Nonpropagation of massive mode on AdS2 in topologically massive gravity

Making use of Achucarro-Ortiz (AO) type of dimensional reduction, we study the topologically massive gravity with a negative cosmological constant on AdS(2) spacetimes. For a constant dilaton, this two-dimensional model also admits three AdS(2) vacuum solutions, which are related to two AdS(3) and warped AdS(3) backgrounds with an identification upon uplifting three dimensions. We carry out the perturbation analysis around these backgrounds to find what is a physically propagating field. However, it turns out that there is no propagating massive mode on AdS(2) background, in contrast to the Kaluza-Klein (KK) type of dimensional reduction. We note that two dimensionally reduced actions are different and thus, the non-equivalence of their on-shell amplitudes is obtained.