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 $n \leq 1/2$.

  • 2
    $\begingroup$ 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 $ds^2 = \frac{4 dz d{\bar z} }{ ( 1 + z {\bar z} )^2 }$. $\endgroup$ – 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.


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