Twisting and localization in supergravity: equivariant cohomology of BPS black holes

Abstract

We develop the formalism of supersymmetric localization in supergravity using the deformed BRST algebra defined in the presence of a supersymmetric background as recently formulated in arxiv:1806.03690. The gravitational functional integral localizes onto the cohomology of a global supercharge $Q_{\text{eq}}$, obeying $Q_{\text{eq}}^2=H$, where $H$ is a global symmetry of the background. Our construction naturally produces a twisted version of supergravity whenever supersymmetry can be realized off-shell. We present the details of the twisted graviton multiplet and ghost fields for the superconformal formulation of four-dimensional N=2 supergravity. As an application of our formalism, we systematize the computation of the exact quantum entropy of supersymmetric black holes. In particular, we compute the one-loop determinant of the $Q_{\text{eq}}\mathcal{V}$ deformation operator for the off-shell fluctuations of the Weyl multiplet around the $Ad{S}_2\times S^2$ saddle. This result, which is consistent with the corresponding large-charge on-shell analysis, is needed to complete the first-principles computation of the quantum entropy.

V2: subsection 4.3 added, typo corrected, accepted version in JHEP; V3: typos corrected; V4: minor typos corrected

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• One-loop determinants for black holes in 4d gauged supergravity
• Duality and Transport for Supersymmetric Graphene from the Hemisphere Partition Function
• Dyonic black hole degeneracies in $\mathcal{N}=4$ string theory from Dabholkar-Harvey degeneracies
• Boundary conditions and localization on AdS: Part 2 General analysis
• Quantum entropy of BMPV black holes and the topological M-theory conjecture
• Localization on $Ad{S}_3\times S^2$ I: the 4d/5d connection in off-shell Euclidean supergravity
• The gravitational path integral for $N=4$ BPS black holes from black hole microstate counting
• An approach to BPS black hole microstate counting in an N=2 STU model
• Boundary Conditions in Topological AdS${}_4$/CFT${}_3$
• Supersymmetric localization: $\mathcal{N}=(2,2)$ theories on S${}^2$ and AdS${}_2$
• Supersymmetric localisation of $\mathcal{N}=(2,2)$ theories on a spindle
• On the exact entropy of $\mathcal{N}=2$ black holes