Universal Dynamics of Heavy Operators in CFT$_2$

TL;DR

<3-5 sentence high-level summary>The paper establishes a universal, finite-c>1 formula for the averaged squared OPE coefficients of heavy operators in unitary compact 2D CFTs, expressing the result as a product of universal building blocks C_0 determined solely by the central charge. This universality is derived by recasting crossing symmetry and modular invariance in terms of Moore–Seiberg crossing kernels (fusion and modular S), enabling the authors to bound or compute OPE asymptotics in three heavy regimes (heavy-light-light, heavy-heavy-light, and heavy-heavy-heavy) without explicit conformal blocks. They connect the universal blocks to Liouville theory via the DOZZ structure constants and discuss holographic interpretations at large c, including BTZ black-hole physics and the Schwarzian limit, thereby linking CFT data to semiclassical gravity and ETH in 2D. The framework provides a unified, kernel-based bootstrap that both reproduces known results in special limits and extends to finite c, with clear implications for chaos, integrability, and universal heavy-state dynamics in CFT2.

Abstract

We obtain an asymptotic formula for the average value of the operator product expansion coefficients of any unitary, compact two dimensional CFT with $c>1$. This formula is valid when one or more of the operators has large dimension or -- in the presence of a twist gap -- has large spin. Our formula is universal in the sense that it depends only on the central charge and not on any other details of the theory. This result unifies all previous asymptotic formulas for CFT$_2$ structure constants, including those derived from crossing symmetry of four point functions, modular covariance of torus correlation functions, and higher genus modular invariance. We determine this formula at finite central charge by deriving crossing kernels for higher genus crossing equations, which give analytic control over the structure constants even in the absence of exact knowledge of the conformal blocks. The higher genus modular kernels are obtained by sewing together the elementary kernels for four-point crossing and modular transforms of torus one-point functions. Our asymptotic formula is related to the DOZZ formula for the structure constants of Liouville theory, and makes precise the sense in which Liouville theory governs the universal dynamics of heavy operators in any CFT. The large central charge limit provides a link with 3D gravity, where the averaging over heavy states corresponds to a coarse-graining over black hole microstates in holographic theories. Our formula also provides an improved understanding of the Eigenstate Thermalization Hypothesis (ETH) in CFT$_2$, and suggests that ETH can be generalized to other kinematic regimes in two dimensional CFTs.

Figures (8)

• Figure 1: The elementary crossing transformations: sphere four-point crossing between $S$ and $T$ channels, and torus one-point crossing between the $\tau$ and $(-1/\tau)$ frames.
• Figure 2: Example of a crossing transformation on the torus two-point function.
• Figure 3: Example of a crossing transformation on the $g=2$ partition function.
• Figure 4: A conformal block decomposition of the torus two-point function $G_{1,2}$, where the kinematic parameters $\sigma$ consist of a complex structure $\tau$ for the torus, and a separation $w$ between operators. We sum over representations in the internal cuffs; for the yellow cuff $i_1$, this corresponds to the operators appearing in the OPE of external operators $e_1,e_2$, and for the blue cuff $i_2$, an insertion of a complete set of states in the thermal trace. $G_{1,2}(\tau,w,\bar{\tau},\bar{w}) = \sum_{i_1}\sum_{i_2} C_{e_1e_2i_1}C_{i_1i_2i_2} \mathcal{F}[P_{e_1},P_{e_2}](P_{i_1},P_{i_2}|w,\tau) \bar{\mathcal{F}}[\bar{P}_{e_1},\bar{P}_{e_2}](\bar{P}_{i_1},\bar{P}_{i_2}|\bar{w},\bar{\tau})$The OPE coefficients $C_{e_1e_2i_1}$, $C_{i_1i_2i_2}$ are associated with the pairs of pants labelled $A,B$ respectively, with $\partial A=(e_1,e_2,i_1)$ and $\partial B=(i_1,i_2,i_2)$.
• Figure 5: The elementary crossing moves relate different pair-of-pants decompositions of the four-punctured sphere and the once-punctured torus, or more generally anywhere that these appear as pieces of any decomposition of a surface. The associated crossing kernels relate Virasoro conformal blocks in the corresponding channels. The fusion kernel (top) relates sphere four-point Virasoro blocks in the S- and T-channels, and the modular kernel (middle) relates torus one-point blocks in modular $S$-transformed frames. In the final line, we show an example relating two channels in the torus two-point function $G_{1,2}$ by composing these moves.