The structure of Zhu's algebras for certain W-algebras

Abstract

We introduce a new approach that allows us to determine the structure of Zhu's algebra for certain vertex operator (super)algebras which admit horizontal $\mathbb{Z}$-grading. By using this method and an earlier description of Zhu's algebra for the singlet W-algebra, we completely describe the structure of Zhu's algebra for the triplet vertex algebra W(p). As a consequence, we prove that Zhu's algebra A(W(p)) and the related Poisson algebra P(W(p)) have the same dimension. We also completely describe Zhu's algebras for the N=1 triplet vertex operator superalgebra SW(m). Moreover, we obtain similar results for the c=0 triplet vertex algebra W_{2,3} important in logarithmic conformal field theory. Because our approach is "internal" we had to employ several constant term identities for purposes of getting right upper bounds on dimension of Zhu's algebras.

arXiv comment27 pages, a few changes, to appear in Advances in Mathematics

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• Logarithmic tensor product theory for generalized modules for a conformal vertex algebra, Part I
• The N=1 triplet vertex operator superalgebras
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Cited by in corpus (6)

• A modular invariant bulk theory for the c=0 triplet model
• Logarithmic W-algebras and Argyres-Douglas theories at higher rank
• The Nichols algebra of screenings
• Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models
• What the characters of irreducible subrepresentations of Jordan cells can tell us about LCFT
• On W-algebra extensions of (2,p) minimal models: p > 3