[Physics] Number of Independent “Cross-ratios or Anharmonic-ratios”

conformal-field-theory

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?

Best 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.

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