Embedding of theories with SU(2|4) symmetry into the plane wave matrix model

1. Embedding of theories with SU(2|4) symmetry into the plane wave matrix model Tsuchiya(Osaka Univ.) in collaboration with G. Ishiki, S. Shimasaki and Y. Takayama hep-th/0610038 JHEP 0610 (2006) 007, hep-th/0605163

2. Gauge/gravity correspondence for theories with SU(2|4) symmetry (1) dimensional reduction (2) embedd-ing (3) plane wave matrix model BMN All these theories have many vacua • Lin-Maldacena developed a method that gives gravity dual of each vacuum • It is predicted that the theory around each vacuum of (1) and (2) is embedded in (3) • We will prove this prediction

3. In the process of proof • We find an extension of Taylor’s compactification (T-duality) in matrix model to that on spheres • We reveal relationships among the spherical harmonics on S３,the monopole harmonics(Wu-Yang,…) andfuzzy sphere harmonics • We give an alternative understanding and a generalization of topologically nontrivial configurationandtopological charges on fuzzy spheres • Our results do not only serve as a nontrivial check of the gauge/gravity correspondence for the SU(2|4) theories, but also shed light on description of curved space and topological inv. in matrix models

4. Contents 1. Introduction -Gauge/gravity correspondence for theories with SU(2|4) symmetry- 2. Dimensional reductions 3. Gravity duals 4. Vacua of the SU(2|4) theories 5. Predictions on relations between vacua 6. Proofs of the predictions 7. Summary and outlook

5. Dimensional reductions cf.)Kim-Klose-Plefka assume all fields are independent of 3d flat space notation dropping derivatives

6. Gravity duals general smooth solution of type IIA SUGRA preserving SU(2|4) : electrostatic potential for axially symmetric system is a b.g. potential and specifies a theory is determined by a config. of conducting disks and specifies each vacuum D2-brane charge NS 5-brane charge

7. Vacua of the SU(2|4) theories space of flat connection ~holonomy U along generator of gauge sym. broken to :monopole charge gauge sym. broken to

8. plane wave matrix model fuzzy spheres radii : representation matrix of spin representation gauge sym. broken to

9. blocks T=3 case (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) (s,t) block (s,t=1,2,3) matrix for and matrix for PWMM

10. Predictions on relations between vacua a) Embedding of into PWMM PWMM

11. b) Embedding of into trivial vacuum

12. Proof of prediction a) Expand around a vacuum angular momentum in the presence of a monopole with magnetic charge q monopole scalar harmonics interaction terms

13. Expand PWMM around a vacuum fuzzy sphere scalar harmonics cf.)Grosse et al., Baez et al.,…. monopole scalar harmonics vectors, fermions and interaction terms are also OK

14. Proof of prediction b) for trivial vacuum of spherical harmonics on S3 Ishiki’s poster for scalar scalar spherical harmonics same relations as monopole scalar harmonics

15. harmonic expansion around trivial vacuum of harmonic expansion around with factor out vectors, fermions and interaction terms are also OK

16. 1. S1 with radius~k S1 with radius~1/ k : winding # T-dual : momentum 2. nontrivial background of gauge fields not S2xS1 but nontrivial S1 fibration over S2S3/Zk 3. trivial vacuum of is embedded into PWMM S3/Zk is realized in PWMM in terms of three matrices fuzzy spheres + S1 on S2

17. Summary 1. We showed that • every vacuum of is embedded into PWMM • the trivial vacuum ofis embedded Into 2. We extended Taylor’s compactification in matrix models to that on spheres 3. We revealed relationships among spherical harmonics on S3, monopole harmonics and fuzzy sphere harmonics (4. We give an alternative understanding and a generalization of the topologically nontrivial configurations and their topological charges on fuzzy spheres)

18. Outlook • Complete the proof of prediction b) for nontrivial vacua of • Realize other fiber bundles in matrix models and find a general recipe • Construct a lattice gauge theory for numerical simulation of AdS/CFT cf.) Kaplan et al.