A universal inequality on the unitary 2D CFT partition function

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 $\frac{c}{12}+\epsilon$ and below the twist $\frac{c}{12}$ is universal in the large $c$ limit for all $\beta_L\beta_R \neq 4\pi^2$. The technique of the proof allows us to derive a one-parameter (with parameter $\alpha\in(0,1]$) 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 $(\beta_L,\beta_R)$ plane. The infinite iteration of this equation gives the boundary of maximal-validity domain, which depends on the parameter $\alpha$ in the inequality. In the $c \to \infty$ limit, with the additional assumption of a sparse spectrum below the scaling dimension $\frac{c}{12} + \epsilon$ and the twist $\frac{\alpha c}{12}$ (with $\alpha \in (0,1]$ 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 $(\beta_L, \beta_R)$ plane, except in the case of $\alpha = 1$. For $\alpha = 1$, this proves the conjecture proposed by [HKS14], and for $\alpha < 1$, 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.