Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

The Cross-ratios or the Anharmonic-ratios are defined as, $${r_{ij}r_{kl}}/{r_{ik}r_{jl}}, \text{ where } r_{ij}=\mod{r_i - r_j}.$$ 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?

share|improve this question
    
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? –  Vibert Jun 13 '13 at 11:28
    
@ Vibert: Can you elaborate for me please? –  layman Jun 13 '13 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 $y=1-x$. –  Marcel Jun 14 '13 at 18:06
add comment

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.

share|improve this answer
    
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. –  layman Jun 15 '13 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 –  Marcel Jun 16 '13 at 16:02
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.