Holographic Description of Quantum Black Hole on a Computer

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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 arXiv1311.5607, M. Hanada, Y.
Hyakutake, G. Ishiki and J. Nishimura arXiv1311.7
526, 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.

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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)
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Conclusion Gauge/gravity correspondence is
correct up to
Plotted curves represent results of quantum
gravity
(internal energy)
(temperature)
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• Plan of the talk
• Introduction and summary
• Black 0-brane and its thermodynamics
• Gauge theory on D0-branes
• Test of gauge/gravity correspondence
• Summary

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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
R-R field
N D0-branes extremal black 0-brane
mass charge
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We rewrite the quantities in terms of dual gauge
theory
t Hooft coupling
typical energy
After taking the decoupling limit ,
the geometry becomes
near horizon geometry.
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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
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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
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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.
?
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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.
(10) dimensional supersymmetric
gauge theory
Then consider thermal theory by Wick rotation of
time direction
Supersymmetry is broken
t Hooft coupling
periodic b.c. anti-periodic b.c.
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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
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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.
T0.07 T0.08, 0.09 T0.10, 0.11 T0.12
N3 ? ? ?
N4 ? ? ? ?
N5 ? ?
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represents a parameter for eigenvalue
distribution of
Bound state
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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.
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for each
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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
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Conclusion Gauge/gravity correspondence is
correct even at finite
(internal energy)
(temperature)
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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.
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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.
Gross and Witten

• 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.
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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.
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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 .
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Equations of motion
seems too hard to solve
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Quantum near-horizon geometry of M-wave
We solved !
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• 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.

Note
Potential barrier
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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 Walds
formula.
By inserting the solution obtained so far, the
entropy is calculated as
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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
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Validity of our analysis
Our analysis is valid when 1-loop terms are
subdominant.
From this we obtain inequalities.