  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 $AdS_2\times S^1$.
