Abstract
We continue the effort of defining and evaluating the quantum entropy function for supersymmetric black holes in 4d mathcal{N} = 2 gauged supergravity, initiated in [1]. The emphasis here is on the missing steps in the previous localization analysis, mainly dealing with one-loop determinants for abelian vector multiplets and hypermultiplets on the non-compact space ℍ2× Σg with particular boundary conditions. We use several different techniques to arrive at consistent results, which have a most direct bearing on the logarithmic correction terms to the Bekenstein-Hawking entropy of said black holes.
Highlights
Corrections to the Bekenstein-Hawking formula for the entropy of black holes
The emphasis here is on the missing steps in the previous localization analysis, mainly dealing with one-loop determinants for abelian vector multiplets and hypermultiplets on the non-compact space H2 × Σg with particular boundary conditions
We are instead interested in the quantum entropy function (QEF) [8] defined on the (Euclidean) near-horizon solution of the supersymmetric (BPS) black holes of interest, 2π dmacro(pI, qI ) := exp 4π qI
Summary
Since we are evaluating a path-integral, we need to first gauge-fix the action and introduce the corresponding ghost fields, as will be discussed in due course Another important observation is that the action of Q2 on the near-horizon spacetime has a fixed “point” on a codimension two submanifold, namely the full S2 (or in general Σg) sitting at the centre of H2. The localized path-integral remains infinite-dimensional, but due to the fact that the classical action was found to depend only on φ+, we could already write the resulting quantum entropy function in the following suggestive form, dmacro(pI , qI ) = This form allowed us to show that the saddle-point approximation, at φI+ = 2X+I where X+I are the attractor values of the scalar fields, agrees with the expected classical entropy function Scl[pI , qI , φI+] used for the leading-order holographic entropy matching [2, 28]. We will briefly comment on it at the end of this paper
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