Universality in asymptotic bounds and its saturation in $2$D CFT

TL;DR

This work reveals universal features in the asymptotics of 2D CFT data by combining Tauberian methods with modular and crossing constraints. It establishes an optimal fixed-spin spectral gap of $\delta\Delta=4$ in the OPE of identical scalars, and demonstrates saturation tendencies in rational CFTs, signaling universality beyond the density-of-states Cardy picture. The authors also connect LLH OPE asymptotics to Virasoro block behavior via numerical Zamolodchikov recursions, and show that heavy-state two-point functions reproduce thermal correlators in large-$c$ limits within an order-one window, while identifying an enigmatic regime that does not dominate. Together, these results sharpen the understanding of ETH-like thermalization and the structure of high-energy CFT data, and provide numerical insights into Virasoro block asymptotics.

Abstract

We study asymptotics of three point coefficients (light-light-heavy) and two point correlators in heavy states in unitary, compact $2$D CFTs. We prove an upper and lower bound on such quantities using numerically assisted Tauberian techniques. We obtain an optimal upper bound on the spectrum of operators appearing with fixed spin from the OPE of two identical scalars. While all the CFTs obey this bound, rational CFTs come close to saturating it. This mimics the scenario of bounds on asymptotic density of states and thereby pronounces an universal feature in asymptotics of 2D CFTs. Next, we clarify the role of smearing in interpreting the asymptotic results pertaining to considerations of eigenstate thermalization in 2D CFTs. In the context of light-light-heavy three point coefficients, we find that the order one number in the bound is sensitive to how close the light operators are from the $\frac{c}{32}$ threshold. In context of two point correlator in heavy state, we find the presence of an enigmatic regime which separates the $AdS_3$ thermal physics and the BTZ black hole physics. Furthermore, we present some new numerical results on the behavior of spherical conformal block.

Figures (9)

• Figure 1: The figure depicts the $(h',\bar{h}')$ plane. The lines parallel to $h'=\bar{h}'$ line are fixed spin lines. The lines perpendicular to fixed spin lines are fixed $\Delta$ lines. Suppose, we find an example such that all the operators appearing in the OPE of two $\mathcal{O}$ operators are separated by $\delta h=\delta\bar{h}=2$. The red dots in the figure depcits them. The Tauberian analysis tells us that the brown circle ($r=2$) and the black ellipse ($\text{semi-minor axis}>\sqrt{2}$) will always have some operators within it, as the center approaches infinity. The black ellipse can be rotated by $\pi/4$, the same result holds. To obtain the red and the blue ellipse, one has to take the black ellipse with semiminor axis of length $\sqrt{2}$ and squeeze the other axis, this can only be achived by using $T$ invariance in the fixed spin analysis. In the limit, the blue ellipse degenerates into a line segment of length (defined as Euclidean distance on the $(h',\bar{h}')$ plane) $2\sqrt{2}$ (or in terms of $\Delta$, the length of the segment is $\delta\Delta=4$). This is the bound on the gap that we obtain from fixed spin Tauberian analysis and this also proves the optimality of our bound, since we can not push the bound to a smaller value. An explicit example of such kind involves 3 copies of Ising, analyzed with respect to the diagonal Virasoro.
• Figure 2: The $n$-dependence of ${\text{sgn}}(c_n)$ for various $h_\mathcal{O}=\frac{c-5}{32},\frac{c-1}{32},\frac{c+5}{32}$, and $\frac{c+10}{32}$ with $c=30$. One can find that the $c_n$ away from the region $h_\mathcal{O} \in [ \frac{c-1}{32}, \frac{c+5}{32} ]$ is always positive, whereas the $c_n$ in the region $h_\mathcal{O} \in [ \frac{c-1}{32}, \frac{c+5}{32} ]$ can be negative.
• Figure 3: The blue dots show the maximal value $n_* (\leq 1500)$$s.t.$$c_{c_*}<0$. The red lines show the special values ${\delta}=-1,5$. Here we fix $c=30$ and define $\frac{c+{\delta}}{32}\equiv h$. The right figure is the zoomed version of the left figure. One can find the plateau in the region ${\delta} \in [-1,5]$. Note that the upper bound $n=1500$ just comes from the limitation of our machine power. We expect that one can see the plateau in ${\delta} \in [-1,5]$ more clearly if we take higher $n$ terms into account.
• Figure 4: The blue dots show the ${\delta}$-dependence of the density $\rho({\delta})$ (see (\ref{['eq:den1']})). The red lines show the special values ${\delta}=-1,5$. Here we also fix $c=30$.
• Figure 5: Upper: The $n$-dependence of $P(n)$ for various ${\delta}=-2,-\frac{9}{5}, -\frac{8}{5}, -\frac{7}{5}, -\frac{6}{5}$, and $-1$ with $c=30$ (see (\ref{['eq:den2']})). Lower: The $n$-dependence of $P(n)$ for various ${\delta}=0,1,2$, and $3$.