Twistor Strings, Gauge Theory and Gravity. Abou Zeid, Hull and Mason hep-th/

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1 Twistor Strings, Gauge Theory and Gravity Abou Zeid, Hull and Mason hep-th/

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3 Amplitudes for YM, Gravity have elegant twistor space structure: Twistor Geometry

4 Amplitudes for YM, Gravity have elegant twistor space structure: Twistor Geometry N=4 SYM, N=8 SUGR: Structure of amplitudes suggest string theory Nair

5 Amplitudes for YM, Gravity have elegant twistor space structure: Twistor Geometry N=4 SYM, N=8 SUGR: Structure of amplitudes suggest string theory Nair QCD String? QCD is stringy at large N

6 Amplitudes for YM, Gravity have elegant twistor space structure: Twistor Geometry N=4 SYM, N=8 SUGR: Structure of amplitudes suggest string theory Nair QCD String? QCD is stringy at large N Dual string for N=4 SYM: Strong coupling: IIB in AdS Weak coupling: String in twistor space?

7 Amplitudes for YM, Gravity have elegant twistor space structure: Twistor Geometry N=4 SYM, N=8 SUGR: Structure of amplitudes suggest string theory Nair QCD String? QCD is stringy at large N Dual string for N=4 SYM: Strong coupling: IIB in AdS Weak coupling: String in twistor space? SU(2,2/4) acts on targets: Twistor space, AdSxS

8 Penrose Transform Space-Time M Twistor Space PT Field, helicity h Solution of massless field equn O(n) Cohomology class H 1 (P T, O(n)) Homogeneous functions of weight n=2h-2 Topological String: Physical states are cohomology classes Witten: Topological B String on SuperTwistor Space N=4 SUSY: PT is Calabi-Yau

9 Twistor Strings Witten: Topological B-model. Open and closed strings Berkovits: Open strings Witten; Mason, Skinner: Heterotic All give same theory: N=4 SYM + Conformal Supergravity 1 g 2 F 2 + W 2 W= Weyl tensor

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11 Can t decouple YM, can t extract YM loop amplitudes as gravitons etc propagate in loops

12 Can t decouple YM, can t extract YM loop amplitudes as gravitons etc propagate in loops Anomaly in SU(4) R-symmetry: only cancels if YM gauge group has dimension DGauge=4 e.g. SU(2)xU(1). Don t yet know how this arises in twistor strings.

13 Can t decouple YM, can t extract YM loop amplitudes as gravitons etc propagate in loops Anomaly in SU(4) R-symmetry: only cancels if YM gauge group has dimension DGauge=4 e.g. SU(2)xU(1). Don t yet know how this arises in twistor strings. Don t yet know how to calculate loops

14 New Twistor Strings Einstein supergravity (or chiral version) + SYM Has decoupling limit giving SYM World-sheet anomalies cancel Interesting spectra. Interactions? 1 g 2 F κ 2 R

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16 3 Classes of twistor strings. Spectra same as those of following theories

17 3 Classes of twistor strings. Spectra same as those of following theories 1) N=4 SYM + SUGRA, any gauge group

18 3 Classes of twistor strings. Spectra same as those of following theories 1) N=4 SYM + SUGRA, any gauge group 2) N=4 SUGRA + four N=4 gravitino multiplets + DGauge vector multiplets. N=8 SUGRA spectrum if DGauge=6

19 3 Classes of twistor strings. Spectra same as those of following theories 1) N=4 SYM + SUGRA, any gauge group 2) N=4 SUGRA + four N=4 gravitino multiplets + DGauge vector multiplets. N=8 SUGRA spectrum if DGauge=6 3) N=0 Self-dual gravity + Self-dual YM + scalar R = 0 F = 0

20 3 Classes of twistor strings. Spectra same as those of following theories 1) N=4 SYM + SUGRA, any gauge group 2) N=4 SUGRA + four N=4 gravitino multiplets + DGauge vector multiplets. N=8 SUGRA spectrum if DGauge=6 3) N=0 Self-dual gravity + Self-dual YM + scalar R = 0 F = 0 4) Any N. SD graviton multiplet, Scalar multiplet, SD YM multiplet, (1, N, N(N 1)/2,..., N, 1) helicities 2,3/2...,2-(N/2) helicities 0,-1/2,-1,...,-N/2 helicities 1,1/2,...,1-(N/2)

21 Yang-Mills d 4 xtr F = da + A A (G µν F µν ɛ 2 Gµν G µν ) G = G 2-form Eliminate auxiliary 2-form 1 2ɛ d 4 xtr Siegel, Chalmers ( F µν (+) F (+)µν) F (±) = 1 (F ± F ) 2 Yang-Mills with Chiral Interactions Tr (G F ) = Tr (G da + G A A) ɛ 0 Field Equations F (+) = 0 D µ G µν = 0 Helicities +1, -1. Chiral Interactions: (+ + ) not (+ )

22 Einstein Action, 1st Order Form e a e b R cd (ω)ε abcd = R (±) bc 1 2 (R bc ± 12 ε bc de R de ) e a e b R (+) ab (ω) + d(t a e a ) ω µab = Ω µab (e) Field Equns Ricci = 0 Ω µ ab (e) e νa [µ e ν] b e νb [µ e ν] a e ρa e σb [ρ e σ]c e µ c Write in terms of SD connection ω ω (+) e a e b ( dω ab + κ 2 ω ac ω c ) b Chiral Limit: κ 0

23 Chiral Gravity Siegel Abou Zeid, Hull e a e b dω ab = e a e b ω ac Ω (+)c b(e) ω µab Lagrange multiplier: Ω (+)a b(e) = 0 R µν ab (Ω (+) ) = 0 Vierbein e: Anti-SD curvature, helicity +2 e b dω ab = 0 Lin. Einstein, helicity -2 Helicities +2, -2. Chiral Interactions: (+ + ) not (+ )

24 Twistor Strings Naively give chiral YM Tr (G F ) Instanton corrections give GG term: Tr (G F ɛ 2 G G ) New Twistor Strings N=4, N=8 have right 3-graviton interaction (+ + ) Non-trivial ineraction! Chiral or full Einstein? Need to find (+ ) If vanishes, chiral gravity. Still good for N=4 SYM!

25 4) Any N. SD graviton multiplet, Scalar multiplet, SD YM multiplet, helicities 2,3/2...,2-(N/2) helicities 0,-1/2,-1,...,-N/2 helicities 1,1/2,...,1-(N/2) Interactions, higher spin? N<4: Self-dual supergravity and superyang-mills N=4: Spectrum of N=4 supergravity and superym N>4: helicities < - 2 Helicity -n/2 Φ A 1 A 2...A n BA 1 ΦA 1 A 2...A n = 0 Consistent for ASD gravity, YM F A B = 0, W A B C D = 0 Higher spin fields linear, but gravity, YM can be chiral

26 Twistor space T Z α = (ω A, π A ) C 4 ProjectiveTwistor space PT Z λz CP 3 ω A = x AA π A

27 Twistor space T Z α = (ω A, π A ) C 4 ProjectiveTwistor space PT Z λz CP 3 ω A = x AA π A Incidence relation, governs relation between twistor space and spacetime

28 Twistor space T Z α = (ω A, π A ) C 4 ProjectiveTwistor space PT Z λz CP 3 ω A = x AA π A Incidence relation, governs relation between twistor space and spacetime A spacetime point x AB determines C 2 C 4 CP 1 CP 3

29 Twistor space T Z α = (ω A, π A ) C 4 ProjectiveTwistor space PT Z λz CP 3 ω A = x AA π A Incidence relation, governs relation between twistor space and spacetime A spacetime point x AB determines C 2 C 4 CP 1 CP 3 Specified by bi-spinor P αβ = P βα

30 Twistor space T Z α = (ω A, π A ) C 4 ProjectiveTwistor space PT Z λz CP 3 ω A = x AA π A Incidence relation, governs relation between twistor space and spacetime A spacetime point x AB determines C 2 C 4 CP 1 CP 3 Specified by bi-spinor P αβ = P βα Spacetime is moduli space of CP 1 CP 3

31 Twistor space T Z α = (ω A, π A ) C 4 ProjectiveTwistor space PT Z λz CP 3 ω A = x AA π A

32 Twistor space T Z α = (ω A, π A ) C 4 ProjectiveTwistor space PT Z λz CP 3 ω A = x AA π A A point in twistor space (ω A, π A ) C 4 determines self-dual 2-plane in space time α-plane

33 Twistor space T Z α = (ω A, π A ) C 4 ProjectiveTwistor space PT Z λz CP 3 ω A = x AA π A

34 Twistor space T Z α = (ω A, π A ) C 4 ProjectiveTwistor space PT Z λz CP 3 ω A = x AA π A SuperTwistor space Z I = (Z α, ψ a ) a = 1,..., N ψ a C 4 N anti-commuting variables Projective SuperTwistor space CP 3 N

35 Twistor space T Z α = (ω A, π A ) C 4 ProjectiveTwistor space PT Z λz CP 3 ω A = x AA π A SuperTwistor space Z I = (Z α, ψ a ) a = 1,..., N ψ a C 4 N anti-commuting variables Projective SuperTwistor space CP 3 N Superconformal group SU(2,2/N) acts linearly on Z Signature 2+2: Superconformal group SL(4,R/N) Can take Z real, real twistor spaces RP 3 N R 4 N

36 Penrose-Ferber Twistor Particle Particle moving in super twistor space ( S = dτ Y I τ Z I ) A(J n) Z I (τ) Y I momentum conjugate to Z I (τ) J = Y I Z I n constant, A gauge field Gauge symmetry Z λz, Y λ 1 Y Gauging: theory on projective space PT Y I Z I J Z I Z I

37 Constraint J-n =0 imposed on wave-functions J Z I Z I ( Z I Z I n ) Φ = 0 Φ(Z) Φ(Z) Homogeneous weight n Φ(Z α, ψ a ) = φ(z α ) + ψ a φ a (Z α ) + ψ a ψ b φ ab (Z α ) +... Weight w=n Helicity h=n/2+1 w=n-1 Helicity h-1/2 w=n-2 Helicity h-1 Supermultiplet, superspin n

38 Berkovits Twistor String (Y I Z I ÃJ ) ) S 0 = d 2 σ S 0 = d 2 σ (ỸJ Z J A J S = S 0 + S 0 + S C J = Y I Z I, J = Ỹ I ZI Z I = (ω A, π A, ψ a ), ZI = ( ω A, π A, ψ a ) Independent real coordinates RP 3 N RP 3 N Real twistors for signature 2+2 Y, Ỹ conjugate momenta A, Ã Gauge fields World-sheet: real coordinates Lorentzian metric σ, σ ds 2 = 2dσd σ

39 Anomaly cancellation Q 2 = 0 Kac-Moody J, J N=4 (or N=D) Virasoro Extra CFT c C = 0 has central charge S C c C = 28 Open Strings Y = Ỹ, Z = Z on boundary Extra CFT j r, j r S C assumed to have Kac-Moody currents for some group G

40 Vertex Operators On open string boundary, primary wrt J,T V φ = j r φ r (Z) φ r weight 0, in lie algebra of G Helicities (1, 1 2, 0, 1 2, 1) SYM multiplet for G V f = Y I f I (Z) f I Weight 1. V g = g I (Z) Z I g I Weight -1

41 Conformal Supergravity V f = Y I f I (Z) Subject to constraints V g = g I (Z) Z I I f I = 0, Z I g I = 0. Gauge equivalences δf I = Z I Λ, δg I = I χ (Changes vertices by BRST trivial operators) f A (Z), f A (Z) each give a graviton multiplet, helicities (+2, + 3 2, +1, +1 2, 0) f a Four gravitino multiplets (+ 3 2, +1, +1 2, 0, 1 2 ) g I = (g A, g A, g a ) 2 graviton multiplets (0, 1 2, 1, 3 2, 2) Four gravitino multiplets (+ 1 2, 0, 1 2, 1, 3 2 )

42 Non-Linear Graviton Curved Twistor Spaces PT Conformally antiself-dual spaces W + = 0 Deform Complex Structure New C.A.S.D. spaces An infinitesimal deformation: holomorphic vector field of weight 1 f I (Z) cf Berkovits vertex In overlap U i U j transition fns Z I i Z I j = f I ij

43 Restriction to Ricci Flat Space-Times PT fibred over CP 1 Ricci-Flat ASD Holomorphic 1-form on CP 1 pulls back to1-form on PT k = I αβ Z α dz β ɛ A B π A dπ B Deformation preserves fibration if hamiltonian: f α = I αβ β h h: weight 2 Poisson structure I αβ = 1 2 ɛαβγδ I γδ INFINITY TWISTOR

44 New Theories Restrict to twistor spaces fibred over CP 1 0 with 1-form k New Current K I IJ Z I Z J Gauge: set K~0 New constraints give Conformal gravity f I = I IJ J h gravity

45 More New Theories Restrict to twistor spaces fibred over CP 1 N with 1-forms k, k a New Currents K I IJ Z I Z J K a Gauge: set K, K a 0 New constraints give Conformal gravity f I = I IJ J h gravity

46 New String Theories S = d 2 σ ( Y Z I I + ỸJ Z J ÃJ A J B Ki i B i K i) + S C Gauging currents K i = (K, K a ) or K i = K T: b,c Ghosts: J: u,v BRST charges Q, Q Q = K: r,s ( ct + vj + s i K i + cu v + cb c + cr i s i i vh i s i r i ) Physical states primary wrt T,J,K Seek theories without anomalies, Q 2 = 0

47 3 Classes of twistor strings. Spectra same as those of following theories 1) N=4 SYM + SUGRA, any gauge group 2) N=4 SUGRA + four N=4 gravitino multiplets + DGauge vector multiplets. N=8 SUGRA spectrum if DGauge=6 3) N=0 Self-dual gravity (R - =0) + Self-dual YM (F - =0) + scalar 4) Any N. SD graviton multiplet, Scalar multiplet, SD YM multiplet, (1, N, N(N 1)/2,..., N, 1) helicities 2,3/2...,2-(N/2) helicities 0,-1/2,-1,...,-N/2 helicities 1,1/2,...,1-(N/2)

48 3) N=0 Self-dual gravity (R - =0) + Self-dual YM (F - =0) + scalar Standard N=0 curved twistor space fibred over CP 1 K = I IJ Z I Z J 4) Any N. PT has dimension 3/N fibred over CP 1 N SD graviton multiplet, Scalar multiplet, SD YM multiplet, K i = (K, K a )

49 2) N=4 SUGRA + four N=4 gravitino multiplets + DGauge vector multiplets. N=8 SUGRA spectrum if DGauge=6 N=4 PT has dimension 3/4 fibred over CP 1 0 K = w(z)i IJ Z I Z J w has weight -2 so K has weight zero e.g. w = 1 and dk=0 δ A B π A π B OK for real Z, but poles if continued to complex Z Choice of w doesn t affect results

50 1) N=4 SYM + SUGRA, any gauge group N=4 PT has dimension 3/4 fibred over CP 1 N K i = (K, K a ) scaled by suitable functions to be weight zero

51 Conclusions New twistor strings: no world-sheet anomalies, spectra of interesting sugras N=4, N=8 have correct MHV 3-graviton interactions Chiral supergravity or full supergravity? Loops? Fixing no. of vector multiplets? Extract YM loop amplitudes for N=4?