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

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 $\mathbb{R}^n$.

• Minor suggestion: Replace $\text{CFT}_2$ in title with 2D CFT, which may be easier to understand. Jun 16, 2018 at 11:22

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 $$x^{\mu}$$.

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

$$2\varepsilon_{\lambda,\mu\nu}~\stackrel{(8)}{=}~g_{\lambda\mu}\partial_{\nu}\omega +g_{\lambda\nu}\partial_{\mu}\omega -g_{\mu\nu}\partial_{\lambda}\omega ~\stackrel{(7)}{=}~\text{constant}.\tag{9}$$ $$\Box$$

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.)

--

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

$$\varepsilon_{\mu}~=~\frac{\omega}{2}x_{\mu}-\frac{x^2}{4}\partial_{\mu}\omega \quad \stackrel{(7)}{\Rightarrow} \quad \varepsilon_{\mu,\nu}~=~\frac{\omega}{2}g_{\mu\nu}+\frac{x_{\mu}}{2}\partial_{\nu}\omega-\frac{x_{\nu}}{2}\partial_{\mu}\omega .\tag{10}$$

Eq. (10) satisfies the CKE (1), which is an inhomogeneous 1st-order linear PDE in $$\varepsilon_{\mu}$$. What other solutions are there? After subtracting the solution (10) from the CKE (1), we get the corresponding homogeneous 1st-order linear PDE, which is 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. The main message is that local conformal deformations are rigid for $$\color{red}{n\geq 3}$$. See also this related Phys.SE post.

• Thank you very much for a precise discussion and references. Although I see why $\mathbb{R}^2$ is special (since $2-2=0$) but it looks so tricky to me. The argument does not show why the conformal group of $\mathbb{R}^2$ has infinite generators, however it shows perhaps conformal group of $\mathbb{R}^2$ is different. I'm really thankful for the answer. At least it provides a reason. Mar 22, 2018 at 14:48