Proving the 6d Cardy formula and matching global gravitational anomalies

A Cardy formula for 6d superconformal field theories (SCFTs) conjectured by Di Pietro and Komargodski in [1] governs the universal behavior of the supersymmetric partition function on S-beta(1) x S-5 in the limit of small beta and fixed squashing of the S-5. For a general 6d SCFT, we study its 5d effective action, which is dominated by the supersymmetric completions of perturbatively gauge-invariant Chern-Simons terms in the small beta limit. Explicitly evaluating these supersymmetric completions gives the precise squashing dependence in the Cardy formula. For SCFTs with a pure Higgs branch (also known as very Higgsable SCFTs), we determine the Chern-Simons levels by explicitly going onto the Higgs branch and integrating out the Kaluza-Klein modes of the 6d fields on S-beta(1). We then discuss tensor branch flows, where an apparent mismatch between the formula in [1] and the free field answer requires an additional contribution from BPS strings. This missing contribution is further sharpened by the relation between the fractional part of the Chern-Simons levels and the (mixed) global gravitational anomalies of the 6d SCFT. We also comment on the Cardy formula for 4d N = 2 SCFTs in relation to Higgs branch and Coulomb branch flows.

Automated summary

The study investigates the Cardy formula for 6D superconformal field theories governing the universal behavior of supersymmetric partition function, focusing on the effects of small beta and squashing of S-5; it also examines the 5D effective action and determines the Chern-Simons levels for SCFTs with a pure Higgs branch. The research further discusses tensor branch flows and the missing contribution from BPS strings in relation to the formula, as well as the connection between the fractional part of Chern-Simons levels and the global gravitational anomalies of 6D SCFT.