Affine Kac-Moody Algebras and the Wess-Zumino-Witten Model

Affine Kac-Moody Algebras and the Wess-Zumino-Witten Model In 1984, Belavin, Polyakov and Zamolodchikov [1] showed how an infinite-dimensional field theory problem could effectively be reduced to a finite problem, by the presence of
an infinite-dimensional symmetry. The symmetry algebra was the Virasoro algebra, or
two-dimensional conformal algebra, and the field theories studied were examples of twodimensional
conformal field theories. The authors showed how to solve the minimal models
of conformal field theory, so-called because they realise just the Virasoro algebra, and they
do it in a minimal fashion. All fields in these models could be grouped into a discrete, finite
set of conformal families, each associated with a representation of the Virasoro algebra.
This strategy has since been extended to a large class of conformal field theories with
similar structure, the rational conformal field theories (RCFT’s) [2]. The new feature is
that the theories realise infinite-dimensional algebras that contain the Virasoro algebra as
a subalgebra. The larger algebras are known as W-algebras [3] in the physics literature. Thus the study of conformal field theory (in two dimensions) is intimately tied to infinitedimensional algebras. The rigorous framework for such algebras is the subject of vertex (operator) algebras [4] [5]. A related, more physical approach is called meromorphic conformal
field theory [6]. Special among these infinite-dimensional algebras are the affine Kac-Moody algebras (or
their enveloping algebras), realised in the Wess-Zumino-Witten (WZW) models [7]. They are the simplest infinite-dimensional extensions of ordinary semi-simple Lie algebras. Much
is known about them, and so also about the WZW models. The affine Kac-Moody algebras
are the subject of these lecture notes, as are their applications in conformal field theory.
For brevity we restrict consideration to the WZW models; the goal will be to indicate how
the affine Kac-Moody algebras allow the solution of WZW models, in the same way that
the Virasoro algebra allows the solution of minimal models, and W-algebras the solution
of other RCFT’s. We will also give a couple of examples of remarkable mathematical
properties that find an “explanation” in the WZW context.
One might think that focusing on the special examples of affine Kac-Moody algebras is
too restrictive a strategy. There are good counter-arguments to this criticism. Affine KacMoody
algebras can tell us about many other RCFT’s: the coset construction [8] builds a
large class of new theories as differences of WZW models, roughly speaking. Hamiltonian
reduction [9] constructs W-algebras from the affine Kac-Moody algebras. In addition,
many more conformal field theories can be constructed from WZW and coset models by
the orbifold procedure [10] [11]. Incidentally, all three constructions can be understood in
the context of gauged WZW models.
Along the same lines, the question “Why study two-dimensional conformal field theory?”
arises. First, these field theories are solvable non-perturbatively, and so are toy models
that hopefully prepare us to treat the non-perturbative regimes of physical field theories.
Being conformal, they also describe statistical systems at criticality [12]. Conformal field
theories have found application in condensed matter physics [13]. Furthermore, they are
vital components of string theory [14], a candidate theory of quantum gravity, that also
provides a consistent framework for unification of all the forces.
The basic subject of these lecture notes is close to that of [15]. It is hoped, however,
that this contribution will complement that of Gawedzki, since our emphases are quite
different.
The layout is as follows. Section 2 is a brief introduction to the WZW model, including
its current algebra. Affine Kac-Moody algebras are reviewed in Section 3, where some
background on simple Lie algebras is also provided. Both Sections 2 and 3 lay the foundation
for Section 4: it discusses applications, especially 3-point functions and fusion rules.
We indicate how a priori surprising mathematical properties of the algebras find a natural
framework in WZW models, and their duality as rational conformal field theories.