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My advisor mentioned to me that he talked to Witten last summer on representation theory, and Witten told him that unitary representations of Kac-Moody algebra are important to working physicists. But he did not explain in details to me.

My question is Why?

Second question: I know there are a lot of people devoting to studying unitary representation of Lie group. But are there papers investigating unitary representations of Kac-Moody algebra?

I am not an expert, so this question might be naive. Thanks

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en.wikipedia.org/wiki/Affine_Lie_algebra might have some information. –  Qiaochu Yuan Jul 4 '10 at 7:52
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After a question on "failures of physicists" (mathoverflow.net/questions/30149) we could hardly hope that a physicist comes and explains the point. :-) –  Wadim Zudilin Jul 4 '10 at 8:06
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My vague understanding is that this is relevant to 2D conformal field theory. Try looking here: en.wikipedia.org/wiki/Conformal_field_theory and here: en.wikipedia.org/wiki/Virasoro_algebra –  Jeff Schenker Jul 4 '10 at 8:13
    
As far as I know they are only interested in the very special affine case. –  Najdorf Jan 9 '11 at 2:35

3 Answers 3

up vote 7 down vote accepted

As others have mentioned, the reasons lie indeed in two-dimensional conformal field theory and in string theory.

The propagation of string on a compact Lie group $G$ is described by the Wess-Zumino-Witten model, whose dynamical variables are maps $g:\Sigma \to G$ from a riemann surface $\Sigma$ to $G$. The quantisation of that model is difficult in terms of $g$ (although see the 1988 papers of Gawedzki and Kupiainen, and also Felder, for a functional integral approach) and one instead chooses to quantise their currents, roughly the (anti)holomorphic components of the pullbacks $g^*\theta_L$ and $g^*\theta_R$ of the Maurer-Cartan forms on $G$. There is a natural action of two copies of the affine Kac-Moody algebra associated to $G$ on the WZW model which preserves the Poisson structure of the WZW model and gives rise to moment mappings which are, essentially, the currents. In other words, the Poisson bracket of the currents is that of two copies of the affine Kac-Moody algebra of $G$. Hence the quantisation naturally leads one to consider unitary, integrable representations of the affine Kac-Moody algebra. The first "modern" reference for this is a 1986 paper of Doron Gepner and Edward Witten String Theory on Group Manifolds; although there are pioneering papers of Halpern, Bardakci,... in from the late 1960s and early 1970s.

At a more abstract level, we can substitute the group $G$ by any (unitary) two-dimensional (super)conformal field theory with the right central charge. This idea of replacing the geometry by a conformal field theory used to be known as "strings without strings", since one loses the description of strings propagating in some geometry. In this context, it is important to have at one's disposal a number of unitary two-dimensional conformal field theories. The natural ones are those coming from from unitary representations of infinite-dimensional Heisenberg and Clifford algebras (so-called free fields) and unitary integrable representations of affine Kac-Moody algebras, but one can also consider constructions (e.g., orbifolds, coset constructions,...) which generate new unitary CFTs from these ones. The first "modern" reference for the coset construction is perhaps the 1986 paper of Peter Goddard, Adrian Kent and David Olive Unitary representations of the Virasoro and superVirasoro algebras.

Finally, I should say that although it's the affine Kac-Moody algebras which seem to have played the more important rôles thus far, there is also the emergence (in the context of M-theory) of more general Kac-Moody algebras. There's work on this in King's College London (West et al.), Brussels (Henneaux et al.) and Potsdam (Nicolai et al.). I'm not very familiar with this, though.

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The following two articles (first), (second) by Louise Dolan survey the main applications of the Kac-Moody algebras in physics. Apart from the well known application in conformal field theories and string theory, the articles describe the role of the Kac-Moody algebras in integrable models and in Yang-Mills theory.

Apart from the applications described in these surveys, there are two more cases that I know of, the first is the work of Juoko Mickelsson on current algeras which includes an attempt to find representations of current algebras in higher dimensions. The key idea of this work is that the (untwisted) Kac-Moody algebra is a central extension of the Lie algebra of mappings from the circle to a Lie group. The latter has no nontrivial unitary representations while the central extention has a rich structure of unitary representations. It is hoped that extensions of higher dimensional current algebras would have nontrivial unitary representations. The second additional application is given in this article by Daboul, Daboul and Slodowy where a dynamical algebra of the full Kepler problem was found to be a twisted Kac-Moody algebra.

A survey of the theory of unitary representations of Kac-Moody algebras is given in the following review by: Antony Wasserman

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Unitarity ensures conservation of probability. For example, the Hamiltonian is Hermitian, so the time evolution operator $U(t,t_0) = e^{iH(t-t_0)}$ is unitarity. The unitarity of $U$ corresponds to conservation of probability because the Hamiltonian being Hermitian does.

I'm sure there has been a lot of work on unitary representations of Kac-Moody algebras. I'm not very familiar with the math literature on this, but a math-for-physicists book I have which discuses this cites, unsurprisingly, Kac and Moody.

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I don't think Shizhuo is asking about the "unitary" part so much as the "Kac-Moody" part. –  Qiaochu Yuan Jul 4 '10 at 7:49
    
It's also a consequence in the Kac-Moody case. From the QFT perspective, it's needed in the no-ghost theorems that prevent unitarity from being violated. If we didn't have unitarity, we would break the positive definiteness of our Hilbert space (we would have 'ghosts'). In the Kac-Moody case, the demonstration of this is a little different than my simple example above; but in either case, you can produce negative-norm states through breaking unitarity, which corresponds to a non-Hermitian Hamiltonian. –  jeremy Jul 4 '10 at 8:37

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