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arXiv:2307.02587 · Annales Henri Poincaré

Lightcone Modular Bootstrap and Tauberian Theory: A Cardy-Like Formula for Near-Extremal Black Holes

Sridip Pal, Jiaxin Qiao

Published May 3, 2024• 7 citations
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Abstract

Abstract We show that for a unitary modular invariant 2D CFT with central charge $$c>1$$ c > 1 and having a nonzero twist gap in the spectrum of Virasoro primaries, for sufficiently large spin J , there always exist spin- J operators with twist falling in the interval $$(\frac{c-1}{12}-\varepsilon ,\frac{c-1}{12}+\varepsilon )$$ ( c - 1 12 - ε , c - 1 12 + ε ) with $$\varepsilon =O(J^{-1/2}\log J)$$ ε = O ( J - 1 / 2 log J ) . We establish that the number of Virasoro primary operators in such a window has a Cardy-like, i.e., $$\exp \left( 2\pi \sqrt{\frac{(c-1)J}{6}}\right) $$ exp 2 π ( c - 1 ) J 6 growth. A similar result is proven for a family of holographic CFTs with the twist gap growing linearly in c and a uniform boundedness condition, in the regime $$J\gg c^3\gg 1$$ J ≫ c 3 ≫ 1 . From the perspective of near-extremal rotating BTZ black holes (without electric charge), our result is valid when the Hawking temperature is much lower than the “gap temperature.”

DOI
10.1007/s00023-024-01441-2

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A bstract We derive Cardy-like formulas for the growth of operators in different sectors of unitary 2 dimensional CFT in the presence of topological defect lines by putting an upper and lower bound on the number of states with scaling dimension in the interval [∆ − δ, ∆ + δ ] for large ∆ at fixed δ . Consequently we prove that given any unitary modular invariant 2D CFT symmetric under finite global symmetry G (acting faithfully), all the irreducible representations of G appear in the spectra of the untwisted sector; the growth of states is Cardy like and proportional to the “square” of the dimension of the irrep. In the Schwarzian limit, the result matches onto that of JT gravity with a bulk gauge theory. If the symmetry is non-anomalous, the result applies to any sector twisted by a group element. For c > 1, the statements are true for Virasoro primaries. Furthermore, the results are applicable to large c CFTs. We also extend our results for the continuous U(1) group.
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Abstract We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central chargec >1 using modular bootstrap. Upper bounds on the gap in the dimension of primary operators of any spin, as well as in the dimension of scalar primaries, are computed numerically as functions of the central charge using semi-definite programming. Our bounds refine those of Hellerman and Friedan-Keller, and are in some cases saturated by known CFTs. In particular, we show that unitary CFTs withc <8 must admit relevant deformations, and that a nontrivial bound on the gap of scalar primaries exists forc <25. We also study bounds on the dimension gap in the presence of twist gaps, bounds on the degeneracy of operators, and demonstrate how “extremal spectra” which maximize the degeneracy at the gap can be determined numerically.
A bstract 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 c12 \frac{c}{12} 12c​ c 12 + ϵ and below the twist c12 \frac{c}{12} 12c​ c 12 is universal in the large c limit for all β L β R ≠ 4 π 2 . 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 ( β L , β R ) 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 c12 \frac{c}{12} 12c​ c 12 + ϵ and the twist αc12 \frac{\alpha c}{12} 12αc​ αc 12 (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 ( β L , β R ) 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.
Abstract We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy interval. They are most revealing at high energies. In this limit we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. We also study partition functions with a UV cutoff. Control over error estimates allows us to probe operators beyond the unity in the modularity condition. We check our results in the 2d Ising model and the Monster CFT and find perfect agreement.