  Quantum field theory on rotating black hole spacetimes is plagued with
technical difficulties. Here, we describe a general method to renormalize and
compute the vacuum polarization of a quantum field in the Hartle-Hawking state
on rotating black holes. We exemplify the technique with a massive scalar field
on the warped AdS3 black hole solution to topologically massive gravity, a
deformation of (2+1)-dimensional Einstein gravity. We use a "quasi-Euclidean"
technique, which generalizes the Euclidean techniques used for static
spacetimes, and we subtract the divergences by matching to a sum over mode
solutions on Minkowski spacetime. This allows us, for the first time, to have a
general method to compute the renormalized vacuum polarization (and, more
importantly, the renormalized stress-energy tensor), for a given quantum state,
on a rotating black hole, such as the physically relevant case of the Kerr
black hole in four dimensions.
