We investigate the quantized scalar field on the Kaluza-Klein spacetimes ofdimensional commutative torus, and S F Z is a noncommutative fuzzy two sphere with a fixed quantized radius. After evaluating the one-loop correction to the spectrum we use the masscorrected term to compute the Casimir energy of the scalar field on the model spacetime. It is seen that, for some values of D and d, the Casimir energy due to vacuum fluctuation in the model spacetimes could give rise a repulsive force to stabilize the commutative torus.

An analysis is presented of the Bianchi type I cosmological models with a bulk viscosity when the universe is filled with the stiff fluid p = ǫ while the viscosity is a power function of the energy density, such as η = α|ǫ| n . Although the exact solutions are obtainable only when the 2n is an integer, the characteristics of evolution can be clarified for the models with arbitrary value of n. It is shown that, except for the n = 0 model that has solutions with infinite energy density at initial state, the anisotropic solutions that evolve to positive Hubble functions in the later stage will begin with Kasner-type curvature singularity and zero energy density at finite past for the n > 1 models, and with finite Hubble functions and finite negative energy density at infinite past for the n < 1 models. In the course of evolution, matters are created and the anisotropies of the universe are smoothed out. At the final stage, cosmologies are driven to infinite expansion state, de Sitter space-time, or Friedman universe asymptotically. However, the de Sitter space-time is the only attractor state for the n < 1/2 models. The solutions that are free of cosmological singularity for any finite proper time are singled out. The extension to the higher-dimensional models is also discussed.

We first apply the transformation of mixing azimuthal and internal coordinate or mixing time and internal coordinate to the 11D M-theory with a stack N M2-branes to find the spacetime of a stack of N D2-branes with magnetic or electric flux in 10 D IIA string theory, after the Kaluza-Klein reduction. We then perform the T duality to the spacetime to find the background of a stack of N D3-branes with magnetic or electric flux. In the near-horizon limit the background becomes the magnetic or electric field deformed AdS 5 × S 5 . We adopt an ansatz to find the classical string solution which is rotating in the deformed S 5 with three angular momenta in the three rotation planes. The relations between the classical string energy and its angular momenta are found and results show that the external magnetic and electric fluxes will increase the string energy. Therefore, from the AdS/CFT point of view, the corrections of the anomalous dimensions of operators in the dual SYM theory will be positive. We also investigate the small fluctuations in these solutions and discuss the effects of magnetic and electric fields on the stability of these classical rotating string solutions. Finally, we find the possible solutions of string pulsating on the deformed spacetimes and show that the corrections to the anomalous dimensions of operators in the dual SYM theory are non-negative.

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The Lagrangian of self-dual gauge theory in various formulations are reviewed. From these results we see a simple rule and use it to present some new non-covariant Lagrangian based on the decomposition of spacetime into D = D 1 + D 2 + D 3 . Our prescription could be easily extended to more complex decomposition of spacetime and some more examples are presented therefore. The self-dual property of the new Lagrangian is proved in detail. We also show that the new non-covariant actions give field equations with 6d Lorentz invariance.