Effective finite temperature partition function for fields on noncommutative flat manifolds

The first quantum correction to the finite temperature partition function for a self-interacting massless scalar field on a D-dimensional flat manifold with p noncommutative extra dimensions is evaluated by means of dimensional regularization, supplemented with zeta-function techniques. It is found that the zeta function associated with the effective one-loop operator may be nonregular at the origin. The important issue of the determination of the regularized vacuum energy, namely the first quantum correction to the energy in such a case, is discussed.