Transcript Expanding (3+1)-dimensional universe from Lorentzian IIB

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
perturbation theory ＋ D-brane
Numerous vacua
There are numerous vacua that are theoretically allowed
space-time dimensions
various
gauge groups
matter contents (generation number)
Use the statistical method or appeal to the anthropic principle
Cosmic (initial) singularity
Liu-Moore-Seiberg (’02), ……
In general, perturbation theory cannot resolve the cosmic singularity
Non-perturbative effects are important at the beginning of the universe
Matrix models
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.
~ lattice QCD for QCD
Proposals
IIB matrix model
Matrix theory
Matrix string theory
Ishibashi-Kawai-Kitazawa-A.T. (’96)
Banks-Fischler-Shenker-Susskind (‘96)
Dijkgraaf-Verlinde-Verlinde (’97)
We study the IIB matrix model.
“Lorentzian” is a key to this project.
0D
1D
2D
10D
U(N)
SYM
Plan of the present talk
1.
2.
3.
4.
5.
6.
Introduction
Review of the IIB matrix model
Defining Lorentzian version
SSB of SO(9) symmetry to SO(3)
Realizing the Standard Model particles
Summary and outlook
IIB matrix model
IIB matrix model
Ishibashi-Kawai-Kitazawa-A.T. (’96)
Hermitian matrices
: 10D Lorentz vector
: 10D Majorana-Weyl spinor
Large-
limit is taken
Space-time does not exist a priori, but is generated dynamically
Time is given a priori in Matrix theory and matrix string theory
Manifest SO(9,1) symmetry, manifest 10D N=2 SUSY
covariant formulation
Matrix theory and matrix string theory are light-cone formulation
Correspondence with world-sheet action
Green-Schwarz action of Schild-type for type IIB superstrig with κ symmetry fixed
matrix regularization
IIB matrix model
2nd quantized
10D N=2 SUSY
Corresponding to 10DN=2SUSY possessed by Green-Schwarz action
dimensional reduction of 10D N=1 SUSY
10D N=2 SUSY
eigenvalues of
are coordinates
suggests that the model includes gravity
translation of
eigenvalues
Interaction between D-branes
graviton
scalar
Light-cone string field theory
Fukuma-Kawai-Kitazawa-A.T. (’97)
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
Euclidean model
Lorentzian model
opposite sign！
Euclideanization
looks quite unstable
Wick rotation
Euclidean model
manifest SO(10) symmetry
: positive definite
Euclidean model is well-defined without cutoffs
Krauth-Nicolai-Staudacher (’98),
Austing-Wheater (’01)
People have been studying the Euclidean model
Space-time in Euclidean model
configurations diagonalizable simultaneously are favored
SSB of SO(10) to SO(4)
10
hermitian matrices
4Ｄ
10Ｄ
low energy effective theory for
~ branched polymer Aoki-Iso-Kawai-Kitazawa-Tada (’99)
If eigenvalues are distributed on 4D hyperplane, SSB of SO(10) to SO(4) occurs
~ dynamical generation of 4D space-time
But not confirmed so far
Defining Lorentzian version
Kim-Nishimura-A. T. (’11)
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
 How to define partition function
natural from a viewpoint of the Wick rotation
of worldsheet
 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
SSB of SO(9) to SO(3)
Kim-Nishimura-A. T. (’11)
Emergence of concept of ``time evolution”
average
these values are
determined dynamically
small
We observe band-diagonal
structure
represents space
structure at fixed time t
small
Determination of block size
We take
SSB of SO(9) symmetry
SSB
symmetric under
we only show
“critical time”
Exponential expansion
inflation
From exponential to linear
6d bosonic model
omit fermions
inflation
Big Bang?
radiation
dominated
universe
Realizing Standard Model particles
Nishimura-A. T. (’12)
Aoki-Nishimura-A. T., work in progress
Cf.) Chatzistavrakidis-Steinnacker-Zoupanos (’11)
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
 present accelerating expansion (dark energy)
understanding of cosmological constant problem
 The Standard Model should appear
 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
is Majorana-Weyl in 10d
Dirac equation in 10d
massless modes
chiral fermions in 4d
Background
(3+1)d Lorentz symmetric background
matrix analog of warp
factor
constructive definition
# of zero modes with
each chirality is the same
and
are different
Warped geometry and chiral fermion
 direct product
vector-like fermions
no-go theorem
 warped geometry
many solutions
# of variables < # of equations
no solution generically
we obtain left-handed chiral fermion in 4d
Example intersecting branes
fuzzy spheres
6
~ D5-brane
5
4
~ D7-brane
4
5
6
7
9
8
Solving 6d Dirac equation
R
L
charge conjugation
~ identified
we solve
zero modes (
) appear at intersection points ~ L and R
Wave functions and warp factor
Left-handed
1st excited state
zero modes
6
4, 5
Right-handed
we obtain one generation of left-handed fermion
6
4, 5
Three generations
we squash
L
R
L
R
L
R
we obtain three generations of left-handed fermions
wave functions are different among generations so are Yukawa coupling
Standard Model fermions and righthanded neutrinoCf.) Chatzistavrakidis-Steinnacker-Zoupanos (’11)
Summary and outlook
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
when
is made diagonal,
structure
have band-diagonal
 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 confirm that the expansion is exponential
~ beginning of inflation
 We observe transition from exponential expansion to linear one
in toy model
 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 gave an example of background where three generations of the
Standard Model particles are realized
 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
 calculation of Yukawa couplings : mass hierarchy , CKM matrix and
MNS matrix
 background at late time : e-foldings, Big Bang, Standard Model
 more efficient Monte Carlo simulation ~ parallelization
 It is important to study the classical solutions and look for a solution
Kim-Nishimura-A. T. (’12)
connected smoothly to the simulation
 use the idea of renormalization group to reach the late time
~ work for a toy model
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
dualities in superstring theory
M
IIA
five superstring theories and M theory
are different descriptions of one theory
Het E8 x E8
Het SO(32)
IIB
I
One can start everywhere with a formulation which
enables one to treat strong coupling dynamics
Lorentzian version
How to define partition function
natural from a viewpoint of correspondence
with worldsheet theory
Wick rotation for worldsheet
Regularization and large-N limit
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
without loss of generality
we put
 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
Avoiding sign problem
（２）
（１）
can give rise to sign problem
（１） Pfaffian coming from integral over fermion
is dominant in the large-N limit
in the Euclidean model
Avoiding sign problem (cont’d)
（２） How to treat
Also problem in field theory on Minkowski space
（analysis of real time dynamics is a notorious problem）
first perform
homogeneous
in
Time evolution of space size
peak at
Symmetric under
We only show
starts to grow for
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
 present accelerating expansion (dark energy)
understanding of cosmological constant problem
 What massless fields appear on it
 prediction for fate of universe big crunch or big rip
Classical approximation as
a complementary approach
 we expect classical approximation is good after late times
because action becomes large due to expansion
 there are infinitely many solutions
 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
Example of solution
 variation function
 EOM
 SL(2,R) solution
(3+1)-dimensional space-time～R×S3
Cosmological implication of SL(2,R) solution
considered to give late time
behaviors in the matrix model
identify t0 with the present time
present accelerating expansion
cosmological constant ~
solve cosmological constant problem
cosmological constant term vanishes in the future
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 times
are 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
Gauge symmetry
if
cf.) Iso-Kawai (’99)
are a classical solution,
is also a classical solution
fluctuations around
local SU(k) symmetry
c.f.) stack of k D-branes
SU(k) gauge theory
GUT
c.f.) H.Aoki PTP 125 (2011) 521
Chatzistavrakidis-Steinacker-Zoupanos
JHEP 09 (2011) 115
minimum
3
2
1
1
1
: bi-fundamental rep. of
: mirror partners
hypercharge can be assigned consistently
is a linear combination of
GUT (cont’d)

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?
c.f.) H.Aoki PTP 125 (2011) 521
Chatzistavrakidis-Steinacker-Zoupanos
JHEP 09 (2011) 115

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
Summary (cont’d)
Kim-Nishimura-A. T. (’12)
 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
 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