Quantum entropy and exact 4D/5D connection

We consider the AdS(2)/CFT1 holographic correspondence near the horizon of rotating five-dimensional black holes preserving four supersyrnmetries in N = 2 supergravity. The bulk partition function is given by a functional integral over string fields in AdS(2) and is related to the quantum entropy via the Sen's proposal. Under certain assumptions we use the idea of equivariant localization to non-rigid backgrounds and show that the path integral of off-shell supergravity on the near horizon background, which is a circle fibration over AdS(2) x S-2, reduces to a finite dimensional integral over n(V) + 1 parameters C-A, where n(V) is the number of vector multiplets of the theory while the C-0 mode corresponds to a normalizable fluctuation of the metric. The localization solutions, which rely only on off-shell supersymmetry, become after a field redefinition, the solutions found for localization of supergravity on AdS(2) x S-2. We compute the renormalized action on the localization locus and show that, in the absence of higher derivative corrections, it agrees with the four dimensional counterpart computed on AdS(2) x S-2. These results together with possible one-loop contributions can be used to establish an exact connection between five and four dimensional quantum entropies.