Century Rain by Alastair Reynolds
Auger, an obsessed and driven archaeologist from the future is recruited into a top-secret project. Humanity now lives in a “habitat ring” in orbit around frozen Earth that has been devastated by nanotechnology plagues, and her regular job is studying the ruins of Paris.
The top-secret project leads her through an artificial wormhole to a “copy” of our Earth, taken at the beginning of World War Two, and preserved in an artificial containment sphere. The Earth-copy’s history immediately started diverging from our timeline, and in that world the Second World War never happened.
Auger is sent into that version of Paris where she meets a likable guy named Floyd, a private investigator working on a murder case. Of course neither he nor anybody else in his world are aware that their surroundings and they themselves are copies of originals that have flowed on in time without them.
The plot is further complicated by a war back in the future between two separate groups of humanity over old Earth. Strange and compelling.