Determinant Formulae of Quasi-Finite Representation of W_{1+\infty} Algebra at Lower Levels

TL;DR

This paper computes the Kac determinant for quasi-finite \\Winf algebra representations, revealing null states at integer central charges and proposing character formulas linked to three-dimensional free fields with unique modular properties.

Contribution

It provides explicit determinant calculations, null state constructions, and character formulas for \\Winf algebra representations at lower levels, advancing understanding of their structure.

Findings

Kac determinant vanishes at integer central charge

Null states are explicitly constructed

Character formulas relate to 3D free fields with exotic modular properties

Abstract

We calculate the Kac determinant for the quasi-finite representation of \Winf algebra up to level 8. It vanishes only when the central charge is integer. We give an algebraic construction of null states and propose the character formulae. The character of the Verma module is related to free fields in three dimensions which has rather exotic modular properties.