Understanding the Segal-Sugawara construction

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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?

This post imported from StackExchange MathOverflow at 2015-11-27 11:59 (UTC), posted by SE-user Jack

asked Nov 22, 2015
recategorized Nov 27, 2015

What is exactly that you want to be clarified? Also, try have a look at http://arxiv.org/abs/math/0301206 where more on the construction is given.

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