3 years ago

# A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions.

Tian Lan, Xiao-Gang Wen

We call a topological order of 3+1-dimensional bosonic systems an all-boson (AB) topological order if all emergent point-like excitations are bosons. It was shown that AB topological orders, $\mathcal C^4_{AB}$, are classified by unitary pointed fusion 2-categories with only trivial 1-morphisms. In fact, AB topological orders can all be realized by Dijkgraaf-Witten gauge theories. In this paper, we consider emergent-fermion (EF) topological orders for 3+1D bosonic systems where some emergent point-like excitations are fermions. We show that all the EF topological orders, $\mathcal C^4_{EF}$, are classified by 2+1D anomalous topological orders $\mathcal A_b^3$ on the canonical boundary of the 3+1D topological order. The anomalous topological orders $\mathcal A_b^3$ is described by a unitary pointed 2-fusion category with only one invertible fermionic 1-morphism for each object. $\mathcal A_b^3$ sometimes also has quantum-dimension-$\sqrt 2$ 1-morphisms (which correspond to Majorana-zero-modes). By studying the mathematical structures of $\mathcal A^3_b$, we show that the EF topological orders $\mathcal C^4_{EF}$ are classified by 3+1D fermionic symmetry protected topological (SPT) orders with a finite unitary symmetry group after gauging all the symmetry group. We further propose that the general classification of 3+1D topological orders with symmetries can be obtained by gauging or partially gauging the finite symmetry group of 3+1D SPT phases of bosonic and fermionic systems.

Publisher URL: http://arxiv.org/abs/1801.08530

DOI: arXiv:1801.08530v1

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