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arXiv:2410.18174 · Journal of High Energy Physics

A universal inequality on the unitary 2D CFT partition function

Indranil Dey, Sridip Pal, Jiaxin Qiao

Published Jul 15, 2025• 4 citations
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Abstract

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 $$ \frac{c}{12} $$ c 12 + ϵ and below the twist $$ \frac{c}{12} $$ 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 $$ \frac{c}{12} $$ c 12 + ϵ and the twist $$ \frac{\alpha c}{12} $$ 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.

DOI
10.1007/jhep07(2025)163

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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.
We consider unitary, modular invariant, two-dimensional CFTs which are invariant under the parity transformation P P . Combining P P with modular inversion S S leads to a continuous family of fixed points of the SP S P transformation. A particular subset of this locus of fixed points exists along the line of positive left- and right-moving temperatures satisfying \beta_L \beta_R = 4\pi^2 β L β R = 4 π 2 . We use this fixed locus to prove a conjecture of Hartman, Keller, and Stoica that the free energy of a large- c c CFT _2 2 with a suitably sparse low-lying spectrum matches that of AdS _3 3 gravity at all temperatures and all angular potentials. We also use the fixed locus to generalize the modular bootstrap equations, obtaining novel constraints on the operator spectrum and providing a new proof of the statement that the twist gap is smaller than (c-1)/12 ( c − 1 ) / 12 when c>1 c > 1 . At large c c we show that the operator dimension of the first excited primary lies in a region in the (h,\overline{h}) ( h , h ¯ ) -plane that is significantly smaller than h+\overline{h}<c/6 h + h ¯ > c / 6 . Our results for the free energy and constraints on the operator spectrum extend to theories without parity symmetry through the construction of an auxiliary parity-invariant partition function.
Two-dimensional conformal field theories exhibit a universal free energy in the high temperature limit T → ∞, and a universal spectrum in the Cardy regime, Δ → ∞. We show that a much stronger form of universality holds in theories with a large central charge c and a sparse light spectrum. In these theories, the free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all Δ≥c6 \Delta \ge \frac{c}{6} Δ≥6c​ . The same is true of three-dimensional quantum gravity; therefore our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics. We also discuss several applications to CFT and gravity, including operator dimension bounds derived from the modular bootstrap, universality in symmetric orbifolds, and the role of non-universal ‘enigma’ saddlepoints in the thermodynamics of 3d gravity.
We prove using invariance under the modular SSS- and STSTST-transformations that every unitary two-dimensional conformal field theory (CFT) of only even-spin operators (with no extended chiral algebra and with central charges c,c~>1c,\tilde{c}>1c,c~>1) contains a primary operator with dimension Δ1Δ_1Δ1​ satisfying 0<Δ1<(c+c~)/24+0.09280...0 < Δ_1 < (c+\tilde{c})/24 + 0.09280...0<Δ1​<(c+c~)/24+0.09280... After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS3_33​/CFT2_22​ dictionary, the bound on Δ1Δ_1Δ1​ proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ<0Λ< 0Λ<0 can be no heavier than 1/(8GN)+O(−Λ)1/(8G_N)+O(\sqrt{-Λ})1/(8GN​)+O(−Λ​); the bounds on the number operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.
Abstract We show that for a unitary modular invariant 2D CFT with central charge c>1c>1c>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 (c−112−ε,c−112+ε)(\frac{c-1}{12}-\varepsilon ,\frac{c-1}{12}+\varepsilon )(12c−1​−ε,12c−1​+ε) ( c - 1 12 - ε , c - 1 12 + ε ) with ε=O(J−1/2log⁡J)\varepsilon =O(J^{-1/2}\log J)ε=O(J−1/2logJ) 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⁡(2π(c−1)J6)\exp \left( 2\pi \sqrt{\frac{(c-1)J}{6}}\right) exp(2π6(c−1)J​​) 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≫c3≫1J\gg c^3\gg 1J≫c3≫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.”
We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central charge c > 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 with c < 8 must admit relevant deformations, and that a nontrivial bound on the gap of scalar primaries exists for c < 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.
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.