1. DIRECT TEST OF THE ADS/CFT CORRESPONDENCE BY MONTE CARLO STUDIES OF N=4 SYM Asato Tsuchiya Shizuoka Univ. @APCTP July 17, 2014

2. References T. Ishii, G. Ishiki, S. Shimasaki and A.T., PRD 78, 106001 (2008), arXiv:0807.2352 G. Ishiki, S. Shimasaki and A.T., JHEP 1111, 036 (2011), arXiv:1106.5590 G. Ishiki, S.-W. Kim, J. Nishimura and A.T., PRL 102, 111601 (2009), arXiv:0810.2884 JHEP 0909, 029 (2009), arXiv:0907.1488 H. Kawai, S. Shimasaki and A.T., IJMP A25, 3389, (2010), arXiv:0912.1456 PRD 81, 085019, (2010), arXiv:1002.2308 M. Honda, G. Ishiki, S.-W. Kim, J. Nishimura and A.T., JHEP 1311, 200 (2013), arXiv:1308.3525

3. Outline 1.Construct a nonperturbative formulation of N=SYM in the planar limit by developing a novel large-N reduction 2.Explain how we can test the AdS/CFT correspondence directly by putting the above formulation on a computer. Show recent results of Monte Carlo simulation cf.) Talk of Sang-Woo Kim Motivations 1. Testing the AdS/CFT correspondence 2. Constructing nonperturbative formulation of （ 4D ） supersymmetric gauge theories 3.Gaining insight into description of curved space-time in matrix models

4. Introduction

5. AdS/CFT correspondence N=4 SU(N) SYM Type IIB string on AdS 5 ×S 5 Maldacena (’97) ’t Hooft limit (planar limit) Loop correction of string suppressed large correction suppressed SUGRA approximation is valid 32 supercharges

6. AdS/CFT correspondence （ cont’d ） 1. One can argue why the correspondence holds by using near horizon limit and open-closed duality The conjecture has not been proved 2. Tests done so far Comparison of physical quantities protected by SUSY Exploiting integrability vev of BPS Wilson loop by localization,…. Pestun (’07),…. spin chain,… Minahan-Zarembo (’02), Bena-Polchinski-Roiban (’03),…. 3.One can perform nontrivial direct test by calculating physical quantities not protected by SUSY in the strongly coupled region of planar N=4 SYM and comparing the result with the prediction form the gravity side One needs a nonperturbative formulation of N=4 SYM such as lattice theory

7. Supersymmetric gauge theory on lattice  It is difficult to formulate SUSY theory on lattice because  It is impossible to realize SUSY fully on lattice It is possible to realize it partially 1 or 2 components of supercharge Sugino, Kaplan-Katz-Unsal, Catterall,….. introduction of non-locality Kawamoto et. al.  It seems like fine tuning of some parameters are needed in proposed lattice theories of N=4 SYM due to fewness of preserved SUSY and instability of moduli Kaplan-Unsal (’05), Elliot-Giedt-Moore (’08), Catterall (’08),…..  Hybrid formulation of N=4 SYM (2D lattice ＋ fuzzy sphere ） Hanada-Matsuura-Sugino (’10)  We have a chance to use the idea of the large-N reduction since we are interested in the planar (’t Hooft) limit

8. Large-N reduction  statement of large-N reduction gauge theory in the planar limit does not depend on volume of the manifold on which the theory is defined In particular, the gauge theory in the planar limit can be described by a matrix model that is obtained by dimensionally reducing it. Eguchi-Kawai (’82)  Size of matrices plays a role of the UV cutoff  First example of emergence of space-time in matrix models  But the original idea does not work for pure YM due to SSB of U(1) D symmetry (background instability) Bhanot-Heller-Neuberger (’82)  introduction of adjoint (massive) fermion in lattice formulation Kovtan-Unsal-Yaffe, Bringoltz-Sharpe, Azeyanagi-Hanada-Unsal But SUSY ？

9. Large-reduction for N=4 SYM N=4 SYM on R 4 N=4 SYM on RxS 3 BFSS model (D0-brane eff. theory) BFSS model (D0-brane eff. theory) plane wave matrix model (BMN matrix model) plane wave matrix model (BMN matrix model) conformal map mass deformation preserving SUSY dimensional reduction to R ? large-N reduction SU(2|4) symmetry (16 supercharges) Ishii-Ishiki-Shimasaki-A.T. (’08)  one obtains N=4 SYM on RxS 3 in the planar limit by expanding PWMM around a particular vacuum consisting of multi fuzzy spheres  a novel large-N reduction extended to a curved space for the first time

10. Large-N reduction for N=4 SYM （ cont’d ）  one resolves the problem of background instability using massiveness and SUSY  a regularization that preserves gauge symmetry and SU(2|4) possessing 16 supercharges 16 is the maximal number that can be preserved because any regularization breaks conformal symmetry  PSU(2,2|4) symmetry is restored in the large-N limit N=4 SYM in the planar limit is obtained without fine tuning  one can perform numerical simulation of N=4 SYM first successful example of numerical simulation of 4D supersymmetric theory  one can perform a direct test of the AdS/CFT by calculating physical quantities in the strongly coupled region of N=4 SYM and comparing the results with the prediction from the gravity side

11. Plan of the present talk 1.Introduction 2.Large-N reduction 3.N=4 SYM on RxS 3 from PWMM 4.Testing the AdS/CFT correspondence 5.Summary and discussion

12. Large-N reduction

13. Large N reduction: Example ……  matrix quantum mechanics : UV cutoff: IR cutoff  introduce a constant matrix  rule of obtaining reduced model : NxN hermitian matrix

14. Large N reduction: Example (cont’d)  Feynman rule momentum carried by(i,j) component Conservation of momentum ~ product of matrices

15. Large N reduction: Example (cont’d) suppressed non- planar No correspondence b/w reduced model and original theory planar  calculation of free energy  reduced model reproduces the planar limit of original theory UV cutoffIR cutoff’t Hooft coupling

16. S 1 case …… : UV cutoff : fixed extracting planar contribution

17. Large-N reduction for YM theory  applying the rule to field strength  reduced model for YM theory is interpreted as a background of The background is unstable due to 0-dim. massless field is D-dim. counterpert of quenching Not compatible with SUSY （ does not work even for non-SUSY case ） Gross-Kitazawa (’82)Bhanot-Heller-Neuberger (’82) dimensional reduction of YM theory to zero dimension

18. N=4 SYM on RxS 3 from PWMM

19. Conformal mapping N=4 SYM on R 4 at a conformal point N=4 SYM on RxS 3 : radius of S 3 scalar field gauge field 1-form fermion field

20. Dimensional reduction of N=4 SYM on RxS 3  S 3 is identified with SU(2 ) isometry of S 3, SO(4)=SU(2)xSU(2), corresponds to left and right translations : right-invariant 1- form  dimensional reduction N=4 SYM on RxS 3 PWMM Kim-Klose-Plefka (’03) : radius of S 3 : Killing vector ~ generator of left translation Maurer-Cartan equation consistent truncation

21. Plane wave (BMN) matrix model  mass deformation of BFSS model  SU(2|4) symmery (16 supercharges) PSU(2,2|4) mass 2 ~ curvature of S 3  Vacua preserve SU(2|4) symmetry, all degenerate multiple fuzzy spheres N-dim. reducible rep. of SU(2)

22. S 3 as Hopf bundle  S 3 : nontrivial S 1 bundle over S 2 S2S2 S1S1 Locally S 2 ×S 1  KK momentum on S 1 =U(1) charge=monopole charge ～ integer or half-integer angular momentum of KK mode with KK momentum : magnetic field has angular momentum  strategy S 1 is obtained by large N reduction S 2 is obtained by continuum limit of fuzzy sphere

23. Regular representation of SU(2) k copies of k-dim. rep. function on S 3 = SU(2) group mfd ~ regular rep. V reg of SU(2) Peter-Weyl’s theorem r label irreducible reps of G : dim of r rep. generalization of Fourier expansion

24. Getting N=4 SYM on RxS 3 ……. ……… : UV cutoff on S 1 : UV cutoff on S 2 Pick up the following vacuum of PWMM and expand the theory around it

25. Getting N=4 SYM on RxS 3 （ cont’d ）  fluctuation of block around the background  SU(2) action on block irreducible decomposition cf.) : UV cutoff on S 1 : UV cutoff on S 2 comparing with KK mode with momentum in

26. Getting N=4 SYM on RxS 3 （ cont’d ）  need to extract planar contribution by taking limit 1) make large-N reduction between S 3 and S 2 hold 2) remove fuzziness of fuzzy spheres  These two cutoffs preserve gauge symmetry and SU(2|4) symmetry  the model is a massive theory, so there is no flat direction background is stable classically furthermore, it is also stable quantum mechanically due to SU(2|4) sym Dasgupta-Sheikh-Jabbari-Raamsdonk(’02)  tunneling to other vacua by instanton effect (Yee-Yi,….) is suppressed in limit  reduced model reproduces the planar limit of N=4 SYM on RxS 3  We can show that interaction is also reproduced correctly

27. Deconfinement transition at finite T known results for N=4 SYM on RxS 3 in the weak coupling limit Aharony-Marsano-Minwalla- Papadodimas- Van Raamsdonk (’03)) In the weak coupling limit at finite temperature, we can integrate out all the massive modes except holonomy around time direction and study the effective theory for the holonomy Ishiki-Kim-Nishimura-A.T. (’08) 1 st order phase transition ~Hawking-Page transition

28. Large-N reduction on group mfd  Our formulation is hybrid in the sense that S 1 is obtained by large-N reduction and S 2 is obtained by continuum limit of fuzzy sphere  Can we obtain whole S 3 by large-N reduction ?  Yes. We can choose another background Li to realize large-N reduction on SU(2) group mfd = S 3  In flat space, we expand the reduced matrix model around in the Fourier basis ~

29. Large-N reduction on group mfd (cont’d) ……… limit We expand PWMM around the regularized version of in the ‘Fourier basis’

30. Large-N reduction on group mfd (cont’d)  One can show that this works including the interaction  In a similar way, one can show that the large-N reduction (at least formally) holds on general group manifold

31. Testing the AdS/CFT correspondence

32. Chiral Primary Operator  chiral primary operator (half-BPS) We focus on  2-pt., 3-pt., 4-pt. correlation functions totally symmetric traceless In free theory

33. Chiral Primary Operator （ cont’d ）  prediction from the gravity side (GKP-Witten relation) from some arguments on the gauge theory side Arutyunov-Frolov (’00) Lee-Minwall-Rangamani-Seiberg (’98)

34. Correlation functions of CPOs from PWMM  CPO on R×S 3  Integral over S 3  corresponding operator in PWMM  correspondence between correlation functions

35. The case of free theory  calculation on R×S 3  calculation in PWMM 3-pt. and 4-pt. also agree ！ Definition of differs slightly

36. 2-pt. function 1 2 continuum limit read off Cf.) Anagnostopoulos-Hanada- Nishimura-Takeuchi (’07)

37. 3-pt. function continuum limit consistent with the prediction of AdS/CFT

38. 4-pt. function Consistent with

39. 4-pt. function （ cont’d ） is obtained by using In particular, agrees well

40. Summary and discussion

41. Summary  Nonperturbative formulation of the planar limit of N=4 SYM on R×S 3 by PWMM  Regularization that preserves gauge symmetry and 16 supersymmetries Overcome difficulty in lattice theory. No need for fine tuning  Calculate correlation functions of CPO by Monte Carlo simulation Compare the results with the prediction from the gravity side Observe reasonable agreement even though matrix size (cutoff) is small  First extension of large N reduction to curved space  We can formulate Chern-Simon theory on S 3 Ishiki-Ohta-Shimasaki-A.T.

42. Discussion  calculate vev of non-BPS Wilson loop  derive integrability of N=4 SYM  N=1 SYM on RxS 3. Gluino condensation.  develop efficient method for numerical simulation parallelization ~ each node treats each block we have succeeded in simulating bosonic IIB matrix model by hybrid parallel computation on Kei supercomputer  large-N reduction on general manifold description of curved space-time in matrix models

43. Large N reduction for N=4 SYM on RxS 3  N=4 SYM on R 4 at conformal point is equivalent to N=4 SYM on RxS 3 through conformal map  one obtains plane wave (BMN) matrix model (PWMM) by dimensionally reducing N=4 SYM on RxS 3 to R Berenstein-Maldacena-Nastase (’02)  One obtains the planar limit of N=4 SYM on RxS 3 by expanding PWMM around a particular vacuum consisting of multi fuzzy spheres  a novel large-N reduction extended to a curved space for the first time  resolve the problem of background instability using massiveness and SUSY Ishii-Ishiki-Shimasaki-A.T. (’08)

44. Strategy  strategy S 1 is obtained by large N reduction S 2 is obtained by continuum limit of fuzzy sphere  fuzzy sphere

45. Large-N reduction on group mfd Large-N reduction holds on general compact semi-simple Lie group Kawai-Shimasaki-A.T. (’09) Expand the theory around generator in regular rep. In practice, we need to regularize regular rep. Free spectrum agree Interaction is also OK regularized regular rep. ~ analog of uniform distribution of eigenvalues