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)

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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

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)

Conclusion : Gauge/gravity correspondence is correct up to

(inte

rnal

ene

rgy)

(temperature)

Plotted curves represent results of [ quantum gravity + ]

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

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

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

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

?

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

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=0.07 T=0.08, 0.09 T=0.10, 0.11 T=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

(inte

rnal

ene

rgy)

(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-waveQuantum 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-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

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 analysisValidity of our analysis