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Space-time picture suggested by the IIB matrix model. YITP workshop “Discretization Approaches to the Dynanics of Space-time and Fields”, Sept.28, 2010 Jun Nishimura (KEK Theory Center & Graduate University for Advanced Studies). 0. Introduction. - PowerPoint PPT Presentation
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Space-time picture suggested by the IIB matrix model YITP workshop “Discretization Approaches to the
Dynanics of Space-time and Fields”, Sept.28, 2010
Jun Nishimura
(KEK Theory Center & Graduate University for Advanced Studies)
0. Introduction
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 3
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
QCD string theory
strong interactions what theory describes all the interactions
including gravity
free quarks perturbation theory 10d space-time
confinement non-perturbative vacuum invisible extra dim. lattice theory non-perturbative formulation matrix models (Wilson ’74) (BFSS,IKKT ’96) properties of hadrons goal black holes, early universe, SM and beyond
Comparing string theory to QCD
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 4
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
QCD string theory
strong interactions what theory describes all the interactions
including gravity
free quarks perturbation theory 10d space-time
confinement non-perturbative vacuum invisible extra dim. lattice theory non-perturbative formulation matrix models (Wilson ’74) (BFSS,IKKT ’96) properties of hadrons goal black holes, early universe, SM and beyond
Comparing string theory to QCD
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 5
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
IKKT matrix model (IIB matrix model)
(Ishibashi-Kawai-Kitazawa-Tsuchiya ’96)
a non-perturbative formulation of type IIB superstring theory in 10 dim. (conjecture)
• Similarity to the Green-Schwarz worldsheet action in the Schild gauge c.f.) Matrix Theory membrane action in the light cone gauge • Interactions between D-branes• Attempt to derive string field theory from SD eqs. for Wilson loops
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 6
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Dynamical generation of 4d space-time
Eigenvalues :
in the limit
The order parameter forthe spontaneous breaking of the SO(10) symmetry
e.g.) SO(10) → SO(4)
c.f.) spontaneous breaking of Lorentz symmetry from tachyonic instability in bosonic SFT Kostelecky and Samuel (1988)
“moment of inertia” tensor
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 7
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Plan of the talk0. Introduction
1. Complex fermion determinant
2. Gaussian expansion method Aoyama-J.N.-Okubo, arXiv:1007.0883[hep-th]
3. Monte Carlo studies (factorization method) Anagnostopoulos-Azuma-J.N., arXiv:1009.4504[cond-mat]
4. Monte Carlo studies of 6d IKKT model (preliminary) Anagnostopoulos-Aoyama-Azuma-Hanada-J.N., work in progress
5. Summary
1. Complex fermion determinant
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 9
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Complex fermion determinant fermion determinant
reweighting method simulate the phase quenched model
cannot be treated as the Boltzmann weight
complex in general
suppressed as
effective sampling becomes difficult“sign problem”
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 10
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Remarkable properties of the phase J.N.-Vernizzi (’00)
Stationarityof the phaseincreasesfor lower d
This effect can compensate the entropy loss for lower d !
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 11
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
This is a dilemma ! Phase of the fermion determinant
important for the possible SSB of SO(10)
difficult to include in Monte Carlo simulation
Gaussian expansion method Section 2 Sugino-J.N. (’00), Kawai et al. (’01),…
New Monte Carlo technique Section 3 Anagnostopoulos-J.N. (’01),…
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 12
Models with similar properties
6d IKKT model
4d toy model (non SUSY)
(SSB of SO(D) expected due to complex fermion det.)
10d IKKT model
J.N. (’01)
Space-time picture suggested by the IIB matrix model
Jun Nishimura (KEK) 10.9.28 YITP workshop
2. Gaussian expansion method
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 14
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Gaussian expansion methode.g.) one-matrix model
Consider the Gaussian action
free parameter
free propagator
interaction vertex
one-loopcounterterm
Perform perturbative expansion using
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 15
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Self-consistency equation
self-consistency eq.:
How to identify the plateau ?
Search for concentration of solutions
plateau
Results of GEM depends on the free parameter
e.g.) free energy of the one-matrix model
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 16
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
GEM applied to 6d IKKT model 6d IKKT model
Gaussian action
Aoyama-J.N.-Okubo,arXiv:1007.0883[hep-th]
Various symmetry breaking patterns
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 17
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Results of GEM for the 6d IKKT model
Krauth-Nicolai-Staudacher (’98)
magnify this region
SO(5) SO(4) SO(3) SO(5) SO(4) SO(3) SO(5) SO(4) SO(3)
Aoyama-J.N.-Okubo,arXiv:1007.0883[hep-th]
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 18
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Results of GEM for the 6d IKKT model (cont’d)
SO(4)
SO(3) SO(4) SO(3)
SO(5)
SO(5)
concentration of solutions identified
SO(6) SO(3) SSB
suggesting :
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 19
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Results of GEM for the 6d IKKT model
SO(5), extended
SO(4), extended
SO(3), extended
extent of the eigenvalue distributionin the extended/shrunk direction
finite in units of
Universal shrunken directions
(cont’d)
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 20
Constant-volume property
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 21
Understanding based on LEET treat them as small fluctuations
and keep only quadratic terms
Aoki-Iso-Kawai-Kitazawa-Tada(’98)
Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00)
branched-polymer-like structure(the reason for constant volume property)
Space-time picture suggested by the IIB matrix model
Jun Nishimura (KEK) 10.9.28 YITP workshop
Shrunken directions dominated by the off-diagonal part
SO(D) inv.
typical scale of the branched polymer
(the reason for the universal shrunken direction)
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 22
SO(2) ansatz
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
d=3 is chosen dynamically in the 6d IKKT model
13 free parameters
Gaussian action
4 free parameters
Cyclic permutations of
Naively, disfavored.
Many solutions at order 5.
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 23
Reconsidering 10d IKKT model
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
(universal shrunken direction)
Free energy is lowerfor d=4 than for d=7
Kawai –Kawamoto-Kuroki-Shinohara (’03)
Sugino-J.N. (’00), Kawai et al. (’01),…
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 24
Constant-volume property
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Consistent with preliminary MC data
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 25
Comparing SO(d) d=2,3,4,5,6,7in 10d IKKT model
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
order 1 order 3
SO(2) 6.49 3.05SO(3) 7.0 -1.36SO(4) 6.15 0.70SO(5) 5.91 1.33SO(6) 5.76 1.54SO(7) 5.52 1.62
ansatz
J.N.-Sugino (’02)
New results (preliminary) J.N.-Okubo-Sugino, work in progress
Old results
3.63[x2] 0.12[x6], 0.11, 0.053.24[x3] 0.10[x6], 0.081.35[x4] 0.14[x6]0.84[x5] 0.11[x3], 0.11, 0.090.67[x6] 0.11[x3], 0.070.57[x7] 0.09[x3]
universal shrunken direction
constant volume property
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 26
Constant volume property in the 10D IKKT model
3. Monte Carlo studies by the factorization method
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 28
The sign problem
VEV w.r.t. phase-quenched model
a general system
reweighting methodcannot be treated as the Boltzmann weight
Exponentially large numbers of configurations are neededto achieve given accuracy.
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 29
Moreover, there is also a general problem in the reweighting method
Region of configuration space sampled bysimulating the phase-quenched model
Region of configuration space that gives important contribution to
Overlap problem
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 30
The basic idea of the factorization method
Control some observables
determine and sample effectively the important region of configuration space
Density of states
normalized observables
Anagnostopoulos-J.N. (’02)Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat]
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 31
Factorization property of the density of states
reweighting formula
constrainedsystem
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 32
The saddle-point equation
effect of the phase
(The constraints enable us to study the important regions.)
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 33
Choice of observables
the remaining overlap problem in evaluating
constrainedsystem
Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat]
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 34
Minimal setAssume that there is no more overlap problem with
Saddle-point eq.
In fact, there is no overlap problem with
Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat]
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 35
The role of the phase
However, one can show that
Note that
Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat]
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 36
A short summary of the methodChoose the set of observables
so that the remaining observables are (approximately) decorrelated with the phase; i.e.,
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 37
GEM results for the 4d toy model
J.N.-Okubo-Sugino (’04)
Space-time picture suggested by the IIB matrix model
Jun Nishimura (KEK) 10.9.28 YITP workshop
J.N. (’01)
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 38
The properties of the fermion determinantIntegrating over fermionic variables, one obtains
analogous to IKKT model !
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 39
Reproduce GEM results by the factorization method
The result for the phase-quenched model
Applying factorization method using , we have checked that the GEM results are indeed solutions to the saddle-point equations.
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 40
Factorization method applied to the toy model
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
1.373(2)
Similar agreement observed also for other equations.
Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat]
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 41
Other possible dangerous observables…
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Remaining overlap problem is small.
4. Monte Carlo studies of 6d IKKT model (preliminary)
Anagnostopoulos-Aoyama-Azuma-Hanada-J.N. work in progress
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 43
Let us recall some GEM results.
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
We will see how these results can be reproduced by Monte Carlo simulation.
constant volume property
Universal shrunken directions
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 44
No SSB in the phase-quenched model
0.6
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 45
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
the normalized observables
The use of the normalized variables enables us tosee the net effects of the phase.
finite N effects
the phase-quenched model :
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 46
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Factorization methodAnagnostopoulos-J.N. (’02)
Distribution of the normalized eigenvalues
has a double-peak structure !
scales ! scales !
L.h.s. is 1/N suppressed !
consistent withbranched polymer
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 47
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Small x behavior of in full 6dIKKT modelphase space suppression :
Large-N extrapolation reveals the existence of a “hard-core potential”
Anagnostopoulos-Aoyama-Azuma-Hanada-J.N., work in progress
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 48
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Determination of the peak position
(at small x)
The extent of the hard core potentialgives the (universal) shrunken direction.
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 49
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Effects of the phase at
The extent of the extended direction is almost decorrelated with the phase.
No need to constrain the large eigenvalues.Constant volume property can be naturally understood.
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 50
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Comparison of the free energy
almost negligible at large N
The difference of the free energy density can beroughly determined by the difference of
e.g.) SO(2) and SO(3)
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 51
Comparison ofvery subtle yet…
More careful analysis will give us a definite conclusion.
5. Summary
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 53
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model
Summary and future prospects IKKT matrix model non-perturbative definition of superstring theory the dynamical origin of space-time dimensionality
6d IKKT model, 4d toy model complex fermion determinant, SSB of SO(D) expected
Gaussian expansion method
4d toy : SO(4) SO(2) SSB
6d IKKT : SO(6) SO(d) SSB
10d IKKT : SO(10) SO(d) SSB
universal extra dimensionconstant volume propertytrue vacuum may be d=3…
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 54
Summary and future prospects (cont’d) MC studies of these models difficult due to the sign problem
factorization method uses the factorization property of the density of states reduces the overlap problem by controlling observables extrapolations possible for the factorized functions
the observables to be controlled have to be chosen appropriately for a general system
demonstration in the 4d toy model 6d IKKT model universal shrunken directions constant volume property
reproduced quantitatively !
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 55
Summary and future prospects (cont’d) Comparison of free energy for SO(d) vacua being pursued using GEM and MC
GEM SO(2) should be studied more carefully comparison of SO(d) d=2,3,4,5,6,7 at the 5th order in 10d IKKT
MC with factorization method so far, no evidence for disfavoring SO(2) large-N extrapolation is important A definite conclusion will be obtained soon.
Jun Nishimura (KEK) 10.9.28 YITP workshop
Space-time picture suggested by the IIB matrix model 56
Summary and future prospects (cont’d) Interpretation of IKKT model
Branched polymer as low energy effective theory SUSY plays an important role in dynamically generating the notion of commutative space-time coordinates certain non-commutativity exists due to the off-diagonal elements “non-commutative extra dimension” Aschieri-Grammatikopoulos-Steinacker-Zoupanos
The ratio R / r seems to be finite. d=3 may be chosen as the true vacuum. What does the IKKT model describe? The state of the early universe? How can we describe time evolution? Matrix Cosmology?
Freedman-Schnabl-Gibbons