The boundary of a topological insulator (more generally a symmetry protected topological phase) cannot exist as itself without the bulk, namely the boundary must be "anomalous" in certain way. In this talk we will discuss two different subjects related to boundary anomalies: (1) We construct a series of bosonic SPT states beyond the group cohomology classification, all these states involve gravitational anomalies at its boundary. Both the classification and field theory description of these states are studied. (2) We discuss strongly interacting fermionic SPT states, and we will demonstrate that our understanding of interacting fermionic SPT states can help us achieve a full regularization of the Standard Model (or a Grand Unified Theory) as a lattice model.
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