We study theories with SU (2|4) symmetry, which include the plane wave matrix model, 2 + 1 SYM on R × S 2 and N = 4 SYM on R × S 3 /Z k . All these theories possess many vacua. From Lin-Maldacena's method which gives the gravity dual of each vacuum, it is predicted that the theory around each vacuum of 2 + 1 SYM on R × S 2 and N = 4 SYM on R × S 3 /Z k is embedded in the plane wave matrix model. We show this directly on the gauge theory side. We clearly reveal relationships among the spherical harmonics on S 3 , the monopole harmonics and the harmonics on fuzzy spheres. We extend the compactification (the T-duality) in matrix models a la Taylor to that on One can see from (D.8) that in the N 0 → ∞ limit, this formula reduces to N 0 tr(

We investigate quantum corrections in non-commutative gauge theory on fuzzy sphere. We study translation invariant models which classically favor a single fuzzy sphere with U (1) gauge group. We evaluate the effective actions up to the two loop level. We find non-vanishing quantum corrections at each order even in supersymmetric models. In particular the two loop contribution favors U (n) gauge group over U (1) contrary to the tree action in a deformed IIB matrix model with a Myers term. We further observe close correspondences to 2 dimensional quantum

We study N = 4 SYM on R × S 3 and theories with 16 supercharges arising as its consistent truncations. These theories include the plane wave matrix model, N = 4 SYM on R × S 2 and N = 4 SYM on R × S 3 /Z k , and their gravity duals were studied by Lin and Maldacena. We make a harmonic expansion of the original N = 4 SYM on R × S 3 and obtain each of the truncated theories by keeping a part of the Kaluza-Klein modes. This enables us to analyze all the theories in a unified way. We explicitly construct some nontrivial vacua of N = 4 SYM on R × S 2 . We perform 1-loop analysis of the original and truncated theories. In particular, we examine states regarded as the integrable SO(6) spin chain and a time-dependent BPS solution, which is considered to correspond to the AdS giant graviton in the original theory.These equations are rewritten in terms of the KK modes (5.6) and (5.7) as (5.26) with no summation over J and m. Here we took the Coulomb gauge ∇ i A i = 0, so that there is no longitudinal mode A l Jm in the above expressions. The equations (5.25) and (5.26) correspond to the longitudinal and transverse components of (5.23), respectively. Figure 3: Region of the regularized summations over J 1 and J 2 variables P = J 1 + J 2 and Q = J 2 − J 1 , which take integers for integer J and half odd integers for half odd integer J. Then, we obtain the regularized expression for (X − f ). − 16   2Λ f −J P =J J Q=−J + J r=−J+1 J Q=r P =2Λ f +2r

We evaluate the effective actions of supersymmetric matrix models on fuzzy S 2 × S 2 up to the two loop level. Remarkably it turns out to be a consistent solution of IIB matrix model. Based on the power counting and SUSY cancellation arguments, we can identify the 't Hooft coupling and large N scaling behavior of the effective actions to all orders. In the large N limit, the quantum corrections survive except in 2 dimensional limits. They are O(N ) and O(N 4 3 ) for 4 and 6 dimensional spaces respectively. We argue that quantum effects single out 4 dimensionality among fuzzy homogeneous spaces.

We study a relation between droplet configurations in the bubbling AdS geometries and a complex matrix model that describes the dynamics of a class of chiral primary operators in dual N = 4 super Yang Mills (SYM). We show rigorously that a singlet holomorphic sector of the complex matrix model is equivalent to a holomorphic part of two-dimensional free fermions, and establish an exact correspondence between the singlet holomorphic sector of the complex matrix model and one-dimensional free fermions. Based on this correspondence, we find a relation of the singlet holomorphic operators of the complex matrix model to the Wigner phase space distribution. By using this relation and the AdS/CFT duality, we give a further evidence that the droplets in the bubbling AdS geometries are identified with those in the phase space of the one-dimensional fermions. We also show that the above correspondence actually maps the operators of N = 4 SYM corresponding to the (dual) giant gravitons to the droplet configurations proposed in the literature.