# 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, \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?

## 1 Answer

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.

• 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. – Qmechanic Aug 5 '19 at 23:53