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Progress of Theoretical Physics Supplement No. 110, 1992 issing Link :between Virasoro and sl 2) Kac Moody Algebras Mitsuhiro KATO and Yasuhiko YAMADA Institute of Physics, University. of Tokyo, Komaba, Tokyo 153 *Nat ional Laboratory fo r High Energy Phy sics KEK), Tsukuba 305 e discuss the representations of the fractional level sl 2) affine Kac-Moody algebra and their relation to the Virasoro representations. A pract ical way of calcula tion of the free fi el d null vectors and explicit relation between those .for both algebras are given with many examples. or the complete degenerate representations some subtle points about the repre sentations on and beyond the boundaries of the conformal grids are clarified. § 1 Introduction 291 Virasoro algebra and sl 2) affine Kac-Moody algebra are the simplest examples of infinite dimensional Lie algebras and have been studied extensively from both mathematical and physical points of view. n important relationship between Virasoro and sl 2) Kac-Moody algebra is ·known as hamiltonian reduction. 1 > a> Starting from the latter with level k, k+2=pfq, y imposing a constraint on one of the sl 2) currents, one obtai n the Virasoro algebra with central charge c=l-6 p-q)2jpq. So far general framework for the correspondence of these algebras is well established. or the representations, however, there seems to be some uncertainty in the literature. ·Let us consider, for instance, minimal 4 > and admissible 5 > 6 > epresentations for each s s q ------------, q -·----------- I I • • • • • • • • I I I • • • • • • • • I I • • • • • • • • I I I. r r a) Virasoro b) sl 2) Fig. 1. Grids for the minimal representations for the Virasoro algebra a), and admissible repre sentations for t h ~ algebra b). b y g u e s t o n J a n u a r y 1 7 , 2 0 1 4 h t t p : / / p t p s . o x f o r d j o u r n a l s . o r g / D o w n l o a d e d f r o m
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