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Proving the 6d Cardy formula and matching global gravitational anomalies

Chi-Ming Chang Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China; Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, CA 95616, US Martin Fluder Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA; Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan; Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA Ying-Hsuan Lin Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA; Hsinchu County Environmental Protection Bureau, Hsinchu, Taiwan Yifan Wang Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA; Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA; Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

arXiv subject: High Energy Physics - Theory (hep-th) mathscidoc:2207.74019

SciPost Physics, 11, (036), 2021.8
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_β×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 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 N=2 SCFTs in relation to Higgs branch and Coulomb branch flows.
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@inproceedings{chi-ming2021proving,
  title={Proving the 6d Cardy formula and matching global gravitational anomalies},
  author={Chi-Ming Chang, Martin Fluder, Ying-Hsuan Lin, and Yifan Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220702143459252547515},
  booktitle={SciPost Physics},
  volume={11},
  number={036},
  year={2021},
}
Chi-Ming Chang, Martin Fluder, Ying-Hsuan Lin, and Yifan Wang. Proving the 6d Cardy formula and matching global gravitational anomalies. 2021. Vol. 11. In SciPost Physics. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220702143459252547515.
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