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We calculate the tree-level partition function of Euclidean BTZ black holewith the end of the world branes (ETW branes) for arbitrary tension and theone-loop partition function of Euclidean thermal $AdS_{3}$ in the presence of atensionless ETW brane with the Neumann boundary condition. At the tree level,our results match with the previous ones in non-rotating cases, but at theone-loop level, our results contain unexpected terms, which contain thefluctuations of the ghost fields. However particularly in the case where thechemical potential is zero, we get a similar form as the original result inAdS/CFT case, but the exponent is slightly changed. In this case the ghostcontribution looks going away and the partition function becomes physical in asense that the coefficients of the expansion in powers of $q$ are positive,which means that the number of states at the level is always positive.

One-Loop Correction to the AdS/BCFT Partition Function in the Three Dimensional Pure Gravity