Kac-Moody groups in two-dimensional field theory

Abstract

Loop groups and their central extensions in relation to Kac-Moody algebras have recently attracted great interest. The theory of loop groups is important among others in understanding completely integrable non-linear systems of KdV type [1], constructing solutions for self-dual Yang-MiIls equations [2], string rnodels in particle physics [3] and anomaly problems in quantum field theory [4]. In this talk the non trivial U(1) bundle P on the space f2G ofloops in a Lie group was studied. This bundle has a Lie group structure such that the corresponding Lie algebra is a Kac-Moody algebra based on the Lie algebra of G (in physics the Lie algebra of P is a current algebra in I + 1 dimensions). The structure of the group P has earlier been described in [1], [5] and [6]. In this talk a new and simple construction of P was explained, [7]. The two-dimensional non-abelian gauge anomaly plays a crucial role in the construction, which makes apparent the connection with the theory of Weyl fermions in two-dimensions. The group P can be written as P = DG x U(1)/f#, where DG is the group of G-valued mappings in the unit disc D (point-wise multiplication) and f# is the gauge group consisting of elements of DG which are equal to one on the boundary S t = ~D. The action of ~# on DG x U(1) is determined by a one-cocycle (= the non-abelian anomaly) and the product in DG x U(1) is a modification of the obvious product by a topological]y trivial but algebraically non-trivial two-cocycle. The construction can be generalized to higher dimensions. There is one important difference, however, as compared to 1 + 1 dimensional field theory. The abelian extension of the current algebra (Hamiltonian formalism, in three space dimensions) is no more central and the Schwinger terms are also functions of the vector potential, [8]. Because of the latter fact the extension of the current algebra is not just by U(1) but by the abelian group consisting of U(1)-valued functions in the space of vector potentials in a three-dimensional space. On the group level the Schwinger terms modify the structure of the group of gauge transformations; a second physically important consequence of the Schwinger terms is that the Gauss law cannot be implemented in the quantized theory.