Finite-temperature Casimir effect on the radius stabilization of non-commutative torus

The one-loop correction to the spectrum of Kaluza-Klein system for the phi (3) model on R-1,R-d x (T-theta(2))(L) is evaluated in the high temperature limit, where the 1 + d dimensions are the ordinary at Minkowski spacetimes and the L extra two-dimensional tori are chosen to be the non-commutative torus with noncommutativity theta. The corrections to the Kaluza-Klein mass formula are evaluated and used to compute the Casimir energy with the help of the Schwinger perturbative formula in the zeta-function regularization method. The results show that the one-loop Casimir energy is independent of the radius of torus if L = 1. However, when L > 1 the Casimir energy could give repulsive force to stabilize the extra non-commutative torus if d - L is a non-negative even integral. This therefore suggests a possible stabilization mechanism of extra radius in high temperature, when the extra spaces are non commutative.