Conformal Supergravity, 4D Scattering Equations (and Monte Carlo Methods) Joe Farrow Based on Farrow, “A Monte Carlo Approach to the 4D Scattering Equations”, 1806.02732 Farrow & Lipstein, “New Worldsheet Formulae for Conformal Supergravity Amplitudes”, 1805.04504 Geyer, Lipstein & Mason, “Ambitwistor Strings in 4 Dimensions”, 1404.6219 26 th September 2018 J. A. Farrow 4D Scattering Equations 1 / 22
Introduction Review of 4D ambitwistor string theory N = 4 conformal supergravity Solving the 4D scattering equations Current work and future directions 26 th September 2018 J. A. Farrow 4D Scattering Equations 2 / 22
4D Ambitwistor Review Witten 2003 considers a string theory where the target space is twistor space 26 th September 2018 J. A. Farrow 4D Scattering Equations 3 / 22
4D Ambitwistor Review Witten 2003 considers a string theory where the target space is twistor space Cachazo, He and Yuan 2013 introduce the scattering equations n k i · k j � = 0 s i − s j j =1 j � = i 26 th September 2018 J. A. Farrow 4D Scattering Equations 3 / 22
4D Ambitwistor Review Geyer, Lipstein and Mason 2014 consider worldsheet action � d 2 σ � � S = Z · ∂W + cZ · W � d 2 σ � � = � µ | ∂ | λ � + [ λ | ∂ | µ ] + c ( � µλ � + [ λµ ]) Amplitudes in field theory are correlation function of worldsheet vertex operators � | λ � � � � µ | � Z = W = | µ ] [ λ | 26 th September 2018 J. A. Farrow 4D Scattering Equations 4 / 22
4D Ambitwistor Review Penrose transform from twistor theory motivates plane-wave vertex opeartors � dt � � | l ] � l | ( σ ) = � lλ ( σ ) � S − 1 V S − ˜ δ 2 e it � µ ( σ ) l � | l ] − t | λ ( σ )] t 2 S − 1 26 th September 2018 J. A. Farrow 4D Scattering Equations 5 / 22
4D Ambitwistor Review Penrose transform from twistor theory motivates plane-wave vertex opeartors � dt � � | l ] � l | ( σ ) = � lλ ( σ ) � S − 1 V S − ˜ δ 2 e it � µ ( σ ) l � | l ] − t | λ ( σ )] t 2 S − 1 � � � dt V S + | r ] � r | ( σ ) = [ rλ ( σ )] S − 1 δ 2 e it [ µ ( σ ) r ] � r | − t � λ ( σ ) | t 2 S − 1 26 th September 2018 J. A. Farrow 4D Scattering Equations 5 / 22
4D Ambitwistor Review Amplitudes are supported on 4D scattering equations refined by MHV degree | r ] � l | � � | l ] = � r | = ( lr ) ( rl ) r ∈ R l ∈ L � � � � � � 26 th September 2018 J. A. Farrow 4D Scattering Equations 6 / 22
4D Ambitwistor Review Amplitudes are supported on 4D scattering equations refined by MHV degree | r ] � l | � � | l ] = � r | = ( lr ) ( rl ) r ∈ R l ∈ L � � � � � � σ i = 1 � 1 � ( ij ) = det( σ i σ j ) s i t i σ = ( σ 1 σ 2 ...σ n ) ∈ Gr (2 , n ) 26 th September 2018 J. A. Farrow 4D Scattering Equations 6 / 22
4D Ambitwistor Review Geyer, Lipstein and Mason 2014 write tree-level S matrices as integrals over these equations � d 2 × n σ � � � � � | r ] � l | 1 A (0) � � � δ 2 |N δ 2 n,L = | l ] − � r | − � GL (2) i ( i i +1) ( lr ) ( rl ) r r l l 26 th September 2018 J. A. Farrow 4D Scattering Equations 7 / 22
4D Ambitwistor Review Geyer, Lipstein and Mason 2014 write tree-level S matrices as integrals over these equations � d 2 × n σ � � � � � | r ] � l | 1 A (0) � � � δ 2 |N δ 2 n,L = | l ] − � r | − � GL (2) i ( i i +1) ( lr ) ( rl ) r r l l � d 2 × n σ GL (2) det ′ H det ′ ˜ M (0) n,L = H � � � � � | r ] � l | � δ 2 |N � δ 2 � | l ] − � r | − ( lr ) ( rl ) r r l l 26 th September 2018 J. A. Farrow 4D Scattering Equations 7 / 22
Conformal Supergravity Berkovits and Witten 2004 consider N = 4 conformal supergravity amplitudes in twistor string framework. Action is schematically d 4 x √− g f ( φ ) W 2 � S = 26 th September 2018 J. A. Farrow 4D Scattering Equations 8 / 22
Conformal Supergravity Berkovits and Witten 2004 consider N = 4 conformal supergravity amplitudes in twistor string framework. Action is schematically d 4 x √− g f ( φ ) W 2 � S = So equations of motion are now fourth order, ie. � 2 φ ( x ) = 0 solved by φ ( x ) = ( A + B · x ) e ik · x 26 th September 2018 J. A. Farrow 4D Scattering Equations 8 / 22
Conformal Supergravity Graviton supermultiplet is Φ − = h − η 1 η 2 η 3 η 4 + η I η J η K ψ IJK + η I η J A IJ + η I ψ I + φ − 26 th September 2018 J. A. Farrow 4D Scattering Equations 9 / 22
Conformal Supergravity Graviton supermultiplet is Φ − = h − η 1 η 2 η 3 η 4 + η I η J η K ψ IJK + η I η J A IJ + η I ψ I + φ − Out-of-MHV amplitudes can now be non-zero M (0) ( − − − ) = δ 8 ( Q ) M (0) ( h − h − φ − ) = � 12 � 4 M (0) ( h − h − h − ) = 0 , M (0) ( h − h − h + ) = 0 So we grade amplitude by both MHV degree and a separate Grassmann degree 26 th September 2018 J. A. Farrow 4D Scattering Equations 9 / 22
Conformal Supergravity 4 types of plane wave vertex operator � dt � � V − ˜ δ 2 | 4 e it � µ ( σ ) l � | l ] � l | ( σ ) = � lλ ( σ ) � | l ] − t | λ ( σ )] t 2 � � � V + ˜ δ 2 | 4 e it � µ ( σ ) l � | l ] � l | ( σ ) = [ λ∂λ ( σ )] | l ] − t | λ ( σ )] dt t 26 th September 2018 J. A. Farrow 4D Scattering Equations 10 / 22
Conformal Supergravity 4 types of plane wave vertex operator � dt � � V − ˜ δ 2 | 4 e it � µ ( σ ) l � | l ] � l | ( σ ) = � lλ ( σ ) � | l ] − t | λ ( σ )] t 2 � � � V + ˜ δ 2 | 4 e it � µ ( σ ) l � | l ] � l | ( σ ) = [ λ∂λ ( σ )] | l ] − t | λ ( σ )] dt t � dt � � V + δ 2 e it ([ µ ( σ ) r ]+ χ ( σ ) · η i ) � r | − t � λ ( σ ) | | r ] � r | ( σ ) = [ rλ ( σ )] t 2 � � � V − ˜ δ 2 e it ([ µ ( σ ) r ]+ χ ( σ ) · η i ) | r ] � r | ( σ ) = � λ∂λ ( σ ) � � r | − t � λ ( σ ) | dt t 26 th September 2018 J. A. Farrow 4D Scattering Equations 10 / 22
Conformal Supergravity Plane wave graviton multiplet S-matrix � d 2 × n σ � � � � � | r ] � l | M (0) � δ 2 | 4 � � δ 2 n,L, Φ − = | l ] − � r | − GL (2) ( lr ) ( rl ) r r l l � � ˜ � � ˜ H l − F l + F r − H r + l + ∈ L ∩ Φ + r + ∈ R ∩ Φ + l − ∈ L ∩ Φ − r − ∈ R ∩ Φ − 26 th September 2018 J. A. Farrow 4D Scattering Equations 11 / 22
Conformal Supergravity Non-plane wave states φ ( x ) = B · xe ik · x = − iB · ∂ ∂ke ik · x 26 th September 2018 J. A. Farrow 4D Scattering Equations 12 / 22
Conformal Supergravity Non-plane wave states φ ( x ) = B · xe ik · x = − iB · ∂ ∂ke ik · x Vertex operators � dt � � � � | l � [ µ ( σ ) | − | λ ( σ ) � ∂ V − ˜ δ 2 | 4 e it � µ ( σ ) l � | l ] � l | ( σ ) = B · | l ] − t | λ ( σ )] t 2 ∂ | l ] � � � ˜ V + | l ] � l | ( σ ) = B · tdt | ∂µ ( σ ) � [ λ ( σ ) | − | µ ( σ ) � [ ∂λ ( σ ) | � � δ 2 | 4 e it � µ ( σ ) l � | l ] − t | λ ( σ )] 26 th September 2018 J. A. Farrow 4D Scattering Equations 12 / 22
Conformal Supergravity M ( h − x h − h + ...h + ) 26 th September 2018 J. A. Farrow 4D Scattering Equations 13 / 22
Conformal Supergravity M ( h − x h − h + ...h + ) � d 2 × n σ ∂ ∂ �� | 1 � ∂ | 2] − | 2 � � � [ rr ′ ] � 12 � ∂ | 1] � = B 1 · ( rr ′ ) GL (2) (12) (12) r ∈ R r ′ ∈ R � [ r ′ r ′′ ] | 1 � [ r | � � � δ ( SE n + L ) ( r ′ r ′′ ) (1 r ) r ∈ R r ′ � = r ∈ R r ′′ ∈ R 26 th September 2018 J. A. Farrow 4D Scattering Equations 13 / 22
Conformal Supergravity M ( h − x h − h + ...h + ) � d 2 × n σ ∂ ∂ �� | 1 � ∂ | 2] − | 2 � � � [ rr ′ ] � 12 � ∂ | 1] � = B 1 · ( rr ′ ) GL (2) (12) (12) r ∈ R r ′ ∈ R � [ r ′ r ′′ ] | 1 � [ r | � � � δ ( SE n + L ) ( r ′ r ′′ ) (1 r ) r ∈ R r ′ � = r ∈ R r ′′ ∈ R � � � � 12 � | 1 � [ r | ∂ = � 12 � 4 B 1 · ψ | 1 �| 2 � � � δ 4 ( P ) + ψ r ′ ,n � 1 r � 2 � 2 r � r,n ∂P 1 r ∈ R r ∈ R r ′ ∈ R,r ′ � = r 26 th September 2018 J. A. Farrow 4D Scattering Equations 13 / 22
Conformal Supergravity M ( h − x h − h + ...h + ) � d 2 × n σ ∂ ∂ �� | 1 � ∂ | 2] − | 2 � � � [ rr ′ ] � 12 � ∂ | 1] � = B 1 · ( rr ′ ) GL (2) (12) (12) r ∈ R r ′ ∈ R � [ r ′ r ′′ ] | 1 � [ r | � � � δ ( SE n + L ) ( r ′ r ′′ ) (1 r ) r ∈ R r ′ � = r ∈ R r ′′ ∈ R � � � � 12 � | 1 � [ r | ∂ = � 12 � 4 B 1 · ψ | 1 �| 2 � � � δ 4 ( P ) + ψ r ′ ,n � 1 r � 2 � 2 r � r,n ∂P 1 r ∈ R r ∈ R r ′ ∈ R,r ′ � = r � � � = � 12 � 4 B 1 · ∂ ψ | 1 �| 2 � δ 4 ( P ) r,n ∂P 1 r ∈ R 26 th September 2018 J. A. Farrow 4D Scattering Equations 13 / 22
Solving the Equations How do we extract amplitudes from worldsheet integrals? � d 2 × n σ GL (2) δ 2 × n ( SE n A (0) n,L = L ) f ( σ ) f ( σ sol ) = δ 4 ( P ) � � ll ′ � − 2 det( J n ll ′ ( σ sol )) L σ sol ∈ solutions 26 th September 2018 J. A. Farrow 4D Scattering Equations 14 / 22
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