Vortices and confinement in hot and cold D = 2+1 gauge theories

We calculate the variation with temperature of the vortex free energy inD=2+1 SU(2) lattice gauge theories. We do so both above and below thedeconfining transition at T=Tc. We find that this quantity is zero at all T forlarge enough volumes. For TTc it is consistent with theconventional expectation of `spatial' linear confinement. In small spatialvolumes this quantity is shown to be non-zero. The way it decreases to zerowith increasing volume is shown to be controlled by the (spatial) stringtension and it has the functional form one would expect if the vortices beingstudied were responsible for the confinement at low T, and for the `spatial'confinement at large T. We also discuss in detail some of the direct numericalevidence for a non-zero spatial string tension at high T, and we show that theobserved linearity of the (spatial) potential extends over distances that arelarge compared to typical high-T length scales.