Holographic Description of Q uantum Black Hole on a Computer

1. 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) Advertisement http://www.nature.com/news/simulations-back-up-theory-that-universe-is-a-hologram-1.14328

2. 1. Introduction and summary One of the remarkable progress in string theory is the realization of holographic principle or gauge/gravity correspondence. Maldacena(1997) • 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. • Information loss paradox will be resolved. However, it is difficult to prove the gauge/gravity correspondence directly. Our works • 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.

3. We considerND0-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) Honda, Ishiki, Kim, Nishimura, Tsuchiya(2013)

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

5. Plan of the talk Introduction and summary Black 0-brane and its thermodynamics Quantum black 0-brane and its thermodynamics Gauge theory on D0-branes Test of gauge/gravity correspondence Summary

6. 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(1998) 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 =

7. 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.

8. 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

9. 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.

10. 3.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, Witten(1986) • Non trivial contributions start from 4-pt amplitudesat tree and 1-loop. • Anomaly cancellation terms can be obtained at 1-loop level. There exist terms like and . From the calculations of scattering amplitudes, it is expected that the effective action takes the form of tree 1-loop 2-loop n-loop In this talk, we consider leading 1-loop correction which is relevant in the low temperature region. c.f. Hanada, Hyakutake, Nishimura, Takeuchi (2008)

11. A part of the effective action up to 1-loop level which is relevant to black 0-brane is given by The structure of the effective action is complicated and difficult to calculate. However, it can be uplifted to 11 dimensions, and the structure of the effective action in 11 dimensions becomes rather simple. Thus analyses should be done in 11 dimensions.

12. Black 0-brane solution is uplifted as follows. Black 0-brane M-wave M-waveis purely geometrical.

13. 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 .

14. Equations of motion seems too hard to solve…

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

16. 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

17. 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

18. 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

19. Validity of our analysis Our analysis is valid when 1-loop terms are subdominant. tree 1-loop 2-loop n-loop From this we obtain inequalities.

20. We considerND0-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. ?

21. 4. 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 t’ Hooft coupling : periodic b.c. : anti-periodic b.c.

22. We fix the gauge symmetry bystatic anddiagonal 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 Anagnostopoulos, Hanada, Nishimura, Takeuchi(2008)

23. 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.

24. represents a parameter for eigenvalue distribution of Bound state

25. 5.Test of the gauge/gravity correspondence We calculated the internal energy from the gravity theory and the result is given by If the gauge/gravity correspondence is true, it is expected that Now we are ready to test the gauge/ gravity correspondence.

26. holds for each

27. 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

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

29. 6.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.

30. c.f. Hanada, Hyakutake, Nishimura, Takeuchi (2008) correction