Dark energy from a Lorentzian matrix - research.kek.jp...
Transcript of Dark energy from a Lorentzian matrix - research.kek.jp...
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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
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1. Introduction
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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
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“critical time”
SSB
Kim-J.N.-Tsuchiya PRL108, 011601 (2012)
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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)
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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
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2. Expanding (3+1)-dimensional
universe from the Lorentzian matrix
model
Kim-J.N.-Tsuchiya PRL108, 011601 (2012)
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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)
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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)
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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
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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
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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
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(continuum limit)
(infinite volume limit)
The theory thus obtained has
no parameters other than one scale paramter !
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Extracting the time evolution
average
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“critical time”
SSB
Kim-J.N.-Tsuchiya PRL108, 011601 (2012)
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Clear large-N scaling behavior observed with
(continuum limit)
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The extent of time increases and
the size of the universe becomes very large at later time.
(infinite volume limit)
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3.Dark energy from the
Lorentzian matrix model
Kim-J.N.-Tsuchiya, in preparation
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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.
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Warming up :
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Using eq. of motion,
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Jacobi identity :
Trivially satisfied.
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How to obtain (3+1)d solutions with no
space-space noncommutativity
We rotate the previous solution as
is also a solution
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So, we need to look for Lie algebras with 3 generators.
SU(2)
SU(1,1)
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For SU(2), we obtain a solution:
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Unitary irreducible representation of SU(1,1)
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unitary irreducible representations can be obtained for
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How to extract the time evolution
The block matrices defined at each time:
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Continuum limit
Space-time noncommutativity
vanishes in the continuum limit.
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This region may describe
the late-time behavior
of the Lorentzian model
dynamically generated scale
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Let us naively interpret:
Cosmological constant problem is naturally solved.c.f.) Kawai-Okada arXiv1110.2303
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5. Summary and discussions
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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
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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
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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
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Thank you for your attention!
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Consider a simpler problem :
solution :
representation matrices of
a compact semi-simple Lie algebra
with d generators
Maximum is achieved for SU(2) algebra