2026/07/17 by Silvia Georgescu, Sameer Murthy, Andrew Svesko
#hep-th
We present saddle-points of the Euclidean Gravitational Path Integral (GPI) corresponding to the supersymmetric index of the D1-D5-P black string. These saddles are complex, supersymmetric, non-extremal solutions of 10-dimensional IIB supergravity theory with arbitrary inverse temperature β. The solutions carry fixed monopole charges Q1, Q5, Qn, and angular momentum JL equal to the extremal supersymmetric black string. Crucially, the solutions have fixed angular velocity ΩR = - 2 πi /β, which is the chemical potential dual to JR. This implements the insertion of (-1)F in the GPI. The Gibbons-Hawking on-shell action is temperature-independent and agrees with the supersymmetric index of the extremal black string carrying the same monopole charges. Upon taking a finite-temperature near-horizon decoupling limit, we obtain solutions of the form S3 fibered over BTZ. Although these near-horizon solutions have finite holomorphic and anti-holomorphic modular parameters, their on-shell action reproduces the Cardy formula of the holomorphic elliptic genus of the dual D1-D5 SCFT2.