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Discreteness and integrality in Conformal Field Theory

Kaidi, Justin and Perlmutter, Eric (2021) Discreteness and integrality in Conformal Field Theory. Journal of High Energy Physics, 2021 (2). Art. No. 64. ISSN 1126-6708. doi:10.1007/JHEP02(2021)064.

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Various observables in compact CFTs are required to obey positivity, discreteness, and integrality. Positivity forms the crux of the conformal bootstrap, but understanding of the abstract implications of discreteness and integrality for the space of CFTs is lacking. We systematically study these constraints in two-dimensional, non-holomorphic CFTs, making use of two main mathematical results. First, we prove a theorem constraining the behavior near the cusp of integral, vector-valued modular functions. Second, we explicitly construct non-factorizable, non-holomorphic cuspidal functions satisfying discreteness and integrality, and prove the non-existence of such functions once positivity is added. Application of these results yields several bootstrap-type bounds on OPE data of both rational and irrational CFTs, including some powerful bounds for theories with conformal manifolds, as well as insights into questions of spectral determinacy. We prove that in rational CFT, the spectrum of operator twists t ≥ c/12 is uniquely determined by its complement. Likewise, we argue that in generic CFTs, the spectrum of operator dimensions Δ > c−1/12 is uniquely determined by its complement, absent fine-tuning in a sense we articulate. Finally, we discuss implications for black hole physics and the (non-)uniqueness of a possible ensemble interpretation of AdS₃ gravity.

Item Type:Article
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Kaidi, Justin0000-0001-6144-0729
Additional Information:© 2021 The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Article funded by SCOAP3. Received 23 October 2020; Accepted 31 December 2020; Published 08 February 2021. We wish to thank Chris Beem, Ying-Hsuan Lin and Arnav Tripathy for helpful discussions, and Alex Belin, Nathan Benjamin and Sunil Mukhi for helpful comments on an earlier version. JK thanks the Mani L. Bhaumik Institute for generous support. EP is supported by Simons Foundation grant 488657 (Simons Collaboration on the Nonperturbative Bootstrap) and by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632.
Group:Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
Simons Foundation488657
Department of Energy (DOE)DE-SC0011632
Subject Keywords:Conformal Field Theory, AdS-CFT Correspondence
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Issue or Number:2
Record Number:CaltechAUTHORS:20200811-133612143
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Official Citation:Kaidi, J., Perlmutter, E. Discreteness and integrality in Conformal Field Theory. J. High Energ. Phys. 2021, 64 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:104915
Deposited By: Joy Painter
Deposited On:11 Aug 2020 20:42
Last Modified:16 Nov 2021 18:37

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