The origin of space-time as seen from matrix model simulationsSeminar at KMI, Nagoya U.,Nov. 8 (Tue.), 2011

Jun Nishimura (KEK,SOKENDAI)

Ref.) M.Hanada, J.N., Y.Sekino, T.Yoneya , Phys.Rev. Lett. 104 (2010) 151601arXiv: 1108.5153

S.-W.Kim, J.N., A.Tsuchiya, arXiv:1108.1540, 1110.4803

Quantum gravity

Superstring theory isa natural candidate for a unified theory

including quantum gravity

Important problems in particle physics:

the hierarchy problem why EW scale is much smaller than the Planck scale(or why gravity is so weak)

the existence of dark energy, dark matterCMB, supernovae, structure formation, …

The testing ground for superstring theory

2 amazing predictions of Einstein’s general relativity

Black hole Big bang

singularitiesQuantum effects of gravity become crucial.

Important developments in the 90s

Gauge-gravity duality (e.g., AdS/CFT correspondence)Maldacena (1997), Gubser-Klebanov-Polyakov, Witten (1998)

Matrix model formulation of superstring/M theoriesBanks-Fischler-Shenker-Susskind (1996),Ishibashi-Kawai-Kitazawa-Tsuchiya (1997)

Gauge theory description of black hole thermodynamics Correspondence at the level of local operators

Dynamical origin of space-time Applications to the physics beyond the Standard Model

Monte Carlo simulation provides an important toolto explore these two directions.

Plan of the talk

1. Introduction2. Black hole thermodynamics from gauge theory3. Direct test of gauge-gravity correspondence4. (3+1)d expanding universe from matrix model

c.f.) Tsuchiya’s talk on Oct.18 (Tue.)5. Expanding universe as a classical solution6. Summary and discussions

Hanada-J.N.-Takeuchi, PRL 99 (’07) 161602Anagnostopoulos-Hanada- J.N.-Takeuchi, PRL 100 (’08) 021601Hanada-Miwa-J.N.-Takeuchi, PRL 102 (’09) 181602Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602

N D0 branes

thorizon

black 0-brane solutionin type IIA SUGRA

1d U(N) SUSYgauge theory

near-extremal black holeat finite T

In the decoupling limit, the D0 brane system describes the black hole microscopically.

Itzhaki-Maldacena-Sonnenschein-Yankielowicz (’98)

Gauge-gravity duality for D0-brane system

SUGRA description : valid

type IIA superstring

quantum description of the states inside the BH

Gauge/gravity duality predicts thatthis should be reproduced by 1d SYM. large-N, low T

microscopic origin of the black hole thermodynamics

Prediction from gauge/gravity duality (I) dual geometry

7.41

Hawking’s theory

black hole thermodynamics

Klebanov-Tseytlin (’96)

Comparison including corrections

corrections

Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602 [arXiv:0811.3102]

M.Hanada, J.N., Y.Sekino, T.Yoneya : Phys.Rev. Lett. 104 (2010) 151601arXiv: 1108.5153

Prediction from gauge/gravity duality (II)

Gubser-Klebanov-Polyakov-Witten relation (’98)

correlation functions in gauge theory

generating functional

operator-field correspondencegauge gravity

SUGRA action evaluated at the classical solutionwith the boundary condition

Correlation functions in 1d SYM theoryPerturbative calculations plagued by severe IR divergence :

require genuinely non-perturbative methods

1) gauge-gravity correspondence

2) Monte Carlo simulation

Sekino-Yoneya (’99)

for operators corresponding to supergravity modes in 10d SUGRA

Hanada-J.N.-Sekino-Yoneya (’09,’11)

Actually, agreement extends to M theory regime !Power-law behavior with the predicted exponent

based on Gubser-Klebanov-Polyakov-Witten relation (’98)

1d gauge theory

p.b.c.p.b.c.

(without loss of generality)

1d SYM with 16 supercharges

The region of validity for the SUGRA analysis

Series of operators (I)

Predicted power lawconfirmed clearly

even beyond the validityregion of 10d SUGRA

Some details of calculations

directly accessible by our Fourier space simulation

Gibbs phenomenon !

Actually,

Removes the Gibbs phenomenon completely.

Series of operators (II)

Comparison in the Fourier space :

(bad UV behavior) Reliable inverse Fourier tr.seems difficult…

Comparison in the Fourier space

polynomials of even powers

Best fit obtained for

Series of operators (III)

IR divergent !

IR divergent correlation function

polynomials of even powers

Best fit obtained for

finite IR cutoff effects

Larger angular momentum

Best fit obtained for

Best fit obtained for

S.-W.Kim, J.N., A.Tsuchiya, arXiv:1108.1540

The action has manifest SO(9,1) symmetry

raised and lowered by the metric

Hermitian matrices

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

Ishibashi-Kawai-Kitazawa-Tsuchiya (’96)

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

interactions between D-branes

string field theory from SD eqs. for Wilson loopsFukuma-Kawai-Kitazawa-Tsuchiya (’98)

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

Evidence for the conjecture :

Aoki-Iso-Kawai-Kitazawa-Tada (’99)

Wick rotation

Euclidean model SO(10) symmetry

opposite sign !

An important feature of the Lorentzian model

A conventional approach was:

Krauth-Nicolai-Staudacher (’98), Austing-Wheater (’01) Partition function becomes finite.

SSB of SO(10) J.N.-Okubo-Sugino, arXiv:1108.1293

Results of the Gaussian expansion methodJ.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 the Euclidean model, but is it really related to the real world ?

extended directions

shrunken directions

connection to the worldsheet theory

Unlike the Euclidean model, the path integral is ill-defined !

Nonperturbative dynamics of the Lorentzian model

studied, for the first time, in Kim-J.N.-Tsuchiya, arXiv:1108.1540

Introduce IR cutoff in both the temporal and spatial directions

They can be removed in the large-N limit. Continuum limit& infinite volume limit

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

The mechanism of SSB : SO(9) -> SO(3)

S.-W.Kim, J.N., A.Tsuchiya, arXiv:1110.4803

Lagrange multipliers corresponding to the IR cutoffs

Classical equations of motion for the Lorentzian model :

We look for a Lie algebraic solution :

c.f.) Euclidean model Chatzistavrakidis arXiv:1108.1107 [hep-th]

Motivated by Monte Carlo results, we restrict ourselves to

and look for solutions with SO(3) symmetry.

From the complete list of real Lie algebras with 4 generators

the one with SO(3) symmetry is UNIQUE !

Others = 0

others = 0

The unitary irreducible representations of

can be classified into 2 categories

1) trivial 1d representations

2) infinite-dimensional representations

the basis of the functional space

Eigenfunctions of the Hamiltonian of a 1d harmonic oscillator

SO(3) symmetric solutions

Using a direct sum of the non-trivial representations,

In what follows,

Compatible with the expanding behavior !

size of the space

(dimensionless) space-time noncommutativity

c.f.) space-time uncertainty principleYoneya (2000)

Speculations

time

classical solution

tcr

Monte Carlosimulation

SO(9)

size of the space

space-space noncommutativity

present time

acceleratingexpansion

space-timenoncommutativity

Space-space NC disappears for some dynamical reason.

symmetry of space

SO(3)

Black hole singularity Big bang singularity

Monte Carlo simulation of supersymmetric gauge theories and matrix models

Quantum effects of gravity become crucial.

Two kinds of singularity predicted by Einstein’s general relativity

Superstring theory

Gauge-gravity correspondence“Emergent space”1d SYM describes the space-time with black hole geometry

Lorentzian matrix modelEmergence of (3+1)d expanding universe

Summary

Future directions Extending the study of supersymmetric gauge theory

to higher dimensions

1d SYM with mass deformation

Large-N equivalenceIshii-Ishiki-Shimasaki-Tsuchiya (2008)

superconformal

“Holographic inflation” (Skenderis)

Connecting the “two ends” in the Lorentzian matrix model Quantum corrections around the classical solutions

Exploring more general Lie-algebraic solutions

The gauge group and the matter contents, power-law expansion

holographic dual of SUSY matrix QM

Itzhaki-Maldacena-Sonnenschein-Yankielowicz ’98

near-extremal 0-brane solution in type IIA SUGRA(string frame)

dilaton :

decoupling limit

with fixed

(’t Hooft coupling)

validity of the SUGRA description :

curvature radius (in string units)

dilaton at the radius U

Black hole thermodynamics

internal energy

Klebanov-Tseytlin ’96

We check this in strongly coupled gauge theory !

corrections to SUGRA action

tree-level scattering amplitudes of the massless modes

low energy effective action of type IIA superstring theory

leading term : type IIA SUGRA action

explicit calculations of 2-pt and 3-pt amplitudes

4-pt amplitudes

Complete form is yet to be determined, but we can still make a dimensional analysis.

Black hole thermodynamics with corrections

curvature radius of the dual geometry

More careful treatment leads to the same conclusion.(Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602

Hanada-J.N.-Takeuchi, PRL 99 (07) 161602 [arXiv:0706.1647]Non-lattice simulation

residual gauge symmetry :

should be fixed by imposing

static diagonal gauge :

Note: Gauge symmetry can be fixed non-perturbatively in 1d.

c.f.) lattice approach : Catterall-Wiseman, JHEP 0712:104,2007

RHMC algorithm can be used efficiently(Fourier acceleration without extra cost etc.)

What is M theory ?hypothetical 11d theory

suggested from string dualities

low-energy effective theory : 11D SUGRA

fundamental d.o.f.: membranesoliton-like objects: M5-brane

compactify the theory on a circle10D type IIA superstring

believed to appear in the strong coupling limitof 10D type IIA superstring

Witten (’95)

Implications on the M theory limit

The exponents obtained from10d SUGRA analysis are valid also in the M-theory limit!

The exponent agrees with the prediction even at N=3.

Surprising aspects of our MC results (2):

M theory limit amounts to: