The one-loop determinants for vector multiplets scale as (v1 chi_V)^{1/4} per multiplet, physical hypermultiplets give the inverse, and these combine into logarithmic corrections to AdS4 black hole entropy with an undetermined gravity-multiplet coefficient a0.
Supersymmetric Localization for BPS Black Hole Entropy: 1-loop Partition Function from Vector Multiplets
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abstract
We use the techniques of supersymmetric localization to compute the BPS black hole entropy in N=2 supergravity. We focus on the n_v+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.
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One-loop determinants for black holes in 4d gauged supergravity
The one-loop determinants for vector multiplets scale as (v1 chi_V)^{1/4} per multiplet, physical hypermultiplets give the inverse, and these combine into logarithmic corrections to AdS4 black hole entropy with an undetermined gravity-multiplet coefficient a0.