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![Page 1: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/1.jpg)
Holographic Description of Quantum Holographic Description of Quantum Black Hole on a ComputerBlack Hole on a Computer
Yoshifumi Hyakutake (Ibaraki Univ.)
Collaboration withM. Hanada ( YITP, Kyoto ) , G. Ishiki ( YITP, Kyoto ) and J. Nishimura ( KEK )ReferencesarXiv:1311.5607, M. Hanada, Y. Hyakutake, G. Ishiki and J. NishimuraarXiv:1311.7526, Y. Hyakutake (to appear in PTEP)
Advertisementhttp://www.nature.com/news/simulations-back-up-theory-that-universe-is-a-hologram-1.14328
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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.
However, it is difficult to prove the gauge/gravity correspondence directly.
• 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.
• 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.
Our work
Maldacena
![Page 3: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/3.jpg)
We consider NN D0-branesD0-branes
Gauge theory on the branes
Thermalized U(N) supersymmetric quantum mechanics
Type IIA supergravity
Non-extremal Charged black hole in 10 dim.
Event horizon
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
(inte
rnal
ene
rgy)
(temperature)
Plotted curves represent results of [ quantum gravity + ]
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Plan of the talkPlan 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
![Page 6: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/6.jpg)
Let us consider D0-branes in type IIA superstring theory and review their thermal properties.
Newton const. dilaton R-R field
N D0-branes ~ extremal black 0-brane
mass = charge =
Low energy limit of type IIA superstring theory ~ type IIA supergravity
2. Black 0-brane and its thermodynamics
Itzhaki, Maldacena, Sonnenschein Yankielowicz
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We rewrite the quantities in terms of dual gauge theory
After taking the decoupling limit , the geometry becomes
near horizon geometry.
‘t Hooft coupling
typical energy
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Entropy is obtained by the area law
Now we consider near horizon geometry of non-extremal black 0-brane.
Horizon is located at , and Hawking temperature is given by
Internal energy is calculated by using
![Page 9: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/9.jpg)
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
![Page 10: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/10.jpg)
We consider NN D0-branesD0-branes
Gauge theory on the branes
Thermalized U(N) supersymmetric quantum mechanics
Type IIA supergravity
Non-extremal Charged black hole in 10 dim.
Event horizon
?
![Page 11: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/11.jpg)
Action for D0-branes is obtained by requiring global supersymmetry with 16 supercharges.
3. Gauge Theory on D0-branes --- How to put on Computer
D0-branes are dynamical due to oscillations of open strings
massless modes : matrices
(1+0) dimensional supersymmetric gauge theory
Then consider thermal theory by Wick rotation of time direction
: periodic b.c.: anti-periodic b.c.
Supersymmetry is broken t’ Hooft coupling
![Page 12: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/12.jpg)
We fix the gauge symmetry by static and diagonal gauge.
static gauge
diagonal gauge
Fourier expansion of
Periodic b.c. Anti-periodic b.c.
UV cut off
By substituting these into the action and integrate fermions, we obtain
![Page 13: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/13.jpg)
##
Since the action is written with finite degrees of freedom, it is possible to analyze the theory on the computer.
3 parameters :
Via Monte Carlo simulation, we obtain histogram of and internal energy of the system.
In the simulation, the parameters are chosen as follows.
T=0.07 T=0.08, 0.09 T=0.10, 0.11 T=0.12
N=3 ○ ○ ○
N=4 ○ ○ ○ ○
N=5 ○ ○
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Bound state
represents a parameter for eigenvalue distribution of
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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.
![Page 16: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/16.jpg)
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
(inte
rnal
ene
rgy)
(temperature)
![Page 19: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/19.jpg)
5. Summary
From the gravity side, we derived the internal energy
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.
c.f. Hanada, Hyakutake, Nishimura, Takeuchi (2008)
It is interesting to study the region of quite low temperature numerically to understand the final state of the black hole evaporation.
correction
![Page 20: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/20.jpg)
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
![Page 21: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/21.jpg)
Black 0-brane
M-wave
M-wave is purely geometrical object and simple.
Thus analyses should be done in 11 dimensions. Black 0-brane solution is uplifted as follows.
![Page 22: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/22.jpg)
In order to solve the equations of motion with higher derivative terms, we relax the ansatz as follows.
Inserting this into the equations of motion and solving these,We obtain and up to the linear order of .
SO(9) symmetry
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Equations of motion seems too hard to solve…
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Quantum near-horizon geometry of M-waveQuantum 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.
Potential barrier
Note:
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Black hole horizon
at the horizon
Temperature of the black hole is given by
From this, is expressed in terms of .
Thermodynamics of the quantum near-horizon geometry of black 0-braneThermodynamics of the quantum near-horizon geometry of black 0-brane
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
![Page 27: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/27.jpg)
Internal energy and specific heat
Finally black hole internal energy is expressed like
Specific heat is given by
Thus specific heat becomes negative when
Instability at quite low temperature via quantum effect
correction
![Page 28: Holographic Description of Quantum Black Hole on a Computer Yoshifumi Hyakutake (Ibaraki Univ.) Collaboration with M. Hanada ( YITP, Kyoto ), G. Ishiki.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e695503460f94b65e37/html5/thumbnails/28.jpg)
Our analysis is valid when 1-loop terms are subdominant.
From this we obtain inequalities.
Validity of our analysisValidity of our analysis