T-DUALITY AND WORLD-SHEET SUPERSYMMETRY

Four-dimensional string backgrounds with local realizations of N = 4 w orld-sheer supersymmetry have, in the presence of a rotational Killing symmetry, only one complex structure which is an SO(2) singlet, while the other two form an SO(2) doublet. Although N = 2 world-sheet super symmetry is always preserved under Abelian T-duality transformations, N = 4 breaks down to N = 2 in the rotational case. A non-local realiza tion of N = 4 supersymmetry emerges, instead, with world-sheet parafer mions. For SO(3)-invariant metrics of purely rotational type, like the Taub-NUT and the Atiyah-Hitchin metrics, none of the locally realized extended world-sheet supersymmetries can be preserved under non-Abeli an duality.