Bound on asymptotics of magnitude of three point coefficients in 2D CFT

TL;DR

This work establishes rigorous lower bounds on the typical magnitude of heavy-light-heavy three-point coefficients in 2D CFTs by adapting complex Tauberian techniques to modular-invariant torus data. It decomposes the spectrum into light and heavy sectors, uses band-limited test functions and HKS-type bounds, and derives universal bounds that reproduce Kraus–Maloney-type exponential suppression in finite central charge under a sparseness condition with $ abla_ ext{χ}<c/12$. The authors further generalize to allow power-law growth, extend the analysis to large central charge with a refined sparseness criterion, and verify the bounds numerically in tensored CFTs including Ising models, providing insight into ETH-like behavior in holographic contexts. They also show that when $ abla_ ext{χ}>rac{c}{12}$, the KM exponential suppression can fail, yielding either sign flips in coefficients or polynomial growth, with implications for the structure of OPE data in composite theories.

Abstract

We use methods inspired from complex Tauberian theorems to make progress in understanding the asymptotic behavior of the magnitude of heavy-light-heavy three point coefficients rigorously. The conditions and the precise sense of averaging, which can lead to exponential suppression of such coefficients are investigated. We derive various bounds for the typical average value of the magnitude of heavy-light-heavy three point coefficients and verify them numerically.

Figures (11)

• Figure 1: $a_\Delta$, the $q$ expansion coefficient of the torus one point function of $\epsilon$ as a function of $\Delta$.
• Figure 2: $q$ expansion coefficient of the torus one point function of $\otimes^{12}\epsilon=\eta^{24}$ and the fact that $|b_N|$ is bounded by $N^{6}$, denoted by the black and blue line.
• Figure 3: Ratio of the weighted three point coefficient and the bound for $4$ copies of Ising model with $c_{eff}=2$, verifying the lower bound. The operator $\mathcal{O}=\epsilon\times\mathbb{I}\times\mathbb{I}\times\mathbb{I}$. We have divided the weighted three point coefficients by the bin width $2\delta$ and we have taken the order one number appearing in the bound to be $0.5$. Technically $0.5$ can only be chosen for $\delta>\frac{1}{\gamma}=2$, for low values of $\delta$, the order one number is less than $0.5$. Nonetheless, if the bound is satisfied with $0.5$, it will be so with any number less than $0.5$.
• Figure 4: Ratio of the weighted three point coefficients and the bound for $2$ copies of Ising model tensored with one copy of Monster such that $c_{eff}=13$. The plot being greater than one verifyies the lower bound. Here we have divided the weighted three point coefficients by the bin width $2\delta$ and we have taken the order one number appearing in the bound to be $0.5$. Technically $0.5$ can only be chosen for $\delta>\frac{1}{\gamma}\simeq 1.06$, for low values of $\delta$, the order one number is less than $0.5$. Nonetheless, if the bound is satisfied with $0.5$, it will be so with any number less than $0.5$.
• Figure 5: The $q$ expansion coefficient as a function of $\Delta$ is bounded by $\Delta^{\Delta_{\mathcal{O}}/2}$, the black lines are $\pm\Delta^{\Delta_{\mathcal{O}}/2}$.