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In the constructions of symmetry-protected topological (SPT) states, we usually decorate lower-dimensional states to higher codimensional domain walls of the system. In this talk, we will argue that domain wall decorations are basically equivalent to spectral sequences in algebraic topology. I will first illustrate this idea in bosonic systems, with explicit formulas for all differentials on all pages in the Lyndon-Hochschild-Serre spectral sequence. These results are useful in bosonic systems with Lieb-Schultz-Mattis (LSM) theorems, SPT-LSM theorems, and symmetry-enriched gauge theories. The second part of the talk will focus on fermionic SPT states. Using domain wall decorations, we will give a systematic construction and classification of fermionic SPT states in 3+1 or lower dimensions. We can obtain the full classifications for arbitrary finite unitary Abelian symmetries and interacting 10-fold way. All the classifications are consistent with known results from other approaches such as point/loop braiding statistics and spin cobordisms.

Zoom: https://harvard.zoom.us/j/977347126