We will discuss "From 3d duality to 2d duality" https://arxiv.org/abs/1710.00926 by Aharony, Razamat and Willett.

Conformal symmetry relates the metric on AdS_2 x 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 \times S^1 a good testing ground to study supersymmetric localization on non-compact spaces. We evaluate the partition function of N=2 supersymmetric Chern-Simons theory on AdS_2 x S^1 using localization, where the radius of S^1 is q times that of AdS_2. With boundary conditions on AdS_2 x S^1 which ensure that all the physical fields are normalizable and lie in the space of square integrable wave functions in AdS_2, we find that the result for the partition function precisely agrees with that of the theory on the q-fold covering of S^3.

AdS2-CFT1 correspondence leads to a prescription for computing the degeneracy of a single centered extremal black hole in terms of path integral over string fields living on the near horizon geometry of the extremal black hole. This prescription is called quantum entropy function. In four dimension the near horizon isometry group SL(2,R) x SU(2) for BPS black hole gets enhanced to full SU(1,1,2). Using these enhanced supersymmetries and localization techniques, we will argue that the path integral receives contribution only from special class of field configurations which are invariant under a particular subgroup of SU(1,1,2). We will identify saddle points which are invariant under this subgroup.