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Recently, it was found by Greene and Levin that the Casimir energy of certaincombinations of massless and massive fields in space with extra dimensions playa crucial role in the accelerated expansion of the late-time universe andtherefore it could serve as a candidate for the dark energy. It also provides amechanism in stabilizing the volume moduli of extra dimensions. However, theshape moduli of the extra dimensions were never taken into account in theprevious work. We therefore study the stabilization mechanism for both volumeand shape moduli due to the Casimir energy in ${\mathbb M}^{1+3} \times{\mathbb T}^{2}$. The result of our study shows that the previously known localminimum is a saddle point. It is unstable to the perturbations in the directionof the shape moduli. The new stable local minima stabilizes all the moduli anddrives the accelerating expansion of the universe. The cosmological dynamicsboth in the bulk and the radion pictures are derived and simulated. Theequations of state for the Casimir energy in a general torus are derived. Shearviscosity in extra dimensions induced by the Casimir density in the late timesis identified and calculated, it is found to be proportional to the Hubbleconstant.

Dark energy and moduli stabilization of extra dimensions in M^{1+3}xT^2 spacetime