Proving the 6d Cardy Formula and Matching Global Gravitational Anomalies

TL;DR

This work proves the 6d Cardy formula for SCFTs with a pure Higgs branch by reducing the theory on S^1 to a 5d effective action dominated by supersymmetric Chern-Simons terms and evaluating them on a rigid squashed S^5. It explicitly relates the CS levels κ_i to perturbative 8-form anomaly coefficients, and shows that on Higgs branches these levels are fixed by free hypermultiplets, while on tensor branches additional BPS-string contributions are required for global anomaly matching. The analysis extends the 4d Cardy program to 6d, connects the Cardy data to moduli-space flows, and highlights a deep link between local CS terms and global gravitational anomalies. The results provide a geometric, RG-invariant framework for understanding Cardy limits in higher-dimensional SCFTs and offer insights into the role of extended objects in anomaly matching.

Abstract

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 $S^1_β\times S^5$ in the limit of small $β$ 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 $β$ 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 $S^1_β$. 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 $\mathcal{N}=2$ SCFTs in relation to Higgs branch and Coulomb branch flows.

Figures (1)

• Figure 1: Diagram depicting the relations among different effective actions. We denote by $\Lambda$ the cutoff of the effective action, and $m_{moduli}$ is the mass scale associated with the moduli scalar vacuum expectation value. The field contents are as follows: the Wilsonian effective action contains light dynamical fields and background fields; the 6d/4d effective action contains massless dynamical fields and background fields; the 5d/3d effective action contains only background fields. On the one hand, the $\downarrow \rightarrow$ direction is in principle correct but difficult to carry out (unless the effective theory is weakly coupled). On the other hand, except on the pure Higgs branch, where the Green-Schwarz/Wess-Zumino type terms relevant for (mixed) gravitational anomalies are absent in the 6d/4d effective action, it is not understood how to perform the dashed arrow on the right. This is the source of the puzzle of Section \ref{['Sec:TB']}.