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Microscopic Black Hole Entropy in Theories with Higher Derivatives

We discuss higher derivative corrections to black hole entropy in theories that allow a near horizon AdS_3 x X geometry. In arbitrary theories with diffeomorphism invariance we show how to obtain the spacetime central charge in a simple way. Black hole entropy then follows from the Euclidean partition function, and we show that this gives agreement with Wald's formula. In string theory there are certain diffeomorphism anomalies that we exploit. We thereby reproduce some recent computations of corrected entropy formulas, and extend them to the nonextremal, nonsupersymetric context. Examples include black holes in M-theory on K3 x T^2, whose entropy reproduces that of the perturbative heterotic string with both right and left movers excited and angular momentum included. Our anomaly based approach also sheds light on why exact results have been obtained in four dimensions while ignoring R^4 type corrections.