  We consider supersymmetric theories on a space with compact space-like
slices. One can count BPS representations weighted by (-1)^F, or, equivalently,
study supersymmetric partition functions by compactifying the time direction. A
special case of this general construction corresponds to the counting of short
representations of the superconformal group. We show that in four-dimensional
N=1 theories the "high temperature" asymptotics of such counting problems is
fixed by the anomalies of the theory. Notably, the combination a-c of the trace
anomalies plays a crucial role. We also propose similar formulae for
six-dimensional (1,0) theories.
