Anomalies of discrete symmetries and Symmetry Protected Topological Phases

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There is a close connection between Symmetry Protected Topological Phases and anomalies: a surface of an SPT phase typically has a global symmetry with a nonvanishing 't Hooft anomaly which is canceled by the anomaly inflow from the bulk. This observation together with the known results about the classification of SPT phases suggest that anomalies are much more ubiquitous than thought previously and do not require chiral fermions We elucidate the physical mechanism of anomalies and give examples of bosonic theories with 't Hooft anomalies in various dimensions.