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Elliptic functions and temperature inversion symmetry on spheres
J. Stuart Dowker and Klaus Kirsten
Finite temperature boson and fermion free field theories on the space-time manifolds $\oR\times$S$^d$ are discussed with one eye on the questions of temperature inversion symmetry and modular invariance. For conformally invariant theories it is shown that the total energy at any temperature for any odd dimension, $d$, is given as a power series in the $d=3$ and $d=5$ energies, for scalars, and the $d=1$ and $d=3$ energies for spinors. Further, these energies can be given in finite terms at specific temperatures associated with singular moduli of elliptic function theory. Some examples are listed and numbers given.