EXPANDING (3+1)-DIMENSIONAL UNIVERSE FROM THE IIB MATRIX MODEL

Asato Tsuchiya (Shizuoka Univ.)

SQS ’2013 @Bogoliubov Laboratory, July 29th, 2013

References

Sang-Woo Kim, Jun Nishimura and A. T. PRL 108 (2012) 011601, arXiv:1108.1540 PRD 86 (2012) 027901, arXiv:1110.4803 JHEP 10 (2012) 147, arXiv:1208.0711 Jun Nishimura and A.T., PTEP 2013, 043B03, arXiv:1208.4910 arXiv: 1305.5547 Hajime Aoki, Jun Nishimura and A. T., in preparation Yuta Ito, Jun Nishimura and A. T., in preparation

Present status of superstring theory

Numerous vacua

Cosmic (initial) singularity

space-time dimensionsgauge groupsmatter contents (generation number)

Liu-Moore-Seiberg (’02), ……

perturbation theory 　＋　 D-brane

various

Use the statistical method or appeal to the anthropic principle

There are numerous vacua that are theoretically allowed

In general, perturbation theory cannot resolve the cosmic singularity

Non-perturbative effects are important at the beginning of the universe

Matrix models

IIB matrix model Ishibashi-Kawai-Kitazawa-A.T. (’96)

Matrix theory Banks-Fischler-Shenker-Susskind (‘96)

Matrix string theory Dijkgraaf-Verlinde-Verlinde (’97)

10D U(N)SYM

0D

1D

2D

There is a possibility that one can actually determine the true vacuum uniquely and resolve the cosmic singularity if one uses a nonperturbative formulation that incorporates full nonperturbative effects.

We study the IIB matrix model. “Lorentzian” is a key to this project.

~ lattice QCD for QCDProposals

Plan of the present talk

1. Introduction2. Review of the IIB matrix model3. Defining Lorentzian version4. SSB of SO(9) symmetry to SO(3)5. Realizing the Standard Model particles6. Summary and outlook

IIB matrix model

IIB matrix model

Space-time does not exist a priori, but is generated dynamicallyTime is given a priori in Matrix theory and matrix string theory

Manifest SO(9,1) symmetry, manifest 10D N=2 SUSY

: 10D Lorentz vector: 10D Majorana-Weyl spinor

Hermitian matrices

Large- limit is taken

covariant formulation Matrix theory and matrix string theory are light-cone formulation

Ishibashi-Kawai-Kitazawa-A.T. (’96)

Correspondence with world-sheet action

Green-Schwarz action of Schild-type for type IIB superstrig with κ symmetry fixed

IIB matrix model

2nd quantized

matrix regularization

eigenvalues of are coordinates

10D N=2 SUSY

dimensional reduction of 10D N=1 SUSY

10D N=2 SUSY

suggests that the model includes gravity

Corresponding to 10DN=2SUSY possessed by Green-Schwarz action

translation of eigenvalues

Interaction between D-branes

graviton scalar

Light-cone string field theory

Schwinger-Dyson equation for on the light front

light-cone string field theory for type IIB superstring

This implies that IIB matrix model reproduces perturbation theory of type IIB superstring

Fukuma-Kawai-Kitazawa-A.T. (’97)

Euclidean model

Wick rotationEuclidean model

manifest SO(10) symmetry

Euclideanization

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

Euclidean model is well-defined without cutoffs

People have been studying the Euclidean model

opposite sign ！ looks quite unstable

Lorentzian model

: positive definite

10 hermitian matrices

Space-time in Euclidean model

Aoki-Iso-Kawai-Kitazawa-Tada (’99)low energy effective theory for 　 ~ branched polymer

10 Ｄ4 Ｄ

configurations diagonalizable simultaneously are favored

SSB of SO(10) to SO(4)

If eigenvalues are distributed on 4D hyperplane, SSB of SO(10) to SO(4) 　 occurs ~ dynamical generation of 4D space-timeBut not confirmed so far

Kim-Nishimura-A. T. (’11)

Defining Lorentzian version

Why Lorentzian version? see time evolution of universe

~ need to study real time dynamics Wick rotation in gravitational theory seems nontrivial, in contrast to

field theory on flat space-time

ex.) causal dynamical triangulation （ CDT) Ambjorn-Jurkiewicz-Loll (’05)

Coleman mechanism in space-time with Lorenztian signature 　　　　　

Kawai-Okada (’11)

Recent results for the Euclidean model in Gaussian expansion method 　　　 Nishimura-Okubo-Sugino (’11) 　　　　　　

suggest dynamical generation of 3-dimensional space-time

Here we study Lorentzian version of the IIB matrix model

　　　

Regularization and large-N limit

By introducing cutoffs, we make time and space directions finite

　 It turns out that these cutoffs can be removed in the large-N limit　　 ~quite nontrivial dynamical property　 Lorentzian version is well-defined　 we expect the effect of the explicit breaking of SO(9,1) and SUSY due to the cutoffs to vanish in the large-N limit need to check　　

How to define partition functionnatural from a viewpoint of the Wick rotationof worldsheet

Kim-Nishimura-A. T. (’11)

SSB of SO(9) to SO(3)

average

Emergence of concept of ``time evolution”

represents space structure at fixed time t

We observe band-diagonal structure

small

small

these values are determined dynamically

We take

Determination of block size

SSB of SO(9) symmetry

“critical time”

SSB

symmetric under

we only show

Exponential expansion

inflation

From exponential to linear

6d bosonic model 　　　 omit fermions

Big Bang?

inflation

radiation dominated universe

Nishimura-A. T. (’12)Aoki-Nishimura-A. T., work in progress

Cf.) Chatzistavrakidis-Steinnacker-Zoupanos (’11)

Realizing Standard Model particles

Late time behaviors

It is likely that we see the beginning of universe (inflation) in the present Monte Carlo simulation we need larger N to see late times At later times, ``well-known” universe should emerge

Do inflation and big bang occur?not phenomenological description by ``inflaton” but first principle description by superstring theory

The Standard Model should appear

present accelerating expansion (dark energy)understanding of cosmological constant problem

we can expect that at late times classical approximation is good and fluctuation around the background is small because the action is large due to expansion

Chiral fermions

Dirac equation in 10d

is Majorana-Weyl in 10d

massless modes

chiral fermions in 4d

and are different

Background

(3+1)d Lorentz symmetric backgroundmatrix analog of warp factor

# of zero modes with each chirality is the same

constructive definition

Warped geometry and chiral fermion

# of variables < # of equationsno solution generically

we obtain left-handed chiral fermion in 4d

many solutions

warped geometry

vector-like fermions no-go theorem

direct product

Example intersecting branes

4

5

6

6

5

9

7

4

8

~ D5-brane

~ D7-brane

fuzzy spheres

Solving 6d Dirac equation

L

R

we solve

charge conjugation ~ identified

zero modes ( ) appear at intersection points ~ L and R

Wave functions and warp factor

Left-handed

Right-handed

zero modes

1st excited state

we obtain one generation of left-handed fermion

6 4, 5

64, 5

Three generations

R L R

LR

L

we squash

we obtain three generations of left-handed fermionswave functions are different among generations so are Yukawa coupling

Standard Model fermions and right-handed neutrino Cf.) Chatzistavrakidis-Steinnacker-Zoupanos (’11)

Summary and outlook

when is made diagonal, have band-diagonal structure

Summary We studied Lorentzian version of the IIB matrix model

We introduced the infrared cutoffs and found that they can be removed in the large-N limit The model thus obtained is well-defined and has no parameters except one scale parameter

This property is expected in nonperturbative string theory the concept of ``time evolution” emerges

After a critical time, SO(9) symmetry of 9 dimension is spontaneously broken down to SO(3) and 3 out of 9 dimensions start to expand rapidly ~ can be interpreted as birth of universe

Summary (cont’d)

We gave an example of background where three generations of the Standard Model particles are realized

We found that matrix analog of warp factor must be introduced to realize chiral fermions, and gave an example of background yielding the chiral fermion

We confirm that the expansion is exponential ~ beginning of inflation We observe transition from exponential expansion to linear one in toy model

There is a possibility that Standard Model can be derived uniquely at low energy structure of extra dimensions and warp factor are determined dynamically ~ to substantiate string theory

Outlook

background at late time : e-foldings, Big Bang, Standard Model

use the idea of renormalization group to reach the late time ~ work for a toy model

It is important to study the classical solutions and look for a solution connected smoothly to the simulation Kim-Nishimura-A. T. (’12)

more efficient Monte Carlo simulation ~ parallelization

calculation of Yukawa couplings : mass hierarchy , CKM matrix and

MNS matrix

Outlook (cont’d)

Does exponential expansion occur? Does big bang occur after that? it should be seen as (the second) phase transition Does it occur at the same time as commutative space-time appears How density fluctuation can be measured to compare with CMB Can lagrangian of GUT be read off from fluctuation around the classical solution? Does standard model appear at low energy? We expect to understand in a unified way various problems in particle physics and cosmology such as mechanism of inflation, cosmological constant problem, hierarchy problem, dark matter, dark energy, baryogenesis

String duality

Het SO(32)

Het E8 x E8

M

IIA

IIB

I

dualities in superstring theory

five superstring theories and M theory are different descriptions of one theory

One can start everywhere with a formulation which enables one to treat strong coupling dynamics

How to define partition function

natural from a viewpoint of correspondencewith worldsheet theory

Wick rotation for worldsheet

Lorentzian version

unlike the case of the Euclidean model, the Lorentzian model is ill-defined as it stands

By introducing cutoffs, we make time and space directions finite

　 It turns out that these cutoffs can be removed in the large-N limit　　 ~quite nontrivial dynamical property　 SO(9,1) symmetry and SUSY are explicitly broken by the cutoffs　 we expect the effect of the explicit breaking to vanish in the large-N limit need to check　　

without loss of generalitywe put 　　　

Regularization and large-N limit

（１）　 Pfaffian coming from integral over fermion

in the Euclidean model

is dominant in the large-N limit

can give rise to sign problem

（２） （１）

Avoiding sign problem

（２） How to treat 　

homogeneousin

first perform

Also problem in field theory on Minkowski space（ analysis of real time dynamics is a notorious problem ）

Avoiding sign problem (cont’d)

Time evolution of space size

peak at starts to grow for

Symmetric under

We only show

Late time behaviors

Seeing later times

It is likely that we see the beginning of universe in the present Monte Carlo simulation At later times, ``well-known” universe should emerge

Do inflation and big bang occur?not phenomenological description by ``inflaton” but first principle description by superstring theory

can be tested by comparison with CMB How commutative space-time appears

What massless fields appear on it

present accelerating expansion (dark energy)understanding of cosmological constant problem

prediction for fate of universe big crunch or big rip

Classical approximation as a complementary approach

if we reach the time by Monte Carlo simulation and uniquely pick up a dominant classical solution connected smoothly to the result in the simulation, we can study late time behaviors

we expect classical approximation is good after late times because action becomes large due to expansion

there are infinitely many solutions

variation function

EOM

Example of solution

SL(2,R) solution

(3+1)-dimensional space-time ～ R×S3

identify t0 with the present time

present accelerating expansion

cosmological constant ~ solve cosmological constant problem

cosmological constant term vanishes in the future

considered to give late time behaviors in the matrix model

Cosmological implication of SL(2,R) solution

Chiral fermions cont’d

nonzero eigenvalues are paired

finite-N matrices ~space of with each chirality has the same dimensions ~number of zero modes with each chirality is the same

Commutative space-time and local fields

results in Monte Carlo simulation

are smaller than 　　　

there are classical solutions representing (3+1)-dimensional commutative space-time

we assume a classical solution 　　　 representing (3+1)-dimensional

commutative space-time is dominant at sufficiently late timesare close to diagonal

space-time points in 3+1 dimensions

are small for with

Commutative space-time and local fields (cont’d)

fluctuations around the classical solution

we further assume that fluctuations corresponding to massless modes are also close to diagonal

are close to each other

same structure in cubic and quartic tems

local field theory

In arXiv:1208.4910, we showed that NG modes for symmetry breaking of Poincare symmetry and SUSY indeed appear as local massless fields

is also a classical solution

cf.) Iso-Kawai (’99)

if 　　　　 are a classical solution,

fluctuations around

c.f.) stack of k D-branes 　　 SU(k) gauge theory

Gauge symmetry

local SU(k) symmetry

is a linear combination of 　　　　　　

3 2 1 1 1

: mirror partners

: bi-fundamental rep. of

hypercharge can be assigned consistently

minimum

c.f.) H.Aoki PTP 125 (2011) 521 Chatzistavrakidis-Steinacker-Zoupanos JHEP 09 (2011) 115

GUT

　　 Given a classical solution, we can read off detail of local field theory from fluctuations around it ~ GUT

　　 Is there a classical solution that allows chiral fermions?　　

　　 Is part of SUSY preserved? (we can answer this question if we are given the classical solution)　　　　　　　 if this is the case, hierarchy problem is OK　　　　　　　　 if this is not, all scalar fields acquire mass of GUT scale due to radiative correction composite Higgs structure of extra dimension in the classical solution is important　　　　　　　　　　　　　　

c.f.) H.Aoki PTP 125 (2011) 521 Chatzistavrakidis-Steinacker-Zoupanos JHEP 09 (2011) 115

GUT (cont’d)

Summary (cont’d)

We gave a general prescription to find solutions and classified the solutions under some ansatzes We gave a general prescription to find solutions and classified the solutions under some ansatzes

We found a solution which represents expanding (3+1)-dimensional

universe and resolves cosmological constant problem

To reach late times, we need larger N, while Classical approximation should be good at late times There are infinitely many solutions we need to know which one is smoothly connected to MC result

Kim-Nishimura-A. T. (’12)