Transcript The origin of space-time as seen from matrix model simulations

The origin of space-time as seen
from matrix model simulations
Seminar 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) 151601
arXiv: 1108.5153
S.-W.Kim, J.N., A.Tsuchiya, arXiv:1108.1540, 1110.4803
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 matter
CMB, supernovae, structure formation, …
Quantum gravity
Superstring theory is
a natural candidate for a unified theory
including quantum gravity
The testing ground for superstring theory
2 amazing predictions of Einstein’s general relativity
Big bang
Black hole
singularities
Quantum 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)
 Gauge theory description of black hole thermodynamics
 Correspondence at the level of local operators
 Matrix model formulation of superstring/M theories
Banks-Fischler-Shenker-Susskind (1996),
Ishibashi-Kawai-Kitazawa-Tsuchiya (1997)
 Dynamical origin of space-time
 Applications to the physics beyond the Standard Model
Monte Carlo simulation provides an important tool
to explore these two directions.
Plan of the talk
1.
2.
3.
4.
Introduction
Black hole thermodynamics from gauge theory
Direct test of gauge-gravity correspondence
(3+1)d expanding universe from matrix model
c.f.) Tsuchiya’s talk on Oct.18 (Tue.)
5. Expanding universe as a classical solution
6. Summary and discussions
Hanada-J.N.-Takeuchi,
Anagnostopoulos-Hanada- J.N.-Takeuchi,
Hanada-Miwa-J.N.-Takeuchi,
Hanada-Hyakutake-J.N.-Takeuchi,
PRL 99 (’07) 161602
PRL 100 (’08) 021601
PRL 102 (’09) 181602
PRL 102 (’09) 191602
Gauge-gravity duality for D0-brane system
type IIA superstring
N D0 branes
Itzhaki-Maldacena-Sonnenschein
-Yankielowicz (’98)
horizon
t
1d U(N) SUSY
gauge theory
at finite T
black 0-brane solution
in type IIA SUGRA
near-extremal black hole
In the decoupling limit, the D0 brane system describes the
black hole microscopically.
SUGRA description : valid
Prediction from gauge/gravity duality (I)
 dual geometry
Hawking’s theory
black hole thermodynamics
7.41
Klebanov-Tseytlin (’96)
Gauge/gravity duality predicts that
this should be reproduced by 1d SYM.
large-N, low T
microscopic origin of the black hole thermodynamics
quantum description of the states inside the BH
Comparison including
corrections
Hanada-Hyakutake-J.N.-Takeuchi,
PRL 102 (’09) 191602 [arXiv:0811.3102]
corrections
M.Hanada, J.N., Y.Sekino, T.Yoneya : Phys.Rev. Lett. 104 (2010) 151601
arXiv: 1108.5153
Prediction from gauge/gravity duality (II)
correlation functions in gauge theory
generating functional
operator-field correspondence
gauge
gravity
Gubser-Klebanov-Polyakov-Witten relation (’98)
SUGRA action evaluated at the classical solution
with the boundary condition
Correlation functions in 1d SYM theory
Perturbative calculations plagued by
severe IR divergence :
require genuinely non-perturbative methods
1) gauge-gravity correspondence
Sekino-Yoneya (’99)
based on Gubser-Klebanov-Polyakov-Witten relation (’98)
for operators
corresponding to supergravity modes in 10d SUGRA
2) Monte Carlo simulation
Hanada-J.N.-Sekino-Yoneya (’09,’11)
Power-law behavior with the predicted exponent
Actually, agreement extends to M theory regime !
1d SYM with 16 supercharges
1d gauge theory
p.b.c.
p.b.c.
The region of validity for the SUGRA analysis
(without loss of generality)
Series of operators (I)
Predicted power law
confirmed clearly
even beyond the validity
region 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)
(bad UV behavior)
Reliable inverse Fourier tr.
seems difficult…
Comparison in the Fourier space :
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
finite IR cutoff effects
Best fit obtained for
Larger angular momentum
Best fit obtained for
Best fit obtained for
S.-W.Kim, J.N., A.Tsuchiya, arXiv:1108.1540
Matrix model proposed as a nonperturbative definition
of type IIB superstring theory in 10 dim.
Ishibashi-Kawai-Kitazawa-Tsuchiya (’96)
Hermitian matrices
raised and lowered by the metric
The action has manifest SO(9,1) symmetry
Evidence for the conjecture :
 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)
Aoki-Iso-Kawai-Kitazawa-Tada (’99)
An important feature of the Lorentzian model
opposite sign !
A conventional approach was:
Wick rotation
Euclidean model SO(10) symmetry
 Partition function becomes finite.
Krauth-Nicolai-Staudacher (’98), Austing-Wheater (’01)
 SSB of SO(10)
J.N.-Okubo-Sugino, arXiv:1108.1293
Results of the Gaussian expansion method
J.N.-Okubo-Sugino (arXiv:1108.1293)
extended directions
shrunken directions
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 ?
Nonperturbative dynamics of the Lorentzian model
studied, for the first time, in Kim-J.N.-Tsuchiya, arXiv:1108.1540
connection to the worldsheet theory
Unlike the Euclidean model, the path integral is ill-defined !
 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
SSB
“critical time”
The mechanism of SSB : SO(9) -> SO(3)
Consider a simpler problem :
solution :
representation matrices of
a compact semi-simple Lie algebra
with d generators
Maximum is achieved for SU(2) algebra
S.-W.Kim, J.N., A.Tsuchiya, arXiv:1110.4803
Classical equations of motion for the Lorentzian model :
Lagrange multipliers corresponding
to the IR cutoffs
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
The unitary irreducible representations of
others = 0
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
Using a direct sum of the non-trivial representations,
SO(3) symmetric solutions
In what follows,
size of the space
Compatible with the expanding behavior !
(dimensionless) space-time noncommutativity
c.f.) space-time uncertainty principle
Yoneya (2000)
Speculations
classical solution
accelerating
expansion
space-space noncommutativity
Monte Carlo
simulation
size of the space
space-time
noncommutativity
time
tcr
SO(9)
symmetry
of space
present time
SO(3)
Space-space NC disappears for some dynamical reason.
Summary
Two kinds of singularity predicted by Einstein’s general relativity
 Black hole singularity
 Big bang singularity
Quantum effects of gravity become crucial.
Monte Carlo simulation of
supersymmetric gauge theories and matrix models
Superstring theory
Gauge-gravity correspondence
“Emergent space”
1d SYM describes the space-time with black hole geometry
Lorentzian matrix model
Emergence of (3+1)d expanding universe
Future directions
 Extending the study of supersymmetric gauge theory
to higher dimensions
superconformal
“Holographic inflation”
(Skenderis)
Large-N equivalence
Ishii-Ishiki-Shimasaki-Tsuchiya (2008)
1d SYM with mass deformation
 Connecting the “two ends” in the Lorentzian matrix model
 Quantum corrections around the classical solutions
The gauge group and the matter contents, power-law expansion
 Exploring more general SO(3) symmetric solutions
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
Klebanov-Tseytlin ’96
internal energy
We check this
in strongly coupled gauge theory !
corrections to SUGRA action
low energy effective action of type IIA superstring theory
tree-level scattering amplitudes of the massless modes
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
Non-lattice simulation
Hanada-J.N.-Takeuchi, PRL 99 (07) 161602 [arXiv:0706.1647]
Note: Gauge symmetry can be fixed non-perturbatively in 1d.
 static diagonal gauge :
 residual gauge symmetry :
should be fixed by imposing
RHMC algorithm can be used efficiently
(Fourier acceleration without extra cost etc.)
c.f.) lattice approach : Catterall-Wiseman, JHEP 0712:104,2007
What is M theory ?
Witten (’95)
hypothetical 11d theory
suggested from string dualities
low-energy effective theory : 11D SUGRA
fundamental d.o.f.: membrane
soliton-like objects: M5-brane
compactify the theory on a circle
10D type IIA superstring
believed to appear in the strong coupling limit
of 10D type IIA superstring
Implications on the M theory limit
M theory limit amounts to:
 Surprising aspects of our MC results (2):
The exponent agrees with the prediction even at N=3.
The exponents obtained from10d SUGRA analysis
are valid also in the M-theory limit!