Duality beyond the first loop

In this article we give a calculation of the two-loop $\sigma$-model corrections to the $T$-duality map in string theory. We use the effective action approach, and analyze two-loop corrections in a specific subtraction scheme. Focusing on backgrounds which have a single Abelian isometry, we find the explicit form for the $O\left({\alpha }^{\prime }\right)$ modifications of the lowest order duality transformations. Rather surprisingly, the manifest two-loop duality depends crucially on the torsion field. In contrast with 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 via dimensional reduction. Our results support the interpretation of $T$ duality as an expansion in the inverse string tension ${\alpha }^{\prime }$, and its order-by-order realization as a manifest symmetry of the full string theory.