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A Cardy formula for 6d superconformal field theories (SCFTs) conjectured byDi Pietro and Komargodski in [1] governs the universal behavior of thesupersymmetric partition function on $S^1_\beta \times S^5$ in the limit ofsmall $\beta$ and fixed squashing of the $S^5$. For a general 6d SCFT, we studyits 5d effective action, which is dominated by the supersymmetric completionsof perturbatively gauge-invariant Chern-Simons terms in the small $\beta$limit. Explicitly evaluating these supersymmetric completions gives the precisesquashing dependence in the Cardy formula. For SCFTs with a pure Higgs branch(also known as very Higgsable SCFTs), we determine the Chern-Simons levels byexplicitly going onto the Higgs branch and integrating out the Kaluza-Kleinmodes of the 6d fields on $S^1_\beta$. We then discuss tensor branch flows,where an apparent mismatch between the formula in [1] and the free field answerrequires an additional contribution from BPS strings. This "missingcontribution" is further sharpened by the relation between the fractional partof the Chern-Simons levels and the (mixed) global gravitational anomalies ofthe 6d SCFT. We also comment on the Cardy formula for 4d $\mathcal{N}=2$ SCFTsin relation to Higgs branch and Coulomb branch flows.

Proving the 6d Cardy Formula and Matching Global Gravitational Anomalies