All solutions of the localization equations for N=2 quantum black hole entropy

TL;DR

This work provides a complete off-shell localization analysis for four-dimensional $\mathcal{N}=2$ supergravity coupled to $n_v$ vector multiplets, including fluctuations of the gravity multiplet and auxiliary fields under $AdS_{2}\times S^{2}$ attractor boundary conditions. Using conformal supergravity and a spinor-bilinear approach to the BPS equations $Q\psi=0$, the authors derive the full localization manifold: with vanishing auxiliary $SU(2)$ field strength the bosonic sector is governed by $n_v+1$ real moduli, while in the general case the manifold expands to include an $SU(2)$ triplet of one-forms and an extra scalar. Turning on gauge fields reveals that the $U(1)$ gauge field is localized away on the locus for vector multiplets, whereas the $SU(2)$ sector remains unconstrained and may entail hypermultiplet contributions for a complete microscopic match. These results extend prior vector-multiplet–only analyses and align the gravity-side localization with microscopic degeneracy calculations of supersymmetric black hole entropy.

Abstract

We find the most general bosonic solution to the localization equations describing the contributions to the quantum entropy of supersymmetric black holes in four-dimensional N=2 supergravity coupled to n_v vector multiplets. This requires the analysis of the BPS equations of the corresponding off-shell supergravity (including fluctuations of the auxiliary fields) with AdS2 \times S2 attractor boundary conditions. Our work completes and extends the results of arXiv:1012.0265 that were obtained for the vector multiplet sector, to include the fluctuations of all the fields of the off-shell supergravity. We find that, when the auxiliary SU(2) gauge field strength vanishes, the most general supersymmetric configuration preserving four supercharges is labelled by n_v+1 real parameters corresponding to the excitations of the conformal mode of the graviton and the scalars of the n_v vector multiplets. In the general case, the localization manifold is labelled by an additional SU(2) triplet of one-forms and a scalar function.