Higher Spin Entanglement and W_N Conformal Blocks

TL;DR

This paper develops techniques to compute entanglement and correlation functions in 2D W_N conformal field theories and their higher spin gravity duals, clarifying when and why these quantities agree across the duality.

Contribution

It introduces a bulk Wilson line approach to compute vacuum conformal blocks and entanglement entropy in higher spin AdS$_3$/CFT$_2$ duality, strengthening the theoretical framework.

Findings

Vacuum conformal blocks are computed by bulk Wilson lines in higher spin backgrounds.

The Wilson line method confirms previous proposals for higher spin entanglement entropy.

Alternative Toda theory computations yield consistent results with the Wilson line approach.

Abstract

Two-dimensional conformal field theories with extended $\cal{W}$-symmetry algebras have dual descriptions in terms of weakly coupled higher spin gravity in AdS$_3\,$ at large central charge. Observables that can be computed and compared in the two descriptions include R\'enyi and entanglement entropies, and correlation functions of local operators. We develop techniques for computing these, in a manner that sheds light on when and why one can expect agreement between such quantities on each side of the duality. We set up the computation of excited state R\'enyi entropies in the bulk in terms of Chern-Simons connections, and show how this directly parallels the CFT computation of correlation functions. More generally, we consider the vacuum conformal block for general operators with $\Delta \sim c\,$. When two of the operators obey ${\Delta \over c} \ll 1\,$, we show by explicit computation that the vacuum conformal block is computed by a bulk Wilson line probing an asymptotically AdS$_3$ background with higher spin fields excited, the latter emerging as the effective bulk description of the excited state produced by the heavy operators. Among other things, this puts a previous proposal for computing higher spin entanglement entropy via Wilson lines on firmer footing, and clarifies its relation to CFT. We also study the corresponding computation in Toda theory and find that this provides yet another independent way to arrive at the same result.