Holographic R\'enyi Entropy of Single Interval on Torus: with W symmetry

In this paper, we extend our study of the holographic R\'enyi entropy of single interval on a torus to the CFT with W symmetry. To read the possible corrections from W fields clearly, we compute the thermal correction to the R\'enyi entropy up to the order $e^{-8\pi /TL}$. In the field theory, this requires us to compute the contribution from all the level 4 states, from both the vacuum module and W primaries. On the gravity side, after computing the eigenvalue of single Schottky generators to the order $e^{-4 \pi/TL}$, we obtain the 1-loop quantum correction to the holographic R\'enyi entropy. We find complete agreement between the bulk and boundary theories.