Three dimensional Einstein gravity with negative cosmological constant -1/\ell^2 deformed by a gravitational Chern-Simons action with coefficient 1/\mu is studied in an asymptotically AdS_3 spacetime. It is argued to violate unitary or positivity for generic \mu due to negative-energy massive gravitons. However at the critical value \mu\ell=1, the massive gravitons disappear and BTZ black holes all have mass and angular momentum related by \ell M=J. The corresponding chiral quantum theory of gravity is conjectured to exist and be dual to a purely right-moving boundary CFT with central charges (c_L,c_R)=(0,3\ell /G).

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We show that specializations of the 4d =2 superconformal index labeled by an integer N is given by TrN where  is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch. We provide evidence that the states enumerated by these limits of the index lead to a family of 2d chiral algebras N. This generalizes the recent results for the N=−1 case which corresponds to the Schur limit of the superconformal index. We show that this specialization of the index leads to the same integrand as that of the elliptic genus of compactification of the superconformal theory on S2×T2 where we turn on 12N units of U(1)r flux on S2.

A decisive question in nonparametric smoothing techniques is the choice of the bandwidth or smoothing parameter. The present paper addresses this question when using local polynomial approximations for estimating the regression function and its derivatives. A fully-automatic bandwidth selection procedure has been proposed by Fan and Gijbels (1995a), and the empirical performance of it was tested in detail via a variety of examples. Those experiences supported the methodology towards a great extend. In this paper we establish asymptotic results for the proposed variable bandwidth selector. We provide the rate of convergence of the bandwidth estimate, and obtain the asymptotic distribution of its error relative to the theoretical optimal variable bandwidth. These asymptotic properties give extra support to the proposed bandwidth selection procedure. It is also demonstrated how the proposed selection