Non-invertible defects on the worldsheet

We consider codimension-one defects in the theory of d compact scalars on a two-dimensional worldsheet, acting linearly by mixing the scalars and their duals. By requiring that the defects are topological, we find that they correspond to a non-Abelian zero-form symmetry acting on the fields as elements of O (d;R) O (d;R), and on momentum and winding charges as elements of O (d, d;R). When the latter action is rational, we prove that it can be realized by combining gauging of non-anomalous discrete subgroups of the momentum and winding U (1) symmetries, and elements of the O (d, d;Z) duality group, such that the couplings of the theory are left invariant. Generically, these defects map local operators into non-genuine operators attached to lines, thus corresponding to a non-invertible symmetry. We confirm our results within a Lagrangian description of the non-invertible topological defects associated to the O (d, d;Q) action on charges, giving a natural explanation of the rationality conditions. Finally, we apply our findings to toroidal compactifications of bosonic string theory. In the simplest non-trivial case, we discuss the selection rules of these non-invertible symmetries, verifying explicitly that they are satisfied on a worldsheet of higher genus.