Topological/Geometrical justification for $\text{CFT}_2$ being special

Question

It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n>2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are from a much wider class of maps, holomorphic/antiholomorphic maps. I was wondering to know if there is any topological or geometrical description for this.

To show what I mean, consider this example: in $\mathbb{R}^n$ for $n>2$ interchanging particles can only change the wave function to itself or its minus. It is related to the fundamental group of $\mathbb{R}^n-x_0$ ($x_0$ is a point in $\mathbb{R}^n$ and $\pi_1(\mathbb{R}^n-\{x_0\})=e$ for $n>2$) but this is not true for $n=2$.

I want to know whether exists any topological invariant or just any geometrical explanation that is related to the fact that I mentioned about conformal maps on $\mathbb{R}^n$.

Answer 1

This is essentially Liouville's rigidity theorem for conformal mappings in $\color{red}{n\geq 3}$ dimensions. Interestingly, the cause is local rigidity [rather than global topological obstructions]. For a proof see Refs. 1 & 2.

Combined with the fact that every mapping in $\color{red}{n=1}$ dimension is automatically conformal, it is perhaps not totally surprising that the border case $\color{red}{n=2}$ is special. In fact, there are infinitely many (dimensions of) local conformal deformations for $\color{red}{1\leq n\leq 2}$.

The main point is the following Lemma.

Lemma. In a coordinate neigborhood where the metric $g_{\mu\nu}$ is constant, the components $\varepsilon^{\mu}$ of every conformal Killing vector field (CKVF) is at most a quadratic polynomial in the coordinates $x^{\nu}$ [i.e. there are only finitely many (dimensions of) local conformal deformations] if $\color{red}{n\geq 3}$.

Proof: Conformal Killing equation (CKE):

$$\omega g_{\mu\nu}~=~\varepsilon_{\mu,\nu}+\varepsilon_{\nu,\mu} .\tag{1}$$

$$n \omega~\stackrel{(1)}{=}~ 2\varepsilon^{\mu}{}_{,\mu}.\tag{2}$$

$$(\color{red}{2-n})\partial_{\mu}\omega ~\stackrel{(1)+(2)}{=}~ 2\Box \varepsilon_{\mu}.\tag{3}$$

$$ (\color{red}{2-n})\partial_{\mu}\partial_{\nu}\omega ~\stackrel{(3)}{=}~ 2\Box\partial_{\mu} \varepsilon_{\nu} .\tag{4}$$

$$ (\color{red}{n-1})\Box \omega~\stackrel{(2)+(4)}{=}~0 \quad \stackrel{\color{red}{n\neq 1}}{\Rightarrow} \quad \Box \omega~=~0.\tag{5}$$

$$ ~(\color{red}{2-n})\partial_{\mu}\partial_{\nu}\omega~\stackrel{(1)+(4)}{=}~g_{\mu\nu} \Box \omega~\stackrel{(5)}{=}~0\quad \stackrel{\color{red}{n\neq 2}}{\Rightarrow} \quad \partial_{\mu}\partial_{\nu}\omega~=~0.\tag{6}$$

Eq. (6) shows that

$$\omega~=~a_{\mu}x^{\mu}+b\tag{7}$$

is an affine function$^1$ of $x^{\mu}$.

$$\varepsilon_{\mu,\nu\lambda}+\varepsilon_{\nu,\mu\lambda}~\stackrel{(1)}{=}~ g_{\mu\nu}\partial_{\lambda}\omega \tag{8}$$

$$2\varepsilon_{\lambda,\mu\nu}~\stackrel{(8)}{=}~g_{\lambda\mu}\partial_{\nu}\omega +g_{\lambda\nu}\partial_{\mu}\omega -g_{\mu\nu}\partial_{\lambda}\omega ~\stackrel{(6)}{=}~\text{constant}.\tag{9}$$ $\Box$

References:

1. P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028; p.5.

2. J. Slovak, Natural Operator on Conformal manifolds, Habilitation thesis 1993; p.46. A PS file is available here from the author's homepage. (Hat tip: Vit Tucek.)

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$^1$ The parameters $a_{\mu}$ and $b$ in eq. (7) correspond to $n$ special conformal transformations and $1$ dilatation, respectively,

$$\varepsilon_{\mu}~=~\frac{\omega}{2}x_{\mu}-\frac{x^2}{4}\partial_{\mu}\omega.\tag{10}$$

Eq. (10) satisfies the CKE (1). What other solutions are there? After subtracting the solution (10) from the CKE (1), we get the Killing equation (KE)

$$\varepsilon_{\mu,\nu}+\varepsilon_{\nu,\mu}~=~0\tag{11}$$

with

$$\omega~=~0.\tag{12}$$

Eqs. (9) & (12) now show that

$$\varepsilon_{\mu}~\stackrel{(9)+(12)}{=}~a_{\mu\nu}x^{\nu}+b_{\mu}\tag{13}$$

are affine functions. Comparing with the KE (11), we see that

$$ a_{\mu\nu}~\stackrel{(11)+(13)}{=}~-a_{\nu\mu} \tag{14}$$

is antisymmetric. The solution (13) correspond to $n(n-1)/2$ rotations and $n$ translations. Altogether we generate nothing but the $(n+1)(n+2)/2$ dimensional (global) conformal algebra. Then main message is that local conformal deformations are rigid for $\color{red}{n\geq 3}$. See also this related Phys.SE post.