Light Cone Bootstrap in General 2D CFTs and Entanglement from Light Cone Singularity

TL;DR

This work advances the light cone bootstrap in general 2D CFTs with c>1 by deriving the explicit asymptotics of Virasoro blocks in the light-cone limit through the fusion matrix. It uncovers a universal total twist at large spin that is saturated at the BTZ threshold, signaling a universal binding energy and potential black hole formation in AdS3, and it connects these results to entanglement dynamics, revealing Renyi phase transitions and universal Regge behavior. The analysis spans both large and finite central charge, providing bounds and phase structures for Renyi entropies in various setups, including global and local quenches, and doubles CFTs. The findings illuminate how Liouville data and bulk AdS3 physics emerge from Virasoro block fusion structures, with implications for holography, thermalisation, and information scrambling in 2D CFTs.

Abstract

The light cone OPE limit provides a significant amount of information regarding the conformal field theory (CFT), like the high-low temperature limit of the partition function. We started with the light cone bootstrap in the {\it general} CFT ${}_2$ with $c>1$. For this purpose, we needed an explicit asymptotic form of the Virasoro conformal blocks in the limit $z \to 1$, which was unknown until now. In this study, we computed it in general by studying the pole structure of the {\it fusion matrix} (or the crossing kernel). Applying this result to the light cone bootstrap, we obtained the universal total twist (or equivalently, the universal binding energy) of two particles at a large angular momentum. In particular, we found that the total twist is saturated by the value $\frac{c-1}{12}$ if the total Liouville momentum exceeds beyond the {\it BTZ threshold}. This might be interpreted as a black hole formation in AdS${}_3$. As another application of our light cone singularity, we studied the dynamics of entanglement after a global quench and found a Renyi phase transition as the replica number was varied. We also investigated the dynamics of the 2nd Renyi entropy after a local quench. We also provide a universal form of the Regge limit of the Virasoro conformal blocks from the analysis of the light cone singularity. This Regge limit is related to the general $n$-th Renyi entropy after a local quench and out of time ordered correlators.

Figures (13)

• Figure 1: The left figure shows the $h_A=h_B$ dependence of the total twist in the OPE between $O_A$ and $O_B$, and the right figure shows the $h_B$ dependence with $h_A \ll h_B$. From both figures, we can find saturation above ${\alpha}_A+{\alpha}_B=\frac{Q}{2}$. In particular, for $h_A\ll h_B$, the transition (saturation) occurs at the BTZ mass threshold $\frac{c-1}{24}$. A similar phenomenon can be seen in the HHLL Virasoro blocks, known as thermalisation.
• Figure 2: This figure shows the implications of the light cone bootstrap on the nature of AdS. In AdS${}_{d\geq 4}$, the interactions between two objects become negligible at large angular momentum. On the other hand, in AdS${}_3$, there exists a universal binding energy $-4\bar{{\alpha}}_A\bar{{\alpha}}_B$ even at large angular momentum. If the total Liouville momentum is above the BTZ threshold ${\alpha}_{\text{BTZ}}$, the binding energy is given by $\frac{c-1}{12}-\tau_A-\tau_B$.
• Figure 3: Potential form in AdS$_3$ between two particles is similar to the form of Coulomb potential. Moreover, this form with Liouville momenta implies that the dynamics of the multiple deficit angle at large angular momentum is completely captured by Liouville CFT.
• Figure 4: Setup of mutual information $I(A,B)$ between two intervals $A$ and $B$ when $A$ is infinitely boosted.
• Figure 5: $c$ dependence of $n_*$. The classical limit $c \to \infty$ matches $n_*=2$.