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

目标是通过计算超重力侧的椭圆属，并将结果与CFT侧涉及Jacobi形式的计算结果进行比较，来执行AdS_3/CFT2的测试。在引力方面，椭圆属是所有场构型上的路径积分，可以使用超对称局域化方法大大简化。该方法首先需要找到所谓的局部解，即符合BPS条件的理论的壳外油田配置。这些场的配置被称为参数化的局部流形。在我们的例子中，我们正在研究一个5d欧几里得超重力理论与向量多重态相结合。我们使用脱壳超共形理论来寻找存在于5d黑弦近视界的BPS构型。该区域的几何形状为AdS3 x S2，我们已经能够在向量和Weyl多胞胎中找到这样的定位解。定位过程的第二步是计算路径积分，这一阶段包括对这些定位解的重整化超重力作用进行评估，并计算与定位流形正交的场构型上的单环。重新规范化的动作本身必须完全确定，这涉及到向大块添加一系列边界项。这些边界项固定了无限的体积散度和/或在定义良好的场上呈现变分原理。最后，超重力计算可以与我们感兴趣的椭圆属展开项进行比较。

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.