Non-Abelian BF theory for 2+1 dimensional topological states of matter

TL;DR

This paper analyzes 2+1 dimensional BF models, both Abelian and non-Abelian, revealing boundary algebra structures and symmetry constraints that influence topological states of matter.

Contribution

It provides a detailed field theoretical analysis of non-Abelian BF models with boundary, highlighting the role of symmetries in boundary current structures.

Findings

Boundary Kač–Moody algebras with opposite chiralities

Time reversal symmetry constrains boundary physics in non-Abelian case

Presence of counter-propagating chiral currents on the boundary

Abstract

We present a field theoretical analysis of the 2+1 dimensional BF model with boundary in the Abelian and the non-Abelian case based on the Symanzik's separability condition. In both cases on the edges we obtain Ka\v{c}--Moody algebras with opposite chiralities reflecting the time reversal invariance of the theory. While the Abelian case presents an apparent arbitrariness in the value of the central charge, the physics on the boundary of the non-Abelian theory is completely determined by time reversal and gauge symmetry. The discussion of the non-Abelian BF model shows that time reversal symmetry on the boundary implies the existence of counter-propagating chiral currents.