Conformal transformations problem in flat worldsheet

Question

In section 2.2 of David Tong's String Theory lecture notes, he claims that conformal transformations on the flat worldsheet are such that $$\sigma^\pm \to \tilde{\sigma}^\pm(\sigma^\pm).\tag{2.10}$$ I'm trying to develop this step-by-step so we consider a line element in the worldsheet with the conformal gauge $h_{ab}=\eta_{ab}$, where $h_{ab}$ is the worldsheet metric in Polyakov action. We have that

\begin{equation} \begin{aligned} ds^2&= \eta_{ab}d\sigma^a d\sigma^b\\ &= -d\tau^2 + d \sigma^2\\ &= -d\sigma^+ d\sigma ^-, \qquad \sigma^\pm = \tau \pm \sigma. \end{aligned} \end{equation} By performing a conformal transformation on the coordinate system $(\sigma^+,\sigma^-)$ we have new coordinates $(\tilde{\sigma}^+,\tilde{\sigma}^-)$ such that

$$ds^2 = - e^{\omega(\tilde{\sigma}^+,\tilde{\sigma}^-)} d\tilde{\sigma}^+d\tilde{\sigma}^- = -d\sigma^+ d\sigma ^-.$$ But we have that

\begin{equation} \begin{aligned} d\tilde{\sigma}^+d\tilde{\sigma}^-&=\left(\partial_+ \tilde{\sigma}^+ d \sigma^+ + \partial_- \tilde{\sigma}^+ d\sigma^- \right)\\ &\times \left(\partial_+ \tilde{\sigma}^- d \sigma^+ + \partial_- \tilde{\sigma}^- d\sigma^- \right)\\ &\propto d \sigma^+ d\sigma^-. \end{aligned} \end{equation}

One of the possible choices and the one chosen by Tong is to consider

$$\partial_+ \tilde{\sigma}^- = \partial_- \tilde{\sigma}^+=0. \tag1$$

However, I could choose instead

$$\partial_+ \tilde{\sigma}^+ = \partial_- \tilde{\sigma}^-=0. \tag2 $$

What kind of argument do I use to eliminate the possibility of $(2)$ and obtain the same result as Tong?

Answer 1

OP has a point. Tong only considers conformal transformations in the 1+1D Minkowski plane $\mathbb{R}^{1,1}$ of the form $\tilde{\sigma}^{\pm}= \tilde{\sigma}^{\pm}(\sigma^{\pm})$ in terms of light cone coordinates. OP's alternative case (2) leads to conformal transformations of the form $\tilde{\sigma}^{\pm}= \tilde{\sigma}^{\pm}(\sigma^{\mp})$, which in principle should also be taken into account. However, a transformation of type (2) is just a transformation of type (1) composed with the map $(\sigma^+,\sigma^-)\mapsto (\sigma^-,\sigma^+)$.

Concerning the global structure of the group(oid) of conformal transformations, see e.g. this Phys.SE post.