Bulk-boundary correspondence of (1+1)D symmetric gapped phases
We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category $\mathcal{C}$, an indecomposable semisimple right $\mathcal{C}$-module category $\mathcal{M}$, a Q-system $Q\in\mathcal{C}$ specifying the bulk phase, and a right $Q$-module $K\in\mathcal{M}_{Q}$, regarded as an object of $\mathcal{M}_{Q}^{\mathrm{op}}$, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor $\mathcal{M}_{Q}^{\mathrm{op}}\to\mathrm{BCond}$ is an equivalence, so simple boundary conditions are classified by simple objects of $\mathcal{M}_{Q}$ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to $(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}$, and the canonical action of the bulk DHR category on it agrees with the categorical action of $Z_1(\mathcal{C}^{\mathrm{rev}})$. Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of $(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}$ on $\mathcal{M}_{Q}^{\mathrm{op}}$. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.