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We show that the linearization of all exact solutions of classical chiralgravity around the AdS3 vacuum have positive energy. Non-chiral andnegative-energy solutions of the linearized equations are infrared divergent atsecond order, and so are removed from the spectrum. In other words, chiralityis confined and the equations of motion have linearization instabilities. Weprove that the only stationary, axially symmetric solutions of chiral gravityare BTZ black holes, which have positive energy. It is further shown thatclassical log gravity-- the theory with logarithmically relaxed boundaryconditions --has finite asymptotic symmetry generators but is not chiral andhence may be dual at the quantum level to a logarithmic CFT. Moreover we showthat log gravity contains chiral gravity within it as a decoupled chargesuperselection sector. We normally evaluate the Euclidean sum over geometriesof chiral gravity and show that it gives precisely the holomorphic extremal CFTpartition function. The modular invariance and integrality of the expansioncoefficients of this partition function are consistent with the existence of anexact quantum theory of chiral gravity. We argue that the problem of quantizingchiral gravity is the holographic dual of the problem of constructing anextremal CFT, while quantizing log gravity is dual to the problem ofconstructing a logarithmic extremal CFT.

Chiral Gravity, Log Gravity and Extremal CFT