Study of Temperature Inversion Symmetry for the Twisted Wess-Zumino

The temperature inversion symmetry, for a non interacting supersymmetric ensemble, at finite volume, is studied. It is found that, the scaled free energy, $f(\xi)$, is antisymmetric under temperature inversion transformation, i.e. $f(\xi)=-\xi ^{d}f(\frac{1}{\xi})$. This occurs for antiperiodic bosons and periodic fermions, in the compact dimension. On the contrary, for periodic bosons and antiperiodic fermions, $f(\xi)=\xi ^{d}f(\frac{1}{\xi})$.