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

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 ?

2

　 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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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Plan of the talk0. 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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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 aseffective sampling becomes difficult

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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 !

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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)

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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 parametere.g.) free energy of the one-matrix model

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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GEM applied to 6d IKKT model Gaussian action

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

Various symmetry breaking patterns

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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)

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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) SSBsuggesting :

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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Results of GEM for the 6d IKKT model

SO(5), extended

SO(4), extendedSO(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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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One-loop effective theory

treat them as small fluctuationsand keep only quadratic terms

valid whenlong distance behavior

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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

due to SUSY0

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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One-loop IKKT model with

No SSB of SO(10) symmetryAmbjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00)

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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One-loop 6d IKKT model with

No SSB of SO(6) symmetryAmbjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00)

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

30

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.

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

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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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

32

Comparison of the peak heights

turned out to be negligible !

The information of comes in onlythrough the determination of

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

33

Effects of the phase at

As far as , the left peak dominates !

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

34

0.6

GEM result translates to

Prediction from GEM

consistent with MC !

5. Summary

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

36

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

Jun Nishimura (KEK) 09.10.1 Imperial College London

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

37

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.