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 [1]. The gravitational functional integral localizes onto the cohomology of a global supercharge Qeq, obeying Q 2eq = 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 \( \mathcal{N} \) = 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 Qeq\( \mathcal{V} \) deformation operator for the off-shell fluctuations of the Weyl multiplet around the AdS2 × S2 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.
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Jeon, I., Murthy, S. Twisting and localization in supergravity: equivariant cohomology of BPS black holes. J. High Energ. Phys. 2019, 140 (2019). https://doi.org/10.1007/JHEP03(2019)140
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DOI: https://doi.org/10.1007/JHEP03(2019)140