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

Question

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, \begin{equation} 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) \end{equation} 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?

Answer 1

Theorem. Every 2D pseudo-Riemannian manifold $(M,g)$ is locally 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 $$g~=~\mathrm{d}u\odot \mathrm{d}u + 2g_{uv}\mathrm{d}u\odot \mathrm{d}v + 2g_{vv}\mathrm{d}v\odot \mathrm{d}v~\stackrel{(2)}{=}~\omega_+\odot \omega_-,\tag{1}$$ 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 $$\omega_+ ~=~\lambda_+\mathrm{d}z, \qquad z ~=~x+iy, \qquad \omega_- ~=~\omega^{\ast}_+ ~=~\lambda^{\ast}_+\mathrm{d}z^{\ast}.\tag{5}$$
     Then the metric tensor (1) reads $$g~\stackrel{(1)+(5)}{=}~|\lambda_+|^2\mathrm{d}z\odot \mathrm{d}z^{\ast}~=~|\lambda_+|^2[\mathrm{d}x\odot \mathrm{d}x+\mathrm{d}y\odot \mathrm{d}y],\tag{6} $$ which is again manifestly real. A Weyl scaling brings the metric on standard Euclidean form. $\Box$

2. 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.