# Diffeomorphism & Weyl transformations in the 2D worldsheet of string theory and the existence of conformal gauge

D. Tong's notes on string theory, chapter 5 (PDF), feature the following in introducing the symmetries used in the Faddeev-Popov method:

We have two gauge symmetries: diffeomorphisms and Weyl transformations. We will schematically denote both of these by $\zeta$. The change of the metric under a general gauge transformation is $g \rightarrow g^{\zeta}$. This is shorthand for, $$g_{\alpha \beta} (\sigma) \rightarrow g_{\alpha \beta}^{\zeta} (\sigma)= e^{2\omega (\sigma)} \frac{\partial \sigma^{\gamma}}{\partial \sigma'^{\alpha}} \frac{\partial \sigma^{\delta}}{\partial \sigma'^{\beta}} g_{\gamma \delta} (\sigma)$$ In two dimensions these gauge symmetries allow us to put the metric into any form that we like — say, $\hat{g}$. This is called the fiducial metric and will represent our choice of gauge fixing.

Is there a proof available that shows that the combined symmetry allows us to "put the metric in any form (locally, of course)?" Is this property restricted to two dimensions?

Theorem. Every 2D pseudo-Riemannian manifold $$(M,g)$$ is conformally flat, i.e. there locally exist isothermal coordinates.$$^1$$

Sketched proof of theorem: Given a point $$p\in M$$. Consider local coordinates $$u,v$$ in a neighborhood of $$p$$.

1. Generic case $$g_{uu}(p)\neq 0$$: Do a Weyl scaling such that $$g_{uu}\equiv 1$$. Then \begin{align}g~=~&\mathrm{d}u\odot \mathrm{d}u + 2g_{uv}\mathrm{d}u\odot \mathrm{d}v + 2g_{vv}\mathrm{d}v\odot \mathrm{d}v\cr ~\stackrel{(2)}{=}~&\omega_+\odot \omega_-,\end{align}\tag{1} where we have defined 2 non-vanishing one-forms $$\omega_{\pm}~:=~\mathrm{d}u + [g_{uv} \pm \sqrt{-\det(g)}]\mathrm{d}v. \tag{2}$$
• Minkowskian case $$\det(g)<0$$: Then the one-forms $$\omega_{\pm}$$ are real. There exist locally 2 real integrating factors $$\lambda_{\pm}\neq 0$$ such that $$\omega_{\pm} ~=~\lambda_{\pm}\mathrm{d}x^{\pm}.\tag{3}$$ Then the metric tensor (1) reads $$g~\stackrel{(1)+(3)}{=}~\lambda_+\lambda_-\mathrm{d}x^+\odot \mathrm{d}x^-.\tag{4}$$ A Weyl scaling brings the metric on light-cone form. $$\Box$$

• Euclidean case $$\det(g)>0$$: Then the one-form $$\omega_{\pm}~=~\omega^{\ast}_{\mp}$$ is complex. There exists locally a complex integrating factor $$\lambda_+\neq 0$$ such that \begin{align}\omega_+ ~=~&\lambda_+\mathrm{d}z, \cr z ~=~&x+iy, \cr \omega_- ~=~&\omega^{\ast}_+ ~=~\lambda^{\ast}_+\mathrm{d}z^{\ast}.\end{align}\tag{5}
Then the metric tensor (1) reads \begin{align}g~\stackrel{(1)+(5)}{=}&~|\lambda_+|^2\mathrm{d}z\odot \mathrm{d}z^{\ast}\cr ~=~&|\lambda_+|^2[\mathrm{d}x\odot \mathrm{d}x+\mathrm{d}y\odot \mathrm{d}y],\end{align}\tag{6} which is again manifestly real. A Weyl scaling brings the metric on standard Euclidean form. $$\Box$$

1. Special case $$g_{uu}(p)=0$$:
• Euclidean case $$\det(g)>0$$: Impossible. $$\Box$$

• Minkowskian case $$\det(g)<0$$: Then $$g_{uv}(p)\neq 0$$. In the Gauss elimination procedure (if we were to bring $$g$$ on diagonal form) this corresponds to a case of vanishing diagonal element. It is possible to perform an affine coordinate transformation $$(u,v) \to (u^{\prime},v^{\prime})$$ so that $$g_{u^{\prime}u^{\prime}}(p)\neq 0$$. Now use the generic case. $$\Box$$

References:

1. M. Nakahara, Geometry, Topology and Physics, 1989; Example 7.32.

2. M. Nakahara, Geometry, Topology and Physics, 2003; Example 7.9.

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$$^1$$ And yes, the theorem is not true in higher dimensions, cf. e.g. this related Phys.SE post.

• Notes for later. 1. Given a non-vanishing real one-form $\omega = f\mathrm{d}x+g\mathrm{d}y$. Existence of real integrating factor $\lambda$ such that $\mathrm{d}(\lambda\omega)=0$. $\quad (g\partial_x-f\partial_y)\ln\lambda=\partial_yf-\partial_xg$ first-order linear PDE. 2. Given a non-vanishing complex one-form $\omega = f\mathrm{d}z+g\mathrm{d}\bar{z}$. Existence of complex integrating factor $\lambda$ such that $\mathrm{d}(\lambda\omega)=0$. View $z$ and $\bar{z}$ as independent complex variables, and essentially proceed as in the real case. Aug 5, 2019 at 23:53