Oscillator Representation of Virasoro Algebra and Kac Determinant

Abstract

An algebraic relation between the harmonic oscillator representation of the 2-dimensional conformal algebra and its generic expression in terms of the Virasoro operators themselves is established utilizing the existence condition of the singular vertex operators which were recently constructed by the present authors. A new derivation of the Kac determinant formula is also given from the viewpoint of the oscillator representation with a variable central charge extension. 2 ) of infinite dimensional affine Lie algebras were initiated by the study of the Virasoro algebra, first discovered in the dual resonance models,3) which also has the 2-D string origin. The Virasoro algebra generates conformal transformations in the 2-D parameter space of a string. In our recent paper 4 ) we studied an oscillator representation of the 2-D conformal algebra (Virasoro algebra) and gave a general construction of null fields making use of the vertex operators. There, the null states are expressed in terms of the mode operators of harmonic oscillators. On the other hand, a null state is expected to be constructed generically from a primary state by multiplying the Virasoro operators to it. To achieve this construction, the null states given by the singular vertex opera­ tors 4 ) must be reexpressed in terms of the Virasoro operators. It is the purpose of the present paper to show explicitly the algebraic relation between the oscillator representation and the generic expression given in terms of the Virasoro operators. In addition, we will derive the Kac 5 ) determinant from the viewpoint of the oscillator representation.