Kac-Moody group representations and generalization of the Sugawara construction of the Virasoro algebra
In the present paper the authors use a previously developed perturbative technique for calculating group laws. The group they apply their techniques to is the infinite-dimensional Lie group with as Lie algebra the semidirect product of the Virasoro algebra V with the affine Kac- Moody algebra of sl(2,\({\mathbb{R}})\) (i.e. the Kac-Moody algebra \(A_ 1^{(1)}).\) They give the commutation relations of \(\tilde V\times A_ 1^{(1)}\), parametrized so as to include all nontrivial central extensions. They classify all possible characteristic subalgebras, leading to different polarizations and then to different representations with a determined vacuum symmetry. The corresponding group law is given as a power series in the parameters (up to fourth order). Making use of this they can give perturbative expressions for the wave functions of representations of the group, and then of the actions of the Lie algebra generators on the wave functions. The highest weight representations considered here are associated with strings on SU(2).
scientific article; zbMATH DE number 4073236
scientific article