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 <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mi>c</mi> <mn>12</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{c}{12} $</EquationSource> </InlineEquation> + ϵ and below the twist <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mi>c</mi> <mn>12</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{c}{12} $</EquationSource> </InlineEquation> is universal in the large c limit for all βLβR ≠ 4π2.The technique of the proof allows us to derive a one-parameter (with parameter α ∈ (0, 1]) family of universal inequalities on the unitary 2D CFT partition function with general central charge c ⩾ 0, using analytical modular bootstrap. We derive an iterative equation for the domain of validity of the inequality on the (βL, βR) plane. The infinite iteration of this equation gives the boundary of maximal-validity domain, which depends on the parameter α in the inequality.In the c → ∞ limit, with the additional assumption of a sparse spectrum below the scaling dimension <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mi>c</mi> <mn>12</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{c}{12} $</EquationSource> </InlineEquation> + ϵ and the twist <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mi mathvariant="italic">αc</mi> <mn>12</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{\alpha c}{12} $</EquationSource> </InlineEquation> (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.

A universal inequality on the unitary 2D CFT partition function

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2025年7月15日

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Indranil Dey,
Sridip Pal,
Jiaxin Qiao

摘要

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 $\frac{c}{12}$ $ \frac{c}{12} $ + ϵ and below the twist $\frac{c}{12}$ $ \frac{c}{12} $ is universal in the large c limit for all β_Lβ_R ≠ 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 (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 $\frac{c}{12}$ $ \frac{c}{12} $ + ϵ and the twist $\frac{αc}{12}$ $ \frac{\alpha c}{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.