Abstract
We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.
Highlights
Fusion categories, works for bosonic systems [2], and important steps were made in [32] for fermionic symmetry enriched topological (SET) orders
We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach
The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories
Summary
We give a classification of 1d anomaly-free SET orders with a finite onsite symmetry G using the idea of gauging the symmetry. Since topological excitations in a 1d anomaly-free SET order can be fused in 1d, they must form a unitary fusion 1-category (without braidings), denoted by A. It must contain all local excitations E as a full subcategory, i.e. a monoidal embedding ηA : E → A. Gauging the symmetry E in 1d amounts to adding more particles to A to form a unitary multi-fusion category M such that the monoidal functor ιM : A −→ ZE(M) induced by the embedding A → M is an equivalence of unitary fusion categories over E (recall [19, Definition 2.7]).
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