4d $$ \mathcal{N} $$ = 2 supergravity observables from Nekrasov-like partition functions

Abstract

We reinterpret the OSV formula for the on-shell action/entropy function of asymptotically flat BPS black holes as a fixed point formula that is formally equivalent to a recent gluing proposal for asymptotically AdS4 black holes. This prompts a conjecture that the complete perturbative answer for the most general gravitational building block of 4d $$ \mathcal{N} $$ N = 2 supergravity at a single fixed point takes the form of a Nekrasov-like partition function with equivariant parameters related to the higher-derivative expansion of the prepotential. In turn this leads to a simple localization-like proposal for a set of supersymmetric partition functions in (UV completed) 4d $$ \mathcal{N} $$ N = 2 supergravity theories. The conjecture is shown to be in agreement with a number of available results for different BPS backgrounds with both Minkowski and AdS asymptotics. In particular, it follows that the OSV formula comes from the unrefined limit of the general expression including only the so-called 𝕎 tower of higher derivatives, while the on-shell action of pure (Euclidean) AdS4 with round S3 boundary comes from the NS limit that includes only the 𝕋 tower. Backgrounds preserving less supersymmetry, such as the under-rotating black holes in flat space, the holographic squashed S3, and the static/rotating twisted and non-twisted Kerr-Newman-like black holes in AdS4 lead to a more general refined version of the corresponding gravitational blocks as dictated by the supersymmetric gluing rules.

The dark (BPS) side of thermodynamics in Minkowski4

“…In particular, the sign between the gravitational blocks in the cubic prepotential is different than the one in the square root prepotential due to the extra factor of i between the two. This means the relative minus sign proposed for the gluing of asymptotically flat black holes in the cubic form [20] flips to a plus when applied here. The original OSV notation using Im is applicable in all cases but is less suggestive about the origin of the formula as made up of two independent contributions.…”

“…The rest of the paper requires more specific supesymmetry knowledge and interest -in section 4 we consider the general STU model, giving more details about the underlying superalgebra that ultimately defines the BPS limit. In section 4.1 we discuss the BPS thermodynamics for the black holes in the STU model, and in section 4.2 we discuss the relation of our results with the OSV formula and the more recent proposal of gravitational block gluing, [19,20]. We finish in section 5 with further discussion and open questions.…”

“…Eq. (4.15) above is in fact the well-known (two-derivartive) OSV formula [18], 20 proposed to capture the entropy of supersymmetric (and extremal) black holes in the mixed ensemble and playing an important role in microstate counting due to its relation with the topological string. We have thus physically derived the OSV formula via BPS thermodynamics in the same sense that [14,15] derived the entropy functions of [39][40][41].…”

“…A more recent development is the explicit interpretation of (4.15) as a fixed-point formula, [20]. More specifically, it was proposed that the OSV formula can be obtained by adding two gravitational blocks by a particular gluing rule, producing 21…”

“…21 Note a subtlety in the gluing rule proposals of [19,20] -the overall signs are explicitly dependent on the form of the prepotential. In particular, the sign between the gravitational blocks in the cubic prepotential is different than the one in the square root prepotential due to the extra factor of i between the two.…”