Non-invertible defects on the worldsheet

Abstract

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 $\text{O}(d;\mathbb{R}) \times \text{O}(d;\mathbb{R})$, and on momentum and winding charges as elements of $\text{O}(d,d;\mathbb{R})$. 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 $\text{U}(1)$ symmetries, and elements of the $\text{O}(d,d;\mathbb{Z})$ 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 $\text{O}(d,d;\mathbb{Q})$ 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.