Cardy Limits of 6d Superconformal Theories

We explore the supersymmetric partition functions of 6d SCFTs on $\mathbb{R}^4\times T^2$ with non-vanishing charges for compatible global symmetries. We utilize the elliptic genera for self-dual strings and compute the free energy of 6d SCFTs in the Cardy limit. For a 6d (2,0) theory on $N$ M5-brane, we obtain the free energy proportional to $N^3$. We find that the origin of $N^3$ comes from the condensation of the self-dual strings, whose total number is proportional to ${N^3-N\over 6}$. We further extend our analysis to the general E-string theory and obtain its Cardy free energy.