# Understanding the Segal-Sugawara construction

I am trying to understand the Segal-Sugawara construction from the book "vertex algebras and algebraic curves" by Frenkel, Ben-Zvi in 3.4.8. As an absolute layman in the area of mathematical physics and vertex algebras, I have difficulties to understand the arguments there, as it seems many details in the proof have been omitted. More concretely, the computations of $S_n.J^{b}_{-1}v_k$ for $n=-1,0,1$ are still obscure for me. Could anyone explain these computations with more details?

• What exactly do you want to be clarified? – Arnold Neumaier Dec 2 '15 at 10:09