## Proving the 6d Cardy formula and matching global gravitational anomalies

Chi-Ming Chang, Martin Fluder, Ying-Hsuan Lin, Yifan Wang

SciPost Phys. 11, 036 (2021) · published 23 August 2021

- doi: 10.21468/SciPostPhys.11.2.036
- Submissions/Reports

### Abstract

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.

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}^{2}Chi-Ming Chang, -
^{3}^{4}^{5}Martin Fluder, -
^{5}^{6}Ying-Hsuan Lin, -
^{4}^{7}Yifan Wang

^{1}University of California, Davis [UCD]^{2}Tsinghua University [THU]^{3}東京大学 / University of Tokyo [UT]^{4}Princeton University^{5}Walter Burke Institute for Theoretical Physics^{6}Hsinchu County Environmental Protection Bureau^{7}Harvard University

- 日本学術振興会 / Japan Society for the Promotion of Science [JSPS]
- National Science Foundation [NSF]
- Scuola Normale Superiore (through Organization: Scuola Normale Superiore di Pisa)
- Simons Foundation
- United States Department of Energy [DOE]