What is the expression for the modular S-matrix of (p,q) minimal model? The Wiki https://en.wikipedia.org/wiki/Minimal_model_(physics) does not provide S-matrix
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$\begingroup$ Could be wrong but it seems that it's given as formula 23b of the original article by Capelli, Itzykson and Zuber in Communications in Mathematical Physics projecteuclid.org/journals/… I think $N=2pq$ and characters are encoded by integers $\lambda$ mod $N$. $\endgroup$– Abdelmalek AbdesselamJul 12, 2021 at 19:34
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$\begingroup$ Sorry Cappelli with two p's. $\endgroup$– Abdelmalek AbdesselamJul 12, 2021 at 21:03
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$\begingroup$ According formula 23b, the matrix elements of S-matrix is just a phase factor/sqrt(N), which seems not right for minimal models. $\endgroup$– Xiao-Gang WenJul 13, 2021 at 0:16
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$\begingroup$ Sound like you are confusing the notion of S-matrix with that of S-matrix. Some S-matrices in the scattering sense are "kinda trivial" when they are diagonal with diagonal elements made of of phase factors. The matrix from 23b is not diagonal. $\endgroup$– Abdelmalek AbdesselamJul 13, 2021 at 0:59
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$\begingroup$ Of course in the first sentence of the last comment I am only kidding. $\endgroup$– Abdelmalek AbdesselamJul 13, 2021 at 1:10
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1 Answer
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The modular S-matrix appears in Section 10.6 of the Big Yellow Book by di Francesco, Mathieu and Sénéchal. For a freely available reference, there are my lecture notes https://arxiv.org/abs/1609.09523 , eq. (A.28).
The Wikipedia article on minimal models was mostly written by someone who does not like the modular bootstrap. But you are free to complete it.
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$\begingroup$ Apart from different notations, your formula A28 seems to the be same as the formula 23b from the CIZ paper I mentioned. $\endgroup$ Jul 13, 2021 at 14:57
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1$\begingroup$ Oh I see, there is indeed a difference due to reduction to a discrete fundamental domain. In the 1997 edition of the yellow book, the formula 23b I mentioned is 10.128 and is a preparatory step for the S-matrix a few lines below, formula 10.132 (recapped and boxed in 10.134) which is your formula A28. $\endgroup$ Jul 13, 2021 at 20:10
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$\begingroup$ Thanks! That is exactly want I was looking for. $\endgroup$ Aug 5, 2021 at 23:10