Topological quantum field theory for Abelian topological phases and loop braiding statistics in $(3+1)$-dimensions

Topological quantum field theory (TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In $2+1d$, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasiparticle braiding statistics, chiral central charge, and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in $3+1d$ and it has been shown that looplike excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and looplike excitation in $3+1d$. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.

Figure 1

Three-loop braiding process.