Boundary conditions and localization on AdS: Part 2 General analysis

We develop the method of Green's function to evaluate the one loop determinants that arise in localization of supersymmetric field theories on $AdS$ spaces. The theories we study have at least ${\cal N}=2$ supersymmetry and normalisable boundary conditions are consistent with supersymmetry. We then show that under general assumptions the variation of the one loop determinant with respect to the localizing background reduces to a total derivative. Therefore it receives contributions only from the origin of $AdS$ and from asymptotic infinity. From expanding both the Greens function and the quadratic operators at the origin of $AdS$ and asymptotic infinity, we show that the variation of the one loop determinant is proportional to an integer. Furthermore, we show that this integer is an index of a first order differential operator. We demonstrate that these assumptions are valid for Chern-Simons theories coupled to chiral multiplets on $AdS_2\times S^1$. Finally we use our results to show that $U(N_c)$ Chern-Simons theory at level $k$ coupled to $N_f$ chiral multiplets and $N_f$ anti-chiral multiplets in the fundamental obeys level-rank duality on $AdS_2\times S^1$.