Supersymmetric localization for BPS black hole entropy: 1-loop partition function from vector multiplets

We use the techniques of supersymmetric localization to compute the BPS black hole entropy in N = 2 supergravity. We focus on the n(nu) + 1 vector multiplets on the black hole near horizon background which is AdS(2) x S-2 space. We find the localizing saddle point of the vector multiplets by solving the localization equations, and compute the exact one-loop partition function on the saddle point. Furthermore, we propose the appropriate functional integration measure. Through this measure, the one-loop determinant is written in terms of the radius of the physical metric, which depends on the localizing saddle point value of the vector multiplets. The result for the one-loop determinant is consistent with the logarithmic corrections to the BPS black hole entropy from vector multiplets.