Cross-Ratio Versus Conformal Ratio | M.E. Irizarry-Gelpí

Sun 01 February 2015
Maths
#conformal symmetry

I recently learned the difference between a cross-ratio invariant and a conformal ratio invariant, thanks to this Wikipedia entry.

Cross-Ratio

The cross-ratio is a projective invariant. Given the coordinates of four points on the complex plane, the cross-ratio is defined as

$$P(1, 2, 3, 4) = \frac{(z_{1} - z_{3}) (z_{2} - z_{4})}{(z_{2} - z_{3}) (z_{1} - z_{4})} .$$

It is invariant under fractional linear transformations of the form

$$ z \longrightarrow \frac{a z + b}{c z + d} .$$

There is always the arbitrariness of how to define the cross-ratio. After all, there are $4! = 24$ permutations of the coordinates, so you can argue that there are 24 possible cross-ratios for a given quartet of coordinates. However, note that if you swap two pairs you again obtain the same cross-ratio. That is,

$$P(1, 2, 3, 4) = P(2, 1, 4, 3) = P(3, 4, 1, 2) = P(4, 3, 2, 1) .$$

This means that the 24 permutations must split into six inequivalent classes, each with four permutations related by the above relation. Representatives of each of these six inequivalent classes are:

$$P(1, 2, 3, 4) = p, $$

$$P(1, 2, 4, 3) = \frac{1}{p}, $$

$$P(1, 3, 4, 2) = \frac{1}{1 - p}, $$

$$P(1, 3, 2, 4) = 1 - p, $$

$$P(1, 4, 3, 2) = \frac{p}{p - 1}, $$

$$P(1, 4, 2, 3) = \frac{p - 1}{p}. $$

As you can see, once the value of $p$ is given, all possible permutations are fixed, so there is really one independent cross-ratio for any given quartet of coordinates.

Conformal Ratio

A conformal ratio is a conformal invariant. Given the coordinates of four points on $D$-dimensional space, a conformal ratio is defined as

$$C(1, 2, 3, 4) = \frac{(x_{1} - x_{3})^{2} (x_{2} - x_{4})^{2}}{(x_{2} - x_{3})^{2} (x_{1} - x_{4})^{2}} ,$$

where $(x_{1} - x_{3})^{2}$ means the square of the magnitude of the $D$-dimensional vector $x_{1} - x_{3}$. In some sense, after a change of basis, the cross-ratio is the square root of a conformal ratio in two dimensions.

Again you encounter the arbitrariness of which of the $4! = 24$ permutations to use for the definition of a conformal ratio. Just like the cross-ratio, you find that a conformal ratio is invariant under the exchange of two pairs of coordinates. So again you find six inequivalent classes. However, unlike the cross-ratio, you find that fixing the value of the conformal ratio for one class does not fix the other five classes:

$$C(1, 2, 3, 4) = u \quad C(1, 2, 4, 3) = \frac{1}{u} \quad C(1, 3, 4, 2) = \frac{1}{v} \quad C(1, 3, 2, 4) = v \quad C(1, 4, 3, 2) = \frac{u}{v} \quad C(1, 4, 2, 3) = \frac{v}{u} .$$

Indeed, now you have two independent conformal ratios for a given quartet of coordinates.