Proving the 6d Cardy formula and matching global gravitational anomalies
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 S¹_β×S⁵ in the limit of small β and fixed squashing of the S⁵. 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 S¹_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.
Additional Information
© 2021 Chang et al. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation. Received 30-04-2021; Accepted 22-07-2021; Published 23-08-2021. We are grateful to Lorenzo Di Pietro for extremely helpful correspondences, and are deeply indebted to Yuji Tachikawa for patiently explaining the various subtleties of global anomaly matching in 6d. We also thank Zohar Komargodski, Kantaro Ohmori, and Amos Yarom for insightful discussions. Finally, we thank Lorenzo Di Pietro, Zohar Komargodski, and Yuji Tachikawa for discerning comments on the draft. CC is supported in part by the U.S. Department of Energy grant DE-SC0009999. The work of MF is supported by the SNS fellowship P400P2-180740, the Princeton physics department, the JSPS Grant-In-Aid for Scientific Research Wakate(A) 17H04837, the WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo, the David and Ellen Lee Postdoctoral Scholarship. YL is supported by the Sherman Fairchild Foundation, and both MF and YL by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632. YW is supported in part by the US NSF under Grant No. PHY-1620059 and by the Simons Foundation Grant No. 488653. YL and YW thank the Bootstrap 2019 conference, and CC, MF and YW thank the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611, for hospitality during the finishing stages of this paper.Attached Files
Published - SciPostPhys_11_2_036.pdf
Submitted - 1910.10151.pdf
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Additional details
- Eprint ID
- 99514
- Resolver ID
- CaltechAUTHORS:20191028-153338282
- DE-SC0009999
- Department of Energy (DOE)
- P400P2-180740
- Swiss National Science Foundation (SNSF)
- Princeton University
- 17H04837
- Japan Society for the Promotion of Science (JSPS)
- Ministry of Education, Culture, Sports, Science and Technology (MEXT)
- University of Tokyo
- David and Ellen Lee Postdoctoral Scholarship
- Sherman Fairchild Foundation
- DE-SC0011632
- Department of Energy (DOE)
- PHY-1620059
- NSF
- 488653
- Simons Foundation
- PHY-1607611
- NSF
- Created
-
2019-10-28Created from EPrint's datestamp field
- Updated
-
2021-11-04Created from EPrint's last_modified field
- Caltech groups
- Walter Burke Institute for Theoretical Physics
- Other Numbering System Name
- CALT-TH
- Other Numbering System Identifier
- 2019-023