1. in collaboration with G. Ishiki, S. Shimasaki
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/
JHEP 0610 (2006) 007, hep-th/

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,…) and fuzzy sphere harmonics
We give an alternative understanding and a generalization of topologically nontrivial configuration and topological 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 (s,t) block (s,t=1,2,3) matrix for and matrix for PWMM
(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
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 S2 S3/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 of is 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.