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Twistors, superarticles, twistor superstrings in various spacetimes Itzhak Bars

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

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Twistors, superarticles, twistor superstrings in various spacetimes Itzhak Bars

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  1. Twistors, superarticles, twistor superstringsin various spacetimesItzhak 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 • AdS5xS5 compactified type-IIB supergravity, KK-towers • Nonlinear sigma model PSU(2,2|4)/SO(4,1xSO(5) versus PSU(2,2|4)/PSU(2|2)xU(1) twistors for AdS5xS5 • 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. What are twistors in 4 flat dimensions?

  3. Physical states in twistor space

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

  5. 3) Duality • 1T solutions of Qij(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. 2T-physics1T 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 Qij(X,P)=0. • Unitarity and causality are satisfied thanks to symmetry. Sp(2,R) gauge choices. Some combination of XM,PM fixed as t,H 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, C2=1-d2/4. 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 Qij(X,P). 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.

  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. 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 Qik(X,P)=0 gauge invariant subspace (eq. of motion for A) Unitarity and causality are satisfied thanks to Sp(2,R) gauge symmetry. Global symmetry determined by form of Qik(X,P). In the example it is SO(d,2). It is gauge invariant since it commutes with Qik. Gauge Symmetry Sp(2,R)

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

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

  10. More examples of gauge fixing

  11. Background fields

  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. 1T solutions of Qik(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. Duality Many emergent spacetimes

  14. 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 Qik(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 C2=1-d2/4. 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 Qik(X,P). Hidden dimensions/symmetries in 1T-physicsand UNIFICATION

  15. 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. Generalizations

  16. SO(d,2) unitary representation unique for a fixed spin=n/2. f (X,P, y) expand in powers of P,y get fields fmn:::…(X). Obtain E&M, gravity, etc. in d dims from background fields f(X,P, y) in d+2 dims. –> holographs.

  17. Twistors emerge in this approach 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 Gd=OSp(1|64)13 4) - Field Theory in 2T(0003100);Standard Model could be obtained as a holograph (Dirac, Salam). -Non-commutative FT f(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

  18. invariant 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

  19. More dualities: 1T images of unique 2T-physics particle via gauge fixing 2T-parent theory has (X,P) and g s-model gauge fix part of (X,P) LMN partly linear Integrate out remaining P e.g. AdS5xS5 sigma model [SU(2,2)xSU(4)]/[SO(4,1)xSO(5)]

  20. Group & Twistor gauge

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

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

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

  24. Spacetime SUSY 2T-superparticle Supergroup Gd contains spin(d,2) and R-symmetry subgroups Local symmetriesOSp(n|2)xGdleftincluding SO(d,2) and kappa Global symmetries:Gdright

  25. Local symmetry embedded in Gleft • 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 Aij

  26. More dualities: 1T images of unique 2T-physics superparticle via gauge fixing 2T-parent theory has Y=(X,P,y) and g s-model gauge fix part of (X,P,y); LMN linear Integrate out remaining P e.g. AdS5xS5 sigma model SU(2,2|4)/SO(4,1)xSO(5)

  27. Spacetime (or particle) gauge

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

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

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

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

  32. Spacetime Supersymmetry For d=10, SO(10,2)  Spin(4,2)xSpin(6), AdS(5)xS(5) Use SU(2,2|4) 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) F(4) contains SO(5,2)xSU(2)

  33. 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 Gd=OSp(1|64)13 : (11,2)=(10,1)+(1,1)

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

  35. 2T-physics, twistor gauge: Supergroups, SO(d,2)< Gd Global symmetry: Gd acting from right side. Conserved current Local symmetry: left sideof g Group/twistor gauge Gd Therefore only coset G/H contains physical degrees of freedom. These must match d.o.f. in lightcone gauge of superparticle

  36. Twistors for d=4 superparticle with N supersymmetries

  37. - super trace - super trace

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

  39. Twistors for d=6 superconformal theory Exactly Bars-Gunaydin doubleton 8q (after kappa) 4creation 4annhilation Lightcone 4 creation ops. -> 23b+23f = 8 bose + 8 fermi states OSp(8|4)supermultiplet 1st & 2nd columns related = Pseudo-real Z from OSp(8|4) SU(2) singlets only in Fock space OSp(8|4) > SO(6,2)xSp(4) > SO(5,1)xSp(4) > SO(4)xSp(4) > SU(2)+xSp(4) SO(4) = SU(2)+xSU(2)- A+[ij]=(3,1,0)

  40. AdS5xS5 as gauge choice in 2T-physics Analog of spherical harmonics Ylm(q,f)

  41. The AdS5xS5 gauge (10,2) (4,2) (6,0)

  42. 2T-superparticle that be gauge fixed to 1T AdS5xS5 superparticle Type-2 coupling, g=SU(2,2|4) coupled to orbital L =SO(4,2)xSO(6) on LEFT side of g Global symmetry on RIGHT side of g = the full SU(2,2|4) g’(t)=g(t)gR • 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 Aij Can remove all bosons from g(t). Any k(t) 4x4=16 complex but only half of them remove gauge d.o.f.

  43. 1T AdS5xS5 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

  44. Coupling Gd , type-2, particle gauge

  45. Twistor gauge for AdSxS Superparticle version has 9x+9p+16q discussed next

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