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

我们的目标是通过计算超重力侧的椭圆亏格并将其结果与CFT侧的Jacobi形式的计算结果进行比较来对AdS_3/CFT 2进行测试。在重力场方面，椭圆亏格是所有场位形上的路径积分，可以用超对称局部化方法大大简化，这种方法首先要找到所谓的局部化解，即满足BPS条件的理论的离壳场位形。这些领域的配置被认为是参数化的局部化流形。在我们的例子中，我们看到的是一个与矢量多重态耦合的5d欧几里德超引力理论。我们使用离壳超共形的形式主义，以找到生活在近视界的5D黑弦的BPS配置。在这个区域的几何是AdS 3 x S2，我们已经能够找到这样的本地化解决方案的矢量和Weyl multiplets.本地化过程的第二步是计算路径积分，这在这个阶段包括评估重整化超引力作用在这些本地化解决方案，并计算一个循环的场配置正交于本地化流形.重整化作用量本身必须被完全确定，这涉及到给块体增加一系列边界项。这些bouquet项固定无限体积发散和/或使变分原理的领域well-defined。最后，超引力计算可以比较我们感兴趣的椭圆亏格展开项。

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.