One-loop correction to the AdS/BCFT partition function in three-dimensional pure gravity

We calculate the tree-level partition function of a Euclidean Banados-Teitelboim-Zanelli (BTZ) black hole in the presence of end of the world branes (ETW branes) and the one-loop partition function of a Euclidean thermal AdS3 in the presence of an ETW brane. At the tree level, our results match with the previous ones for static BTZ black holes. Interestingly, at the one-loop level, our results contain novel terms, which reflect the existence of an ETW brane. The ETW brane has a consistent profile for a static thermal AdS and, in this case, we showed that the spectrum obtained from the one-loop partition function takes a physically sensible form.