Exotic invertible phases with higher-group symmetries

We investigate a family of invertible phases matter with higher-dimensional exotic excitations in even spacetime dimensions, which includes and generalizes the Kitaev’s chain 1+1d. The excitation has \mathbb{Z}_2 ?2 higher-form symmetry that mixes Lorentz to form higher group symmetry. focus on loop topological phase 3+1d. This is protected by one-form time-reversal symmetry, surface thermal Hall conductance not realized conventional symmetric ordinary bosonic systems without local fermion particles loops. describe UV realization order using SO(3)_- display="inline">SO(3)? gauge theory unit discrete theta parameter, enjoys same two-group discuss several applications including analogue ``fermionization’’ for theories non-anomalous internal