Derivative expansion of the heat kernel at finite temperature

The method of covariant symbols of Pletnev and Banin is extended to space-times with topology R-n x S-1 X center dot center dot center dot X S-1. By means of this tool, we obtain explicit formulas for the diagonal matrix elements and the trace of the heat kernel at finite temperature to fourth order in a strict covariant derivative expansion. The role of the Polyakov loop is emphasized. Chan's formula for the effective action to one-loop is similarly extended. The expressions obtained apply formally to a larger class of spaces, h-spaces, with an arbitrary weight function h(p) in the integration over the momentum of the loop.