# Fermionic SPT phases in higher dimensions and bosonization

## TL;DR

This paper explores fermionic SPT phases across various dimensions, proposing a classification scheme that extends existing models and connects bosonic and fermionic phases through condensation processes.

## Contribution

It introduces a new classification scheme for 3D fermionic SPT phases with arbitrary symmetries, generalizing supercohomology and linking bosonic shadows to fermionic phases.

## Key findings

- Classification scheme for 3D fermionic SPT phases with arbitrary symmetry G.
- Connection between bosonic shadows and fermionic phases via condensation.
- Emphasis on the role of cohomological parameters and higher-form symmetries.

## Abstract

We discuss bosonization and Fermionic Short-Range-Entangled (FSRE) phases of matter in one, two, and three spatial dimensions, emphasizing the physical meaning of the cohomological parameters which label such phases and the connection with higher-form symmetries. We propose a classification scheme for fermionic SPT phases in three spatial dimensions with an arbitrary finite point symmetry G. It generalizes the supercohomology of Gu and Wen. We argue that the most general such phase can be obtained from a bosonic "shadow" by condensing both fermionic particles and strings.