Higher Spin Entanglement and W_N Conformal Blocks

TL;DR

This work develops a coherent framework for computing Rényi and entanglement entropies and correlation functions in 2d ${\cal W}_N$ CFTs via their AdS$_3$ higher-spin duals. It unifies conformal-block monodromy methods in the CFT with bulk Chern-Simons holonomies, and demonstrates that vacuum ${\cal W}_N$ blocks are captured efficiently by bulk Wilson lines, with perturbative checks for ${N=2}$ and ${N=3}$. A parallel Toda-theory perspective is presented, linking bulk CS data to semiclassical Toda correlators and clarifying the emergence of the vacuum block in this regime. The results strengthen the holographic higher-spin entanglement entropy proposals and illuminate the connections between microstate geometries, Wilson lines, and boundary conformal blocks, with extensions to multiple intervals and genus explored.

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 Rényi 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 Rényi 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 $Δ\sim c\,$. When two of the operators obey ${Δ\over c} \ll 1\,$, 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.