Notes from the bulk: metric dependence of the edge states of Chern-Simons theory

TL;DR

This paper investigates how the edge states of abelian Chern-Simons theory on a curved cylindrical background depend on the bulk metric, revealing a local chiral velocity influenced by the boundary's induced metric.

Contribution

It demonstrates that the chiral velocity of edge modes in Chern-Simons theory depends locally on the boundary metric, extending previous flat bulk results to curved backgrounds.

Findings

Ka-Moody algebra with constant central charge persists on curved boundaries.

The boundary theory remains topologically protected as a Luttinger liquid.

Edge mode velocity varies locally with the boundary's induced metric.

Abstract

The abelian Chern-Simons theory is considered on a cylindrical spacetime $\mathbb{R} \times D$, in a not necessarily flat Lorentzian background. As in the flat bulk case with planar boundary, we find that also on the radial boundary of a curved background a Ka\c{c}-Moody algebra exists, with the same central charge as in the flat case, which henceforth depends neither on the bulk metric nor on the geometry of the boundary. The holographically induced theory on the 2D boundary is topologically protected, in the sense that it describes a Luttinger liquid, no matter which the bulk metric is. The main result of this paper is that a remnant of the 3D bulk theory resides in the chiral velocity of the edge modes, which is not a constant like in the flat bulk case, but it is local, depending on the determinant of the induced metric on the boundary. This result may provide a theoretical framework for the recently observed accelerated chiral bosons on the edge of some Hall systems.