Unitary Representations of $W$ Infinity Algebras

Abstract

We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of $W$ infinity algebras ($W_{\infty }$, $W_{1+\infty }$, $W_{\infty }^{1,1}$, $W_{\infty }^M$, $W_{1+\infty }^N$, $W_{\infty }^{M,N}$) with central charges ($2$, $1$, $3$, $2M$, $N$, $2M+N$). The characters of these representations are computed. We construct a new extended superalgebra $W_{\infty }^{M,N}$, whose bosonic sector is $W_{\infty }^M\oplus W_{1+\infty }^N$. Its representations obtained from a free field realization with central charge $2M+N$, are classified into two classes: continuous series and discrete series. For the former there exists a supersymmetry, but for the latter a supersymmetry exists only for $M=N$.

20 pages

Cited by in corpus (12)

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• Free Fields and Quasi-Finite Representation of $W_{1+\infty }$ Algebra
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• Quasifinite Highest Weight Modules over Super $W_{1+\infty }$ Algebra
• Subalgebras of $W_{1+\infty }$ and Their Quasifinite Representations
• The $\mathcal{N}=2$ Supersymmetric $w_{1+\infty }$ Symmetry in the Two-Dimensional SYK Models
• The $\mathcal{N}=4$ Coset Model and the Higher Spin Algebra
• The $\mathcal{N}=4$ Higher Spin Algebra for Generic $\mu$ Parameter
• The Structure of the $\mathcal{N}=4$ Supersymmetric Linear $W_{\infty }[\lambda ]$ Algebra