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 mathod to embed varasoro algebra into completions of universal enveloping algebras of affine algras ? what special properties do this kind of embeding have? In soliton theorey, 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?


The origin of the Sugawara construction in Physics is, not surprisingly, the 1968 paper A field theory of currents by Hirotaka Sugawara. (There was also work of Sommerfield at about the same time.) The context of that work was to find a theory of strong interactions. (Recall that the currently favoured theory of strong interactions (QCD) would not be discovered until the mid 1970s.) There was a feeling at the time that there might not be a field theory describing the strong interactions, whose dynamical fields are associated to the particles themselves. (QCD is such a theory, of course.) Hence Sugawara's idea was to quantise the theory using "currents" (field bilinears) as the elementary dynamical variables. His theory is a four-dimensional theory and I think it is fair to say that it is now just of historical interest.

What most people call the Sugawara construction is what Marty Halpern insisted in calling the affine Sugawara construction. This is a construction in two-dimensional conformal field theory by which a Virasoro element is constructed in the conformal field theory associated to an affine Kac-Moody algebra. The construction has a long history of papers converging to the correct formula. I forgot the actual sequence of papers, but it probably starts with a 1971 paper of Bardakçi and Halpern, who introduced the construction, and ends with Knizhnik's and Zamolodchikov's 1984 paper. (But I stand to be corrected on this.)

A possibly physical interpretation, in the context of string theory, is that this is a quantisation of string propagation on a Lie group (with a bi-invariant metric), whose classical action is given by the Wess-Zumino-Witten model. In that model the dynamical fields are maps $g:\Sigma \to G$, where $\Sigma$ is a Riemann surface and $G$ is a Lie group with a bi-invariant metric. Just as in the original four-dimensional Sugawara construction, it is simpler to quantise the currents $g^{-1}\partial g$ and $\bar\partial g g^{-1}$ than the actual fields $g$. This was done by Edward Witten in his celebrated paper Non-abelian bosonization in two dimensions. The Sugawara construction serves to prove the exact quantum conformal invariance of the Wess-Zumino-Witten model to all orders in perturbation theory, something which is not possible to do by quantising the original fields $g$.

Mathematically, it embeds the Virasoro algebra in the vertex algebra of an affine Kac-Moody algebra (actually this fails at the so-called critical level) in such a way that any module of the Kac-Moody algebra (of noncritical level) is also a Virasoro module. If $\mathfrak{g}$ is a simple Lie algebra and $\widehat{\mathfrak{g}}_\ell$ the corresponding (untwisted) affine Kac-Moody algebra at level $\ell$, then the central charge of the Sugawara Virasoro element is given by $$c = \frac{\ell \dim\mathfrak{g}}{\ell + h^\vee}$$ where $h^\vee$ the dual Coxeter number. (So $\ell = - h^\vee$ is the critical level.)

The defining property of the Sugawara construction is that the currents are primary fields (of weight 1) of the Virasoro element.

This same construction extends to the affinisation of any metric Lie algebra; that is, admitting an ad-invariant metric.

  • $\begingroup$ Hi José, how exactly does the Sugawara construction prove conformal invariance to all orders? In Witten's paper I can only see a calculation in one-loop-order. $\endgroup$ – Konrad Waldorf Feb 25 '10 at 20:15
  • $\begingroup$ If you define the quantisation of the WZW model in terms of the currents, so that the Hilbert space of the theory consists of all the integrable highest weight representations (up the relevant level) of the corresponding affine Kac-Moody algebra (I'm talking about the case of $\mathfrak{g}$ simple), then the Sugawara construction defines on those representations the structure of a Virasoro module. This, by definition, is an exact quantum conformal field theory. $\endgroup$ – José Figueroa-O'Farrill Feb 25 '10 at 22:19
  • $\begingroup$ @JoséFigueroa-O'Farrill Why do we only use the integrable highest weight representations? $\endgroup$ – Soap Oct 20 at 16:11
  • $\begingroup$ @Soap The mathematical condition of "highest weight" is equivalent to the physical condition of "minimal energy"; that is, representations where the energy is bounded below, which is a desirable physical property. $\endgroup$ – José Figueroa-O'Farrill Oct 20 at 17:13
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    $\begingroup$ @Soap In their paper String theory on group manifolds, Gepner and Witten show that any correlation function containing a highest-weight vector of a non-integrable representation vanishes. Hence one does not lose anything by restricting to integrable highest weight representations. $\endgroup$ – José Figueroa-O'Farrill Oct 20 at 17:55

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 curioous early occurences 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 overmore 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)

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    $\begingroup$ Sugawara's original theory does not calculate the constant in font of the bilinear expression. I think you are confusing this with the affine (two-dimensional) construction, to which Sugawara did not contribute. As in my answer, this probably started with Bardakçi-Halpern. There was some initial confusion between two notions of normal ordering which accounted for the wrong constant. I am not sure which was the first paper to get it right, though. $\endgroup$ – José Figueroa-O'Farrill Feb 25 '10 at 16:08

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