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Anomalous Symmetry Fractionalization and Surface Topological Order

In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways , leading to a variety of symmetry enriched topological (SET) phases. While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems. Instead, certain `anomalous' SETs can only occur on the surface of a 3D symmetry protected topological (SPT) phase. In this paper we describe a procedure for determining whether an SET of a discrete, onsite, unitary symmetry group $G$ is anomalous or not. The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes. Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions are captured by the fourth cohomology group $H^4( G, \,U(1))$, which also precisely labels the set of 3D SPT phases, with symmetry group $G$. We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3d bulk SPT whose surface termination realizes it. We illustrate this idea using the chiral spin liquid ($U(1)_2$) topological order with a reduced symmetry $\mathbb{Z}_2 \times \mathbb{Z}_2 \subset SO(3)$, which can act on the semion quasiparticle in an anomalous way. We construct exactly solved 3d SPT models realizing the anomalous surface terminations, and demonstrate that they are non-trivial by computing three loop braiding statistics. Possible extensions to anti-unitary symmetries are also discussed.