the origin of space-time as seen from matrix model simulations · the origin of space-time as seen...
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The origin of space-time as seen from matrix model simulationsSeminar 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) 151601arXiv: 1108.5153
S.-W.Kim, J.N., A.Tsuchiya, arXiv:1108.1540, 1110.4803
Quantum gravity
Superstring theory isa natural candidate for a unified theory
including quantum gravity
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 matterCMB, supernovae, structure formation, …
The testing ground for superstring theory
2 amazing predictions of Einstein’s general relativity
Black hole Big bang
singularitiesQuantum 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)
Matrix model formulation of superstring/M theoriesBanks-Fischler-Shenker-Susskind (1996),Ishibashi-Kawai-Kitazawa-Tsuchiya (1997)
Gauge theory description of black hole thermodynamics Correspondence at the level of local operators
Dynamical origin of space-time Applications to the physics beyond the Standard Model
Monte Carlo simulation provides an important toolto explore these two directions.
Plan of the talk
1. Introduction2. Black hole thermodynamics from gauge theory3. Direct test of gauge-gravity correspondence4. (3+1)d expanding universe from matrix model
c.f.) Tsuchiya’s talk on Oct.18 (Tue.)5. Expanding universe as a classical solution6. Summary and discussions
Hanada-J.N.-Takeuchi, PRL 99 (’07) 161602Anagnostopoulos-Hanada- J.N.-Takeuchi, PRL 100 (’08) 021601Hanada-Miwa-J.N.-Takeuchi, PRL 102 (’09) 181602Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602
N D0 branes
thorizon
black 0-brane solutionin type IIA SUGRA
1d U(N) SUSYgauge theory
near-extremal black holeat finite T
In the decoupling limit, the D0 brane system describes the black hole microscopically.
Itzhaki-Maldacena-Sonnenschein-Yankielowicz (’98)
Gauge-gravity duality for D0-brane system
SUGRA description : valid
type IIA superstring
quantum description of the states inside the BH
Gauge/gravity duality predicts thatthis should be reproduced by 1d SYM. large-N, low T
microscopic origin of the black hole thermodynamics
Prediction from gauge/gravity duality (I) dual geometry
7.41
Hawking’s theory
black hole thermodynamics
Klebanov-Tseytlin (’96)
Comparison including corrections
corrections
Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602 [arXiv:0811.3102]
M.Hanada, J.N., Y.Sekino, T.Yoneya : Phys.Rev. Lett. 104 (2010) 151601arXiv: 1108.5153
Prediction from gauge/gravity duality (II)
Gubser-Klebanov-Polyakov-Witten relation (’98)
correlation functions in gauge theory
generating functional
operator-field correspondencegauge gravity
SUGRA action evaluated at the classical solutionwith the boundary condition
Correlation functions in 1d SYM theoryPerturbative calculations plagued by severe IR divergence :
require genuinely non-perturbative methods
1) gauge-gravity correspondence
2) Monte Carlo simulation
Sekino-Yoneya (’99)
for operators corresponding to supergravity modes in 10d SUGRA
Hanada-J.N.-Sekino-Yoneya (’09,’11)
Actually, agreement extends to M theory regime !Power-law behavior with the predicted exponent
based on Gubser-Klebanov-Polyakov-Witten relation (’98)
1d gauge theory
p.b.c.p.b.c.
(without loss of generality)
1d SYM with 16 supercharges
The region of validity for the SUGRA analysis
Series of operators (I)
Predicted power lawconfirmed clearly
even beyond the validityregion 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)
Comparison in the Fourier space :
(bad UV behavior) Reliable inverse Fourier tr.seems difficult…
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
Best fit obtained for
finite IR cutoff effects
Larger angular momentum
Best fit obtained for
Best fit obtained for
S.-W.Kim, J.N., A.Tsuchiya, arXiv:1108.1540
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 loopsFukuma-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, arXiv:1108.1293
Results of the Gaussian expansion methodJ.N.-Okubo-Sugino (arXiv:1108.1293)
Minimum of the free energyoccurs at d=3
Extent of space-timefinite 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, arXiv:1108.1540
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
“critical time”
SSB
Consider a simpler problem :
solution :
representation matrices ofa compact semi-simple Lie algebrawith d generators
Maximum is achieved for SU(2) algebra
The mechanism of SSB : SO(9) -> SO(3)
S.-W.Kim, J.N., A.Tsuchiya, arXiv:1110.4803
Lagrange multipliers corresponding to the IR cutoffs
Classical equations of motion for the Lorentzian model :
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
others = 0
The unitary irreducible representations of
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
SO(3) symmetric solutions
Using a direct sum of the non-trivial representations,
In what follows,
Compatible with the expanding behavior !
size of the space
(dimensionless) space-time noncommutativity
c.f.) space-time uncertainty principleYoneya (2000)
Speculations
time
classical solution
tcr
Monte Carlosimulation
SO(9)
size of the space
space-space noncommutativity
present time
acceleratingexpansion
space-timenoncommutativity
Space-space NC disappears for some dynamical reason.
symmetry of space
SO(3)
Black hole singularity Big bang singularity
Monte Carlo simulation of supersymmetric gauge theories and matrix models
Quantum effects of gravity become crucial.
Two kinds of singularity predicted by Einstein’s general relativity
Superstring theory
Gauge-gravity correspondence“Emergent space”1d SYM describes the space-time with black hole geometry
Lorentzian matrix modelEmergence of (3+1)d expanding universe
Summary
Future directions Extending the study of supersymmetric gauge theory
to higher dimensions
1d SYM with mass deformation
Large-N equivalenceIshii-Ishiki-Shimasaki-Tsuchiya (2008)
superconformal
“Holographic inflation” (Skenderis)
Connecting the “two ends” in the Lorentzian matrix model Quantum corrections around the classical solutions
Exploring more general Lie-algebraic solutions
The gauge group and the matter contents, power-law expansion
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
internal energy
Klebanov-Tseytlin ’96
We check this in strongly coupled gauge theory !
corrections to SUGRA action
tree-level scattering amplitudes of the massless modes
low energy effective action of type IIA superstring theory
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
Hanada-J.N.-Takeuchi, PRL 99 (07) 161602 [arXiv:0706.1647]Non-lattice simulation
residual gauge symmetry :
should be fixed by imposing
static diagonal gauge :
Note: Gauge symmetry can be fixed non-perturbatively in 1d.
c.f.) lattice approach : Catterall-Wiseman, JHEP 0712:104,2007
RHMC algorithm can be used efficiently(Fourier acceleration without extra cost etc.)
What is M theory ?hypothetical 11d theory
suggested from string dualities
low-energy effective theory : 11D SUGRA
fundamental d.o.f.: membranesoliton-like objects: M5-brane
compactify the theory on a circle10D type IIA superstring
believed to appear in the strong coupling limitof 10D type IIA superstring
Witten (’95)
Implications on the M theory limit
The exponents obtained from10d SUGRA analysis are valid also in the M-theory limit!
The exponent agrees with the prediction even at N=3.
Surprising aspects of our MC results (2):
M theory limit amounts to: