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Solvable lattice model for (2+1)D bosonic topological insulator

We construct an exactly sovable commuting projector Hamiltonian for (2+1)D bosonic topological insulator which is one of symmetry-protected topological (SPT) phases protected by U(1) and time-reversal $\mathbb{Z}_2^T$ symmetry, where the symmetry group is U(1)$\rtimes\mathbb{Z}_2^T$. The model construction is based on the decorated domain-wall interpretation of the $E_{\infty}$-page of a spectral sequence of a cobordism group that classifies the SPT phases in question. We demonstrate nontriviality of the model by showing an emergence of a Kramers doublet when the system is put on a semi-infinite cylinder $(-\infty,0]\times S^1$ with an inserted $π$-flux. The surface anomaly manifests itself as a non-onsite representation of the U(1)$\rtimes\mathbb{Z}_2^T$ symmetry. Anomaly matching on a boundary is discussed within a simple boundary theory.