Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions

Talk at KEK for String Advanced Lectures, 2011.9.12 (Tue.)Jun Nishimura (KEK, Sokendai)

Ref.) Kim-J.N.-Tsuchiya arXiv:1108.1540

Answers from a nonperturbative formulationof superstring theory in (9+1)-dimensions

fundamental questions concerning our universe

1.Why (3+1)-dimensions ?

2.Why expanding ?

Previous proposal

Euclideanized model with SO(10) symmetry

SSB of SO(10) studied by Gaussian Expansion Method J.N.-Okubo-Sugino, arXiv:1108.1293

free energy of SO(d) symmetric vacua (d=2,3,4,5,6,7) minimum at d=3 extent of space-time finite in all directions

motivated us to reconsider the formulation

Ishibashi-Kawai-Kitazawa-Tsuchiya (’96)

Our new proposal :

regularize SO(9,1) Lorentzian symmetric model without Wick rotation

Monte Carlo simulation

3 out of 9 spatial directions start to expand at some “critical time”

Comments on related works

Quantum Cosmology with minisuperspace approx.

Causal Dynamical Traingulation

Matrix Cosmology Freedman-Gibbons-Schnabl (’05) Matrix Big Bang Craps-Sethi-Verlinde (’05) Emergent gravity Steinacker (’11), Yang (’10)

Vilenkin (’82,’84), Hartle-Hawking (’83)

Ambjorn-Jurkiewicz-Loll (’05)

We expect that what we are doing is essentiallya first-principle calculation of the unified theory including quantum gravity.

Plan of the talk

1. Introduction2. Matrix model for superstrings3. Matrix model with SO(9,1) symmetry4. Monte Carlo results5. Mechanism of the SSB6. Removing the cutoffs7. Summary and discussions

IKKT model

matrix regularization of the Green-Schwarz worldsheet action in the Schild gauge

interactions between D-branes

string field theory from SD eqs. for Wilson loops

Ishibashi-Kawai-Kitazawa-Tsuchiya (’96)

proposed as a nonperturbative definition of type IIB superstring theory in 10 dim.

Fukuma-Kawai-Kitazawa-Tsuchiya (’98)

c.f.) Matrix Theory Banks-Fischler-Shenker-Susskind (’96)

10 Ｄ4 Ｄ

10 　 Hermitian matrices

Derivation of low-energy effective theory branched-polymer-like system Aoki-Iso-Kawai-Kitazawa-Tada (’99)

Explicit calculations by the Gaussian expansion methodJ.N.-Sugino (’02), J.N.-Okubo-Sugino,Kawai, Kawamoto, Kuroki, Matsuo, Shinohara, Aoyama, Shibusa,…

Does our 4-dimensional space-time appear from 10-dimensions ?

Results of the Gaussian expansion method J.N.-Okubo-Sugino (arXiv:1108.1293)

Minimum of the free energyoccurs at d=3

Extent of space-timefinite in all directions

SSB of SO(10) : interesting dynamical property of IKKT model,but we still seem to miss some important ingredient…

extended directions

shrunken directions

Matrix model with SO(9,1) symmetry

raised and lowered by the metric

　 Hermitian matrices

Wick rotation

IKKT model SO(10) symmetry

Partition function of the Lorentzian model

connection to the worldsheet theory

opposite sign !Unlike IKKT model, the path integral is ill-defined !

c.f.) complex in the case of IKKT model the phase factor induces SSB of SO(10)

J.N.-Vernizzi (’00), Anagnostopoulos-J.N.(’02)

Krauth-Nicolai-Staudacher (’98), Austing-Wheater (’01)

Regularizing the Lorentzian model

In order to separate space and time,we “gauge fix” the boost invariance.

(1) IR cutoff in the temporal direction

SO(9) symmetry is still manifest.

Regularizing oscillating functions

inserting unity

convergence factor

Cure this divergence by imposing :

(2) IR cutoff in the spatial direction

Thus we arrive at

Yoneya ('97)

Monte Carlo simulation : Rational Hybrid Monte Carlo algorithm no sign problem unlike in IKKT model

Extracting time evolution

“critical time”

SSB

Consider a simpler problem :

solution :

representation matrices ofa compact semi-simple Lie algebrawith d generators

Maximum is achieved for SU(2) algebra

(continuum limit)

(infinite volume limit)

The theory thus obtained hasno parameters other than one scale paramter !

Clear large-N scaling behavior observed with

(continuum limit)

The extent of time increases andthe size of the universe becomes very large at later time.

(infinite volume limit)

A new proposal for the nonperturbative formulation of type IIB superstring theory in ten dimensions.

instead of making Wick rotation, we introduce the IR cutoffs for both temporal and spatial directions

after “gauge fixing” the boost symmetry

The two cutoffs can be removed in the large-N limit.

The theory thus obtained has no parameters other than one scale parameter.

Summary

Integrating over the scale factor first, we obtain a model without sign problem.

c.f.) Monte Carlo studies of IKKT model difficult due to sign problem

Monte Carlo simulation revealed SSB of SO(9) down to SO(3) at some critical time. mechanism is totally different from that for the SSB in the IKKT model

the size of the 3d space increases with time

Cosmological singularity is naturally avoided due to noncommutativity.

Crucial role played by SUSY

c.f.) bosonic model

no expansion and no SSB !

Future directions Does a local field theory on a commutative space-time appear at later time ?

How do 4 fundamental interactions and the matter fields appear at later time ?