There seems to be a simple method to transfer a CFT's correlations from $\mathbb{R}^3$ to $S^3$ but I am not understanding why it is supposed to work.

The idea is that somehow because, $ds^2_{S^3} = \frac{4}{(1 + \vert \vec{r} \vert^2 )^2}ds^2_{\mathbb{R}^3 }$ it follows that all that one needs to do is that replace all occurences of $\vec{r} - \vec{r}'$ by $\frac{2(\vec{r} - \vec{r}') }{\sqrt{ (1 + \vert \vec{r} \vert ^2 )(1 + \vert \vec{r}' \vert ^2 ) } } $

  • Why is this supposed to work?

  • What is the necessary relationship between the two CFTs for this to work?

    (like if one uses this on the free scalar field theory then it will turn out that w.r.t the Laplacians on $\mathbb{R}^3$ or $S^3$ the two 2-point functions satisfy different P.D.Es respectively and hence they come from two different Lagrangians - since we anyway knew that the Lagrangian for the conformally coupled scalar on the flat space is not the same as the Lagrangian for the conformally coupled scalar on $S^3$)

  • Is it part of some general idea that would work between other pairs of manifolds?

  • $\begingroup$ I suppose one could use a stereographic projection from $S^3$ to $R^3$, this map being conformal. $\endgroup$ – Trimok Dec 13 '13 at 10:38
  • $\begingroup$ @Trimok Can you explain this a bit more? How did one come up with this map? And for what kind of CFTs between what kind of manifolds would such a thing be permissible? $\endgroup$ – user6818 Dec 13 '13 at 18:19
  • $\begingroup$ I had no precise idea but I note that the $ds^2_{S^3}$ metrics you gave corresponds to a stereographic projection : $\phi(x,y,z,w) = (\frac{x}{1-w},\frac{y}{1-w} ,\frac{z}{1-w})$, and $\phi^{-1}(a,b,c) = (\frac{2a}{1+r^2}, \frac{2b}{1+r^2}, \frac{2c}{1+r^2}, \frac{-1+r^2}{1+r^2})$, with $r^2=a^2+b^2+c^2$ $\endgroup$ – Trimok Dec 14 '13 at 9:51

Both $\mathbb{R}^3$ and $S^3$ are rank 1 symmetric spaces explicitly, as a homogeneous spaces they are given by:

$$\mathbb{R}^3 = ISO(3)/SO(3)$$


$$S^3 = SO(4)/SO(3)$$

The significance of their being rank-1 symmetric spaces is that there is only one "two-point" invariant on them, i.e., any function of two points $r_1$ and $r_2$ invariant under the automorphism group ($ISO(3)$ in the case of $\mathbb{R}^3$ and $SO(4)$ in the case of $S^3$) must be a function of a single two point invariant, which can be taken as the geodesic distance:

$$ d(r_1, r_2) = |r_1-r_2|$$

in the case of $\mathbb{R}^3$ and

$$ d(r_1, r_2) = \frac{|r_1-r_2|}{\sqrt{1+\frac{r_1}{R}^2}\sqrt{1+\frac{r_2}{R}^2}}$$

In the stereographic projection coordinates of $S^3$ ($R$ is the sphere's radius).

Thus this replacement is the natural one to adopt to retain the invariance. Also in the limit where the sphere's radius tends to infinity, the $\mathbb{R}^3$ functions are obtained

More deeply, $ISO(3)$ can be obtained from $SO(4)$ by a deformation process called the Wigner-İnönü Contraction. Please see the following expository article by Shu-Heng Shao. This is a singular limit and one cannot expect a smooth mappings from $R \leq \infty$ to $R = \infty$ after all the topology is different.

Due to this reason one cannot expect the contraction of group representations to be one to one, please see this recent article by B. Cahen on the contraction of group representations. For example unitary irreducible representations of noncompact groups are infinite dimensional, while for compact groups finite dimensional, but it is known that the discrete representations of the noncompact groups go in the limit to the (always discrete) representations of the compact groups.

Now, two point functions can be decomposed into sums of eigenfunctions of the Laplacian carrying group representations.

A famous example is the heat kernel, which can be decomposed by the eigenvalues of the Laplacian, depends in the semiclassical limit solely on the geodesic distance:

$$ K(r_1, r_2, \beta) = \sum_{n} \psi_n^*(r_1)\psi_n(r_2) e^{-\beta E_n} \sim \mathrm{Hessian}(d(r_1,r_2)^2)e^{-\frac{d(r_1,r_2)^2}{4\beta}}$$

Where $\psi_n$ is the eigenfunction of the Laplacian corresponding to the energy $E_n$. These functions carry representations of the automorphism group which have limits upon contraction.

Thus in order to get the analogous two point function on the sphere, the harmonic functions should be replaced by the corresponding eigenfunctions of the Laplacian on the sphere. In this case one gets automatically the geodesic distance on the sphere in the limit.

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