A universal inequality on the unitary 2D CFT partition function
- Creators
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Dey, Indranil1
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Pal, Sridip2
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Qiao, Jiaxin3
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1.
Tata Institute of Fundamental Research
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2.
California Institute of Technology
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3.
École Polytechnique Fédérale de Lausanne
Abstract
We prove the conjecture proposed by Hartman, Keller and Stoica (HKS) [1]: the grand-canonical free energy of a unitary 2D CFT with a sparse spectrum below the scaling dimension + ϵ and below the twist is universal in the large c limit for all βLβR ≠ 4π2.
The technique of the proof allows us to derive a one-parameter (with parameter α ∈ (0, 1]) family of universal inequalities on the unitary 2D CFT partition function with general central charge c ⩾ 0, using analytical modular bootstrap. We derive an iterative equation for the domain of validity of the inequality on the (βL, βR) plane. The infinite iteration of this equation gives the boundary of maximal-validity domain, which depends on the parameter α in the inequality.
In the c → ∞ limit, with the additional assumption of a sparse spectrum below the scaling dimension + ϵ and the twist (with α ∈ (0, 1] fixed), our inequality shows that the grand-canonical free energy exhibits a universal large c behavior in the maximal-validity domain. This domain, however, does not cover the entire (βL, βR) plane, except in the case of α = 1. For α = 1, this proves the conjecture proposed by HKS [1], and for α < 1, it quantifies how sparseness in twist affects the regime of universality. Furthermore, this implies a precise lower bound on the temperature of near-extremal BTZ black holes, above which we can trust the black hole thermodynamics
Copyright and License
© 2025, The Author(s). 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.
Acknowledgement
We are grateful to Nathan Benjamin, Anatoly Dymarsky, Victor Gorbenko, Shiraz Minwalla, Mukund Rangamani, Ashoke Sen, Sandip Trivedi and Joaquin Turiaci for insightful discussions. We especially thank Tom Hartman, Mrunmay Jagadale, Hirosi Ooguri, Eric Perlmutter, Slava Rychkov and Allic Sivaramakrishnan for their valuable feedback on the draft. JQ would also like to thank Queen Mary University of London for their hospitality during the final stages of the draft preparation. ID acknowledges support from the Government of India, Department of Atomic Energy, under Project Identification No. RTI 4002, and from the Quantum Space-Time Endowment of the Infosys Science Foundation. SP is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632, and by the Walter Burke Institute for Theoretical Physics. JQ is supported by Simons Foundation grant 994310 (Simons Collaboration on Confinement and QCD Strings).
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Additional details
- Department of Atomic Energy
- RTI 4002
- Office of High Energy Physics
- DE-SC0011632
- California Institute of Technology
- Walter Burke Institute for Theoretical Physics -
- Simons Foundation
- Simons Collaboration on Confinement and QCD Strings 994310
- Caltech groups
- Division of Physics, Mathematics and Astronomy (PMA), Walter Burke Institute for Theoretical Physics
- Publication Status
- Published