TITLE: How to interpret the Sugawara construction from a physical or mathematical viewpoint?
QUESTION [16 upvotes]: In theoretical physics, the Sugawara theory is a set of formulae and theorems that allow one to construct a stress-energy tensor of a specific type of conformal field theory from a bilinear expression involving currents.
How to interpret the Sugawara construction from a physical or mathematical viewpoint?
Sugawara construction is a kind of method to embed Virasoro algebra into completions of universal enveloping algebras of affine algebras ?
what special properties do this kind of embedding have?
In soliton theory, I know there is a boson-fermion correspondence which realize free boson algebra in the completion of free fermion algebra.
I wonder if there are some common principles under them?

REPLY [4 votes]: Just a historical comment (from my memory; I am at the moment rather far from the subject)
Ivan Todorov gave and wrote few times lectures about the history of Sugawara construction to which he himself also contributed. He even mentions some works in late 1940-s in field theory of photons among curious early occurrences and then the intricate history in 1960's and 1970's, involving Kac, Feĭgin-Fuks, Sugawara, Segal, Todorov etc. Sugawara is in any case not the first in that sequence and he had a wrong proportionality constant calculated, what is corrected in 1970s.
It is also curious that the (affine) Sugawara construction is not confined to working over complex numbers, there is a more general version over more general fields explained by Faltings. I think I have seen it in his paper (I do not have it here so I can not check)
Gerd Faltings, A proof for the Verlinde formula.  J. Algebraic Geom.  3  (1994),  no. 2, 347--374. MR1257326 (95j:14013)