I want to understand the relation of projective structure and the respective projective invariants between the Riemann sphere $\mathbb{CP}^1$ with coordinates $z$ and the (bi-infinite) cylinder with coordinates $w$.
Context: The reason for comparing the two is, that they are conformally equivalent in the following sense. In standard conformal field theory language, one calls the holomorphic maps of $\mathbb{CP}^1$ onto itself the global conformal maps and the remaining conformal maps, i.e. those which are not holomorphic, the locally conformal ones, see for example page 17 in S. Ribaults review of CFT on the plane. The coordinate transformation above is in this sense a locally conformal transformation and I wonder what this type of conformality destroys or conserves.
Properties of Maps Between Riemann Sphere and Cylinder
First a short summary of what I know about their difference and similarities in terms of maps between them. The map $\mathbb{CP}^1 \to \mathbb{C}/\mathbb{Z}$, $z = \text{e}^w$ , i.e. the map $z \mapsto w = \text{ln}(z)$, translates between Riemann sphere and cylinder. As $\text{ln}(z)$ has a branch point at $z=0$, this map is only meromorphic. This is clear as the cylinder is only homeomorphic to the punctured plane $\mathbb{C}^* = \mathbb{CP}^1/\{0,\infty\}$, furthermore, the map from $\mathbb{CP}^1$ to the cylinder cannot be holomorphic as the latter is compact.
Cross-Ratio on the Cylinder
The symmetry- or automorphism group of $\mathbb{CP}^1$ is the Möbius group $SL(2,\mathbb{C})/\mathbb{Z}_2 \sim PSL(2,\mathbb{Z})$. Its elements are the holomorphic maps of the Riemann sphere onto itself. This projective symmetry group allows us to introduce the projective invariant (cross-ratio), i.e. to fix three points on the Riemann sphere which are usually taken to be $\{0,1,\infty\}$.
According to Wikipedia the cross-ratio on the Riemann sphere is defined as \begin{equation} (z_1, z_2; z_3, z_4) = \frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_2)(z_4 - z_1)} = \frac{z_{31} z_{42}}{z_{32} z_{41}}\,. \end{equation} for four points ${z_1,z_2,z_3,z_4}$. This concept does not pose any problems on the cylinder when we just define the cross-ratio as the above equation but with every $z_i$ substituted by the corresponding $\text{e}^{w_i}$, but this feels like cheating as our objects seem to still "live" on the Riemann sphere?
The above definition does not work on the cylinder if we just substitute every $z_i$ by an $w_i$, as we have both $\infty$ and $-\infty$. However, we can consider the automorphism group of the cylinder and find that is spanned by rotations and inversions and probably would keep a cross-ratio on the cylinder invariant (as those transformations also keep the cross-ratio on $\mathbb{CP}^1$ invariant). What is the right definition of a cross-ratio on the cylinder, if there is one?
Schwarzian Derivative and Cross-Ratios
There seems to be a peculiar connection between cross ratios and the Schwarzian Derivative, as the latter can be constructed as the limit of the four points in the cross-ratios approaching one point, i.e. it is the infinitesimal version of the projective invariant. As I have discussed above, the map $w = \text{ln}(z)$ is only meromorphic and a characteristic of the Schwartzian derivative is, that it only sends Möbius transformations to zero, this means that the Schwartzian derivative of our coordinate change does not vanish. What does this mean for the symmetries on the cylinder and the cross-ratio on the cylinder?
Connection Between Cross-Ratios Cylinder and Torus
To maybe looking from a "reverse" angle in terms of glueing: We can glue the torus from the cylinder and know that we have there the modular $\lambda$ function, which is the cross-ratio of the branch points that we get on $\mathbb{CP}^1$ by viewing the torus as a double ramified covering of the Riemann sphere. We then can find a rational function of $\lambda$ that is invariant under the modular group on the torus, which is the $j(z)$-invariant. Is there a cross-ratio on the cylinder that is connected to this cross-ratio on the torus? What is their relation?
Summarized, my questions are:
- Very broadly speaking, what is "conserved" by switching to the cylinder, what is "lost"?
- Is there still projective symmetry on the (bi-infinite) cylinder? Is it enough to still have the notion of a projective invariant? Does this have anything to do with the fact that the Schwartzian derivative does not vanish for the change of coordinates?
- How does this object carry over to the torus?
I know that the above is a lot, as I feel all of them belong together and probably rely on the same misunderstanding, I decided to ask them in the same context. If someone thinks that it would be better to ask them in different threads, I'd gladly do that. I would be happy for any help on the above questions, even just some hints with regard to literature would go a long way!