Remarks on Boundaries, Anomalies, and Noninvertible Symmetries

TL;DR

This work develops a systematic framework for understanding how boundary conditions in 1+1d and 2+1d quantum systems transform under non-invertible global symmetries, distinguishing weakly symmetric boundaries (defect lines ending on boundaries) from strongly symmetric boundaries (boundary states eigenunder defect actions). It shows that these notions coincide for invertible symmetries but can bifurcate for non-invertible fusion-category symmetries, linking their existence to anomalies and generalized gauging via algebra objects. The authors provide concrete 1+1d and 2+1d examples (Ising, Fibonacci, Rep$(S_3)$, Rep$(H_8)$, Maxwell, toric code, and Chern-Simons theories) and derive implications for bulk and boundary renormalization group flows, including cases where non-anomalous symmetries still forbid trivially gapped symmetric boundaries. They extend the discussion to higher dimensions, introducing TQFT-valued boundary coefficients and partially Dirichlet boundaries arising from non-invertible condensation defects, illustrating a broad and versatile structure for boundaries and symmetries in QFT and TQFT settings.

Abstract

What does it mean for a boundary condition to be symmetric with respect to a non-invertible global symmetry? We discuss two possible definitions in 1+1d. 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 non-invertible symmetries. We discuss the relation to anomalies, where we observe that it is sometimes possible to gauge a non-invertible 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 non-invertible condensation defects on the boundaries of $U(1)$ gauge theory and several TQFTs. Starting from the Dirichlet boundary of free Maxwell theory, the non-invertible symmetries generate infinitely many boundary conditions that are neither Dirichlet nor Neumann.

Figures (31)

• Figure 1: We call a boundary condition strongly symmetric if the corresponding boundary state is an eigenstate under the action of the symmetry, as shown in the figure on the left (closed string channel). Here $\langle {\cal L}\rangle\geq1$ is the quantum dimension of the topological line. On the other hand, we call a boundary condition weakly symmetric if the topological defect lines can topologically end on the boundary, as shown in the figure on the right (open string channel).
• Figure 2: Left: A topological junction operator $v$ in $\mathrm{Hom}_{\mathcal{C}}(\mathcal{L}_i\otimes\mathcal{L}_j,\mathcal{L}_k)$. Right: By a conformal map, it is mapped to a dimension-zero state in the $S^1$ Hilbert space twisted by $\mathcal{L}_i$, $\mathcal{L}_j$, and $\mathcal{L}_k$.
• Figure 3: Left: A topological junction operator $\tilde{v}$ in $\mathrm{Hom}_{\mathcal{M}}(\mathcal{L}_i\otimes \mathcal{B}_a,\mathcal{B}_b)$. Right: By a conformal transformation, it is mapped to a dimension zero state in the interval Hilbert space with the boundary conditions $\mathcal{B}_a$ and $\mathcal{B}_b$ imposed at the two ends, and twisted by the line $\mathcal{L}_i$.
• Figure 4: A topological line ${\cal L}_i$ ends topologically on a weakly symmetric boundary ${\cal B}_a$, giving a conserved operator that commutes with the Hamiltonian on an interval.
• Figure 5: Fusion of two topological lines $\mathcal{L}_i$ and $\mathcal{L}_j$ on the interval with boundary conditions $\mathcal{B}_a$ at both ends depends on the junction vectors $v$, $\bar{u}$, $v'$, and $\bar{u}'$. It is written as a linear combination of topological lines $\mathcal{L}_k$ on the interval with junction vectors $v"$ and $\bar{u}"$ with, possibly, non-integer coefficients $C_{i,v;~j,v';~\bar{u}"}^{\bar{u};~ \bar{u}';~ k,v"}$.