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 extention.