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Proving the 6d Cardy Formula and Matching Global Gravitational Anomalies

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

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Submission summary

As Contributors: Chi-Ming Chang
Preprint link: scipost_202104_00035v2
Date accepted: 2021-07-22
Date submitted: 2021-07-16 12:14
Submitted by: Chang, Chi-Ming
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


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.

Published as SciPost Phys. 11, 036 (2021)

List of changes

1. The reference [88] in the previous version removed.
2. A sentence below (2.2) added to specify the reality property of the twisting parameters.

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