These edge modes have played a key role in understanding various condensed-matter properties, such as the quantum Hall effect [5, 6].

Our list of references is not complete; this is partly due to our limited knowledge and partly due to space constraints.

See also [13, 14, 15] for the attempts of explaining the entropy of BTZ black hole in terms of the would-be gauge mode.

Remark that the above procedure breaks the manifest 2D Lorentz symmetry in the intermediate step, but it is also possible to devise an improved procedure which preserve Lorentz symmetry manifestly [8, 50].

We used the group decomposition (16) in order to simplify the expressions (17) where we set ${{\tilde{f}}^{\pm }}_1=\mp e^{-r}{\mathcal{L}}_{\pm }$ by the $D_{\pm }$ Stueckelberg symmetry. This choice of $\tilde{A}$ corresponds to the background metric ${\tilde{g}}_{\mu \nu }$ given by $\mathrm{d}{s}^2=\mathrm{d}{r}^2-(e^r\mathrm{d}{x}^{+}+e^{-r}{\mathcal{L}}_{-}\mathrm{d}{x}^{-})(e^r\mathrm{d}{x}^{-}+e^{-r}{\mathcal{L}}_{+}\mathrm{d}{x}^{+})$ with $x^{\pm }=\frac{1}{2}(x^0\pm x^1)$.