A Cardy formula for 6d superconformal field theories (SCFTs) conjectured by Di Pietro and Komargodski governs the universal behavior of the supersymmetric partition function on $S^1_\beta \times S^5$ 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 $S^1_\beta$. We then discuss tensor branch flows, where an apparent mismatch between the formula in Di Pietro-Komargodski 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 $\mathcal{N}=2$ SCFTs in relation to Higgs branch and Coulomb branch flows.

中文翻译：

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证明6d Cardy公式并匹配全球引力异常

Di Pietro 和 Komargodski 推测的 6d 超共形场理论 (SCFTs) 的 Cardy 公式控制了 $S^1_\beta \times S^5$ 上的超对称配分函数在小 $\beta$ 和固定的极限上的普遍行为压扁 $S^5$。对于一般的 6d SCFT，我们研究了它的 5d 有效作用，它由小 $\beta$ 限制中微扰规范不变陈-西蒙斯项的超对称完成支配。明确评估这些超对称完成给出了 Cardy 公式中精确的挤压依赖性。对于具有纯 Higgs 分支的 SCFT（也称为非常 Higgsable SCFT），我们通过明确进入希格斯分支并整合 $S^1_\beta$ 上 6d 场的 Kaluza-Klein 模式来确定 Chern-Simons 能级. 然后我们讨论张量分支流，其中 Di Pietro-Komagodski 中的公式与自由场答案之间的明显不匹配需要 BPS 字符串的额外贡献。陈-西蒙能级的小数部分与 6d SCFT 的（混合）全球引力异常之间的关系进一步加剧了这种“缺失的贡献”。我们还评论了 4d $\mathcal{N}=2$ SCFT 与希格斯支流和库仑支流相关的卡迪公式。