G + ∂(ψµΓνµν
)G = G − gνσ∂
(ψµΓµνσ
)Great, but now G, G no longer worldsheet primaries.
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 26 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 27 / 35
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]
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 28 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 29 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 30 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 31 / 35
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
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 31 / 35
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...
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 32 / 35
The only anomaly cancellation conditions are:
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 .
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 33 / 35
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!
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 34 / 35
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!
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 34 / 35
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 )
T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 35 / 35