Casimir Effect on the Radius Stabilization of the Noncommutative Torus

We evaluate the one-loop correction to the spectrum of Kaluza-Klein system for the ${\varphi }^3$ model on $R^{1,d}\times (T_{\theta }^2{)}^L$, where $1+d$ dimensions are the ordinary flat Minkowski spacetimes and the extra dimensions are the L two-dimensional noncommutative tori with noncommutativity $\theta$. The correction to the Kaluza-Klein mass spectrum is then used to compute the Casimir energy. The results show that when $L>2$ the Casimir energy due to the noncommutativity could give repulsive force to stabilize the extra noncommutative tori in the cases of $d=4n-2$, with $n$ a positive integral.