Topological non-linear $\sigma$-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems

TL;DR

This paper constructs exactly soluble topological non-linear sigma models to realize and classify all 3+1D bosonic topological orders, including those with emergent fermions, using higher gauge theory and homotopy groups.

Contribution

It introduces a framework for realizing all 3+1D bosonic topological orders via topological non-linear sigma models based on specific homotopy groups, extending previous models.

Findings

Realizes all 3+1D bosonic topological orders without fermions using sigma models with finite G.

Constructs models with emergent fermions using specific homotopy groups, including Z2 in pi_2.

Identifies a subset corresponding to 2-gauge theories for classifying topological orders with fermions.

Abstract

A discrete non-linear $\sigma$-model is obtained by triangulate both the space-time $M^{d+1}$ and the target space $K$. If the path integral is given by the sum of all the complex homomorphisms $\phi: M^{d+1} \to K$, with an partition function that is independent of space-time triangulation, then the corresponding non-linear $\sigma$-model will be called topological non-linear $\sigma$-model which is exactly soluble. Those exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects (i.e. fluctuations described by homomorphisms $\phi$) usually produce a topologically ordered state and are topological phase transitions, while phase transitions induced by fluctuations with all the topological defects give rise to trivial product states and are not topological phase transitions. If $K$ is a space with only non-trivial first homotopy group $G$ which is finite, those topological non-linear $\sigma$-models can realize all 3+1D bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group $\pi_1(K)=G$. Here, we show that the 3+1D bosonic topological orders with emergent fermions can be realized by topological non-linear $\sigma$-models with $\pi_1(K)=$ finite groups, $\pi_2(K)=Z_2$, and $\pi_{n>2}(K)=0$. A subset of those topological non-linear $\sigma$-models corresponds to 2-gauge theories, which realize and classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections. The classification of 3+1D bosonic topological orders may correspond to a classification of unitary fully dualizable fully extended topological quantum field theories in 4-dimensions.