Quantum gravity is a lot like bookkeeping: every possible trajectory is an entry in the ledger, and to tally the accounts you have to sum them all up. But what about a black hole that doesn't sit still — one that spins, and in imaginary time at that?
The authors found special "saddle points"—field configurations that give the dominant contribution to the sum over histories. These points describe a black string in ten dimensions with three types of charges (think of them as monopole charges Q1, Q5, and Qn) and an angular momentum JL. The trick is that the angular velocity ΩR is purely imaginary and locked to the inverse temperature β: ΩR = -2πi/β. This isn't a gimmick; it's a way to insert the operator (-1)^F into the path integral, which distinguishes particles from their superpartners.
The Gibbons-Hawking action for these solutions is independent of temperature. It exactly reproduces the supersymmetric index of the extremal black string with the same charges. The receipt stays the same no matter how long the purchase takes.
As you approach the event horizon at finite temperature, the geometry simplifies to a familiar construction: a three-sphere ($S^3$) fibered over BTZ space — a black hole in a three-dimensional anti-de Sitter universe. The parameters of this construction (the modular parameters) are finite, and the action again yields Cardy's formula — the famous result connecting black hole entropy to the number of microstates in the dual two-dimensional conformal field theory.
So an imaginary angular velocity becomes a tool that closes the loop between gravity and quantum field theory. The ledger balances.
The supersymmetric index is invariant under continuous deformations of the theory, so you can compute it in a convenient limit (say, weak coupling) and then carry the result to strong coupling, where gravity dominates.