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.
Highlights
Entanglement entropy has emerged as an interesting new observable in quantum field theory, yielding information that goes beyond that provided by correlation functions of local operators
To the extent that the Ryu-Takayanagi formula is an important clue to understanding the mechanism underlying holography, it is of great interest to generalize to theories that go beyond ordinary Einstein gravity
Recall that on the CFT side we are computing a contribution to a vacuum four-point function, or equivalently a two-point function evaluated in an excited state
Summary
Entanglement entropy has emerged as an interesting new observable in quantum field theory, yielding information that goes beyond that provided by correlation functions of local operators. In the W-symmetry case we face the additional difficulty that the non-vacuum blocks of the WN algebra are not as well understood as those of the Virasoro algebra, and it would be interest to further explore this issue in the future Barring this technical assumption, the validity of the Wilson line proposal for computing entanglement entropy in an excited state of the higher spin theory is on the same footing as the validity of the Ryu-Takayanagi formula for an excited state in ordinary gravity. We can replace the twist operators by general operators that carry both a scaling dimension and a higher spin charge, and think of computing the corresponding vacuum block This more general setup again can be formulated as a monodromy problem, both in Chern-Simons theory and in the CFT. The appendices collect useful formulas and conventions, and some complementary material
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