Complex BPS Black Holes in AdS$_3\times S^3$

Abstract: The correct gravitational representation of the supersymmetric index is a smooth complex solution, rather than the naïve Euclidean continuation of the Lorentizan BPS black hole. We construct such saddles for black holes with (\mathrm{AdS}_3\times S^3) boundary conditions in the STU model. We arrive at the same finite-temperature BTZ (\times S^3) geometries in two independent ways: from two-center four-dimensional BPS solutions with complex dipole charges, and from six-dimensional black strings after imposing the BPS relation among conserved charges. We analyze the resulting smoothness and periodicity conditions and show that they are precisely those required for globally well-defined supersymmetry and for the thermodynamic potentials of the supersymmetric index.