1. 1 Twistors, superarticles, twistor superstrings in various spacetimes Itzhak Bars Twistors in 4 flat dimensions; Some applications. –Massless particles, constrained phase space (x,p) versus twistors –Wavefunctions for massless spinning particles in twistor space –Simplifications in super Yang-Mills theory Introduction to 2T-physics and derivation of 1T-physics holographs –Sp(2,R) gauge symmetry, constraints, solutions and (d,2) –Holography, duality, 1T-images and physical interpretation –SO(d,2) global symmetry, 1T-interpretations: conformal symmetry and others –covariant quantization and SO(d,2) singleton. Supersymmetric 2T-physics, gauge symmetries & twistor gauge. –Coupling X,P,g (g=group element containing spinors); Gauge symmetries, global symmetries. –Twistor gauge: twistors and supertwistors in various dimensions as holographs dual to phase space. –Quantization, constrained generators, and representations of some superconformal groups Supertwistors and superparticle spectra in d=3,4,5,6,10 –Super Yang-Mills d=3,4; Supergravity d=3,4 –Self-dual supermultiplet and conformal theory in d=6 –AdS 5 xS 5 compactified type-IIB supergravity, KK-towersAdS 5 xS 5 –Nonlinear sigma model PSU(2,2|4)/SO(4,1xSO(5) versus PSU(2,2|4)/PSU(2|2)xU(1) twistors for AdS 5 xS 5 –Constrained twistors and their spectra – oscillator formalism for non-compact supergroups. Twistor superstrings –2T-view; worldsheet anomalies and quantization of twistor superstring –Spectra, vertex operators for twistor superstrings in d=3,4,6,10 –Computing amplitudes of the twistor superstring in d=4 (SYM, conformal supergravity, gravity). –Open problems.

2. 2 What are twistors in 4 flat dimensions?

3. 3 Physical states in twistor space

4. 4 Penrose Homework: find the correct wavefunctions with definite momentum and helicity

5. 5 2T-physics 2T-physics 2T-physics 1T spacetimes & dynamics (time, Hamiltonian) are emergent concepts from 2T phase space The same 2T system in (d,2) has many 1T holographic images in (d-1,1), obey duality Each 1T image has hidden symmetries that reveal the hidden dimensions (d,2) 1) Gauge symmetry Fundamental concept is Sp(2,R) gauge symmetry: Position and momentum (X,P) are indistinguishable at any instant. This symmetry demands 2T signature (-,-,+,+,+,…,+) to have nontrivial gauge invariant subspace Q ij (X,P)=0. Unitarity and causality are satisfied thanks to symmetry. 2) Holography 1T-physics is derived from 2T physics by gauge fixing Sp(2,R) from (d,2) phase space to (d-1,1) phase space. Can fix 3 pairs of (X,P), fix 2 or 3. The perspective of (d-1,1) in (d,2) determines “time” and H in the emergent spacetime. The same (d,2) system has many 1T holographic images with various 1T perspectives. 5) Unification Different observers can use different emergent (t,H) to describe the same 2T system. This unifies many emergent 1T dynamical systems into a single class that represents the same 2T system with an action based on some Q ij (X,P). 3) Duality 1T solutions of Q ij (X,P)=0 are dual to one another; duality group is gauge group Sp(2,R). Simplest example (see figure): (d,2) to (d-1,1) holography gives many 1T systems with various 1T dynamics. These are images of the same “free particle” in 2T physics in flat 2T spacetime. 4) Hidden symmetry (for the example in figure) The action of each 1T image has hidden SO(d,2) symmetry. Quantum: SO(d,2) global sym realized in same representation for all images, C 2 =1-d 2 /4. 6) Generalizations found Spinning particles: OSp(2|n); Spacetime SUSY Interactions with all backgrounds (E&M, gravity, etc.) 2T field theory; 2T strings/branes ( both incomplete) Twistor superstring 7) Generalizations in progress New twistor superstrings in higher dimensions. Higher unification, powerful guide toward M-theory 13D for M-theory (10,1)+(1,1)=(11,2) suggests OSp(1|64) global SUSY. Sp(2,R) gauge choices. Some combination of X M,P M fixed as t,Hgauge choices

6. 6 2T-physics 1T spacetimes & dynamics (time, Hamiltonian) are emergent concepts from 2T phase space The same 2T system in (d,2) has many 1T holographic images in (d-1,1). The images are dual to each other. Each 1T image has hidden symmetries that reveal the hidden dimensions (d,2).

7. 7 Gauge Symmetry Sp(2,R) Fundamental concept is Sp(2,R) gauge symmetry: Position and momentum (X,P) are indistinguishable at any instant. This symmetry demands 2T signature (-,-,+,+,+,…,+) to have nontrivial solutions of Q ik (X,P)=0 gauge invariant subspace (eq. of motion for A) signature Unitarity and causality are satisfied thanks to Sp(2,R) gauge symmetry. Global symmetry determined by form of Q ik (X,P). In the example it is SO(d,2). It is gauge invariant since it commutes with Q ik.

8. 8 Spacetime signature determined by gauge symmetry EMERGENT DYNAMICS AND SPACE-TIMES return

9. 9 Some examples of gauge fixing 2 gauge choices made.  reparametrization remains. 3 gauge choices made. Including  reparametrization.

10. 10 More examples of gauge fixing

11. 11 Background fields

12. 12 Holography and emergent spacetime 1T-physics is derived from 2T physics by gauge fixing Sp(2,R) from (d,2) phase space to (d-1,1) phase space. Can fix 3 pairs of (X,P): 3 gauge parameters and 3 constraints. Fix 2 or 3. The perspective of (d-1,1) in (d,2) determines “time” and Hamiltonian in the emergent spacetime. The same (d,2) system has many 1T holographic images with various 1T perspectives.

13. 13 Duality 1T solutions of Q ik (X,P)=0 (holographic images) are dual to one another. Duality group is gauge group Sp(2,R): Transform from one fixed gauge to another fixed gauge. Simplest example (figure): (d,2) to (d-1,1) holography gives many 1T systems with various 1T dynamics. These are images of the same “free particle” in 2T physics in flat 2T spacetime. Many emergent spacetimes

14. 14 Hidden dimensions/symmetries in 1T-physics and UNIFICATION Hidden dimensions/symmetries There is one extra time and one extra space. The action of each 1T image has hidden SO(d,2) symmetry in the flat case, or global symmetry of Q ik (X,P) in general case. The symmetry is a reflection of the underlying bigger spacetime. Example, conformal symmetry SO(d,2). Also H-atom, etc. Quantum: SO(d,2) global symmetry is realized for all images in the same unitary irreducible representation, with Casimir C 2 =1-d 2 /4. This is the singleton. Unification Different observers can use different emergent (t,H) to describe the same 2T system. This unifies many emergent 1T dynamical systems into a single class that represents the same 2T system with an action based on some Q ik (X,P).

15. 15 Generalizations Generalizations obtained Spinning particles: use OSp(2|n) Spacetime SUSY: special supergroups Interactions with all backgrounds (E&M, gravity, etc.) 2T field theory; 2T strings/branes ( both incomplete) Twistors in d=3,4,6,10,11 Twistor superstring in d=4 In progress New twistor superstrings in higher dimensions: d=3,4,6,10 Higher unification, powerful guide toward M-theory (hidden symmetries, dimensions) 13D for M-theory (10,1)+(1,1)=(11,2) suggests OSp(1|64) global SUSY.

16. 16 SO(d,2) unitary representation unique for a fixed spin=n/2. Obtain E&M, gravity, etc. in d dims from background fields  (X,P,  ) in d+2 dims. –> holographs.  (X,P,  ) expand in powers of P,  get fields    X).

17. 17 4) - Field Theory in 2T (0003100); Standard Model could be obtained as a holograph (Dirac, Salam). - Non-commutative FT  (X,P) (0104135, 0106013) similar to string field theory, Moyal star. 5) - String/brane theory in 2T (9906223, 0407239). -Twistor superstring in 2T (0407239, 0502065 ) both 4 & 5 need more work If D-branes admitted, then more general (super)groups can be used, in particular a toy M-model in (11,2)=(10,1)+(1,1) with G d =OSp(1|64) 13 Twistors emerge in this approach

18. 18 Particle gauge: Use local SO(d,2) to set g=1. Then we have only X,P. Action reduces to 2T particle in flat space. It gives the previous holographs. The global SO(d,2) current J reduces to orbital L invariant

19. 19 More dualities: 1T images of unique 2T-physics particle via gauge fixing Spacetime gauge -eliminate all bosons from g -fix (X,P) (d,2) to (d-1,1) (x,p): 1T particle (& duals) group/twistor gauge kill (X,P) completely keep only g constrained twistors/oscillators 2T-parent theory has (X,P) and g  -model gauge fix part of (X,P) L MN partly linear Integrate out remaining P e.g. AdS 5 xS 5 sigma model [SU(2,2)xSU(4)]/[SO(4,1)xSO(5)]

20. 20 Group & Twistor gauge

21. 21 Two SO(6,2) spinors one SO(4,2) spinor Only ONE block row of g ONE block column of I=1,2,3,4, SU(4)=SO(6) in SO(6,2)

22. 22 Compare two gauges through the gauge invariant J and relate the twistor variables Z to phase space variables x,p

23. 23 u is any kxk unitary matrix except for overall scale

24. 24 Spacetime SUSY 2T-superparticle Local symmetries OSp(n|2)xG d left including SO(d,2) and kappa Global symmetries: G d right Supergroup G d contains spin(d,2) and R-symmetry subgroups

25. 25 Local symmetry embedded in G left local spin(d,2) x R acts on g from left as spinors acts on (X,P) as vectors Local kappa symmetry (off diagonal in G) acts on g from left acts also on sp(2,R) gauge field A ij

26. 26 More dualities: 1T images of unique 2T-physics superparticle via gauge fixing 2T-parent theory has Y=(X,P,  ) and g  -model gauge fix part of (X,P,  ); L MN linear Integrate out remaining P e.g. AdS 5 xS 5 sigma model SU(2,2|4)/SO(4,1)xSO(5)

27. 27 Spacetime (or particle) gauge

28. 28

29. 29 Quantum states of d=4 superparticle with N supersymmetries N=4 gives SYM, N=8 gives SUGRA 2T-physics tells us that : singleton

30. 30 Quantum states of d=4 superparticle with N supersymmetries N=4 gives SYM, N=8 gives SUGRA 2T-physics tells us that :

31. 31 D=4, N=4 SYM, superconformal

32. 32 D=4 N=8 SUGRA SU(2,2|8)

33. 33 For d=10, SO(10,2)  Spin(4,2)xSpin(6), AdS(5)xS(5) Use SU(2,2|4) F(4) contains SO(5,2)xSU(2) For d=11, SO(11,2)  Spin(6,2)xSpin(5): AdS(7)xS(4) Spin(3,2)xSpin(8): AdS(4)xS(7) Use OSp(8|4) Spacetime Supersymmetry

34. 34 SMALLEST BOSONIC GROUP G THAT CONTAINS spin(d,2) For d>6 contains D-brane-like generators If D-branes admitted, then more general (super)groups can be used, in particular a toy M-model with G d =OSp(1|64) 13 : (11,2)=(10,1)+(1,1)

35. 35 Twistor (group) gauge Coupling of type-1

36. 36

37. 37 Group/twistor gauge G d 2T-physics, twistor gauge: SupergroupsSupergroups, SO(d,2)< G d Global symmetry: G d acting from right side. Conserved current Local symmetry: left side of g Therefore only coset G/H contains physical degrees of freedom. These must match d.o.f. in lightcone gauge of superparticle

38. 38 Twistors for d=4 superparticle with N supersymmetries

39. 39 - super trace

40. 40 Values of  different than N/2 are non-unitary of d=4 N=4 model. Consider N=4 and  =0,4.

41. 41 Twistors for d=6 superconformal theory SO(4) = SU(2) + xSU(2) - A + [ij] =(3,1,0) OSp(8|4) > SO(6,2)xSp(4) > SO(5,1)xSp(4) > SO(4)xSp(4) > SU(2) + xSp(4) Exactly Bars-Gunaydin doubleton 8  (after kappa) 4creation 4annhilation Lightcone 4 creation ops. -> 2 3 b +2 3 f = 8 bose + 8 fermi states SU(2) singlets only in Fock space OSp(8|4) supermultiplet 1 st & 2 nd columns related = Pseudo-real Z from OSp(8|4)

42. 42

43. 43 AdS 5 xS 5 as gauge choice in 2T-physics Analog of spherical harmonics Y lm (  )

44. 44

45. 45

46. 46 The AdS 5 xS 5 gauge (10,2) (4,2) (6,0)

47. 47 2T-superparticle that be gauge fixed to 1T AdS 5 xS 5 superparticle Type-2 coupling, g=SU(2,2|4) coupled to orbital L =SO(4,2)xSO(6) on LEFT side of g local SU(2,2) x SU(4) or SO(4,2) x SO(6) in SU(2,2|4) acts on (X,P) as vectors, and on g from left as spinors, Local kappa symmetry (off diagonal in G) acts on g from left, also on sp(2,R) gauge field A ij Any  4x4=16 complex but only half of them remove gauge d.o.f. Can remove all bosons from g(  ). Global symmetry on RIGHT side of g = the full SU(2,2|4) g’(t)=g(t)g R

48. 48 1T AdS 5 xS 5 superparticle (a gauge) Use Sp(2,R) to gauge fix (X,P) to AdSxS as in purely bosonic case. Use local SU(2,2)xSU(4), to eliminate all bosons in g. Use all of the kappa gauge symmetry to eliminate half of the fermions in g. Remaining degrees of freedom = superparticle on AdS5xS5, with 16 real fermionic degrees of freedom. Quantum superparticle: Clifford algebra for the fermions (8 creation, 8 annihilation), and (x,p in AdSxS space that satisfy Spectrum = |AdSxS, 128 bosons + 128 fermions> (II-B SUGRA) The symmetry group that classifies states is the original SU(2,2|4), The states = Kaluza-Klein towers = unitary represent. of SU(2,2|4) distinguished by the Casimir of the subgroup SU(4)=SO(6) = l(l+4) Through the 2T superparticle we see that the spectrum of 10D type II-B SUGRA is related to a 2T-theory in (10,2) dimensions. Tests of the hidden aspects of the extra dimensions can be performed (example all Casimirs vanish for all the KK states – comes directly from the 12D constraints P.P=X.X=X.P=0

49. 49 Coupling G d, type-2, particle gauge

50. 50 Twistor gauge for AdSxS discussed next Superparticle version has 9x+9p+16 

51. 51

52. 52 Gauge invariant algebra of physical observables True in any gauge (all holographs). Spectrum determined as the representation space for this symmetry algebra

53. 53 Twistors & constraints in various dimensions

54. 54 conclusions 2T-physics in (d,2) is used as a tool to find twistor representations of 1T-physics systems in d=3,4,5,6,10,11 The twistors provide a hologram of the 2T-theory in (d,2) dimensions. The twistor hologram is dual to any of the other 1T-physics holograms. The new twistors lead to twistor formulations of SYM d=4 N=4, SUGRA d=4 N=8, CFT d=6 N=4, KK towers of AdS 5 xS 5 type-IIB d=10 SUGRA, new TWISTOR SUPERSTRINGS.

55. 55

56. 56

57. 57

58. 58

59. 59 Twistor superstring in d=4 (Berkovits, Berkovits & Witten) Signature SO(2,2), then Y,Z are real, can have different dimensions

60. 60 Vertex operators for SYM Must have dimension 2 under T, and must be gauge invariant under J To satisfy the conditions,  must be homogeneous of degree 0 under scalings of Z Expanding in the fermion gives all helicity states of SYM This gives all MHV amplitudes for (++-------) or permutations. Gluons, gluinos, etc. Similarly, vertex operators for conformal SUGRA

61. 61 If we insist on signature SO(3,1) and not SO(2,2), then Y is complex conjugate of Z and must have same dimension.

62. 62 Computations See Berkovits and Witten for explicit computations in string theory See Cachazo & Svrcek 0504194 for an overall review of computations in SYM using the new diagramatic rules. See also Prof. Zhu for SYM computations

63. 63

64. 64 gamma matrices except

65. 65 Gauge fixing 2T superparticle

66. 66 Spacetime (or particle) gauge

67. 67 Twistor gauge for coupling type-1

68. 68

69. 69 AdSxS sigma model