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: , M. Hanada, Y. Hyakutake, G. Ishiki and J. Nishimura arXiv: , Y. Hyakutake (to appear in PTEP) Advertisement

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

N D0-branes We consider N D0-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)

Conclusion : Gauge/gravity correspondence is correct up to (internal energy) (temperature) Plotted curves represent results of [ quantum gravity + ]

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

Let us consider D0-branes in type IIA superstring theory and review their thermal properties. Newton const.dilatonR-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

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

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

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

N D0-branes We consider N D0-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 ?

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 brokent’ Hooft coupling

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

#### 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=0.07T=0.08, 0.09T=0.10, 0.11T=0.12 N=3○○○ N=4○○○○ N=5○○

Bound state represents a parameter for eigenvalue distribution of

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

Conclusion : Gauge/gravity correspondence is correct even at finite (internal energy) (temperature)

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

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

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.

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

Equations of motion seems too hard to solve…

Quantum near-horizon geometry of M-wave We solved !

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:

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-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

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

Our analysis is valid when 1-loop terms are subdominant. From this we obtain inequalities. Validity of our analysis