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

Understand

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.