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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$_Λ$) 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$_Λ$ with identification upon uplifting three dimensions. A massive propagating scalar mode $δF$ 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, $δF$ 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 $δF_{\rm ml}$ and its logarithmic partner $δF^{\rm 4th}_{\rm log}$.