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

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.

arXiv commentYITP/K-1054, UT-669, SULDP-1994-1, (11 pages, LaTeX file)

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• Non-relativistic Fermions, Coadjoint Orbits of \winf\ and String Field Theory at $c=1$
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• Unifying W-Algebras
• Determinant and Character of W-infinity algebra
• Crystal Melting and Black Holes
• W(1+infinity) algebra as a symmetry behind AGT relation
• The $W_{1+\infty }$ effective theory of the Calogero- Sutherland model and Luttinger systems.
• Generalized MacMahon G(q) as q-deformed CFT Correlation Function
• Quasifinite Highest Weight Modules over Super $W_{1+\infty }$ Algebra
• Subalgebras of $W_{1+\infty }$ and Their Quasifinite Representations
• W-algebras from Heisenberg categories
• Trace of the twisted Heisenberg Category
• Effective short distance interaction in Calogero-Sutherland quantum fluids
• Quantum dynamics of the effective field theory of the Calogero-Sutherland model
• Swimming of Microorganism and the String- and Membrane- like Algebra
• Quantum Mechanics for the Swimming of Micro-Organism in Two Dimensions