A bstract We study superconformal interfaces between $ mathcal{N}=left( {1,1} ight) $ supersymmetric sigma models on tori, which preserve a $ widehat{u}{(1)^{2d }} $ current algebra. Their fusion is non-singular and, using parallel transport on CFT deformation space, it can be reduced to fusion of defect lines in a single torus model. We show that the latter is described by a semi-group extension of $ Oleft( {d,dleft| mathbb{Q} ight.} ight) $ ), and that (on the level of Ramond charges) fusion of interfaces agrees with composition of associated geometric integral transformations. This generalizes the well-known fact that T-duality can be geometrically represented by Fourier-Mukai transformations. Interestingly, we find that the topological interfaces between torus models form the same semi-group upon fusion. We argue that this semi-group of orbifold equivalences can be regarded as the α ′ deformation of the continuous O ( d , d ) symmetry of classical supergravity.
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