Highest weight representations of the virasoro algebra

Abstract

The Virasoro algebra plays a fundamental role in modern research areas such as condensed matter physics or quantum gravity. Applying the theory requires a deep understanding of the underlying mathematical structure. Here, we define the Virasoro algebra as the unique central extension to the Witt algebra. The investigation of the Virasoro algebra leads to Verma modules and the Hermitian Shapovalov form which is used to define unitary highest weight representations. Subsequently, we investigate the Hermitian form and compute an explicit expression for the Kac-determinant. We use the determinant formula to obtain first results about the classification of unitary highest weight representations. To complete the classification, we explicitly construct unitary highest weight representations of the Virasoro algebra from factor algebras of affine Lie algebras. We conclude the investigation with some calculations regarding the Ising model and a short introduction to the applications of the Virasoro algebra in condensed matter physics and gravity.