# A question about Witten's paper on QFT and Jones polynomial

So I have been reading the famous paper on quantum field theory and Jones polynomial and have the following questions:

1. On P.31 (381), it was said that the eigenvalues of $$B$$ are $$λ_i = ±\exp(i\pi(2h_R - h_{E_i})).\tag{4.18}$$ How is this formula derived? I know only a little conformal field theory, so please explain in more details.

2. On P.40 (390), it was stated the matrix element of $$S$$ are $$S_{mn} = \sqrt{\dfrac{2}{k+2}}\sin\left(\dfrac{(m+1)(n+1)\pi}{k+2}\right).\tag{4.39}$$ Where does this equation come from?

• anyone thinks I should ask the same question on Math Stack Exchange? Because this question is quite mathematical... Dec 16, 2018 at 14:29
• As for 2, it's the modular S-matrix. In order to derive it, you consider the $SU(2)_k$ characters $\chi_j(\tau)$ expressed in terms of level $k$ $\theta$ functions and perform an $S$-transformation on them, i.e you will get $\chi_j(-1/\tau) = \sum_l S_{jl} \chi_l (\tau)$ Jan 27, 2020 at 17:30