Entanglement Entropy, OTOC and Bootstrap in 2D CFTs from Regge and Light Cone Limits of Multi-point Conformal Block

TL;DR

<3-5 sentence high-level summary>The paper develops a fusion-matrix and monodromy framework to study light-cone and Regge limits of n-point Virasoro blocks in c>1 pure CFTs, enabling analytic control of 1/c corrections and applications to entanglement and chaos. By deriving explicit light-cone and Regge singularities and connecting them to bulk AdS3 physics via deficit-angle interpretations, it obtains universal late-time logarithmic growth of the nth Renyi entropy after local quenches and exponential late-time decay of OTOCs with distinct regimes determined by external dimensions. The Liouville CFT is analyzed within the same formalism, revealing a late-time constant OTOC and richer phase structure, while RCFTs and large-c holographic limits yield Cardy-type or BH-type entropies; the results illuminate deep connections between Virasoro blocks, bootstrap data, and bulk gravity. Together, these results provide analytic tools for the conformal bootstrap and hint at broader implications for AdS3/CFT2 holography and information dynamics in quantum gravity.

Abstract

We explore the structures of light cone and Regge limit singularities of $n$-point Virasoro conformal blocks in $c>1$ two-dimensional conformal field theories with no chiral primaries, using fusion matrix approach. These CFTs include not only holographic CFTs dual to classical gravity, but also their full quantum corrections, since this approach allows us to explore full $1/c$ corrections. As the important applications, we study time dependence of Renyi entropy after a local quench and out-of-time ordered correlator (OTOC) at late time. We first show that, the $n$-th ($n>2$) Renyi entropy after a local quench in our CFT grows logarithmically at late time, for any $c$ and any conformal dimensions of excited primary. In particular, we find that this behavior is independent of $c$, contrary to the expectation that the finite $c$ correction fixes the late time Renyi entropy to be constant. We also show that the constant part of the late time Renyi entropy is given by a monodromy matrix. We also investigate OTOCs by using the monodromy matrix. We first rewrite the monodromy matrix in terms of fusion matrix explicitly. By this expression, we find that the OTOC decays exponentially in time, and the decay rates are divided into three patterns, depending on the dimensions of external operators. We note that our result is valid for any $c>1$ and any external operator dimensions. Our monodromy matrix approach can be generalized to the Liouville theory and we show that the Liouville OTOC approaches constant in the late time regime. We emphasize that, there is a number of other applications of the fusion and the monodromy matrix approaches, such as solving the conformal bootstrap equation. Therefore, it is tempting to believe that the fusion and monodromy matrix approaches provide a key to understanding the AdS/CFT correspondence.

Figures (12)

• Figure 1: We probe the growth of entanglement between $A$ and $A^C$ after a local quench in this setup.
• Figure 2: This figure shows the growth of the Renyi entanglement entropy compared to the vacuum, ${\Delta} S^{(n)}_A(t)$. (Left) From the numerical calculations in Kusuki2018b, we shed light on the $n$ and $h_O$ dependence of ${\Delta} S^{(n)}_A(t)$ in the purple, green and red regions. However, we do not have any information in the white region. (Right) Our new findings in this paper. Here we introduce the Liouville notations; $h_a={\alpha}(Q-{\alpha})$ and $c=1+6Q^2$.
• Figure 3: The monodromy matrix is given by three steps; (1) fusion transformation, (2) picking up a trivial phase factor, and (3) fusion transformation.
• Figure 4: The coefficients $c_n$ for small $n$ can be calculated numerically. On the other hand, the coefficients for large $n$ can be analytically evaluated by using the fusion or the monodromy matrix approaches.
• Figure 5: The simple interpretation of the light cone limit and Regge limit singularities. The left upper figure shows the trivial OPE singularity $z^{h_p-2h_A}$. The upper right figure shows the light cone singularity (\ref{['eq:LC']}), which predicts that the interaction between $O_A$ and $O_B$ is characterized by the linear combination of their deficit angles $\{\phi_A, \phi_B\}$, or equivalently, by the fusion rule of the Liouville CFT (\ref{['eq:fusion rule']}). The lower figure shows the Regge limit singularity (\ref{['eq:Reggesing']}).