4. Affine Lie (super-)algebras

Abstract

The theory and applications of Kac-Moody algebras thrived in the last several decades on the interface between mathematics and physics. Kac-Moody algebras were identified first in the stationary Einstein-Maxwell equations [384], in two-dimensional (2D) principal- (or super-) chiral models [192], in 2D sigma models on coset spaces [113], in 2D Heisenberg model [208], in the (anti-) self-dual sector on pure Yang-Mills theory [114], in supersymmetric (N = 4) Yang-Mills theory [565], in completely integrable systems [528], in Toda systems [481] and also as current algebras in two- [570, 124, 439] and three-dimensional [448] models. Naturally, in these pioneer attempts little was used from the representation theory of these algebras (cf. [354] and the references therein). One motivation for our research came from the remarkable papers by the Sato followers [528] which use for the hierarchies of completely integrable systems the most basic representations-the so-called fundamental modules. We recall that every fundamental module appears as the irreducible subquotient of an indecomposable Verma module, the latter being a member of an infinite set of other partially equivalent indecomposable Verma modules. Such a set is called as in the semisimple Lie algebra (SSLA) case a multiplet [153]. However, there are other multiplets besides those containing the fundamental modules. Using an approach developed earlier for semisimple Lie groups or algebras [153] and adapted here for Kac-Moody algebras we classify and parametrize all such multiplets for A⁽¹⁾_ℓ and give them explicitly for A⁽¹⁾₁ and A⁽¹⁾₂ . One motivation for this was the search for new hierarchies of completely integrable systems arising from multiplets which do not contain the fundamental modules. This seems very natural since the setting we describe contains infinitely many differential invariant operators; namely, the intertwining operators which give the partial equivalences in the multiplets become differential operators when we realize the HWM as spaces of functions. Every invariant operator between function spaces gives rise to an invariant equation. These invariant operators and equations were very useful as we know from previous chapters and volumes. Such differential equations also led to the successful treatment of some statistical physics models by Belavin, Polyakov and Zamolodchikov [57] by considering another infinite-dimensional Lie algebra, the Virasoro one. We mention this not only because such a line of research is another of our motivations but also because the representation theory of the Virasoro algebra and the Kac-Moody A₁⁽¹⁾are closely related. The exact correspondence is discussed. A more general relation is that every HWM of a Kac-Moody algebra may be extended to the semidirect product with the Virasoro algebra [533]. The elements of the latter can be realized as bilinear combinations (normally ordered in some sense) of elements of the Kac-Moody algebras, i. e. as elements of the latter universal enveloping algebras. This in turn provides explicit constructions of the Kac-Moody fundamental modules [356]. Thus there is rich ground for interplay between the differential operators invariant under the two algebras. Possible applications of such interplays, e. g. for finding exact formulae for anomalous dimensions and differential equations for the correlation functions were first mentioned in lectures of Polyakov in Shumen (Bulgaria) in August 1984 [496] (see also Todorov [554]). Furthermore, in Knizhnik-Zamolodchikov [392] the simplest one was realized involving a first order linear differential equation. The Kac-Moody algebras encountered by Witten [570] (for even central charge) and by Craigie-Nahm [124] were also good starting points for such applications.