Localization on AdS2 × S1

Abstract

Conformal symmetry relates the metric on AdS2 × S1 to that of S3. This implies that under a suitable choice of boundary conditions for fields on AdS2 the partition function of conformal field theories on these spaces must agree which makes AdS2 × S1 a good testing ground to study localization on non-compact spaces. We study supersymmetry on AdS2 × S1 and determine the localizing Lagrangian for N = 2 supersymmetric Chern-Simons theory on AdS2 × S1. We evaluate the partition function of N = 2 supersymmetric Chern-Simons theory on AdS2 × S1 using localization, where the radius of S1 is q times that of AdS2. With boundary conditions on AdS2 × S1 which ensure that all the physical fields are normalizable and lie in the space of square integrable wave functions in AdS2, the result for the partition function precisely agrees with that of the theory on the q-fold covering of S3.