Space-time picture suggested by the IIB matrix model

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


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





　　　　　 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





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





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





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





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”





Remarkable properties of the phase J.N.-Vernizzi (’00)

Stationarityof the phaseincreasesfor lower d

This effect can compensate the entropy loss for lower d !





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),…



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)



2. Gaussian expansion method





Gaussian expansion methode.g.) one-matrix model

Consider the Gaussian action

free parameter

free propagator

interaction vertex

one-loopcounterterm

Perform perturbative expansion using





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





GEM applied to 6d IKKT model 6d IKKT model

Gaussian action

Aoyama-J.N.-Okubo,arXiv:1007.0883[hep-th]

Various symmetry breaking patterns





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]





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 :





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)



Constant-volume property





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)



Shrunken directions dominated by the off-diagonal part

SO(D) inv.

typical scale of the branched polymer

(the reason for the universal shrunken direction)



SO(2) ansatz



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.



Reconsidering 10d IKKT 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),…



Constant-volume property



Consistent with preliminary MC data



Comparing SO(d) d=2,3,4,5,6,7in 10d IKKT 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



Constant volume property in the 10D IKKT model

3. Monte Carlo studies by the factorization method



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.



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



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]



Factorization property of the density of states

reweighting formula

constrainedsystem



The saddle-point equation

effect of the phase

(The constraints enable us to study the important regions.)



Choice of observables

the remaining overlap problem in evaluating

constrainedsystem

Anagnostopoulos-Azuma-J.N. arXiv:1009.4504 [cond-mat]



Minimal setAssume that there is no more overlap problem with

Saddle-point eq.

In fact, there is no overlap problem with




The role of the phase

However, one can show that

Note that




A short summary of the methodChoose the set of observables

so that the remaining observables are (approximately) decorrelated with the phase; i.e.,



GEM results for the 4d toy model

J.N.-Okubo-Sugino (’04)



J.N. (’01)



The properties of the fermion determinantIntegrating over fermionic variables, one obtains

analogous to IKKT model !



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.



Factorization method applied to the toy model



1.373(2)

Similar agreement observed also for other equations.




Other possible dangerous observables…



Remaining overlap problem is small.

4. Monte Carlo studies of 6d IKKT model (preliminary)

Anagnostopoulos-Aoyama-Azuma-Hanada-J.N. work in progress



Let us recall some GEM results.



We will see how these results can be reproduced by Monte Carlo simulation.

constant volume property

Universal shrunken directions



No SSB in the phase-quenched model

0.6





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 :





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





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





Determination of the peak position

(at small x)

The extent of the hard core potentialgives the (universal) shrunken direction.





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.





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)



Comparison ofvery subtle yet…

More careful analysis will give us a definite conclusion.

5. Summary





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…



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 !



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.



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

Space-time picture suggested by the IIB matrix model

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