Dark energy from a Lorentzian matrix

model for superstring theory

Talk at KEK Theory Workshop, 2012.3.5-7Jun Nishimura (KEK, Sokendai)

Ref.) Kim-J.N.-Tsuchiya arXiv:1108.1540PRL 108, 011601 (2012)

Kim-J.N.-Tsuchiya, in preparation

1. Introduction

The birth of our Universe

from a Lorentzian matrix model

IKKT matrix model (Ishibashi,Kawai,Kitazawa,Tsuchiya ’96)

Monte Carlo studies of the Lorentzian matrix model

with SO(9,1) symmetry

Kim-J.N.-Tsuchiya PRL108, 011601 (2012)

Surprisingly, 3 out of 9 directions start to expand

at some critical time !

A nonperturbative definition of

type IIB superstring theory in (9+1) dimensions

“critical time”

SSB

Kim-J.N.-Tsuchiya PRL108, 011601 (2012)

Here we would like to study the behavior at later times.

As a complementary approach to Monte Carlo sim.,

we study classical equations of motion.

(3+1)-dimensonal commutative space-time

the time-dependence compatible with expanding universe

dark energy, natural solution to cosmological const. problem

(expected to be valid at later times)

Plan of the talk

1. Introduction

2. Expanding (3+1)-dimensional Universe from

the Lorentzian matrix model

3. Dark energy from the Lorentzian matrix

model

4. Summary and discussions

2. Expanding (3+1)-dimensional

universe from the Lorentzian matrix

model

Kim-J.N.-Tsuchiya PRL108, 011601 (2012)

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 loops

Fukuma-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, JHEP 1110 (2011) 135

Results of the Gaussian expansion methodJ.N.-Okubo-Sugino JHEP 1110 (2011) 135

Minimum of the free energy

occurs at d=3

Extent of space-time

finite 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 PRL108, 011601 (2012)

Introduce IR cutoff in both the temporal and spatial directions

(continuum limit)

(infinite volume limit)

The theory thus obtained has

no parameters other than one scale paramter !

Extracting the time evolution

average

“critical time”

SSB

Kim-J.N.-Tsuchiya PRL108, 011601 (2012)

Clear large-N scaling behavior observed with

(continuum limit)

The extent of time increases and

the size of the universe becomes very large at later time.

(infinite volume limit)

３．Dark energy from the

Lorentzian matrix model

Kim-J.N.-Tsuchiya, in preparation

Lagrange multipliers corresponding

to the IR cutoffs

Classical equations of motion for the Lorentzian model :

Let us construct (3+1)-dimensional solutions

with NO space-space noncommutativities.

Warming up :

Using eq. of motion,

Jacobi identity :

Trivially satisfied.

How to obtain (3+1)d solutions with no

space-space noncommutativity

We rotate the previous solution as

is also a solution

So, we need to look for Lie algebras with 3 generators.

SU(2)

SU(1,1)

For SU(2), we obtain a solution:

Unitary irreducible representation of SU(1,1)

unitary irreducible representations can be obtained for

How to extract the time evolution

The block matrices defined at each time:

Continuum limit

Space-time noncommutativity

vanishes in the continuum limit.

This region may describe

the late-time behavior

of the Lorentzian model

dynamically generated scale

Let us naively interpret:

Cosmological constant problem is naturally solved.c.f.) Kawai-Okada arXiv1110.2303

5. Summary and discussions

Summary

IKKT matrix model

type IIB superstring theory in (9+1)-dimensions

Previous results on the Euclidean model motivated

us to study the Lorentzian model

Monte Carlo studies(probes only the early times due to small matrix size)

SO(9) -> SO(3) at some “critical time”

“the birth of our Universe”

Classical solutions (expected to be valid at later times)

Expanding (3+1)-dim. commutative Universe

Dark energy

Speculations

time

classical solution

tcr

Monte Carlo

simulation

SO(9) SO(3)

size of the space

space-space noncommutativity

present time

accelerating

expansion

space-time

noncommutativity

Space-space NC disappears for some dynamical reason.

symmetry of space

We hope the Lorentzian matrix model

provides a new perspective on

particle physics beyond the standard model

cosmological models for inflation, modified gravity, etc..

Future directions

Increase N and study later times by Monte Carlo sim.

studies in the bosonic model with quenched

Exploring other (3+1)dim. Classical solutions

gauge group and the matter content

the structure in the extra dimensions is important

c.f.) intersecting D-brane models

Quantum corrections around the classical solution

validity of the classical solution

power-law expansion at earlier time

Thank you for your attention!

Consider a simpler problem :

solution :

representation matrices of

a compact semi-simple Lie algebra

with d generators

Maximum is achieved for SU(2) algebra