The interplay between θ 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 ofF strongly suggests that a reduced number of degrees of freedom contributes to the free energy in the non-planar sector,F_np, 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 ofF_np 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.