Twistors and Conformal Higher-Spin. Theory. Tristan Mc Loughlin Trinity College Dublin

Transcription

1 Twistors and Conformal Higher-Spin Tristan Mc Loughlin Trinity College Dublin Theory Based on work with Philipp Hähnel & Tim Adamo ,

2 Given the deep connections between twistors, the conformal group and conformal geometry it natural to ask if there is a twistor description of conformal higher-spin theories.

3 Main idea of twistor theory is to relate differential field equations in space-time to holomorphic structures on a complex manifold

4 Starting point is the twistor equation A, A = 0,1 are two-component spinor indices. r AA 0 = r a Twistor Theory r (A A 0! B) =0 is the covariant derivative where we make use of the bi-spinor notation for vectors. The metric is: g ab = AB A 0 B 0 Equation is conformally invariant. Under we have so that ˆ! A =! A ˆr(A A 0 ˆ! B) = 1 r (A A 0! B) g ab 7! ĝ ab = 2 g ab Only consistent for: ABCD! D =0 where C abcd = ABCD A 0 B 0 C 0 D 0 + A 0 B 0 C 0 D 0 AB CD The space of solutions form a four dimensional vector space over the complex numbers.

5 Flat Twistor Space In Minkowski space the twistor equation has non-trivial solutions:! A =! A + ix AA0 A 0 A = 0 A 0! A, A 0 are constant spinor fields defining a four complexdimensional vector space of solutions called twistor space,. We denote elements of this solution space W and we can represent them in a non-conformally invariant fashion by the pair of fields W =( A 0,! A ) T ' C 4 We can similarly define dual twistors as solutions of =(µ A0, A) r (A0 A µb0) =0

6 Twistor Theory Projective Twistor space PT = CP 3 corresponds to the identification t, t 2 C We identify points in CM with complex projective lines in PT via the incidence relation PT CM L x = CP 1 x µ A0 = ix A0 A A

7 Twistor Theory One application of twistor theory gives an identification between complex data on Twistor space and solutions of linear massless equations [Hitchen 80, Eastwood et al =0 r A0 1 A 1 A 1...A 2h =0 is a (0,1)-form with values in. O( 2h 2) A 1...A 2h is a space-time field of helicity 2h. via the Penrose transform A 1...A 2h = 1 2 i where D = AB Ad B h d i is the volume form on CP 1. Can prove [Hitchen 80, Atiyah 79] L x A 1... A 2h ^ D H 0,1 (PT; O( 2h 2)) = space of solutions on CM of r A0 1 A 1 A 1...A 2h =0

8 Generalized Weyl C s µ(s) (s) Self-dual Actions For YM and Weyl Gravity (and linearized HS) Schematically we can write this as curvature tensors can be decomposed as: C (s) A(s)A 0 (s) B(s)B 0 (s) = A I B I e A 0 I B 0 I + A 0 I B0 I A I B I S s [ ]= 1 2" 2 d 4 x S s [ ]= 1 " 2 d 4 x A I B I A I B I + e A 0 I B0 I e A 0 I B0 I + Boundary term or introducing Lagrange multiplier S s [,G]= d 4 xg AI B I A I B I " 2 2 d 4 xg AI B I G A I B I. which has E.o.M.: A I B I = " 2 G AI B I, r A I A 0 I GAI B I =0. Provides an expansion about self-dual sector: " =0

9 Complex curved projective twistor spaces PT Penrose/Ward Self-dual space-times: ABCD =0 Form curved twistor spaces T by deforming the complex f + where the deformation is a (1,0)-vector valued (0,1)-form f = f To be well-defined on projective space, PT, must have homogeneity 1 under coordinate rescalings. Also has the invariance f = f + The condition for the deformation to give rise to an integrable complex structures f 2 + f f )@ =0 f

10 Write an action, [Berkovits/Witten 04], by introducing a Lagrange multiplier 2 f + f f )@ =0

11 Write an action, [Berkovits/Witten 04], by introducing a Lagrange multiplier field S S.D. = ^ g ^ + f f ) PT

12 Write an action, [Berkovits/Witten 04], by introducing a Lagrange multiplier field S S.D. = ^ g ^ + f f ) PT Holomorphic (3,0) volume form of homogeneity 4. In terms of homogeneous coordinates D 3 = d 4

13 Write an action, [Berkovits/Witten 04], by introducing a Lagrange multiplier field S S.D. = ^ g ^ + f f ) PT (0,1)-form with homogeneity -5 satisfying constraint g =0

14 Write an action, [Berkovits/Witten 04], by introducing a Lagrange multiplier field S S.D. = ^ g ^ + f f ) Equation of motion for tensor f g =0 PT P.T. r C A 0 r D B 0 G ABCD =0 Describes a helicity-(-2) particle moving in a self-dual background. Corresponds to self-dual sector of Weyl gravity S S.D. = d 4 x p gg ABCD ABCD

15 Write an action, [Berkovits/Witten 04], by introducing a Lagrange multiplier field S S.D. = ^ g ^ + f f ) Equation of motion for tensor f g =0 PT P.T. r C A 0 r D B 0 G ABCD =0 Describes a helicity-(-2) particle moving in a self-dual background. Corresponds to self-dual sector of Weyl gravity S S.D. = d 4 x p gg ABCD ABCD To include the anti-self-dual interactions we add term S A.S.D. = d 4 x p gg ABCD G ABCD This is equivalent to usual Weyl action up to topological terms.

16 Write an action, [Berkovits/Witten 04], by introducing a Lagrange multiplier field S S.D. = ^ g ^ + f f ) Equation of motion for tensor f g =0 PT P.T. r C A 0 r D B 0 G ABCD =0 Describes a helicity-(-2) particle moving in a self-dual background. Corresponds to self-dual sector of Weyl gravity S S.D. = d 4 x p gg ABCD ABCD To include the anti-self-dual interactions we add twistor term constructed by using the Penrose transform involving the deformation which gives rise to a MHV vertex like expansion. Can be used to very efficiently calculate all Einstein gravity MHV scattering amplitude.

17 Following Maldacena s argument for truncating conformal gravity to Einstein gravity with a non-zero cosmological constant, Adamo and Mason gave a twistor prescription for extracting EG amplitudes by restricting the fields to a Unitary sector : f = h g = I h } h, h describe ±2 helicity gravitons Use infinity twistor for AdS space, I AB 0 = 0 A0 B 0 plane-wave wave-functions for h and h and produces AdS analogue of scattering amplitudes which in the flat limit reproduce Hodges formula for MHV amplitudes after accounting for overall powers of. Can we generalise this to Higher Spin Fields?

18 Higher Spin Deformations Consider deformations of Dolbeault operator for a curved twistor space corresponding to an arbitrary self-dual space @ f + f

19 Consider deformations of Dolbeault operator for a curved twistor space corresponding to an arbitrary self-dual space time using a multi-index notation. Higher Spin f + f I f I is a rank-n symmetric tensor valued (0,1)-form of homogeneity n on T. In order to be defined on PT it must have the invariance f 1... n! f 1... n + ( n ) Spin-( I +1)

20 Consider deformations of Dolbeault operator for a curved twistor space corresponding to an arbitrary self-dual space time using a multi-index notation. Higher Spin f + f I f I is a rank-n symmetric tensor valued (0,1)-form of homogeneity n on T. In order to be defined on PT it must have the invariance f 1... n! f 1... n + ( n ) Spin-( I +1) symmetric tensor of homogeneity n-1 ( 1... n 1 )

21 Higher Spin Deformations Consider deformations of Dolbeault operator for a curved twistor space corresponding to an arbitrary self-dual space time using a multi-index f + f I Spin-( I +1) We impose the equation of motion I + f I I f I 2 =0 and can write an action function by introducing a Lagrange multiplier field S S.D. = ^ g I ^ I + f I I f I ) PT

22 Higher Spin Deformations Consider deformations of Dolbeault operator for a curved twistor space corresponding to an arbitrary self-dual space time using a multi-index f + f I Spin-( I +1) We impose the equation of motion I + f I I f I 2 =0 and can write an action function by introducing a Lagrange multiplier field S S.D. = ^ g I ^ I + f I I f I ) PT g I = g ( 1... n ) is a (0,1)-form of homogeneity -n-4 s.t. g 1... n 1 =0

23 Linearized Deformations At a linearized level the deformation and the Lagrange multiplier define elements of cohomology groups H 0,1 (PT ; O( n 4)) & H 0,1 (PT ; O(n)) Via the Penrose transform for homogeneous tensors [Eastwood, Mason] we 1... n =0 P.T. r A0 1 A 1...r A0 n A n G A1...A n B 1...B n 1... n =0 r A0 1 (A 1...r A0 n A n B 1...B n )A A0 n =0

24 Linearized Deformations At a linearized level the deformation and the Lagrange multiplier define elements of cohomology groups H 0,1 (PT ; O( n 4)) H 0,1 (PT ; O(n)) & Straightforward to calculate flat-space on-shell spectrum á la Witten & Berkovits : e.g. spin = 12 on-shell states

25 Linearized Conserved Charges There is a nice trick for calculating conserved charges in a linearised spin-n/2 massless theory. Given a field satisfying r A 1A 0 1 and a solution of the twistor equation A 1...A n =0 r B0 1 (B 1 B 1...B n 2 ) =0 we can form a solution to Maxwell equation A 1 A 2 = A1...A n A 3...A n and so a complex charge if + F S µ + iq = 1 4 I S AB A0 B 0 dx AA0 ^ dx BB0 Hence for every solution of the twistor equation we find two conserved charges (half are actually zero).

26 Linearized Conserved Charges For conformal higher-spin theory we can do a similar construction. Given a conformal higher-spin field we can use a symmetric trace-free G 1... s 1 B1...B s+1 [ s s] -twistor Ŵ 1... s s 1 to form a solution of the Maxwell equation B 1 B 2 = G 1... s 1 B1B2...B s+1 W 1... s 1 B 3...B s+1 Each charge corresponds to a solution of the twistor equation. Easy to count in flat-space that there are 1 12 s2 (s + 1) 2 (2s + 1) 2 This is essentially the number of conformal Killing tensors and matches with the calculation of Eastwood. Here it is easy to generalise to arbitrary self-dual backgrounds.

27 At linearized level twistor action appears to describe conformal higherspin theory. What about non-linear terms?

28 Infinite-Spin Deformations We define a deformed Dolbeault operator involving all f 1X + f J then demand that each term vanishes J 2 f I + I X 1X C K J f ( J K K f I J ) =0 J =0 K =0 Lower-spin fields source higher-spin fields and can t truncate to just e.g. spin-3 fields. Self-dual action is simply the sum of constraints. Defines a holomorphic structure on the infinite jet bundle of symmetric product of dual tangent bundles of twistor space. Can straightforwardly write an action by introducing Lagrange multiplier fields.

29 Unitary Self-Interactions On-shell flat space spectrum forms a non-diagonalizable representation of Poincaré algebra. We can identify a unitary sub-sector analogous to EG in CG f 1... n = I I n 1...@ n h s h s & h s g 1... n = I I n n 1... n hs describe helicity ±s states. Choose AdS background, evaluate action on plane-wave solutions to find the AdS analogue of 3-pt MHV-bar amplitudes fm 3, 1 ( s 1, +s 2, +s 3 )= s 1 1 en (s 1,s 2,s 3 ) [2 3] s 1+s 2 +s 3 [1 2] s 1+s 3 s 2 [3 1] s 1 +s 2 s 3 (1 + P ) s (P ) Thus accounting for powers of we reproduce the unique flat space answer consistent with Poincaré symmetry and helicity constraints.

30 ASD interactions So far we have only considered the self-dual sector to motivate the interactions consider the quadratic case S spin n = d 4 xg A I d 4 xg A I G AI A I P.T. 1 2 P.T. x AA0 M It is convenient to picture real, Euclidean space-times. In this case PT is fibered over the space-time M thus we see the action corresponds to a fibre wise product of twistor spaces PT M PT

31 ASD interactions We can use the twistor formulation to propose a full ASD action but it s rather involved. The interaction term for arbitrary negative helicity CHS fields on a conformal gravity background is relatively pleasing S (2) int [f (2),g]= PT M PT ^ 0 1 X I=0 I 0 I g I () g I ( 0 ) and can be expanded to compute arbitrary (-s, -s, 2, 2, 2,...) MHV amplitudes in a flat-space limit: lim!0 fm n,0 / 4 (P ) h12i2s+2 h1ii 2 h2ii 2 12i 12i, where ij = [ij] hiji, ii = for i 6= j However here it is defined for arbitrary self-dual backgrounds and so provides a general quadratic coupling of CHS fields to conformal gravity. X j6=i [ij] hiji h ji 2 h ii 2.

32 Conclusions/Outlook Described a twistor action to describe higher-spin fields on arbitrary self-dual backgrounds. Flat space-time action/spectrum is that of conformal higher-spin theory and produces linearised charges. Interactions involve one copy of all spins s>0. Have interpretation of holomorphic structure for infinite jet bundle of homogeneous tensors. Identified a unitary sub-sector, analogues of EG inside CG, which can be used to reproduce MHV and MHV-bar 3-pt amplitudes up powers of cosmological constant. Can be used to calculate n-point (-s, -s, 2, 2, 2,...) MHV amplitude. What is the full non-linear interacting theory. CHS theory of Segal? What is the unitary subsector? Can we calculate something interesting? All MHV amplitudes? Correlation functions? Partition functions? Classical solutions? Instantons? Black Holes?