A holographic view of topological stabilizer codes

We study boundaries of topological stabilizer codes and the constraints imposed on them by the emergent conservation laws that govern the bulk topological order. We show ---at the level of the boundary operator algebra without referring to a particular boundary Hamiltonian--- that these constraints forbid the boundary from being realized via a local tensor product Hilbert space. Furthermore, we demonstrate that the different ways in which the boundary Hilbert space fails to be a tensor product directly encode topological properties of the bulk. In particular, we find quantities of the boundary operator algebra that are directly related to the self and mutual statistics of bulk excitations. We demonstrate this explicitly in a variety of topological stabilizer codes, including Type I and II fracton codes.