The Interplay Between $\theta$ and T

TL;DR

This paper investigates how noncommutative geometry affects the free energy's temperature dependence, revealing a reduced number of degrees of freedom in the non-planar sector at high temperatures, independent of UV sensitivity.

Contribution

It extends previous work on free energy dependence on noncommutativity to theories with different UV sensitivities, showing a universal reduction in degrees of freedom at high temperature.

Findings

Non-planar free energy decreases with temperature due to fewer degrees of freedom.

High-temperature behavior can be modeled with classical statistical mechanics.

The phenomenon is independent of UV sensitivity and dimensionality.

Abstract

We extend a recent computation of the dependence of the free energy, F, on the noncommutative scale $\theta$ 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, $F_{\rm np}$, 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 of $F_{\rm np}$ 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.