Quantum entropy and exact 4d/5d connection

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We consider the AdS₂/CFT₁ holographic correspondence near the horizon of rotating five-dimensional black holes preserving four supersymmetries in N=2 supergravity. The bulk partition function is given by a functional integral over string fields in AdS₂ 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 AdS₂xS², reduces to a finite dimensional integral over nᵥ+1 parameters Cᵃ, where nᵥ is the number of vector multiplets of the theory while the C⁰ 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 AdS₂xS². 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 AdS₂x S². These results together with possible one-loop contributions can be used to establish an exact connection between five and four dimensional quantum entropies.