Duality between supercohomology fermionic SPT (symmetry-protected-topological) phases and higher-group bosonic SPT phases

The first part of this talk will introduce generalized Jordan–Wigner transformation on arbitrary triangulation of any simply connected manifold in arbitrary dimensions. This gives a duality between any fermionic systems and a new class of Z₂ lattice gauge theories. This map preserves the locality and has an explicit dependence on the second Stiefel–Whitney class and a choice of spin structure on the manifold. In the Euclidean picture, this mapping is exactly equivalent to adding topological terms, Steenrod square, to the spacetime action. The second part of the talk is the application of this boson-fermion duality on SPT phases. By the boson-fermion duality, we are able to show the equivalent between any supercohomology fermionic SPT and some higher-group bosonic SPT phase in arbitrary dimensions. Particularly in (3+1)D, we will show a unitary quantum circuit for any supercohomology fermionic SPT state with gapped boundary construction.