Abstract
In this paper we investigate the holographic Rényi entropy of two disjoint intervals on complex plane with small cross ratio x for conformal field theory with W symmetry in the ground state, which could be dual to a higher spin AdS3 gravity. We focus on the cases of W 3 and W 4 symmetries. In order to see the nontrivial contributions from the W fields, we calculate the Rényi entropy in the expansion of x to order x 8 in both the gravity and the CFT sides. In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions not only from massless gravitons, but also from massless higher spin fields. In the CFT side we still use the operator product expansion of twist operators in the small interval limit, but now we need to consider the quasiprimary fields constructed from W fields, besides the ones from Virasoro Verma module. In the large central charge limit, we obtain the classical, 1-loop, 2-loop, and 3-loop parts of the Rényi entropy. The classical and 1-loop results in the gravity and the CFT sides are in exact match. This confirms the higher spin gravity/CFT correspondence, and also supports the holographic computation of Rényi entanglement entropy, including the quantum correction, in both the AdS3 gravity and the higher spin AdS3 gravity.
Highlights
The entanglement entropy and the Renyi entropy are related by SA = limn→1 SA(n)
In this paper we investigate the holographic Renyi entropy of two disjoint intervals on complex plane with small cross ratio x for conformal field theory with W symmetry in the ground state, which could be dual to a higher spin AdS3 gravity
In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions from massless gravitons, and from massless higher spin fields
Summary
We calculate the classical and 1-loop parts of the holographic Renyi entropy for two intervals with small cross ratio in CFT with W symmetry. When the gauge group is SL(4, R), it describe both the spin-4 and spin-3 fields interacting with the gravity, which is dual to a CFT with W (2, 3, 4) symmetry. One may obtain only the spin-4 field interacting with the gravity by choosing the gauge group to be SO(5) or Sp(4) [36] This truncated spin-4 gravity is conjectured to be dual to a CFT with W (2, 4) symmetry. In all these cases, the dual CFT has the same central charge as the one for pure AdS3 gravity, so that all the higher spin fields could be set to vanish without spoiling the underlying correspondence. All the classical gravitational configurations in [30] are still the classical solutions of higher spin gravity and their bulk classical actions would not be changed by the presence of higher spin fields
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