Two and three loop alpha corrections to T-duality: Kasner and Schwarzschild

Two and three loop alpha corrections are calculated for Kasner and Schwarzschild metrics, and their T-duals, in the bosonic string theory. These metrics are used to obtain the two and three loop alpha corrections to T-duality. It is noted in particular that the inclusion of alpha corrections and the requirement of consistency with the alpha-corrected T-duality for the Kasner and Schwarzschild metrics enables one to fix uniquely the covariant form of the T-duality rules at three loops. As a generalization of the T-dual of the Schwarzschild geometry a class of massless geometries is presented.