Critical gravity on AdS2 spacetimes

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We study the critical gravity in two dimensional AdS (AdS₂) spacetimes, which was obtained from the cosmological topologically massive gravity (TMGΛ) in three dimensions by using the Kaluza-Klein dimensional reduction. We perform the perturbation analysis around AdS₂, which may correspond to the near-horizon geometry of the extremal BTZ black hole obtained from the TMGΛ 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 ml and its logarithmic partner δF 4th log.