Critical gravity on AdS2 spacetimes

We study the critical gravity in two-dimensional anti-de Sitter (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 Banados, Teitelboim, and Zanelli (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-ml and its logarithmic partner delta F-log(4th).