Using the Hopf fibration and starting from a four dimensional noncommutative Moyal plane, R 2 θ × R 2 θ , we obtain a star-product for the noncommutative (fuzzy) R 3Furthermore, we show that there is a projection function which allows us to reduce the functions on R 3 λ to that of the fuzzy sphere, and hence we introduce a new star-product on the fuzzy sphere. We will then briefly discuss how using our method one can extract information about the field theory on fuzzy sphere and R 3 λ from the corresponding field theories on R 2 θ × R 2 θ space.

We propose CFT descriptions of the D1/D5 system in a class of freely acting Z 2 orbifolds/orientifolds of type IIB theory, with sixteen unbroken supercharges. The CFTs describing D1/D5 systems involve N = (4, 4) or N = (4, 0) sigma models on (where the action of Z 2 is diagonal and its precise nature depends on the model. We also discuss D1(D5)-brane states carrying non-trivial KaluzaKlein charges, which correspond to excitations of two-dimensional CFTs of the type (The resulting multiplicities for twocharge bound states are shown to agree with the predictions of U-duality. We raise a puzzle concerning the multiplicities of three-charge systems, which is generically present in all vacuum configurations with sixteen unbroken supercharges studied so far, including the more familiar type IIB on K3 case: the constraints put on BPS counting formulae by U-duality are apparently in contradiction with any CFT interpretation. We argue that the presence of RR backgrounds appearing in the symmetric product CFT may provide a resolution of this puzzle. Finally, we show that the whole tower of D-instanton corrections to certain "BPS saturated couplings" in the low energy effective actions match with the corresponding one-loop threshold corrections on the dual fundamental string side.

We compute the one loop partition function of type IIB string in plane wave R-R 5-form background F 5 using both path integral and operator formalisms and show that the two results agree perfectly. The result turns out to be equal to the partition function in the flat background. We also study the Tadpole cancellation for the unoriented closed and open string model in plane wave R-R 5-form background studied in hep-th/0203249 and find that the cancellation of the Tadpole requires the gauge group to be SO(8).

We analyze non-supersymmetric four-dimensional open string models of type IIB string theory compactified on T 2 × K3 with Scherk-Schwarz deformation acting on an S 1 of the T 2 torus. We find that there are always two solutions to the tadpole conditions that are shown to be connected via Wilson lines in an non-trivial way. These models although non-supersymmetric, are free of R-R and NS-NS tadpoles. 