From Chern-Simons to Tomonaga-Luttinger

TL;DR

This paper demonstrates how a boundary in the 3D Chern-Simons model naturally leads to a 2D edge theory that embodies the bosonization of Weyl fermions and aligns with the Tomonaga-Luttinger model, revealing deep connections between topological field theories and condensed matter physics.

Contribution

It establishes a unique boundary condition in the Chern-Simons model that encodes bosonization and edge state dynamics, linking topological field theory to Luttinger liquid behavior.

Findings

Boundary condition enforces chirality and bosonization.

Derived boundary symmetry determines effective 2D action.

Boundary equations match Tomonaga-Luttinger theory.

Abstract

A single-sided boundary is introduced in the three-dimensional Chern-Simons model. It is shown that only one boundary condition for the gauge fields is possible, which plays the twofold role of chirality condition and bosonization rule for the two-dimensional Weyl fermion describing the degrees of freedom of the edge states of the Fractional Quantum Hall Effect. It is derived the symmetry on the boundary which determines the effective two dimensional action, whose equation of motion coincides with the continuity equation of the Tomonaga-Luttinger theory. The role of Lorentz symmetry and of discrete symmetries on the boundary is also discussed.