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In this article we give a calculation of the two-loop $\sigma$-modelcorrections to the T-duality map in string theory. We use the effective actionapproach, and analyze two-loop corrections in a specific subtraction scheme.Focusing on backgrounds which have a single Abelian isometry, we find theexplicit form for the $O(\alpha')$ modifications of the lowest order dualitytransformations. Rather surprisingly, the manifest two-loop duality dependscrucially on the torsion field. In contrast to the dilaton and metric fields,which are merely passive spectators, the torsion plays a more active role,because of the anomalous couplings to the gauge fields that arise viadimensional reduction. Our results support the interpretation of T-duality asan expansion in the inverse string tension $\alpha'$, and its order-by-orderrealization as a manifest symmetry of the full string theory.Comment: 26 pages, latex, no figures. Several typos corrected, and references  to Nucl. Phys. B405 (1993) 109 by E. Kiritsis and Nucl. Phys. B411 (1994) 487  by A. Giveon and E. Kiritsis adde

Duality beyond the first loop