# Number of Independent "Cross-ratios or Anharmonic-ratios"

The Cross-ratios or the Anharmonic-ratios are defined as, $$\frac{r_{ij}r_{kl}}{r_{ik}r_{jl}}, \text{ where } r_{ij}=|\mathbf{r}_i - \mathbf{r}_j|.$$ Now the claim is: conformal symmetry implies that for computing $$N$$ point correlation function there will be $$N(N-3)/2$$ number of independent cross-ratios.

I can't prove this claim. I have seen the Ginsparg's explanation on this claim but I can't understand that. I need the proof. Can anyone help me?

• Hi Palash, what don't you understand? The Ginsparg proof is the only one I know of. If the general $N$ case is too difficult, did you try $N=4$ and $N=5$ and see how those work out? Jun 13, 2013 at 11:28
• @ Vibert: Can you elaborate for me please? Jun 13, 2013 at 12:59
• That you can form $N(N-3)/2$ cross ratios is pure combinatorics. I would like to remark that in 1D there are just $N-3$ independent cross ratios (I am not taking the modulus, though), for example for $N=4$ the two cross ratios are related by $y=1-x$. Jun 14, 2013 at 18:06

## 1 Answer

The number of $r_{ij}$'s is $N(N-1)/2$. A general monomial $\prod_{1\leq i<j\leq N} r_{ij}^{\mu_{ij}}$ is conformally invariant if and only if $d_i = \sum_{j=1}^{i-1} \mu_{ji} + \sum_{j=i+1}^N \mu_{ij} =0$ for all $i=1,\ldots, N$. This are $N$ equations. So we get $N(N-1)/2-N=N(N-3)/2$ cross-ratios.

But note also if your dimension is $D$ then the dimension of the conformal group is $(D+2)(D+1)/2$ and because the cross-ratios itself just depend on $D\cdot N$ variables and there are $(D+2)(D+1)/2$ constraints the number of "algebraically independent" variables can actually be just $D\cdot N-(D+2)(D+1)/2$, i.e. if the number of cross ratios is bigger than this there are have to be (complicated) relations between the cross ratios. This you can see in the example for $D=1$, see comment above.

• Thank you very much. But how can I prove that "A general monomial is conformally invariant if and only if d_i =0"? is it true for any dimension? Can you give me some reference? Again there is no notion of conformal transformation. For D=2 there is infinite conformal group parameters and we get more constraints on correlation function. I am only interested in D>2. Jun 15, 2013 at 3:33
• It follows from the fact that $(x_i-x_j)^2 \mapsto \frac{(x_i-x_j)^2}{\omega(g,x_i)\omega(g,x_j)}$ under a conformal transformation $g$, $d_i=0$ is precisely that all $\omega(g,x_i)$ cancel. See for example: arxiv.org/abs/hep-th/0009004 Jun 16, 2013 at 16:02
• @Marcel, Hi Marcel. does that there are $N(N-1)/2$ pairs of $r_{ij}$ means that there are $N(N-1)$ general monomial $\prod_{1\leq i<j\leq N} r_{ij}^{\mu_{ij}}$? Otherwise, why do you subtract the number of constraints from $N(N-1)/2$? Jul 23, 2016 at 19:24