  We consider the AdS_2/CFT_1 holographic correspondence near the horizon of
big four-dimensional black holes preserving four supersymmetries in toroidally
compactified Type-II string theory. The boundary partition function of CFT_1 is
given by the known quantum degeneracies of these black holes. The bulk
partition function is given by a functional integral over string fields in
AdS_2. Using recent results on localization we reduce the infinite-dimensional
functional integral to a finite number of ordinary integrals over a space of
localizing instantons. Under reasonable assumptions about the relevant terms in
the effective action, these integrals can be evaluated exactly to obtain a bulk
partition function. It precisely reproduces all terms in the exact Rademacher
expansion of the boundary partition function as nontrivial functions of charges
except for the Kloosterman sum which can in principle follow from an analysis
of phases in the background of orbifolded instantons. Our results can be
regarded as a step towards proving `exact holography' in that the bulk and
boundary partition functions computed independently agree for finite charges.
Since the bulk partition function defines the quantum entropy of the black
hole, our results enable the evaluation of perturbative as well as
nonperturbative quantum corrections to the Bekenstein-Hawking-Wald entropy of
these black holes.
