Critical gravity on AdS2 spacetimes

We study the critical gravity in two dimensional AdS (AdS$_2$) spacetimes, which was obtained from the cosmological topologically massive gravity (TMG$_\Lambda$) in three dimensions by using the Kaluza-Klein dimensional reduction. We perform the perturbation analysis around AdS$_2$, which may correspond to the near-horizon geometry of the extremal BTZ black hole obtained from the TMG$_\Lambda$ with identification upon uplifting three dimensions. A massive propagating scalar mode $\delta F$ satisfies the second-order differential equation away from the critical point of $K=l$, whose solution is given by the Bessel functions. On the other hand, $\delta F$ satisfies the fourth-order equation at the critical point. We exactly solve the fourth-order equation, and compare it with the log-gravity in two dimensions. Consequently, the critical gravity in two dimensions could not be described by a massless scalar $\delta F_{\rm ml}$ and its logarithmic partner $\delta F^{\rm 4th}_{\rm log}$.