As far as I'm aware, there are no quantum gravity theories that deal directly with closed timelike curves. Some of them (like canonical quantum gravity, causal dynamical triangulation and loop quantum gravity) forbid them outright, others merely seem to not discuss the topic. I've found quite a variety of QFT behaviour in classical spacetimes with closed timelike curves, including string theory in a CTC background, but I can't really think of any paper where the metric wavefunction (or sum of metric histories or whatever else) might run over acausal states.
The obvious candidate for this would be one of the variant of path integrals like Euclidian gravity, Lorentzian gravity, Regge calculus, etc. But there seem to always be this assumption that the boundary conditions (if present) will always be on spacelike hypersurfaces, which, while it does not make it impossible to have closed timelike curves on such spaces, certainly restricts their numbers (I suspect that CTCs in such a case only arises from the topology and not the metric).
String theory might also work out, as I am not aware of any theorems forbidding the strings to reduce to CTC solutions in the classical limit, but I do not know that much about string theory unfortunately.
Are there any papers discussing such topics? Are they even possible in the context of any of those theories as we currently understand them?
Best Answer
Exotic objects known as "negative branes" can be constructed (at least perturbatively) within string theory. Once they are introduced, many exotic aspects of timelike compactification, closed timelike curves or even "emergent timelike directions" can be studied. The relevant paper is Negative Branes, Supergroups and the Signature of Spacetime. For an overview see this talk.