  We study the role of boundary conditions on the one loop partition function
of ${\cal N}=2$ chiral multiplet of R-charge $\Delta$ on $AdS_2\times S^1$. The
chiral multiplet is coupled to a background vector multiplet which preserves
supersymmetry. We implement normalizable boundary conditions in $AdS_2$ and
develop the Green's function method to obtain the one loop determinant. We
evaluate the one loop determinant for two different actions: the standard
action and the $Q$-exact deformed positive definite action used for
localization. We show that if there exists an integer $n$ in the interval $D: (
\frac{\Delta-1}{2L}, \frac{\Delta}{2L} )$, where $L$ being the ratio of radius
of $AdS_2$ to that of $S^1$, then the one loop determinants obtained for the
two actions differ. It is in this situation that fields which obey normalizable
boundary conditions do not obey supersymmetric boundary conditions. However if
there are no integers in $D$, then fields which obey normalizable boundary
conditions also obey supersymmetric boundary conditions and the one loop
determinants of the two actions precisely agree. We also show that it is only
in the latter situation that the one loop determinant obtained by evaluating
the index of the $D_{10}$ operator associated with the localizing action agrees
with the one loop determinant obtained using Green's function method.
