Twistor strings for N =8. supergravity

Transcription

1 Twistor strings for N =8 supergravity David Skinner - IAS & Cambridge Amplitudes MPI Ringberg

2 Twistor space is CP 3 CP 3, described by co-ords R 3,1 Z a rz a X Y y x CP 1 in twistor space Point in space-time Two lines intersect Separation is null X ab = Z [a 1 Zb] 2 Y cd = Z [c 3 Zd] 4 ɛ(1, 2, 3, 4) (x y) 2

3 To define a metric, not just a conformal structure, we need to choose an infinity twistor I ab = I [ab] For flat space-time with twistor coords Z a =(µ α, λ α ) I ab = ( ɛ α β ) I ab = ( ) ɛ αβ (x y) 2 = ɛ(1, 2, 3, 4) λ α =0 x I X ı 0 I ab Z a Z b = µ α µ α I ab Z a 1 Z b 2 = 12

4 [Penrose;Atiyah,Hitchin,Singer] For self-dual Einstein gravity, deform C-structure + V V H 0,1 (PT,T PT ) For deformation to preserve the holomorphic Poisson structure, V should be Hamiltonian wrt the infinity twistor: V = {h, } = I ab h Z a Z b where h H 0,1 (PT, O(2)) For N =8supergravity, extend to CP 3 8. Then h(z, χ) =h(z)+χ A ψ A (Z)+ +(χ) 8 h(z) describes the N =8 supermultiplet on twistor space

5 Scattering amplitudes & the infinity twistor

6 The infinity twistor plays an important role in scattering amplitudes - conf ml breaking is explicit in twistor space! g-loop, n-pt Feynman diagram dimensions balanced by kinematics G N n+2 2g! parity exchanges I ab [, ] with I ab, infinity twistor The n-particle, g-loop amplitude with n ± external gravitons of helicity ±2 scales with n + + g 1 powers of [, ] parity n + g 1 and powers of, n +2g 2

7 3-pt Yang-Mills amplitudes δ 4 8 ( p) δ 4 ( p) δ 0 4 (η 1 [23] + cyc) [12][23][31] 3-pt gravitational amplitudes δ 4 16 ( p) ( ) 2 δ 4 ( p) δ 0 8 (η 1 [23] + cyc) ([12][23][31]) 2

8 3-pt Yang-Mills amplitudes δ 2 4 (Z 1, Z 2, Z 3 ) δ 3 4 (Z 1, Z 2 ) δ 3 4 (Z 1, Z 3 ) 3-pt gravitational amplitudes 23 δ 2 8 (Z 1 ; Z 2, Z 3 ) [ 2 3 ] δ 3 8 (Z 1 ; Z 2 ) δ 3 8 (Z 1 ; Z 3 )

9 Hodges representation of the MHV tree amplitude M MHV n = δ 4 16( pi ) H ijk rst ij jk ki rs st tr where H is the n n symmetric matrix H ij = [ij] ij H ii = j i H ij pj qj pi qi Permutation symmetric without explicit sum!! determinant suggests correlator of fermion bilinears rk(h) = (n 3), and is an (n 3) minor H ijk rst! provides required power of [, ]! suggests fixing of some residual fermionic symmetry

10 The worldsheet theory

11 { The worldsheet is a 1 2-dimensional supermanifold! local coords (x, θ a ). has conf weight -!, charge 1 { θ a! global holomorphic measure d 1 2 x of charge -2 { X D Σ Vector fields of the form V a (x, θ) θ a D T X generate SL(1 2) algebra Gauge this by introducing ghost C Ω 0 (X, D) C a (x, θ) =γ a (x)+θ b N a b(x)+θ 2 ν a (x) and antighost multiplet B Ω 0 (X, Ber(X) D ) B a (x, θ) =µ a (x)+θ b M ab (x)+θ 2 β a (x)

12 The matter field Z Ω 0 (X, C 4 N L) describes a map to N -extended supertwistor space. It has components Z I (x, θ) =Z I (x)+θ a ρ I a(x)+θ 2 Y I (x) After gauge fixing, the worldsheet action is S = d 1 2 x Z, Z + B C a a X while the (classically) nilpotent BRST operator is Q = d 1 2 x Z,C a a Z 1 2 B a[c, C] a Q depends on,, breaking conformal invariance.

13 The worldsheet theory is chiral, so there are potential anomalies in these gauge transformations GL(1) anomaly: ( 1) F i qi 2 = (4 N )+2+2 i YZ βγ µν SL(2) anomaly: i ( 1) F i AutΓ i tr R (t t) = 3 i 4 (N 8) The total Virasoro central charge is c = 2(4 N ) + (4 N ) = 3(8 N ) YZ ρρ βγ MN µν! not a string theory - no bc ghosts! mixed Vir - GL(1) anomaly also vanishes iff N =8

14 Positively charged fields have zero modes: Z I : d +1 g fermionic components give MHV level selection rule n = d +1 g γ a : d +2 2g zero modes of bosonic ghost - fermionic symmetry needs fixing #γ zm = n #[, ] µ a : d zero modes of bosonic antighost - fermionic moduli (handle by PCOs) #µ zm =#,

15 Matter vertex operators are similar to RNS string: cδ 2 (γ)h(z) or U for fixed vertex operators. Integrated operators are [ ] [ V d 2 h θ h(z) = Z,Y ρ I Z I ρ, h ] Z and deform the worldsheet action. Σ Σ δ 2 (γ) h(z)! h represents an N =8multiplet as before Picture changing operators associated to µ zm are Υ [Q, Θ(µ a )] = δ 2 (µ) ρ,z ρ I Z I + a=1,2

16 The classical S-matrix

17 All tree-level amplitudes in N =8supergravity come from the g =0twistor string correlator All [, ] d+2 cu 1 cu 2 cu 3 dependence lives here All i=4, U i n j=d+3 V j d k=1 dependence lives here Υ fixed vertex op integrated vertex op PCO! degree of curve determines MHV level! required powers of [, ] and, from V and Υ

18 Correlator of PCOs is independent of insertion points d k=1 Υ(x k ) = R(λ α ) δ 2 (µ) ρz ρz the resultant of the two components of λ α Z : Σ PT [Cachazo] I R(λ α )=0 λ α (x )=0 for some x Σ λ α =0 is the line I at infinity The amplitude is thus supported Z(Σ) on holomorphic curves in PT I - the inside of space-time [Casali,DS]

19 involves d+2 k=1 Φ ij = 1 x ij δ 2 (γ)h k [ µ i µ j n l=d+3 ] ([ Y, h l µ (n d 2) (n d 2) ] +ρ Z minor of Φ ii = j i Φ ij ρ ρ contractions correction from YZ system [ ρ, h l µ Φ d a=0 ]) y a x j y a x i

20 involves d+2 k=1 Φ ij = 1 x ij δ 2 (γ)h k [ µ i µ j n l=d+3 (n d 2) (n d 2) ] ([ Y, h l µ ] +ρ Z minor of Φ ii = j i [ Φ ij ρ, h l µ Φ d a=0 ]) y a x j y a x i Fixed vertex operators rows & columns that are removed in computing minor, so we obtain Φ r 1 r d+2 c 1 c d+2 det ω j (x rk ) det ω l (x cm ) where {ω j (x)} form a basis of H 0 1 (Σ,T 2 Σ L), i.e. the space of γ zero modes! remove rows/cols independently by different gauge fixing! extends Hodges representation to all N k MHV [Cachazo,DS]

21 Kirchoff matrix-tree theorem: [Feng,He; Adamo,Mason] Hodges determinant sum over rooted trees [Nguyen,Spradlin,Volovich,Wen; Bern,Dixon,Perelstein,Rozowsky] [Casali,DS] fixed vertex op Y Z propagator integrated vertex op ρ ρ propagator! matrix-tree theorem due to worldsheet susy! there s a tree formula at N k MHV on worldsheet

22 What do these worldsheet trees actually mean? instead of computing d+2 k=1 δ 2 (γ)h k n l=d+3 [ Y, h l µ ] +ρ Z using the original free action, [ ρ, h l µ ] we can equivalently compute d+2 k=1 δ 2 (γ)h k (Z) using the nonlinear action S = Σ ( Y I Z I + I IJ h Z J ) + fermions describing holomorphic maps to deformed twistor space

23 Path integral over Y imposes ( + {h, })Z I (x) = 0! define Z (x) by Z (x) =( + {h, })Z(x)! Jacobian for [DZ] [DZ ] provided by path integral over fermions: Nicolai map Z I = Z I 1 (I IJ J h)! expanding each h(z(z )) grows a rooted tree The integrated vertex operators perturbatively construct a nonlinear graviton background

24 Combining all the ingredients, the g =0 twistor string is simply the statement that all tree amplitudes in N =8supergravity are supported on degree d holomorphic maps M n,d = Γ d r=0 d4 8 Z r vol(gl(2)) Φ r 1 r d+2 c 1 c d+2 det ω j (x rk ) det ω l (x cm ) R(λ α) n i=1 h i (x i )dx i to curved twistor space with infinity removed! precisely agrees with a representation of the classical gravitational S-matrix discovered last year [Cachazo,DS]

25 A key test: multi-particle factorization! in twistor space, statement that poles of amplitude ( occur whenever = ( = ( subset p i) 2 =0 D 3 8 Z becomes Z corresponding to boundary divisor in M 0,n (PT,d) [Gukov,Motl,Neitzke;Vergu;DS]! also has correct asymptotics under BCFW shift, so satisfies BCFW recursion relations [Cachazo,Mason,DS] We ve also checked parity, soft limits, and various analytical and numerical checks for low d or low n [Cachazo,Mason,DS;Bullimore;He]

26 Conclusions

27 I have presented a twistor string theory that computes the classical S-matrix of maximal supergravity! anomaly free when N =8! spectrum describes N =8 twistor supermultiplet! weighted matrix-tree theorem from supersymmetry There are many open questions! convincing way to couple to worldsheet gravity?! other states?! relation to N =2superstring? [Berkovits,Ooguri,Siegel,Vafa]! other backgrounds? (e.g. boundary correlators in AdS4)

28 More generally, for Yang-Mills the twistor string is one of many inter-related approaches to scattering amplitudes! as yet, no good MHV diagram approach for gravity [Bjerrum-Bohr,Dunbar,Ita,Perkins,Risager;Bianchi,Elvang,Freedman;Rajabi,Penante,Sizov]! gravitational BCFW recursion works... [Cachazo,Svrcek; Bedford,Brandhuber,Spence,Travaglini] = p 2 L ˆ1 ˆn 1... but resulting expressions not understood [Drummond,Spradlin,Volovich,Wen]! what is the role of colour/kinematics duality here? [Bern,Carrasco,Johansson;Bjerrum-Bohr,Damgaard,Monteiro,O Connell;Boels,Isermann; Sondergaard,Vanhove;Broedel,Dixon;Oxburgh,White;Du,Feng,Fu;Cachazo,Geyer]