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Twistors, superarticles, twistor superstringsin 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– AdS5xS

5 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.

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What are twistors in 4 flat dimensions?

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Physical states in twistor space

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Penrose

Homework: find the correct wavefunctions with definite momentum and helicity

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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 isSp(2,R) gauge symmetry: Position and momentum (X,P) are indistinguishableat any instant.• This symmetry demands 2T signature (-,-,+,+,+,…,+)to have nontrivial gaugeinvariant subspace Qij(X,P)=0.• Unitarity and causality aresatisfied thanks to symmetry.

2) Holography• 1T-physics is derived from 2T physics by gauge fixingSp(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 hasmany 1T holographic images with various 1T perspectives.

5) Unification• Different observers can usedifferent emergent (t,H) todescribe the same 2T system.• This unifies many emergent 1T dynamical systems into asingle class that represents the same 2T system with anaction based on some Qij(X,P).

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 givesmany 1T systems with various1T dynamics. These are imagesof the same “free particle” in 2T physics in flat 2T spacetime.

4) Hidden symmetry(for the example in figure)• The action of each 1T imagehas hidden SO(d,2) symmetry.• Quantum: SO(d,2) global symrealized in same representationfor all images, C2=1-d2/4.

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 XM,PM fixed as t,H

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2T-physics2T-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).

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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 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.

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Spacetime signature determined by gauge symmetry

EMERGENT DYNAMICS AND SPACE-TIMES

return

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Some examples of gauge fixing

2 gauge choices made. reparametrization remains.

3 gauge choices made. Including reparametrization.

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More examples of gauge fixing

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Background fields

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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.

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Duality • 1T solutions of Qik(X,P)=0

(holographic images) are dual to one another. Duality group isgauge 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

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Hidden dimensions/symmetries in 1T-physicsand 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 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).

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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.

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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).

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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 Gd=OSp(1|64)13

Twistors emerge in this approach

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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

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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 gaugekill (X,P) completely

keep only g

constrained twistors/oscillators

2T-parent theory has (X,P) and g

-model gaugefix 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)]

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Group & Twistor gauge

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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)

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

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u is any kxk unitary matrix except for overall scale

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Spacetime SUSY 2T-superparticle

Local symmetries OSp(n|2)xGd

left including SO(d,2)

and kappa Global

symmetries: Gdright

Supergroup Gd contains spin(d,2) and R-symmetry subgroups

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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

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More dualities: 1T images of unique 2T-physics superparticle via gauge fixing

Spacetime gauge-eliminate all bosons from g

keep only ½ fermi part: -fix Y=(X,P,) (d,2) to (d-1,1)

(x,p,)1T superparticle (& duals)

group/twistor gaugekill Y=(X,P,) completely

keep only g

constrained twistors/oscillators

2T-parent theory has Y=(X,P,) and g

-model gaugefix part of (X,P,); LMN linear

Integrate out remaining P

e.g. AdS5xS5 sigma modelSU(2,2|4)/SO(4,1)xSO(5)

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Spacetime (or particle) gauge

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Quantum states of d=4 superparticle with N supersymmetries N=4 gives SYM, N=8 gives SUGRA

2T-physics tells us that :

singleton

singleton

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Quantum states of d=4 superparticle with N supersymmetries N=4 gives SYM, N=8 gives SUGRA

2T-physics tells us that :

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D=4, N=4 SYM, superconformal

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

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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

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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)

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Twistor (group) gaugeCoupling of type-1

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Group/twistor gauge Gd

2T-physics, twistor gauge:

Supergroups, SO(d,2)< Gd

Global symmetry: Gd 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

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Twistors for d=4 superparticle with N supersymmetries

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- super trace

- super trace

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Values of different than N/2 are non-unitary

of d=4 N=4 model.

Consider N=4 and =0,4.

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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. -> 23b+23

f

= 8 bose + 8 fermi states

SU(2) singlets only in Fock space

OSp(8|4) supermultiplet

1st & 2nd columns related = Pseudo-real Z from OSp(8|4)

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AdS5xS5 as gauge choice in 2T-physics

Analog of spherical harmonics Ylm()

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The AdS5xS5 gauge

(10,2) (4,2) (6,0)

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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

• 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

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)gR

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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

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Coupling Gd , type-2, particle gauge

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Twistor gauge for AdSxS

discussed next

Superparticle version has 9x+9p+16

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Gauge invariant algebra of physical observablesTrue in any gauge (all holographs).

Spectrum determined as the representation space for this

symmetry algebra

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Twistors & constraints in various dimensions

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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 AdS5xS5 type-IIB d=10 SUGRA, new TWISTOR SUPERSTRINGS.

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Twistor superstring in d=4 (Berkovits, Berkovits & Witten)

Signature SO(2,2), then Y,Z are real, can have different dimensions

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Vertex operators for SYMMust 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

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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.

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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

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gamma matrices

except

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Gauge fixing 2T superparticle

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Spacetime (or particle) gauge

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Twistor gauge for coupling type-1

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AdSxS sigma model