Localization on AdS$_{2} \times$ S$^{1}$

Conformal symmetry relates the metric on AdS$_{2}$ × S$^{1}$ to that of S$^{3}$. This implies that under a suitable choice of boundary conditions for fields on AdS$_{2}$ the partition function of conformal field theories on these spaces must agree which makes AdS$_{2}$ × S$^{1}$ a good testing ground to study localization on non-compact spaces. We study supersymmetry on AdS$_{2}$ × S$^{1}$ and determine the localizing Lagrangian for $ \mathcal{N} $ = 2 supersymmetric Chern-Simons theory on AdS$_{2}$ × S$^{1}$. We evaluate the partition function of $ \mathcal{N} $ = 2 supersymmetric Chern-Simons theory on AdS$_{2}$ × S$^{1}$ using localization, where the radius of S$^{1}$ is q times that of AdS$_{2}$. With boundary conditions on AdS$_{2}$ × S$^{1}$ which ensure that all the physical fields are normalizable and lie in the space of square integrable wave functions in AdS$_{2}$, the result for the partition function precisely agrees with that of the theory on the q-fold covering of S$^{3}$.