Conformal Gauge and Residual Symmetry

On a two-dimensional Lorentzian worldsheet patch where the auxiliary metric is written $h=e^{2\varphi }\eta$, with $\eta =\mathrm{diag}(-1,+1)$, classical conformal gauge is the local diffeomorphism choice that makes the metric conformally flat followed by the Weyl choice that sets $\varphi =0$. Locally, the two distinct null directions of a smooth Lorentzian metric give transverse one-dimensional foliations; coordinates $u,v$ constant along the opposite null directions make the metric have only a nonzero $dudv$ term. Reversing one coordinate if needed writes it as $h=-e^{2\varphi }dudv$, and $\tau =(u+v)\mathrm{/}2,\sigma =(u-v)\mathrm{/}2$ gives the displayed conformal form. Weyl rescaling removes $e^{2\varphi }$ on that patch.

Use light-cone coordinates ${\sigma }^{\pm }=\tau \pm \sigma$. An orientation- and time-orientation-preserving residual conformal transformation has the form ${\sigma }^{+}\mapsto f_{+}({\sigma }^{+})$, ${\sigma }^{-}\mapsto f_{-}({\sigma }^{-})$, where $f_{\pm }:\mathbb{R}\to \mathbb{R}$ are smooth increasing diffeomorphism lifts with positive derivatives. Conversely, each such pair preserves the conformal class because it pulls back $-d{\sigma }^{+}d{\sigma }^{-}$ to $-f_{+}^{\mathrm{\prime }}{f}_{-}^{\mathrm{\prime }}d{\sigma }^{+}d{\sigma }^{-}$.

For the closed cylinder, $\sigma \sim \sigma +2\pi$ means ${\sigma }^{+}\mapsto {\sigma }^{+}+2\pi$ and ${\sigma }^{-}\mapsto {\sigma }^{-}-2\pi$; a map descends to the cylinder exactly when both lifts satisfy $f_{\pm }(u+2\pi )=f_{\pm }(u)+2\pi$. The lift relation is illustrated by the quotient square:

Here $q(u)=[u]$, and the period condition is exactly what makes the square commute. For the open strip $\sigma \in [0,\pi ]$, preservation of the boundary $\sigma =0$ forces $f_{+}=f_{-}=f$; preservation of $\sigma =\pi$ then requires $f(u+2\pi )=f(u)+2\pi$. This definition is local and does not quotient conformal Killing stabilizers or global moduli, nor claim a global conformal-gauge slice.

Flat Bosonic String Model

Bosonic Polyakov Model