Realizing anomalous anyonic symmetries at the surfaces of 3d gauge theories

Abstract

The hallmark of a 2 dimensional topologically ordered phase is the existence of deconfined `anyon' excitations that have exotic braiding and exchange statistics, different from those of ordinary bosons or fermions. As opposed to conventional Landau-Ginzburg-Wilson phases, which are classified on the basis of the spontaneous breaking of an underlying symmetry, topologically ordered phases, such as those occurring in the fractional quantum Hall effect, are absolutely stable, not requiring any such symmetry. Recently, though, it has been realized that symmetries, which may still be present in such systems, can give rise to a host of new, distinct, many-body phases, all of which share the same underlying topological order. A useful approach to classifying SETs is to determine all possible such symmetry actions on the anyons that are algebraically consistent with the anyons' statistics. Remarkably, however, there exist symmetry actions that, despite being algebraically consistent, cannot be realized in any physical system, and hence do not lead to valid 2d SETs. One class of such `anomalous' SETs, characterized by certain disallowed symmetry fractionalization patters, finds a physical interpretation as an allowed surface state of certain 3d short-range entangled phases, but another, characterized by some seemingly valid but anomalous permutation actions of the symmetry on the anyons, has so far eluded a physical interpretation. In this work, we find a physical realization for these anomalously permuting SETs as surface theories of certain 3d long-range entangled phases, completing our understanding of general anomalous SETs in 2 dimensions.