Download Brown-Henneaux`s Canonical Approach to Topologically Massive

Holographic Description of Quantum
Black Hole on a Computer
Yoshifumi Hyakutake (Ibaraki Univ.)
Collaboration with
M. Hanada（YITP, Kyoto）, G. Ishiki （YITP, Kyoto） and J. Nishimura（KEK）
References
arXiv:1311.5607, M. Hanada, Y. Hyakutake, G. Ishiki and J. Nishimura
arXiv:1311.7526, Y. Hyakutake (to appear in PTEP)
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1. Introduction and summary
One of the remarkable progress in string theory is the realization of
holographic principle or gauge/gravity correspondence.
Maldacena
• Lower dimensional gauge theory corresponds to higher
dimensional gravity theory.
• Strong coupling limit of the gauge theory can be studied by
the classical gravity.
• Applied to QCD or condensed matter physics.
However, it is difficult to prove the gauge/gravity correspondence directly.
Our work
• Take account of the quantum effect in the gravity side.
• Execute numerical study in the gauge theory side.
• Compare the both results and test the gauge/gravity correspondence.
We consider N D0-branes
Gauge theory on the branes
Type IIA supergravity
Event horizon
Thermalized U(N) supersymmetric
quantum mechanics
Non-extremal Charged black
hole in 10 dim.
It is possible to evaluate internal energy from both sides.
By comparing these, we can test the gauge/gravity correspondence.
cf. Gubser, Klebanov, Tseytlin (1998)
Conclusion : Gauge/gravity correspondence is correct up to
(internal energy)
Plotted curves represent results of [ quantum gravity +
(temperature)
]
Plan of the talk
1. Introduction and summary
2. Black 0-brane and its thermodynamics
3. Gauge theory on D0-branes
4. Test of gauge/gravity correspondence
5. Summary
2. Black 0-brane and its thermodynamics
Let us consider D0-branes in type IIA superstring theory and review their
thermal properties.
Itzhaki, Maldacena, Sonnenschein Yankielowicz
Low energy limit of type IIA superstring theory ~ type IIA supergravity
Newton const.
dilaton
N D0-branes ~ extremal black 0-brane
mass = charge =
R-R field
We rewrite the quantities in terms of dual gauge theory
‘t Hooft coupling
typical energy
After taking the decoupling limit
near horizon geometry.
, the geometry becomes
Now we consider near horizon geometry of non-extremal black 0-brane.
Horizon is located at
, and Hawking temperature is given by
Entropy is obtained by the area law
Internal energy is calculated by using
Note that supergravity approximation is valid when
curvature radius at horizon
Out of this range, we need to take into account quantum corrections
to the supergravity. We skip the details but the result of the 1-loop
correction becomes
leading quantum correction
We consider N D0-branes
Gauge theory on the branes
Type IIA supergravity
Event horizon
Thermalized U(N) supersymmetric
quantum mechanics
?
Non-extremal Charged black
hole in 10 dim.
3. Gauge Theory on D0-branes --- How to put on Computer
D0-branes are dynamical due to oscillations of open strings
massless modes :
matrices
Action for
D0-branes is obtained by requiring global
supersymmetry with 16 supercharges.
(1+0) dimensional supersymmetric
gauge theory
Then consider thermal theory by Wick rotation of time direction
Supersymmetry is broken
: periodic b.c.
: anti-periodic b.c.
t’ Hooft coupling
We fix the gauge symmetry by static and diagonal gauge.
static gauge
diagonal gauge
UV cut off
Fourier expansion of
Periodic b.c.
Anti-periodic b.c.
By substituting these into the action and integrate fermions, we obtain
Since the action is written with finite degrees of freedom, it is possible to
analyze the theory on the computer.
Via Monte Carlo simulation, we
obtain histogram of and
internal energy
of the system.
#
#
3 parameters :
In the simulation, the parameters are chosen as follows.
T=0.07
N=3
N=4
N=5
○
T=0.08, 0.09
T=0.10, 0.11
T=0.12
○
○
○
○
○
○
○
○
represents a parameter for
eigenvalue distribution of
Bound state
4. Test of the gauge/gravity correspondence
We calculated the internal energy from the gravity theory and the result is
If the gauge/gravity correspondence is true, it is expected that
Now we are ready to test the gauge/ gravity correspondence.
for each
We fit the simulation data by assuming
Then
is plotted like
This matches with the result from the gravity side.
Furthermore
is proposed to be
(internal energy)
Conclusion : Gauge/gravity correspondence is correct even at finite
(temperature)
5. Summary
From the gravity side, we derived the internal energy
c.f. Hanada, Hyakutake, Nishimura, Takeuchi (2008)
correction
The simulation data is nicely fitted by the above function up to
Therefore we conclude the gauge/gravity correspondence is
correct even if we take account of the finite contributions.
It is interesting to study the region of quite low temperature numerically
to understand the final state of the black hole evaporation.
A. Quantum black 0-brane and its thermodynamics
The effective action of the superstring theory can be derived so
as to be consistent with the S-matrix of the superstring theory.
• Non trivial contributions start from 4-pt amplitudes.
• Anomaly cancellation terms can be obtained at 1-loop level.
There exist terms like
and
.
A part of the effective action up to 1-loop level which is relevant to
black 0-brane is given by
This can be simplified in 11 dimensions.
Gross and Witten
Thus analyses should be done in 11 dimensions.
Black 0-brane solution is uplifted as follows.
Black 0-brane
M-wave
M-wave is purely geometrical object and simple.
In order to solve the equations of motion with higher derivative terms,
we relax the ansatz as follows.
SO(9) symmetry
Inserting this into the equations of motion and solving these,
We obtain
and
up to the linear order of .
Equations of motion
seems too hard to solve…
Quantum near-horizon geometry of M-wave
We solved !
Note:
• The solution is uniquely determined by imposing the
boundary conditions at the infinity and the horizon.
• Quantum near-horizon geometry of black 0-brane is
obtained via dimensional reduction.
• Test particle feels repulsive force near the horizon.
Potential barrier
Thermodynamics of the quantum near-horizon geometry of black 0-brane
Black hole horizon
at the horizon
Temperature of the black hole is given by
From this,
is expressed in terms of
.
Black hole entropy
Black hole entropy is evaluated by using Wald’s formula.
By inserting the solution obtained so far, the entropy is calculated as
Internal energy and specific heat
Finally black hole internal energy is expressed like
correction
Specific heat is given by
Thus specific heat becomes negative when
Instability at quite low temperature
via quantum effect
Validity of our analysis
Our analysis is valid when 1-loop terms are subdominant.
From this we obtain inequalities.