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] : � n � � 1 | z 1 z 2 z 3 | k i · k j ¯ Pf ′ ( M ) Pf ′ ( � M n , 0 = δ M ) vol SL (2 , C ) z i − z j d z 1 d z 2 d z 3 i =4 j � = i 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] : � n � � 1 | z 1 z 2 z 3 | k i · k j ¯ Pf ′ ( M ) Pf ′ ( � M n , 0 = δ M ) vol SL (2 , C ) z i − z j d z 1 d z 2 d z 3 i =4 j � = i { z i } ⊂ Σ ∼ = CP 1 , { k i } null momenta, � � A � − C T d z i d z j Pf ′ ( M ) = ( − 1) i + j Pf ( M ij M = , ij ) , C B z i − z j � � � d z i d z j d z i d z j d z i d z j A ij = k i · k j , B ij = ǫ i · ǫ j , C ij = ǫ i · k j z i − z j z i − z j z i − z j � C ij √ A ii = B ii = 0, C ii = − d z i j � = i d z i d z j T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 9 / 35
This representation of M n , 0 manifests (gauge) 2 =(gravity), and related to BCJ duality All integrals over M 0 , n fixed by delta functions, imposing the scattering equations [Fairlie-Roberts, Gross-Mende, Witten] : � k i · k j i ∈ { 4 , . . . , n } , = 0 z i − z j j � = i So the locations { z i } ⊂ Σ are fixed by the scattering equations. T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 10 / 35
This representation of M n , 0 manifests (gauge) 2 =(gravity), and related to BCJ duality All integrals over M 0 , n fixed by delta functions, imposing the scattering equations [Fairlie-Roberts, Gross-Mende, Witten] : � k i · k j i ∈ { 4 , . . . , n } , = 0 z i − z j j � = i So the locations { z i } ⊂ Σ are fixed by the scattering equations. Structure of M n , 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 Ψ µ − e ∂ X µ + Ψ µ ¯ ∂ Ψ µ − χ P µ Ψ µ + � Ψ µ − � P µ ¯ Ψ µ ¯ ∂ � χ P µ � 2 P 2 2 π Σ Ψ µ ∈ ΠΩ 0 (Σ , K 1 / 2 ) P µ ∈ Ω 0 (Σ , K ) and Ψ µ , � T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 11 / 35
Worldsheet theory, I Consider worldsheet action [Mason-Skinner] : � S = 1 Ψ µ − e ∂ X µ + Ψ µ ¯ ∂ Ψ µ − χ P µ Ψ µ + � Ψ µ − � P µ ¯ Ψ µ ¯ ∂ � χ P µ � 2 P 2 2 π Σ Ψ µ ∈ ΠΩ 0 (Σ , K 1 / 2 ) P µ ∈ Ω 0 (Σ , K ) and Ψ µ , � � gauge-fixing → 1 ∂ X µ + Ψ µ ¯ ∂ Ψ µ + � Ψ µ + S gh P µ ¯ Ψ µ ¯ ∂ � − − − − − − − − − 2 π Σ where fixing e = 0 enforces the constraint P 2 = 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, P µ becomes meromorphic : n � ¯ k i µ δ 2 ( z − z i ) . ∂ P µ ( z ) = 2 π i d z ∧ d ¯ z i =1 T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 12 / 35
Scattering equations from the worldsheet In the presence of vertex operator insertions, P µ becomes meromorphic : n � ¯ k i µ δ 2 ( z − z i ) . ∂ P µ ( z ) = 2 π i d z ∧ d ¯ z i =1 Likewise, quadratic differential P 2 becomes meromorphic, with residues: � k i · k j Res z = z i P 2 ( z ) = k i · P ( z i ) = d z i z i − z j j � = i Setting Res z = z i P 2 ( z ) = 0 for i = 4 , . . . , n is sufficient to set P 2 ( z ) = 0 globally on Σ. T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 12 / 35
But these are the scattering equations! � k i · k j P 2 ( z ) = 0 Res z = z i P 2 ( z ) = 0 = ↔ i ∈ { 4 , . . . , n } z i − z j j � = i T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 13 / 35
But these are the scattering equations! � k i · k j P 2 ( z ) = 0 Res z = z i P 2 ( z ) = 0 = ↔ i ∈ { 4 , . . . , n } z i − z j j � = i The condition P 2 ( z ) = 0 globally on Σ defines the scattering equations for any genus worldsheet [TA-Casali-Skinner] ( n − 3) × Res z = z i P 2 ( z ) = 0 g = 0 P 2 ( z 1 ) = 0 ( n − 1) × Res z = z i P 2 ( z ) = 0 , g = 1 (3 g − 3) × P 2 ( z r ) = 0 n × Res z = z i P 2 ( z ) = 0 , g ≥ 2 T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 13 / 35
This theory has a BRST-charge � c T m + : bc ∂ c : +˜ c 2 P 2 + γ P µ Ψ µ + ˜ Ψ µ , γ P µ � Q = which is nilpotent Q 2 = 0 provided the space-time has d = 10. T Adamo (DAMTP) Scattering Eqns + Curved Space 22 September 2014 14 / 35
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