The interplay between ? and T

We extend a recent computation of the dependence of the free energy, F, on the non-commutative scale θ to theories with very different UV sensitivity. The temperature dependence of F strongly suggests that a reduced number of degrees of freedom contributes to the free energy in the non-planar sector, Fnp, at high temperature. This phenomenon seems generic, independent of the UV sensitivity, and can be traced to modes whose thermal wavelengths become smaller than the noncommutativity scale. The temperature dependence of Fnp can then be calculated at high temperature using classical statistical mechanics, without encountering a UV catastrophe even in large number of dimensions. This result is a telltale sign of the low number of degrees of freedom contributing to F in the non-planar sector at high temperature. Such behavior is in marked contrast to what would happen in a field theory with a random set of higher derivative interactions.