Remarks on boundaries, anomalies, and noninvertible symmetries

Abstract

What does it mean for a boundary condition to be symmetric with respect to a noninvertible global symmetry We discuss two possible definitions in 1+1D QFTs and lattice models. On the one hand, we call a boundary weakly symmetric if the symmetry defects can terminate topologically on it, leading to conserved operators for the Hamiltonian on an interval (in the open string channel). On the other hand, we call a boundary strongly symmetric if the corresponding boundary state is an eigenstate of the symmetry operators (in the closed string channel). These two notions of symmetric boundaries are equivalent for invertible symmetries, but bifurcate for noninvertible symmetries. We discuss the relation to anomalies, where we observe that it is sometimes possible to gauge a noninvertible symmetry in a generalized sense even though it is incompatible with a trivially gapped phase. The analysis of symmetric boundaries further leads to constraints on bulk and boundary renormalization group flows. In 2+1D, we study the action of noninvertible condensation defects on the boundaries of U(1) gauge theory and several TQFTs. Starting from the Dirichlet boundary of free Maxwell theory, the noninvertible symmetries generate infinitely many boundary conditions that are neither Dirichlet nor Neumann.