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# Proving the 6d Cardy Formula and Matching Global Gravitational Anomalies

Chang, Chi-Ming and Fluder, Martin and Lin, Ying-Hsuan and Wang, Yifan (2019) Proving the 6d Cardy Formula and Matching Global Gravitational Anomalies. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20191028-153338282

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## 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 $S^1_\beta \times S^5$ 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 [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 flow.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1910.10151arXivDiscussion Paper
ORCID:
AuthorORCID
Fluder, Martin0000-0002-0780-8550
Lin, Ying-Hsuan0000-0001-8904-1287
Wang, Yifan0000-0001-9965-9777
Additional Information: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.
Group:Walter Burke Institute for Theoretical Physics
Funders:
Funding AgencyGrant Number
Department of Energy (DOE)DE-SC0009999
David and Ellen Lee Postdoctoral ScholarshipUNSPECIFIED
Department of Energy (DOE)DE-SC0011632
Sherman Fairchild FoundationUNSPECIFIED
NSFPHY-1620059
Simons Foundation488653
NSFPHY-1607611
Swiss National Science Foundation (SNSF)P400P2-180740
Princeton UniversityUNSPECIFIED
Japan Society for the Promotion of Science (JSPS)17H04837
Other Numbering System:
Other Numbering System NameOther Numbering System ID
CALT-TH2019-023
Record Number:CaltechAUTHORS:20191028-153338282
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20191028-153338282
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:99514
Collection:CaltechAUTHORS
Deposited By: Joy Painter
Deposited On:28 Oct 2019 22:46