Transcript Twistors, superarticles, twistor superstrings in various spacetimes

2 Time Physics and Field theory
Kuo, Yueh-Cheng
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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)
3) Duality
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
• 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.
4) Hidden 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.
6) Generalizations found
• Spinning particles: OSp(2|n); Spacetime SUSY
• Interactions with all backgrounds (E&M, gravity, etc.)
• 2T field theory (standard model)
•2T string/p-brane
• Twistor superstring
(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.
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).
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.
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• Basic 2T physics
• Generalizations
* Spinning particles
* Supersymmetric particles and Twistors
* 2T field theory
• Outlook
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Heuristic Motivation for 2 Time
• 1985 Witten
low energy, strong coupling
limit of type 2A string  10+1
dim supergravity
suggest a quantum theory
(with N=1 supersymmetry) in
10+1 dim whose classical limit
is supergravity
• The same supersymmetry can
also be realized in 10+2 dim
spacetime
Note:
11+1 D spinor: chiral and complex
10+2 D spinor: chiral and real
Q ,Q
32 33
2
p
528
C
11 10
2 1
Q ,Q
32 33
2
55
C
528
12 11
2 1
C
11 10 9 8 7
462
5 4 3 2 1
C
Self-dual
66
12 11 10 9 8 7
1
6 5 4 3 2 1
2
462
But how about ghosts arising from one more
time?
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Spacetime signature determined by gauge symmetry
EMERGENT DYNAMICS AND SPACE-TIMES
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return
Some examples of gauge fixing
Covariant quantization:
2
0
d
XM
X
1
M
2
2
0
2
2
0
3
3
X
2
2
X
1
' X
2
X
2
parametrize : X
x
F
d 2
M
X
M
M
M
2
d
M
F
2
2
'
0
4
F
X
2
2
F
0
5
'
x
2
2
X
M
M
4
5
2 gauge choices made.
t reparametrization remains.
F X2
0
d 2
2
x ,
x2
2
0
How could one obtain the three constraints from
a Lagrangian of scalar field?
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Some examples of gauge fixing
3 gauge choices made.
Including t reparametrization.
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More examples of gauge fixing
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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. SO(d,2) global symmetry (note:
generators of SO(d,2) commute with those of Sp(2,R)) is realized for all
images in the same unitary irreducible representation, with Casimir C2=1-d2/4.
This is the singleton.
•
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. Different observers can use different emergent (t,H) to
describe the same 2T system.
•
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 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
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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 strings/branes (incomplete)
•
2T field theory (new progresses
recently)
•
Twistors in d=3,4,6,10,11 ; Twistor
superstring in d=4
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SO(d,2) unitary representation
unique for a fixed spin=n/2.
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M
Gauge Fixing: (for example, n=1)
X
P
'
M
1,
'
x2
2
S
XM
0, X 2
XMPM
0, XM PM
MN
M
M
, x
M
0, x p , p
M
0, x
2
0,
M
PM
0, P 2
0
A ab Y M a Y N b
,
remaining constraint : p 2
0,
p
0
,
S
x p
A 22
2
p
C
p
2
2
Dirac Equ. for massless spin 1 2 particle
One could choose other gauges or do covaraint quantization.
These on-shell condition will coincide with those constraints imposed from
considering spinning 2T particle.
Obtain E&M, gravity, etc. in d dims from background fields f(X,P, y) in d+2 dims. –> holographs.
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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) – 2T Field Theory
.
- Non-commutative FT f(X,P) (0104135, 0106013)
-
similar to string field theory, Moyal star.
BRST 2T Field theory
- Standard Model
5) - String/brane theory in 2T (9906223,. 0407239)
-Twistor superstring in 2T
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(0407239, 0502065 )
both 4 & 5 need more work
Spacetime SUSY: 2T-superparticle
Supergroup Gd contains spin(d,2)
R-symmetry subgroups
and
Local symmetries
OSp(n|2)xGdleft
including SO(d,2) and kappa
Global symmetries:
Gdright
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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
2T-parent theory
has Y=(X,P,y) and g
Spacetime gauge
group/twistor gauge
s-model gauge
-eliminate all bosons from g
kill Y=(X,P,y) completely
keep only ½ fermi part: q,
-fix Y=(X,P,y) (d,2) to (d-1,1)
keep only g
fix part of (X,P,y); LMN linear
Integrate out remaining P
(x,p,q)1T superparticle (& duals)
constrained twistors/oscillators
e.g. AdS5xS5 sigma model
SU(2,2|4)/SO(4,1)xSO(5)
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Spacetime (or particle) gauge
Residual local sym: reparametrization and K sym
Global sym: superconformal
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Twistor (group) gauge
Coupling
of type-1
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Twistors for d=4 superparticle with N supersymmetries
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2T field theory
hep-th/xxxxxxx by I. Bars and Y. C. Kuo
BRST operator
i
cmfm
Q
f
1
1
f
2
4
1
f
3
4
1
4
X, c
m nkc
2
P
2
P
2
P
M
X
2
X
2
m
cnbk
f
1
f
2
f
3
0
0
0
0
0
m
X PM
X
c
m
m
X
cc
m
m
X
ccc
0
X
,
are pure gauge
0
M
Action
Fix gauge:
dd
S
2
X
dd x d x
a
S
0
m,
XM
0
Physical states lies in Q cohomology
0 f 22
'
d
0
2
Q
d 2
d
dd
2
X
3
Xd c
0
gauge sym :
S
0
Q
fm
a
Q
m
m
X, c
X, c
fm
Q
0
m nk
m
fn
k
i
m
d
dd x
x
dd
2
'
x2
X
P2
0
d 2
0
2
d
p2
0
p2
0
2
0
m
X, c
0
22
2
0
2T field theory
No Interaction terms maintaining the gauge sym of free field
and without giving trivial physics ( 0 0 ) can be written
down.
One could try to modify the BRST operators and hence the
corresponding gauge sym
X2
f 11
2
P2
f 22
0
2
1
f 12
PM XM
4
0
0
XMPM
0
0
S
constant
dd x
d
4
Interaction
0
2
d 2
p2
0
0
2
0
0
0
0
d
d 2
4
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Outlook
• Standard Model as a 2T field theory
• 2T string/brane (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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