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Does 4d space-time emerge dynamically from 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)
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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.)
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0. Introduction
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 4 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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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 5 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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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 6 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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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 7 Dynamical generation of 4d space-time Eigenvalues : in the limit The order parameter for the 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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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 8 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
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1.Complex fermion determinant
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 10 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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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 11 Remarkable properties of the phase J.N.-Vernizzi (’00) Stationarity of the phase increases for lower d This effect compensates the entropy loss for lower d !
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 12 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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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 13 Cousins of the IKKT matrix model 6d IKKT model 4d IKKT model complex fermion determinant 10d IKKT model We mostly focus on this model for technical simplicity
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2. Gaussian expansion method
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 15 Gaussian expansion method e.g.) one-matrix model Consider the Gaussian action free parameter free propagator interaction vertex one-loop counterterm Perform perturbative expansion using
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 16 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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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 17 GEM applied to 6d IKKT model Gaussian action Aoyama-J.N.-Okubo, in prep. Various symmetry breaking patterns
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 18 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)
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 19 Results of GEM for the 6d IKKT model (cont ’ d) SO(4) SO(3) SO(4)SO(3) SO(5) concentration of solutions identified SO(6)SO(3) SSB suggesting :
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 20 Results of GEM for the 6d IKKT model SO(5), extended SO(4), extended SO(3), extended extent of the eigenvalue distribution in the extended/shrunk direction finite in units of shrunk directions (cont ’ d) 0.18
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3. Monte Carlo studies of one-loop effective theory
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 22 One-loop effective theory treat them as small fluctuations and keep only quadratic terms valid when long distance behavior
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 23 Branched-polymer-like structure Aoki-Iso-Kawai-Kitazawa-Tada(’98) due to SUSY 0
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 24 One-loop IKKT model with No SSB of SO(10) symmetry Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00)
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 25 One-loop 6d IKKT model with No SSB of SO(6) symmetry Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(’00)
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 26 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 effects of the phase
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 27 The behavior of the correction factor no. of d.o.f. favours collapsed configs. Anagnostopoulos-J.N. (’02) one-loop 6d IKKT model
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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 28 Factorization method Anagnostopoulos-J.N. (’02) Distribution of the normalized eigenvalues has a double-peak structure ! scales ! L.h.s. is 1/N suppressed ! consistent with branched polymer
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4. Full Monte Carlo studies (preliminary)
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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 model phase 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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Jun Nishimura (KEK) 09.10.1 Imperial College London Does 4d space-time emerge dynamically from the IKKT matrix model ? 31 Determination of the peak position (at small x) Precise position of the peak can be obtained by more careful large-N extrapolation of
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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 only through the determination of
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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 !
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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 !
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5. Summary
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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
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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.
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