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

TL;DR

The paper applies supersymmetric localization to compute the exact quantum entropy contribution of $n_v+1$ abelian vector multiplets for an extremal BPS black hole in ${\cal N}=2$ supergravity, with near-horizon geometry $AdS_2 \times S^2$. Localization yields saddle points labeled by two real parameters per vector multiplet, and the authors compute the one-loop determinant around these saddles via an equivariant index, first with a naive measure and then with a scale-invariant measure that uses the physical radius $\ell_{P}$. The key result is that the vector multiplet contribution to the quantum entropy encodes a universal logarithmic correction $-\frac{n_v+1}{6} \log \ell_{P}$, consistent with known microscopic results, once the proper measure is used. This provides a concrete microscopic-entropy check for the quantum entropy function program and clarifies how to incorporate measure factors to obtain background-independent, physical results. The work lays the groundwork for including Weyl multiplets and hypermultiplets to complete the exact quantum entropy for ${\cal N}=2$ (and higher) supergravities.

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.