Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in $(3+1)$-Dimensions

TL;DR

This work develops a dynamical, diffeomorphism-invariant TQFT framework to study Abelian topological phases in $(3+1)$ dimensions, where both point-like particles and loop-like excitations exist. By coupling 1-form and 2-form gauge fields and imposing generalized gauge transformations, the authors derive and classify topological terms—notably $A\wedge A\wedge dA$ and $B\wedge B$—and extract explicit particle and loop braiding statistics via canonical quantization and membrane operators. The paper computes three-loop braiding invariants $e^{i\Theta_{IJ,K}}$ and related half-braiding phases, establishing quantization conditions and showing how these invariants depend on coefficients $M_{IJK}$ and $K_{IJ}$; it also highlights deconfinement and potential fermionic excitations in certain sectors. These results provide a systematic framework for understanding 3+1D Abelian topological orders and set the stage for extensions to non-Abelian cases and matter couplings, with potential applications to topological phase transitions and gapped boundary phenomena.

Abstract

Topological qauntum field theory(TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In $2+1$D, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasi-particle 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 $3+1$D and it has been shown that loop like 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 loop like excitation in $3+1$D. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.

Figures (1)

• Figure 1: Three-loop braiding process.