I am going to talk about how to derive an exact partition function of supersymmetric field theories on a supersymmetry-preserving quotient of Euclidean AdS_3 by localization. This is one of first examples about localization on non-compact spaces. I will describe the situation by comparing it to a situation in compact spaces. Then, I will explain how to perform a localization procedure on the manifold, and finally, I will discuss the features of the partition function we derived.

We study the free energy of the three dimensional N=2 supersymmetric Chern-Simons-Matter theory dual to AdS4 times toric seven dimensional Sasaki-Einstein manifold. Although it is not known in general how to construct the CSM theory from the geometric data of the 7d SE manifold (which is called toric diagram), the free energy in the large N limit may be derived. We present the free energy as a quartic polynomial of trial R-charges for generic toric diagram with up to 6 vertices, and some particular diagrams with 8 vertices. The free energy reproduces the Martelli-Sparks-Yau volume function.