Cardy formulae for SUSY theories in d = 4 and d = 6

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 $$ \mathcal{N}=1 $$ 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