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 a Lie group to the circle. 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

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