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We use the techniques of supersymmetric localization to compute the BPS blackhole entropy in N=2 supergravity. We focus on the n_v+1 vector multiplets onthe black hole near horizon background which is AdS_2 x S^2 space. We find thelocalizing saddle point of the vector multiplets by solving the localizationequations, and compute the exact one loop partition function on the saddlepoint. Furthermore, we propose the appropriate functional integration measure.Through this measure, the one loop determinant is written in terms of theradius of the physical metric, which depends on the localizing saddle pointvalue of the vector multiplets. The result for the one loop determinant isconsistent with the logarithmic corrections to the BPS black hole entropy fromvector multiplets.

Supersymmetric Localization for BPS Black Hole Entropy: 1-loop Partition Function from Vector Multiplets