Localization on AdS₂ × S¹

Abstract

Conformal symmetry relates the metric on AdS₂ × S¹ 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 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 S ₁ is q times that of AdS₂. 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 AdS₂, the result for the partition function precisely agrees with that of the theory on the q-fold covering of S ³.