Localization on $AdS_2\times S^1$ (1609.07443v1)
Abstract: Conformal symmetry relates the metric on $AdS_2 \times S{1}$ to that of $S3$. 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 localization on non-compact spaces. We study supersymmetry on $AdS_2\times S1$ and determine the localizing Lagrangian for ${\cal N}=2$ supersymmetric Chern-Simons theory on $AdS_2\times S1$. We evaluate the partition function of ${\cal N}=2$ supersymmetric Chern-Simons theory on $AdS_2 \times S1$ using localization, where the radius of $S1$ is $q$ times that of $AdS_2$. With boundary conditions on $AdS_2\times S1$ 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 $S3$.