Tauberian-Cardy formula with spin

TL;DR

This work delivers a rigorous 2D Tauberian framework for 2D CFTs to extract the asymptotic density of states on the (h, h̄) plane, including spin dependence and twist-driven corrections. By combining modular invariance with carefully constructed test functions, the authors derive precise leading behavior and controlled error terms, uncovering how the twist gap and areal gaps constrain the spectrum. They extend the Cardy regime to scenarios with finite twist, large spin, and large central charge, establishing universal inequalities and bounds that connect spectral gaps to geometric areas in weight space. The results have broad implications for holographic CFTs and the bootstrap program, providing rigorous footing for spin-sensitive Cardy-like growth and guiding future explorations of OPE data and chaos in 2D theories.

Abstract

We prove a $2$ dimensional Tauberian theorem in context of $2$ dimensional conformal field theory. The asymptotic density of states with conformal weight $(h,\bar{h})\to (\infty,\infty)$ for any arbitrary spin is derived using the theorem. We further rigorously show that the error term is controlled by the twist parameter and insensitive to spin. The sensitivity of the leading piece towards spin is discussed. We identify a universal piece in microcanonical entropy when the averaging window is large. An asymptotic spectral gap on $(h,\bar{h})$ plane, hence the asymptotic twist gap is derived. We prove an universal inequality stating that in a compact unitary $2$D CFT without any conserved current $Ag\leq \frac{π(c-1)r^2}{24}$ is satisfied, where $g$ is the twist gap over vacuum and $A$ is the minimal "areal gap", generalizing the minimal gap in dimension to $(h',\bar{h}')$ plane and $r=\frac{4\sqrt{3}}π\simeq 2.21$. We investigate density of states in the regime where spin is parametrically larger than twist with both going to infinity. Moreover, the large central charge regime is studied. We also probe finite twist, large spin behavior of density of states.

Figures (15)

• Figure 1: Approaching to infinity on $(h',\bar{h}')$ plane: The blue lines denote how the number of operators with $(h',\bar{h}')$ is counted such that $h'+\bar{h}'\leq \Delta$ and then we let $\Delta\to\infty$, this is given by the function $F^{\text{MZ}}(\Delta)$, originally calculated in Baur. On the other hand, the black lines denote how the number of operator is counted upto some value of $h=\bar{h}=\Delta/2$ i.e. $h'\leq\Delta/2,\bar{h}'\leq\Delta/2$ and then we let $\Delta\to\infty$. This approach to infinity is captured by the function $F(\Delta/2,\Delta/2)$, calculated in this paper. We see that the square of size $\Delta/2$ with one vertex at origin and another one at $(\Delta/2,\Delta/2)$ are always contained within the rightangled triangular region, created by $h'$ axis, $\bar{h}'$ axis and $h'+\bar{h}'=\Delta$ line. This is consistent with the observation that leading behavior of $F(\Delta/2,\Delta/2)$ is suppressed compared to $F^{\text{MZ}}(\Delta)$.
• Figure 2: $(h',\bar{h}')$ plane : asymptotically, the rectangular region (blue shaded) contains exponentially less number of operators compared to square region (red shaded). They are contained within the rightangled triangle, created by $h'$ axis, $\bar{h}'$ axis and $h'+\bar{h}'=\Delta$ line. Here $\Delta=12$.
• Figure 3: Operator spectrum of chiral Monster CFT tensored with its antichiral avatar on $(h',\bar{h}')$ plane: each vertex in the lattice represents the presence of operators. One can always circumscribe a square of unit length by a circle of radius $\frac{1}{\sqrt{2}}$. Thus any circle centered at $(h,\bar{h})$ and of radius $\frac{1}{\sqrt{2}}+\epsilon_g$ with $\epsilon_g>0$ would contain at least an operator. This is consistent with our result that asymptotically any circle of radius $\frac{\gamma r}{\sqrt{2}}+\epsilon_g$ with $\gamma\geq1$ will contain at least one operator where $\gamma$ is defined as the fourth root of asymptotic ratio of $\text{max}(h,\bar{h})$ and $\text{min}(h,\bar{h})$. We show $r=\frac{4\sqrt{3}}{\pi}>1$ and suspect that $r$ can be made to $1$. Here we have shown three such examples of containment. For each choice, two circles with different radius have been drawn to show that a circle of $\frac{\gamma}{\sqrt{2}}+\epsilon_g$ will contain at least one operator, as long as $\epsilon_g>0$. On the diagonal we have $\gamma=1$. The red dots denote the contained operators. The black circle shows that one can not ensure containment with circle of radius less than $\frac{1}{\sqrt{2}}$, showing optimality along the $h=\bar{h}$ curve if one can show $r=1$. We remark that as we have only shown that $r=\frac{4\sqrt{3}}{\pi}>1$ (the dashed black circle), we have not yet reached the optimal bound. The blue circle suggests that the optimal bound should be $\frac{1}{\sqrt{2}}$ even for the case where $h\neq\bar{h}$. Assuming a twist gap makes this bound insensitive to the line of approach towards infinity, but sensitive to the gap. See fig. \ref{['fig:monster2']}.
• Figure 4: Assuming twist gap $g$: operator spectrum of chiral Monster CFT tensored with its antichiral avatar on $(h',\bar{h}')$ plane: each vertex in the lattice represents the presence of operators. One can always circumscribe a square of unit length by a circle of radius $\frac{1}{\sqrt{2}}$. Thus any circle centered at $(h,\bar{h})$ and of radius $\frac{\kappa}{\sqrt{2}}+\epsilon_g$ with $\epsilon_g>0,\kappa\geq 1$ would contain at least an operator. This is consistent with our result that asymptotically any square of side length $\frac{4\sqrt{3}}{\pi}$, hence any circle of radius $\frac{\sigma r}{\sqrt{2}}+\epsilon_g$ will contain at least one operator, where $\sigma=\text{max}\left(\sqrt{\frac{c-1}{12g}},1\right)$ and $r=\frac{4\sqrt{3}}{\pi}>1$. Again we suspect that $r$ can be made to $1$. In the Monster example $g=4$, hence $\sigma=1$.
• Figure 5: We consider three kind of strips: the red vertical one is $H_2(\bar{h})$, the blue horizontal one is $H_1(h)$ and the black one is $H_3(\Delta)$. In each cases, we see that there is a minimum width of the strip such that the strips contain at least one operator. If we tensor chiral Monster CFT with its anti-chiral avatar, we can not make the width of the horizontal and vertical strip less than $1$. Thus the bound we find (which is $\sqrt{2}$) might not be optimal. A technical remark is that if it is not optimal, that would hint that one could have estimated the heavy part of the partition function in a better way.