Does 4d space-time emerge dynamically from the IKKT matrix model ? Numerical Approaches to AdS/CFT,...

Does 4d space-time emerge dynamicallyfrom the IKKT matrix model ?

Numerical Approaches to AdS/CFT, Large N and Gravity,Sep 28-Oct 2, 2009,Imperial College, London

Jun Nishimura (KEK Theory Center)

Jun Nishimura (KEK) 09.10.1 Imperial College London

Does 4d space-time emerge dynamically from the IKKT matrix model ?

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　 based on collaborations with

Konstantinos Anagnostopoulos (National Technical University, Athens)

Tatsumi Aoyama (KEK)

Takehiro Azuma 　（ Setsunan U. ）

Masanori Hanada (Weizmann Inst.)

Toshiyuki Okubo (Meijo U.)

Fumihiko Sugino (Okayama Inst.)

0. Introduction



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　　　　　 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, derivation of SM

Comparing string theory to QCD



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　　　　　 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, derivation of SM

Comparing string theory to QCD



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IKKT 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



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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



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Plan of the talk

0. Introduction

1. Complex fermion determinant

2. Gaussian expansion method

3. Monte Carlo studies of one-loop effective theory

4. Full Monte Carlo studies (preliminary)

5. Summary

1. Complex fermion determinant



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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



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

Stationarityof the phaseincreasesfor lower d

This effect compensates the entropy loss for lower d !



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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)



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Cousins of the IKKT matrix model

6d IKKT model

4d IKKT model

complex fermion determinant

10d IKKT model

We mostly focus on this modelfor technical simplicity

2. Gaussian expansion method



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Gaussian expansion methode.g.) one-matrix model

Consider the Gaussian action

free parameter

free propagator

interaction vertex

one-loopcounterterm

Perform perturbative expansion using



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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



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GEM applied to 6d IKKT model Gaussian action

Aoyama-J.N.-Okubo, in prep.

Various symmetry breaking patterns



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Results of GEM for the 6d IKKT model Aoyama-J.N.-Okubo, in prep.

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)



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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 :



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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

shrunk directions

(cont’d)

0.18

3. Monte Carlo studies ofone-loop effective theory



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One-loop effective theory

treat them as small fluctuationsand keep only quadratic terms

valid when

long distance behavior



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Branched-polymer-like structure Aoki-Iso-Kawai-Kitazawa-Tada(’98)

due to SUSY0



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One-loop IKKT model with

No SSB of SO(10) symmetry

Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00)



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One-loop 6d IKKT model with

No SSB of SO(6) symmetry

Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00)



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Distribution of the normalized eigenvalues in the phase-quenched model

in the full model

How to include the effects of the phase

In order to see the net effects of the phase, we use the normalized variables:

finite N effects

Results in the phase-quenched model :

represents the effectsof the phase



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The behavior of the correction factor

no. of d.o.f.

favourscollapsed configs.

Anagnostopoulos-J.N. (’02)

one-loop6d IKKT model



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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

4. Full Monte Carlo studies (preliminary)



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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., in prog.



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Determination of the peak position

(at small x)

Precise position of the peak can be obtainedby more careful large-N extrapolation of



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Comparison of the peak heights

turned out to be negligible !

The information of comes in onlythrough the determination of



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Effects of the phase at

As far as , the left peak dominates !



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0.6

GEM result translates to

Prediction from GEM

consistent with MC !

5. Summary



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Summary and future prospects IKKT matrix model non-perturbative definition of superstring the dynamical origin of space-time dim. 6d IKKT model complex fermion determinant Gaussian expansion method

MC studies with factorization method large-N extrapolation for each factor of the distribution so far, consistent with GEM prediction Extension to IKKT model, straightforward

SO(6) SO(3) SSB



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Some comments D0 brane system Hanada and Wiseman’s talk

The phase of the fermion determinant can be neglected. agreement with the gauge/gravity duality the reason needs to be understood

SSB of SO(9) seems unlikely to happen… studies based on GEM Aoyama-J.N.-Okubo-Takeuchi, in prog IKKT model appearance of 4d gravity Kawai’s talk

spontaneous breaking of SUSY ? Sugino’s talk 　 How to think about the Euclidean time Ambjorn’s talk How to read off the Standard Model d.o.f.

Does 4d space-time emerge dynamically from the IKKT matrix model ? Numerical Approaches to AdS/CFT,...

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