The Scattering Equations in Curved Space Tim Adamo DAMTP, - - PowerPoint PPT Presentation

The Scattering Equations in Curved Space

Tim Adamo DAMTP, 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?

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

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?

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

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] 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

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 background field/perturbative expansion required) See non-linear field equations as some sort of anomaly cancellation condition

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=i

ki · kj zi − zj   Pf′(M) Pf′( M)

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=i

ki · kj zi − zj   Pf′(M) Pf′( M) {zi} ⊂ Σ ∼ = CP1, {ki} null momenta, M = A −C T C B

• ,

Pf′(M) = (−1)i+j

• dzi dzj

zi − zj Pf(Mij

ij ) ,

Aij = ki · kj

• dzi dzj

zi − zj , Bij = ǫi · ǫj

• dzi dzj

zi − zj , Cij = ǫi · kj

• dzi dzj

zi − zj Aii = Bii = 0, Cii = −dzi

• j=i

Cij

√

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 to BCJ duality All integrals over M0,n fixed by delta functions, imposing the scattering equations [Fairlie-Roberts, Gross-Mende, Witten] : i ∈ {4, . . . , n} ,

• j=i

ki · kj zi − zj = 0 So the locations {zi} ⊂ Σ are fixed by the scattering equations.

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 10 / 35

This representation of Mn,0 manifests (gauge)2=(gravity), and related to BCJ duality All integrals over M0,n fixed by delta functions, imposing the scattering equations [Fairlie-Roberts, Gross-Mende, Witten] : i ∈ {4, . . . , n} ,

• j=i

ki · kj zi − 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 ¯ ∂X µ + Ψµ ¯ ∂Ψµ − χPµΨµ + Ψµ ¯ ∂ Ψµ − χPµ Ψµ − e 2P2 Pµ ∈ Ω0(Σ, K) and Ψµ, Ψµ ∈ ΠΩ0(Σ, K 1/2)

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 ¯ ∂X µ + Ψµ ¯ ∂Ψµ − χPµΨµ + Ψµ ¯ ∂ Ψµ − χPµ Ψµ − e 2P2 Pµ ∈ Ω0(Σ, K) and Ψµ, Ψµ ∈ ΠΩ0(Σ, K 1/2) gauge-fixing − − − − − − − − − → 1 2π

• Σ

Pµ ¯ ∂X µ + 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 ∧ d¯ z

n

• i=1

ki µ δ2(z − zi).

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 ∧ d¯ z

n

• i=1

ki µ δ2(z − zi). Likewise, quadratic differential P2 becomes meromorphic, with residues: Resz=ziP2(z) = ki · P(zi) = dzi

• j=i

ki · kj zi − zj Setting Resz=ziP2(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=ziP2(z) = 0 =

• j=i

ki · kj zi − zj i ∈ {4, . . . , n}

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 13 / 35

But these are the scattering equations! P2(z) = 0 ↔ Resz=ziP2(z) = 0 =

• j=i

ki · kj zi − zj i ∈ {4, . . . , n} The condition P2(z) = 0 globally on Σ defines the scattering equations for any genus worldsheet [TA-Casali-Skinner] g = 0 (n − 3) × Resz=ziP2(z) = 0 g = 1 (n − 1) × Resz=ziP2(z) = 0 , P2(z1) = 0 g ≥ 2 n × Resz=ziP2(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 T m+ : bc∂c : +˜

c 2P2 + γPµΨµ + ˜ γPµ Ψµ , which is nilpotent Q2 = 0 provided the space-time has d = 10.

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 14 / 35

This theory has a BRST-charge Q =

• c T m+ : bc∂c : +˜

c 2P2 + γPµΨµ + ˜ γPµ Ψµ , which is nilpotent Q2 = 0 provided the space-time has d = 10. Fixed and integrated vertex operators: c˜ cδ(γ)δ(˜ γ) U ,

• Σ

¯ δ

• ReszP2

V for U ∈ Ω0(Σ, K), V ∈ Ω0(Σ, K 2).

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 14 / 35

This theory has a BRST-charge Q =

• c T m+ : bc∂c : +˜

c 2P2 + γPµΨµ + ˜ γPµ Ψµ , which is nilpotent Q2 = 0 provided the space-time has d = 10. Fixed and integrated vertex operators: c˜ cδ(γ)δ(˜ γ) U ,

• Σ

¯ δ

• ReszP2

V for U ∈ Ω0(Σ, K), V ∈ Ω0(Σ, K 2). Anomalies in BRST-closure ↔ double contractions between currents P2 , PµΨµ , Pµ Ψµ , and U , V . For momentum eigenstates, this constrains: QU = QV = 0 ⇔ k2 = 0 = ǫ · k = ˜ ǫ · 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 in d = 10 exactly Gives scattering equations in the form P2 = 0 Enforces the linearized Einstein equations about flat space on vertex

• perators via BRST-closure

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 in d = 10 exactly Gives scattering equations in the form P2 = 0 Enforces the linearized Einstein equations about flat space on vertex

• perators via BRST-closure

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

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 16 / 35

Once more, analogy with strings:

String theory

Tree-level S-matrix α′→0 − − − → supergravity linearized EFEs ↔ anomalous conformal weights

Worldsheet theory

Exact supergravity tree-level S-matrix linearized EFEs ↔ anomalies w/ currents

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 17 / 35

Once more, analogy with strings:

String theory

Tree-level S-matrix α′→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 ¯ ∂X µ + ¯ ψµ ¯ Dψµ + Sgh = 1 2π

• Σ

Pµ ¯ ∂X µ + ¯ ψµ

• δµ

ν ¯

∂ + Γµ

νρ ¯

∂X ρ ψν + Sgh with complex fermion ψµ = Ψµ + i Ψµ to make life easier. Why this way?

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 18 / 35

Field redefinition

Make the redefinition Πµ ≡ Pµ + Γρ

µν ¯

ψρψν so action becomes: S = 1 2π

• Σ

Πµ ¯ ∂X µ + ¯ ψµ ¯ ∂ψµ .

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 19 / 35

Field redefinition

Make the redefinition Πµ ≡ Pµ + Γρ

µν ¯

ψρψν so action becomes: S = 1 2π

• Σ

Πµ ¯ ∂X µ + ¯ ψµ ¯ ∂ψµ . Free action and OPEs: X µ(z) Πν(w) ∼ δµ

ν

z − w , ψµ(z) ¯ ψν(w) ∼ δµ

ν

z − w .

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 19 / 35

Field redefinition

Make the redefinition Πµ ≡ Pµ + Γρ

µν ¯

ψρψν so action becomes: S = 1 2π

• Σ

Πµ ¯ ∂X µ + ¯ ψµ ¯ ∂ψµ . Free action and OPEs: X µ(z) Πν(w) ∼ δµ

ν

z − w , ψµ(z) ¯ ψν(w) ∼ δµ

ν

z − w . Covariance non-manifest, due to transformation: ˜ Πµ = ∂X ν ∂ ˜ X µ Πν + ∂2X κ ∂ ˜ X µ∂ ˜ X ν ∂ ˜ X ν ∂X σ ¯ ψκψσ

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 19 / 35

Classical currents

Action has fermionic symmetries generated by: G◦ = ψµΠµ , ¯ G◦ = gµν ¯ ψµ

• Πν − Γρ

νσ ¯

ψρψσ .

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 20 / 35

Classical currents

Action has fermionic symmetries generated by: G◦ = ψµΠµ , ¯ G◦ = gµν ¯ ψµ

• Πν − Γρ

νσ ¯

ψρψσ . Classically, obey the algebra {G◦, G◦} = { ¯ G◦, ¯ G◦} = 0 , {G◦ , ¯ G◦} = H◦ with H◦ = gµν Πµ − Γρ

µσ ¯

ψρψσ Πν − Γκ

νλ ¯

ψκψλ − 1 2ψµψν ¯ ψρ ¯ ψσRρσ

µν

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 20 / 35

Classical currents

Action has fermionic symmetries generated by: G◦ = ψµΠµ , ¯ G◦ = gµν ¯ ψµ

• Πν − Γρ

νσ ¯

ψρψσ . Classically, obey the algebra {G◦, G◦} = { ¯ G◦, ¯ G◦} = 0 , {G◦ , ¯ G◦} = H◦ with H◦ = gµν Πµ − Γρ

µσ ¯

ψρψσ Πν − Γκ

νλ ¯

ψκψλ − 1 2ψµψν ¯ ψρ ¯ ψσRρσ

µν

These are analogues of the flat space currents: ψµPµ → G , gµν ¯ ψµPν → ¯ G , P2 → H

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 20 / 35

BRST charge

Gauge these currents ⇒ Q =

• c T m+ : bc∂c : +˜

c 2 H◦ + ¯ 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

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 at the 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) X µ(w) ∼ V µ z − w , OV (z) ψµ(w) ∼ ∂νV µ ψν z − w , OV (z) ¯ ψµ(w) ∼ −∂µV ν ¯ ψν z − w , Ov(z) Πµ(w) ∼ −∂µV ν Πν + ∂µ∂νV ρ ¯ ψρψν z − w

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) X µ(w) ∼ V µ z − w , OV (z) ψµ(w) ∼ ∂νV µ ψν z − w , OV (z) ¯ ψµ(w) ∼ −∂µV ν ¯ ψν z − w , Ov(z) Πµ(w) ∼ −∂µV ν Πν + ∂µ∂νV ρ ¯ ψρψν z − w Implemented by: OV = −

• V µΠµ + ∂νV µ ¯

ψµψν

T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 24 / 35

Quantum currents

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

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(LV F) z − w + · · · But our currents G◦, ¯ G◦ don’t obey this. (double contractions!)

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(LV F) 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◦ + ∂

• ψµΓν

µν

• ¯

G = ¯ G◦ − gνσ∂ ¯ ψµΓµ

νσ

• 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 ∂X µ − 1 2 ¯ ψµ ∂ψµ − 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 curved βγ-systems [Nekrasov, Witten] See also math literature, sheaves of chiral algebras, chiral de Rham complex [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 built from ghosts and the currents:

Quantum Currents

G = ψµΠµ + ∂

• ψµΓν

µν

• ¯

G = gµν ¯ ψµ

• Πν − Γρ

νσ ¯

ψρψσ − gνσ∂ ¯ ψµΓµ

νσ

• T

= −Πµ ∂X µ − 1 2 ¯ ψµ ∂ψµ − 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) ∼ H z − 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 ¯ ψµ ¯ ψν ¯ ψρψσ z − w R µνρ

σ

+ ∂ ¯ ψµ ¯ ψνRµν z − w + 2 ¯ ψµ ¯ ψν∂X σ z − w

• Γν

αβRβαµ σ + Γα σβ(Rµβν α + Rνβµ α)

• G(z) ¯

G(w) ∼ 2 (z − w)3 R + 2(Γµ

σν∂X σ + ψµ ¯

ψν) (z − w)2 Rµ

ν +

H z − 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!

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 6Hµνρψµψνψρ + ∂

• ψµΓν

µν

• − 2∂ (ψµ∂µΦ)

¯ G = gµν ¯ ψµ

• Πν − Γρ

νσ ¯

ψρψσ + 1 6Hµνρ ¯ ψµ ¯ ψν ¯ ψρ −gνσ∂ ¯ ψµΓµ

νσ

• − 2∂

¯ ψµgµν∂νΦ

• T

= −Πµ ∂X µ − 1 2 ¯ ψµ ∂ψµ − 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 , ∇ρHρ

µν − 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 become G(z) ¯ G(w) ∼ 0.

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 become G(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 the field 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 )

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