I am working on exact results in black hole-related physics. In particular, I would like to apply methods from holography to compute the partition fun

The goal is to perform a test of AdS_3/CFT2 by computing the elliptic genus on the supergravity side and comparing the result to a calculation involving Jacobi forms on the CFT side. On the gravity side, the elliptic genus is a path integral over all field configurations, which can be drastically simplified using the method of supersymmetric localization.
This method involves first finding so-called localizing solutions, i.e. off-shell field configurations of the theory that obey the BPS conditions. These field configurations are said to parametrize the localizing manifold . In our case, we are looking at a 5d Euclidean supergravity theory coupled to vector multiplets. We are using the off-shell superconformal formalism in order to find BPS configurations living in the near horizon of the 5d black string. The geometry in this region is AdS3 x S2, and we have been able to find such localizing solution in both the vector and Weyl multiplets.
The second step of the localization procedure is to compute the path integral, which at this stage consists of evaluating the renormalized supergravity action on these localizing solutions, and to compute the one-loop over the field configurations orthogonal to the localizing manifold. The renormalized action itself must be fully determined, which involves adding to a bulk piece a series of boundary terms. These boudary terms fix infinite volume divergences and/or render the variational principle on the fields well-defined.
Finally, the supergravity calculation can be compared to our elliptic genus expansion term of interest.