Higher spins and twistor theory

Transcription

1 Higher spins and twistor theory Tim Adamo Imperial College London New Horizons in Twistor Theory 5 January 2017 Work with P. Haehnel & T. McLoughlin [arxiv: ] T Adamo (Imperial) Higher spins + twistors 5 January / 32

2 No-go theorems for s > 2 Many familiar constraints on interacting higher-spins Low-energy: soft limits + gauge/lorentz invariance [Weinberg] S-matrix symmetries: Poincaré + local symmetries trivially [Coleman-Mandula] Curved background: covariant local action + gauge invariance [Aragone-Deser] BCFW: tree-level S-matrix constructable by BCFW recursion [Benincasa-Cachazo] T Adamo (Imperial) Higher spins + twistors 5 January / 32

3 No-go theorems for s > 2 Many familiar constraints on interacting higher-spins Low-energy: soft limits + gauge/lorentz invariance [Weinberg] S-matrix symmetries: Poincaré + local symmetries trivially [Coleman-Mandula] Curved background: covariant local action + gauge invariance [Aragone-Deser] BCFW: tree-level S-matrix constructable by BCFW recursion [Benincasa-Cachazo] Of course, these are only as good as their assumptions (flat space, finite spectrum, unitarity, manifest covariance, etc.) T Adamo (Imperial) Higher spins + twistors 5 January / 32

4 Why care about higher spins? String theory contains infinite tower of massive higher spins Required by causality if GR is deformed by higher-derivatives [Camanho-Edelstein-Maldacena-Zhiboedov] Related to CFTs in general dimension [Klebanov-Polyakov, Giombi-Yin, Maldacena-Zhiboedov, Tseytlin] Good UV or quantum properties [Fradkin-Tseytlin, Beccaria-Tseytlin, Giombi-Klebanov-Pufu-Safdi-Tarnopolsky] T Adamo (Imperial) Higher spins + twistors 5 January / 32

5 Today... Focus on conformal higher-spin (CHS) theory in d = 4 T Adamo (Imperial) Higher spins + twistors 5 January / 32

6 Today... Focus on conformal higher-spin (CHS) theory in d = 4 Escapes no-go theorems via higher-derivative e.o.m.s (non-unitary) Like higher-spin generalization of Yang-Mills & conformal gravity Renormalizable (UV finite provided anomalies vanish) T Adamo (Imperial) Higher spins + twistors 5 January / 32

7 Today... Focus on conformal higher-spin (CHS) theory in d = 4 Escapes no-go theorems via higher-derivative e.o.m.s (non-unitary) Like higher-spin generalization of Yang-Mills & conformal gravity Renormalizable (UV finite provided anomalies vanish) Basic Idea: Conformal symmetry twistors unitary truncation T Adamo (Imperial) Higher spins + twistors 5 January / 32

8 Linear CHS theory Let φ (µ1 µ s) := φ µ(s) be a spin s gauge field. S s [φ] = 1 ε 2 d 4 x φ µ(s) P µ(s)ν(s) φ ν(s) [Fradkin-Tseytlin] P µ(s)ν(s) O( 2s ), ε dimensionless T Adamo (Imperial) Higher spins + twistors 5 January / 32

9 Linear CHS theory Let φ (µ1 µ s) := φ µ(s) be a spin s gauge field. S s [φ] = 1 ε 2 d 4 x φ µ(s) P µ(s)ν(s) φ ν(s) [Fradkin-Tseytlin] P µ(s)ν(s) O( 2s ), ε dimensionless Invariant under generalized conformal transformations: δφ µ(s) = (µ1 ɛ µ(s 1)) + η (µ1 µ 2 α µ(s 2)) Equations of motion: P µ(s)ν(s) φ ν(s) ( 2s φ) µ(s) = 0 T Adamo (Imperial) Higher spins + twistors 5 January / 32

10 Alternatively, define linearized Weyl curvatures [Fradkin-Linetsky, Damour-Deser, Marnelius, Vasiliev] C (s) µ(s)ν(s) ( s φ) µ(s)ν(s) C (s) = 0 φ µ(s) conformally trivial T Adamo (Imperial) Higher spins + twistors 5 January / 32

11 Alternatively, define linearized Weyl curvatures [Fradkin-Linetsky, Damour-Deser, Marnelius, Vasiliev] C (s) µ(s)ν(s) ( s φ) µ(s)ν(s) C (s) = 0 φ µ(s) conformally trivial S s [φ] = 1 ε 2 d 4 x C (s) 2 Generalized Bach equations: B (s) µ(s) = ν(s) C (s) µ(s)ν(s) = 0 T Adamo (Imperial) Higher spins + twistors 5 January / 32

12 Alternatively, define linearized Weyl curvatures [Fradkin-Linetsky, Damour-Deser, Marnelius, Vasiliev] C (s) µ(s)ν(s) ( s φ) µ(s)ν(s) C (s) = 0 φ µ(s) conformally trivial S s [φ] = 1 ε 2 d 4 x C (s) 2 Generalized Bach equations: B (s) µ(s) = ν(s) C (s) µ(s)ν(s) = 0 s = 1 Maxwell, s = 2 lin. Weyl gravity,... T Adamo (Imperial) Higher spins + twistors 5 January / 32

13 Expansion around SD sector Decompose C (s) into SD & ASD parts: Ψ α(s) β(s), Ψ α(s)β(s) S s [φ] = 1 2ε 2 d 4 x ( Ψ 2 + Ψ 2) Bach equations: α(s) α(s) Ψ α(s)β(s) = 0 = α(s) α(s) Ψ α(s) β(s) T Adamo (Imperial) Higher spins + twistors 5 January / 32

14 Expansion around SD sector Decompose C (s) into SD & ASD parts: Ψ α(s) β(s), Ψ α(s)β(s) S s [φ] = 1 2ε 2 d 4 x ( Ψ 2 + Ψ 2) Bach equations: α(s) α(s) Ψ α(s)β(s) = 0 = α(s) α(s) Ψ α(s) β(s) S s [φ] = 1 ε 2 d 4 x Ψ 2 + total derivative T Adamo (Imperial) Higher spins + twistors 5 January / 32

15 Expansion around SD sector Decompose C (s) into SD & ASD parts: Ψ α(s) β(s), Ψ α(s)β(s) S s [φ] = 1 2ε 2 d 4 x ( Ψ 2 + Ψ 2) Bach equations: α(s) α(s) Ψ α(s)β(s) = 0 = α(s) α(s) Ψ α(s) β(s) S s [φ] = 1 ε 2 d 4 x Ψ 2 + total derivative Lagrange multiplier G α(s)β(s) to obtain Chalmers-Siegel type Lagrangian: S s [φ, G] = d 4 x G Ψ ε2 d 4 x G 2 2 T Adamo (Imperial) Higher spins + twistors 5 January / 32

16 Linearized Field Equations Ψ α(s)β(s) = ε 2 G α(s)β(s), α(s) α(s) G α(s)β(s) = 0 ε controls expansion around SD (Ψ α(s)β(s) = 0) sector T Adamo (Imperial) Higher spins + twistors 5 January / 32

17 Linearized Field Equations Ψ α(s)β(s) = ε 2 G α(s)β(s), α(s) α(s) G α(s)β(s) = 0 ε controls expansion around SD (Ψ α(s)β(s) = 0) sector Clear analogy with well-studied examples of Yang-Mills & conformal gravity T Adamo (Imperial) Higher spins + twistors 5 January / 32

18 Linearized Field Equations Ψ α(s)β(s) = ε 2 G α(s)β(s), α(s) α(s) G α(s)β(s) = 0 ε controls expansion around SD (Ψ α(s)β(s) = 0) sector Clear analogy with well-studied examples of Yang-Mills & conformal gravity Question: can we formulate these linear theories in twistor space? T Adamo (Imperial) Higher spins + twistors 5 January / 32

19 Free CHS fields on twistor space Idea: use Penrose transform to encode free e.o.m. [Baston-Eastwood, Mason] PT open subset of P 3, Z A = (µ α, λ α ) homog. coords. T Adamo (Imperial) Higher spins + twistors 5 January / 32

20 Free CHS fields on twistor space Idea: use Penrose transform to encode free e.o.m. [Baston-Eastwood, Mason] PT open subset of P 3, Z A = (µ α, λ α ) homog. coords. Let g A(s 1) (Z) Ω 0,1 (PT, O( s 3)) for s 1 (section of s 1 TPT ) Γ A(s 1) β(s+1) = Dλ λ β1 λ βs+1 g A(s 1) X X T Adamo (Imperial) Higher spins + twistors 5 January / 32

21 Free CHS fields on twistor space Idea: use Penrose transform to encode free e.o.m. [Baston-Eastwood, Mason] PT open subset of P 3, Z A = (µ α, λ α ) homog. coords. Let g A(s 1) (Z) Ω 0,1 (PT, O( s 3)) for s 1 (section of s 1 TPT ) Γ A(s 1) β(s+1) = Dλ λ β1 λ βs+1 g A(s 1) X X β β Γ A(s 1) β(s+1) = 0 g A(s 1) H 0,1 (PT, O( s 3)) where acts on Γ via the local twistor connection T Adamo (Imperial) Higher spins + twistors 5 January / 32

22 Impose gauge condition Z A g AA(s 2) = 0. all components of Γ solved for in terms of derivatives of G α(s 1)β(s+1) = G (α(s 1)β(s+1)), obeying α(s) α(s) G α(s)β(s) = 0 T Adamo (Imperial) Higher spins + twistors 5 January / 32

23 Impose gauge condition Z A g AA(s 2) = 0. all components of Γ solved for in terms of derivatives of G α(s 1)β(s+1) = G (α(s 1)β(s+1)), obeying α(s) α(s) G α(s)β(s) = 0 Similar story (potentials modulo gauge) for f A(s 1) H 0,1 (PT, O(s 1)) Ψ α(s)β(s) = 0 T Adamo (Imperial) Higher spins + twistors 5 January / 32

24 Upshot Solutions to free, spin-s Bach equations given by f A(s 1) H 0,1 (PT, O(s 1)), g A(s 1) H 0,1 (PT, O( s 3)) T Adamo (Imperial) Higher spins + twistors 5 January / 32

25 Upshot Solutions to free, spin-s Bach equations given by f A(s 1) H 0,1 (PT, O(s 1)), g A(s 1) H 0,1 (PT, O( s 3)) Free CHS action (in Chalmers-Siegel form) equivalent to: [Haehnel-McLoughlin] S s [f, g] = PT D 3 Z g AI f A I ε2 2 PT M PT D 3 Z 1 D 3 Z 2 Z A I 1 Z B I 2 g B I (Z 1 ) g AI (Z 2 ) where g AI g A( I ) and I = s 1, etc. T Adamo (Imperial) Higher spins + twistors 5 January / 32

26 Degrees of freedom Spin s gauge invariances & constraints f A 1 A s 1 f A 1 A s 1 + Z (A 1 Λ A 2 A s 1 ), Z A 1 g A1 A s 1 = 0 T Adamo (Imperial) Higher spins + twistors 5 January / 32

27 Degrees of freedom Spin s gauge invariances & constraints f A 1 A s 1 f A 1 A s 1 + Z (A 1 Λ A 2 A s 1 ), Z A 1 g A1 A s 1 = 0 D.o.f. in f A(s 1) : 1 s (s + 1) (s + 2) } 6 {{} degrees of freedom T Adamo (Imperial) Higher spins + twistors 5 January / 32

28 Degrees of freedom Spin s gauge invariances & constraints f A 1 A s 1 f A 1 A s 1 + Z (A 1 Λ A 2 A s 1 ), Z A 1 g A1 A s 1 = 0 D.o.f. in f A(s 1) : 1 s (s + 1) (s + 2) } 6 {{} degrees of freedom 1 s (s 1) (s + 1) 6 } {{ } constraints T Adamo (Imperial) Higher spins + twistors 5 January / 32

29 Degrees of freedom Spin s gauge invariances & constraints f A 1 A s 1 f A 1 A s 1 + Z (A 1 Λ A 2 A s 1 ), Z A 1 g A1 A s 1 = 0 D.o.f. in f A(s 1) : 1 s (s + 1) (s + 2) } 6 {{} degrees of freedom 1 s (s 1) (s + 1) 6 } {{ } constraints = 1 s (s + 1) 2 T Adamo (Imperial) Higher spins + twistors 5 January / 32

30 Degrees of freedom Spin s gauge invariances & constraints f A 1 A s 1 f A 1 A s 1 + Z (A 1 Λ A 2 A s 1 ), Z A 1 g A1 A s 1 = 0 D.o.f. in f A(s 1) : 1 s (s + 1) (s + 2) } 6 {{} degrees of freedom 1 s (s 1) (s + 1) 6 } {{ } constraints = 1 s (s + 1) 2 + same for g A(s 1) T Adamo (Imperial) Higher spins + twistors 5 January / 32

31 Degrees of freedom Spin s gauge invariances & constraints f A 1 A s 1 f A 1 A s 1 + Z (A 1 Λ A 2 A s 1 ), Z A 1 g A1 A s 1 = 0 D.o.f. in f A(s 1) : 1 s (s + 1) (s + 2) } 6 {{} degrees of freedom 1 s (s 1) (s + 1) 6 } {{ } constraints = 1 s (s + 1) 2 + same for g A(s 1) So s (s + 1) on-shell d.o.f. for each s Correct! [Riegert, Fradkin-Tseytlin] T Adamo (Imperial) Higher spins + twistors 5 January / 32

32 OK, so we can describe linear CHS theory in twistor space... So what? Want to compute something with twistors that traditional methods can t get at T Adamo (Imperial) Higher spins + twistors 5 January / 32

33 Interacting CHS theory Let χ i be N free, massless complex scalars. Infinite tower of symmetry currents J µ(s) = χ i µ(s) χ i. Interacting CHS as induced theory: [Tseytlin, Segal, Bekaert-Joung-Mourad] ( S CHS [φ] log det 2 + φ µ(s) J µ(s)) UV s=0 div. T Adamo (Imperial) Higher spins + twistors 5 January / 32

34 Interacting CHS theory Let χ i be N free, massless complex scalars. Infinite tower of symmetry currents J µ(s) = χ i µ(s) χ i. Interacting CHS as induced theory: [Tseytlin, Segal, Bekaert-Joung-Mourad] ( S CHS [φ] log det 2 + φ µ(s) J µ(s)) UV s=0 div. Infinite tower of higher-spin fields (s = 1, 2,... in d = 4) Vanishing partition function + conformal anomalies on M, S 4 Flat space vacuum, single dimensionless coupling constant Quadratic action S CHS [φ] quad = s=1 S s [φ] T Adamo (Imperial) Higher spins + twistors 5 January / 32

35 Gives local (UV finite) interacting higher-spin theory, but... Induced definition hard to work with Formulation in terms of gauge fields φ µ(s) instead of curvatures C (s) µ(s)ν(s) T Adamo (Imperial) Higher spins + twistors 5 January / 32

36 Gives local (UV finite) interacting higher-spin theory, but... Induced definition hard to work with Formulation in terms of gauge fields φ µ(s) instead of curvatures C (s) µ(s)ν(s) Example: Conjecture [Joung-Nakach-Tseytlin, Beccaria-Nakach-Tseytlin] The S-matrix of CHS theory, restricted to two-derivative external states, is trivial order-by-order in perturbation theory Strong motivation, but very difficult to show...only 4-point done explicitly T Adamo (Imperial) Higher spins + twistors 5 January / 32

37 Twistor theory to the rescue! free CHS interacting CHS natural in twistor space infinite classes of tree amplitudes easily computed (MHV & MHV) restriction to two-derivative subsector straightforward T Adamo (Imperial) Higher spins + twistors 5 January / 32

38 Twistor theory to the rescue! free CHS interacting CHS natural in twistor space infinite classes of tree amplitudes easily computed (MHV & MHV) restriction to two-derivative subsector straightforward Will raise several intriguing questions about CHS on space-time and higher-spins more generally T Adamo (Imperial) Higher spins + twistors 5 January / 32

39 Interacting CHS on Twistor Space Recall s = 2 case: + f A A, PT PT [Penrose, Mason, TA-Mason] T Adamo (Imperial) Higher spins + twistors 5 January / 32

40 Interacting CHS on Twistor Space Recall s = 2 case: + f A A, PT PT [Penrose, Mason, TA-Mason] Now, define f := + I =0 f A I AI f almost complex structure on infinite jet bundle of PT T Adamo (Imperial) Higher spins + twistors 5 January / 32

41 Interacting CHS on Twistor Space Recall s = 2 case: + f A A, PT PT [Penrose, Mason, TA-Mason] Now, define f := + I =0 f A I AI f almost complex structure on infinite jet bundle of PT Integrability: N A I = f A I + I J =0 K =0 ( ) J + K f B K (A J BK f A I J) = 0 J T Adamo (Imperial) Higher spins + twistors 5 January / 32

42 Interacting CHS on Twistor Space Recall s = 2 case: + f A A, PT PT [Penrose, Mason, TA-Mason] Now, define f := + I =0 f A I AI f almost complex structure on infinite jet bundle of PT Integrability: N A I N A I = f A I + I J =0 K =0 ( ) J + K f B K (A J BK f A I J) = 0 J suggest higher-spin generalization of Newlander-Nirenberg I = 0, 1: truncations self-consistent (Yang-Mills & CG) I 2: N A I contains source terms for all lower spins full infinite tower of CHS fields T Adamo (Imperial) Higher spins + twistors 5 January / 32

43 SD Sector This gives natural interacting generalization of SD action: [Haehnel-McLoughlin] S SD [f, g] = PT Ω g AI N A I I =0 with e.o.m.s N A I = 0, f g AI = 0, I = 0, 1,... T Adamo (Imperial) Higher spins + twistors 5 January / 32

44 SD Sector This gives natural interacting generalization of SD action: [Haehnel-McLoughlin] S SD [f, g] = PT Ω g AI N A I I =0 with e.o.m.s N A I = 0, f g AI = 0, I = 0, 1,... For restriction to I = 0, 1, equivalent to SD Yang-Mills and conformal gravity T Adamo (Imperial) Higher spins + twistors 5 January / 32

45 ASD Interactions Add terms to action for quadratic ASD fields on SD background Example: coupling to SD s = 2 background S (2) int [f (2), g] = PT M PT Ω 1 Ω 2 I =0 Z A I 1 ZB I 2 g B I (Z 1 ) g AI (Z 2 ). T Adamo (Imperial) Higher spins + twistors 5 January / 32

46 ASD Interactions Add terms to action for quadratic ASD fields on SD background Example: coupling to SD s = 2 background S (2) int [f (2), g] = PT M PT Ω 1 Ω 2 I =0 Z A I 1 ZB I 2 g B I (Z 1 ) g AI (Z 2 ). PT M PT = M P 1 P 1, M defined by f (2) = + f A A for each x M, f (2)Z A (x, σ) = 0 Z A (x, σ) = f A (Z(x, σ)) T Adamo (Imperial) Higher spins + twistors 5 January / 32

47 ASD Interactions Add terms to action for quadratic ASD fields on SD background Example: coupling to SD s = 2 background S (2) int [f (2), g] = PT M PT Ω 1 Ω 2 I =0 Z A I 1 ZB I 2 g B I (Z 1 ) g AI (Z 2 ). PT M PT = M P 1 P 1, M defined by f (2) = + f A A for each x M, f (2)Z A (x, σ) = 0 Z A (x, σ) = f A (Z(x, σ)) Note: interactions for s > 2 background not well understood T Adamo (Imperial) Higher spins + twistors 5 January / 32

48 So full interacting theory in twistor space: S CHS [f, g] = S SD [f, g] ε2 2 s=1 S (s) int [f, g] T Adamo (Imperial) Higher spins + twistors 5 January / 32

49 So full interacting theory in twistor space: A few obvious questions: S CHS [f, g] = S SD [f, g] ε2 2 s=1 S (s) int [f, g] T Adamo (Imperial) Higher spins + twistors 5 January / 32

50 So full interacting theory in twistor space: A few obvious questions: S CHS [f, g] = S SD [f, g] ε2 2 s=1 S (s) int [f, g] Question 1 Is this theory equivalent to the induced CHS theory on space-time? T Adamo (Imperial) Higher spins + twistors 5 January / 32

51 So full interacting theory in twistor space: A few obvious questions: S CHS [f, g] = S SD [f, g] ε2 2 s=1 S (s) int [f, g] Question 1 Is this theory equivalent to the induced CHS theory on space-time? Question 2 Is there a formulation of CHS theory on space-time in terms of conformal curvatures, and does it have a pertubative expansion around a SD sector? T Adamo (Imperial) Higher spins + twistors 5 January / 32

52 Answering these questions obviously interesting & important but for now... let s just compute with the twistor action and see what we find! T Adamo (Imperial) Higher spins + twistors 5 January / 32

53 Tree amplitudes in twistor space Our theory has a MHV decomposition; two classes of amplitude easily accessed: 3-point MHV cubic part of S SD n-point MHV on CG background generated by S (2) [f (2), g] T Adamo (Imperial) Higher spins + twistors 5 January / 32

54 Tree amplitudes in twistor space Our theory has a MHV decomposition; two classes of amplitude easily accessed: 3-point MHV cubic part of S SD n-point MHV on CG background generated by S (2) [f (2), g] External states encoded using helicity raising/lowering operators: g AI = B AI ψ ( 3 s), f A I = A A I ψ (s 1) ψ ( 3 s) H 0,1 (PT, O( 3 s)), ψ (s 1) H 0,1 (PT, O(s 1)) T Adamo (Imperial) Higher spins + twistors 5 January / 32

55 Using dual twistor wavefunctions = B AI dt t I +3 e t W Z, f A I = A A I g AI dt W Z et t I +1 can obtain explicit (if opaque) expressions... T Adamo (Imperial) Higher spins + twistors 5 January / 32

56 Using dual twistor wavefunctions = B AI dt t I +3 e t W Z, f A I = A A I g AI dt W Z et t I +1 can obtain explicit (if opaque) expressions... Example: M 3, 1 ( s 1, +s 2, +s 3 ) = N (s 1,s 2,s 3 ) ( (s 2 1)! (s 1 s 2 )! (B 1 A 2 ) s 1 s 3 (B 1 A 3 ) s3 1 (A 2 W 1 ) s ( 1) s (s 3 1)! (s 1 s 3 )! (B 1 A 2 ) s2 1 (B 1 A 3 ) s 1 s 2 (A 3 W 1 ) s ) 23 1 for s ij k := s i + s j s k 1, N (s 1,s 2,s 3 ) := [23]s 2+s 3 +1 [12] s 3 [31] s 2 δ 2 ([23] µ 1 + [31] µ 2 + [12] µ 3 ) 1 s 23 1! (s 1 s 3 )! (s 1 s 2 )! T Adamo (Imperial) Higher spins + twistors 5 January / 32

57 The unitary subsector Want to test Tseytlin s conjecture. Need unitary external states. T Adamo (Imperial) Higher spins + twistors 5 January / 32

58 The unitary subsector Want to test Tseytlin s conjecture. Need unitary external states. Conformal invariance broken with infinity twistor I AB = ( Λɛ α β 0 0 ɛ αβ Λ the cosmological constant of (A)dS 4 ) ( ), I AB ɛ α β 0 = 0 Λɛ αβ T Adamo (Imperial) Higher spins + twistors 5 January / 32

59 The unitary subsector Want to test Tseytlin s conjecture. Need unitary external states. Conformal invariance broken with infinity twistor I AB = ( Λɛ α β 0 0 ɛ αβ Λ the cosmological constant of (A)dS 4 External States: ) ( ), I AB ɛ α β 0 = 0 Λɛ αβ g AI I AI B I Z B I h ( I +1) (Z), f A I I A I B I BI h ( I +1) (Z) h (s) Ω 0,1 (PT, O( 2s 2)), h (s) Ω 0,1 (PT, O(2s 2)) T Adamo (Imperial) Higher spins + twistors 5 January / 32

60 Find: M 3, 1 ( s 1, +s 2, +s 3 ) unit. O(Λ s 1 1 ) M n,0 ( s, s, +2,...) unit. O(Λ) T Adamo (Imperial) Higher spins + twistors 5 January / 32

61 Find: M 3, 1 ( s 1, +s 2, +s 3 ) unit. O(Λ s 1 1 ) M n,0 ( s, s, +2,...) unit. O(Λ) So for Λ 0, all such amplitudes vanish, as desired! Much easier than computing from induced action on space-time T Adamo (Imperial) Higher spins + twistors 5 January / 32

62 Interesting fact... Non-vanishing answers for Λ 0 if re-scaled: M 3, 1 ( s 1, +s 2, +s 3 ) lim Λ 0 Λ s = 1 1 Ñ (s 1,s 2,s 3 ) δ 4 (P) M n,0 ( s, s, +2,...) lim δ 4 (P) Λ 0 Λ [2 3] s 1+s 2 +s 3 [1 2] s 1+s 3 s 2 [3 1] s 1 +s 2 s s+2 12i Φ 12i 1 i 2 2 i 2 T Adamo (Imperial) Higher spins + twistors 5 January / 32

63 Interesting fact... Non-vanishing answers for Λ 0 if re-scaled: M 3, 1 ( s 1, +s 2, +s 3 ) lim Λ 0 Λ s = 1 1 Ñ (s 1,s 2,s 3 ) δ 4 (P) M n,0 ( s, s, +2,...) lim δ 4 (P) Λ 0 Λ [2 3] s 1+s 2 +s 3 [1 2] s 1+s 3 s 2 [3 1] s 1 +s 2 s s+2 12i Φ 12i 1 i 2 2 i 2 Ñ (s 1,s 2,s 3 ) := 1+( 1)s Γ(s 1 +s 2 s 3 ) Γ(s 1 s 2 +s 3 ) (s 1 s 2 )! (s 1 s 3 )! Γ(s 2 +s 3 s 1 ) Φ the n n Hodges matrix T Adamo (Imperial) Higher spins + twistors 5 January / 32

64 Consider SD action, make unitary truncation at level of action. T Adamo (Imperial) Higher spins + twistors 5 January / 32

65 Consider SD action, make unitary truncation at level of action. S (s) SD [h, h (s) ] = PT + (s 1)! (2s 2)! S SD [f, g] unit. = s=1 ( D 3 Z h (s) h (s) s s 3 =1 s 2 =1+s s 3 Λ s 1 S (s) SD [h, h (s) ], { } ) Ñ (s,s2,s3) h (s2), h (s 3), (s 2 +s 3 s 1) T Adamo (Imperial) Higher spins + twistors 5 January / 32

66 Consider SD action, make unitary truncation at level of action. S (s) SD [h, h (s) ] = PT + (s 1)! (2s 2)! S SD [f, g] unit. = s=1 ( D 3 Z h (s) h (s) s s 3 =1 s 2 =1+s s 3 Λ s 1 S (s) SD [h, h (s) ], { } ) Ñ (s,s2,s3) h (s2), h (s 3), (s 2 +s 3 s 1) {f, g} (k) := I A I B I AI f BI g, I = k 1 T Adamo (Imperial) Higher spins + twistors 5 January / 32

67 Compare for example: SD Yang-Mills theory (s = 1) [Mason] S[h (1), h ( (1) ] = D 3 Z Tr [ h (1) h (1) + h (1) h (1))] PT T Adamo (Imperial) Higher spins + twistors 5 January / 32

68 Compare for example: SD Yang-Mills theory (s = 1) [Mason] S[h (1), h ( (1) ] = D 3 Z Tr [ h (1) h (1) + h (1) h (1))] PT SD Einstein gravity (s = 2) [Mason-Wolf] S[h (2), h (2) ] = D 3 Z h (2) PT ( h (2) + 1 { h (2), h (2)} ) 2 (1) T Adamo (Imperial) Higher spins + twistors 5 January / 32

69 Twistor higher-spin theories What are the properties of S (s) [h, h (s) ]? T Adamo (Imperial) Higher spins + twistors 5 January / 32

70 Twistor higher-spin theories What are the properties of S (s) [h, h (s) ]? Linear theory: T Adamo (Imperial) Higher spins + twistors 5 January / 32

71 Twistor higher-spin theories What are the properties of S (s) [h, h (s) ]? Linear theory: Lin. e.o.m.s: h(s) = 0, h (s) = 0, k = 1, 2,... Spectrum: one negative helicity, one positive helicity: ±s Massless (2-derivative) higher spin fields T Adamo (Imperial) Higher spins + twistors 5 January / 32

72 Twistor higher-spin theories What are the properties of S (s) [h, h (s) ]? Linear theory: Lin. e.o.m.s: h(s) = 0, h (s) = 0, k = 1, 2,... Spectrum: one negative helicity, one positive helicity: ±s Massless (2-derivative) higher spin fields Non-linear theory: Requires infinite tower of higher spin fields. So we should really consider S[h, h] = S (s) [h, h] s=1 Consistent s = 1, 2 truncations SD Maxwell & CG [Mason, Mason-Wolf] T Adamo (Imperial) Higher spins + twistors 5 January / 32

73 What s going on here? S[h, h] is well-defined in the Λ 0 limit T Adamo (Imperial) Higher spins + twistors 5 January / 32

74 What s going on here? S[h, h] is well-defined in the Λ 0 limit Aren t we violating no-go theorems? T Adamo (Imperial) Higher spins + twistors 5 January / 32

75 What s going on here? S[h, h] is well-defined in the Λ 0 limit Aren t we violating no-go theorems? Get out of jail: Lack of manifest covariance on space-time (e.g., s = 2) c.f., related work [Metsaev, Bengtsson, Ponomarev-Skvortsov] T Adamo (Imperial) Higher spins + twistors 5 January / 32

76 What s going on here? S[h, h] is well-defined in the Λ 0 limit Aren t we violating no-go theorems? Get out of jail: Lack of manifest covariance on space-time (e.g., s = 2) c.f., related work [Metsaev, Bengtsson, Ponomarev-Skvortsov] More generally, reminiscent of Einstein CG story [Maldacena, TA-Mason] T Adamo (Imperial) Higher spins + twistors 5 January / 32

77 More obvious questions: Question 3 Is there a unitary truncation of CHS theory on space-time which is related to a (non-conformal) HS theory at the level of couplings by G s 1 c s Λ s 1, where c s is the dimensionless CHS coupling and G s is the dimensionful HS coupling? T Adamo (Imperial) Higher spins + twistors 5 January / 32

78 More obvious questions: Question 3 Is there a unitary truncation of CHS theory on space-time which is related to a (non-conformal) HS theory at the level of couplings by G s 1 c s Λ s 1, where c s is the dimensionless CHS coupling and G s is the dimensionful HS coupling? Question 4 Can we prove an analogue of the non-linear graviton construction for (C)HS fields? T Adamo (Imperial) Higher spins + twistors 5 January / 32

79 Lots to do... Much left to understand, both on space-time and twistor space: Clarify relation between twistor and space-time constructions Understand all ASD interactions & consider their unitary truncation Beyond MHV & MHV sectors? Connected prescription? Relationship between unitary actions & light cone approaches Partition functions T Adamo (Imperial) Higher spins + twistors 5 January / 32