A universal inequality on the unitary 2D CFT partition function

TL;DR

The paper proves a universal bound on the grand-canonical free energy of unitary 2D CFTs with sparse low-lying spectra, using an analytic modular bootstrap approach and a tunable parameter $\alpha\in(0,1]$. It develops a general iterative inequality for modular-invariant partition functions, yielding a maximal-validity domain $\mathcal{D}_\alpha$ in the $(\beta_L,\beta_R)$ plane and an explicit bound on the error term $\mathcal{E}(\beta_L,\beta_R)$ in terms of low- and high-twist contributions. In particular, for $\alpha=1$ the result proves the Hartman–Keller–Stoica conjecture: the large-$c$ free energy is universal away from the self-dual line $\beta_L\beta_R=4\pi^2$, and holographically corresponds to a sharp black-hole vs. thermal-AdS competition; for $\alpha<1$, the universality region shrinks, quantifying the role of twist sparseness. The analysis is extended to Virasoro-primary partition functions, delivering strengthened universality results under weaker sparseness and elucidating the structure of the large-$c$ phase diagram, including implications for near-extremal BTZ black holes. Overall, the work provides a rigorous, quantitative bridge between modular invariance, large-$c$ CFT universality, and holographic black-hole thermodynamics.

Abstract

We prove the conjecture proposed by Hartman, Keller and Stoica [HKS14]: the grand-canonical free energy of a unitary 2D CFT with a sparse spectrum below the scaling dimension $\frac{c}{12}+ε$ and below the twist $\frac{c}{12}$ is universal in the large $c$ limit for all $β_Lβ_R \neq 4π^2$. The technique of the proof allows us to derive a one-parameter (with parameter $α\in(0,1]$) family of universal inequalities on the unitary 2D CFT partition function with general central charge $c\geqslant 0$, 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 $c \to \infty$ limit, with the additional assumption of a sparse spectrum below the scaling dimension $\frac{c}{12} + ε$ and the twist $\frac{αc}{12}$ (with $α\in (0,1]$ 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 [HKS14], 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.

Figures (8)

• Figure 1: An example, showing the set of points $(\beta_L^{(i)},\beta_R^{(i)})$, obtained using the algorithm, appearing in fig. \ref{['fig:algorithmbeta']}, for $\alpha=1$, starting from an initial choice of $(\beta_L,\beta_R)=(\frac{41\pi}{25},\frac{5\pi}{2})$. We begin with $(\beta_L^{(1)},\beta_R^{(1)})=(\frac{41\pi}{25},\frac{5\pi}{2})$ and the recursion terminates at $N=5$. The numbers near the points denote at which step they are reached, added by $1$.
• Figure 2: The large-$c$ phase diagrams corresponding to $\alpha=1$ and $\alpha=0.7$. The grey shaded region is dominated by black hole while the blue shaded region is dominated by thermal AdS. For $\alpha=1$, the union of the two regions covers the whole plane except for the hyperbola $\beta_L\beta_R=4\pi^2$, depicted as the black solid curve in the figure on the left. For $\alpha<1$ (the figure on the right), white region, where we do not have any universality, starts to emerge. The boundary of the white region is analytically given by the union of curves: $\frac{\beta_L}{2\pi}=f^{(2)}_\alpha\!(\frac{\beta_R}{2\pi})$, $\frac{\beta_R}{2\pi}=f^{(2)}_\alpha\!(\frac{\beta_L}{2\pi})$ (the red curves) and $\frac{2\pi}{\beta_L}=f^{(2)}_\alpha\!(\frac{2\pi}{\beta_R})$, $\frac{2\pi}{\beta_R}= f^{(2)}_\alpha\!(\frac{2\pi}{\beta_L})$ (the blue curves), where $f^{(2)}_\alpha$ is defined in \ref{['twosolutions']}. For $\alpha>1/2$, we show that the white region must cap off at the intersection point of $\beta_L\beta_R=4\pi^2$ and $\text{Max}(\beta_L,\beta_R)=\frac{2\pi}{2\alpha-1}$, which is $5\pi$ for $\alpha=0.7$ and diverges as $\alpha\to 1/2$ from above. For $\alpha=0.7$, we have shown this feature with dotdashed line at $\text{Max}(\beta_L,\beta_R)=5\pi$. The black dotted curve in the figure on the right is $\beta_L\beta_R=4\pi^2$.
• Figure 3: The large-$c$ phase diagrams corresponding to $\alpha=0.5$ and $\alpha=0.3$. The grey shaded region is dominated by black hole while the blue shaded region is dominated by thermal AdS. The free energy is not universal in white region. The boundary of these white regions are analytically given by the union of curves: $\frac{\beta_L}{2\pi}=f^{(2)}_\alpha\!(\frac{\beta_R}{2\pi})$, $\frac{\beta_R}{2\pi}=f^{(2)}_\alpha\!(\frac{\beta_L}{2\pi})$ and $\frac{2\pi}{\beta_L}=f^{(2)}_\alpha\!(\frac{2\pi}{\beta_R})$, $\frac{2\pi}{\beta_R}= f^{(2)}_\alpha\!(\frac{2\pi}{\beta_L})$, where $f^{(2)}_\alpha$ is defined in \ref{['twosolutions']}. For $\alpha\leq 1/2$, the white region extends to infinity. The black dotted curve is $\beta_L\beta_R=4\pi^2$
• Figure 4: The algorithm for generating $(\beta_{L}^{(i)},\beta_{R}^{(i)})$ ($i=1,2,\ldots,N$). For any $(\beta_{L},\beta_{R})\in\mathcal{D}_\alpha$, this iteration takes finite steps.
• Figure 5: The algorithm for generating $(x_i,y_i)$ ($i=1,2,\ldots,N$). For any $(x,y)\in\Omega_\alpha$, this iteration takes finite steps.

Theorems & Definitions (21)

• Theorem 2.1
• Remark 2.2
• Lemma 2.3
• Conjecture 2.4
• Corollary 2.5
• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof