# 2D global conformal transformations and the $z= \frac1w$ argument

For instance in Blumenhagen's CFT, there is a standard argument which determines that globally defined conformal transformations on the Riemann sphere where

$$l_n = -z^{n+1} \partial_z$$ is an element of the Witt algebra. In this argument we note that $$l_n$$ is non-singular at $$z=0$$ only for $$n\geq -1$$. Also substituting $$z=-\frac1w$$, we find

$$l_n = -\left(-\frac1w\right)^{n-1}\partial_w$$ is non-singular at $$w=0$$ only for $$n\leq +1$$. Therefore the global transformations are generated by $$\{l_{-1},l_0,l_1\}$$.

Why is the substitution $$z=-\frac1w$$ special? For instance if I use $$z = -\frac{1}{w^2}$$ I could repeat the argument above and conclude $$n \leq 1/2$$.

• The transformation $(z,{\bar z}) \to \left( - \frac{1}{w} , - \frac{1}{ {\bar w} } \right)$ is an isometry of the Riemann sphere whose metric is $ds^2 = \frac{4 dz d{\bar z} }{ ( 1 + z {\bar z} )^2 }$. – Prahar Dec 19 '18 at 15:32

The minus sign is not essential, so let's remove it for simplicity. An atlas of the Riemann sphere, say $$M$$ ( which is a 1-dimensional complex manifold), is given by two coordinate charts
1. $$z:M\to\mathbb{C}$$, which covers the whole $$M$$ except the "infinity" $$i\in M$$.
2. $$w:M\to\mathbb{C}$$, which covers the whole $$M$$ except the "origin" $$o\in M$$.
There is a holomorphic transition map: $$w=1/z$$ in the overlap $$M\backslash\{i,o\}$$. In polar coordinates $$z=re^{i\theta}$$, we have $$w=e^{-i\theta}/r$$. Therefore the monodromy $$z\to e^{2\pi i}z$$ leads to the correct corresponding opposite monodromy $$w\to e^{-2\pi i}w$$ on the other chart, i.e. if we circle around the origin $$o$$ one time in one chart, it corresponds to circle around the infinity $$i$$ one time in the opposite direction in the other chart.
In contrast, a transition map of the form $$z =1/w^m$$, where $$m\in \mathbb{N}\backslash\{1\}$$, as OP suggests, would lead to wrong monodromy properties $$w\to e^{-2\pi i/m}w$$, and would hence not correspond topologically to a Riemann sphere.