1. 1 AdS/CFT correspondence and generation of space-time in Matrix models March 05 2008 at KEK Hikaru Kawai arXiv:0706.1163, 0708.2463, with T. Suyama arXiv:0708.1732, with M.Sato

2. 2 Introduction Alday and Maldacena High energy scattering in D=4 N=4 large-N super Yang Mills theory using AdS/CFT correspondence. D3-branes → string in AdS space ↓ gravity dual ↓T-duality D-instanton → string in dual AdS space In the T-dual picture, the AdS/CFT correspondence becomes a relation between IIB matrix model and gravity. It might provide a clue to understanding the dynamics of IIB matrix model.

3. 3 High energy scattering in D=4 N=4 large-N super Yang Mills theory planar diagrams

4. 4 IR regularization massive states Colors of external particles are taken from the N’×N’ matrices.

5. 5 AdS/CFT correspondence as a consequence of scale invariance (1) X Y L

6. 6 AdS/CFT correspondence as a consequence of scale invariance (2)

7. 7 Proof of the scale invariance (1) n holes = We want to show the invariance for such F. Basic assumptions for : has a finite convergence radius. analytically continued to positive real axis. Need to check (1) Boundary condition is invariant. (2) World sheet action is not invariant but the variation is negligible, if is large enough.

8. 8 Proof of the scale invariance (2) - boundary condition Naively is invariant under Subtlety Need a regularization, but in superstring theory this factor cancels out.

9. 9 Proof of the scale invariance (3) – world sheet action ( ⅰ ) z = LSZ like formula

10. Proof of the scale invariance (4) – world sheet action ( ⅱ ) = z n holes L |z||z| L’ = max(L,1) Slowly changes. Large-N Plausible assumption: G is also defined by analytic continuation of perturbation series: ⇒ (for any positive )

11. 11 Proof of the scale invariance (5) – world sheet action ( ⅲ ) Here we have assumed F has a mild behavior in large.

12. 12 Proof of the scale invariance (6) Consistent with the gravity limit.

13. Comment 1: off-shell operators (i) 13 Wilson loops can be discussed similarly. D5-brane D3-brane × × × × × ×

14. off-shell operators (ii) 14 Wilson loop with arbitrary shape? Local operators like ? Multi trace operators? ⇒ closed string states ⇒ Multi string states? ↓ scale trans. ↓ Anomalous dimensions are given by string Hamiltonian.

15. Comment 2: Meaning of perturbation series Gravity limit 15 Weak gravity expansion has finite convergence radius. Analytically continued to positive real axis. AdS cave Flat space-time

16. -finite temperature (i) 16

17. 17 -finite temperature (ii) Perturbation series has finite convergence radius. ⇒ Horizon appears after analytic continuation.

18. Comment 3: finite or expansion 18 × × × × × × = + massive states L String in D-brane background = string in AdS + expansion of NL-σ model gives a correct asymptotic series.

19. 19 T- duality (1) Alday-Maldacena D=4 N=4 large-N super Yang Mills theory → D3-branes → string in AdS space → string in dual AdS space T- duality X Y boundary horizon

20. 20 T- duality (2) Emission vertex horizon boundary horizon Minimal surface

21. 21 T-duality from the beginning (1) Sphere with holes 1 2 n …

22. 22 T-duality from the beginning (2) N D3-branes N D- instantons T-duality (Large-N) Low energy effective theory 4D Yang-Mills theoryQuenched reduced model

23. 23 T-duality from the beginning (3) -SUGRA background y=0 is boundary of AdS. Consistent with the former analysis. X Y 4D hyper plane Uniformly distributed D-instantons

24. 24 T-duality from the beginning (4) -scale invariance ( ⅰ ) Scale transformation 1 2 n … is invariant if. (1) Boudary condition is invariant. (2) World sheet action is not invariant but the variation is negligible.

25. 25 T-duality from the beginning (4) -scale invariance ( ⅱ ) D-instantons scale transformation boundary of AdS Matrix model is a good approximation. Classical string in AdS is good if

26. Remarks: Scale invariance of quenched reduced model 26

27. 27 Summary and discussions AdS/CFT correspondence can be understood as a consequence of scale invariance. In T-dual picture, the ordinary AdS/CFT correspondence becomes a relation between matrix model and gravity, which is obtained by a simple scale invariance. It would be interesting to compare this with the appearance of gravity in IIB matrix model.