On <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>-cofiniteness of parafermion vertex operator algebras

The Bershadsky-Polyakov algebra is the W-algebra associated to sl 3 with its minimal nilpotent element f θ . For notational convenience we define W ℓ = W ℓ−3/2 (sl 3 , f θ ). The simple quotient of W ℓ is denoted by W ℓ , and for ℓ a positive integer, W ℓ is known to be C 2 -cofinite and rational. We prove that for all positive integers ℓ, W ℓ contains a rank one lattice vertex algebra V L , and that the coset C ℓ = Com(V L , W ℓ ) is isomorphic to the principal, rational W(sl 2ℓ )-algebra at level (2ℓ + 3)/(2ℓ + 1) − 2ℓ. This was conjectured in the physics literature over 20 years ago. As a byproduct, we construct a new family of rational, C 2 -cofinite vertex superalgebras from W ℓ .

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Schur–weyl Duality for Heisenberg Cosets

Creutzig

Kanade

Linshaw

et al. 2018

Transformation Groups

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Let V be a simple vertex operator algebra containing a rank n Heisenberg vertex algebra H and let C = Com (H,V) be the coset of H in V. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of C are found and every simple C-module is shown to be contained in at least one V-module. A corollary of this is that if V is rational and C 2 -cofinite and CFT-type, and Com (C,V) is a rational lattice vertex operator algebra, then so is C. These results are illustrated with many examples and the C 1 -cofiniteness of certain interesting classes of modules is established. is exact, for M ′′ ∼ = (J ⊠ M)/M ′ = 0. But, fusion is right-exact [HLZ, Prop. 4.26], sois exact. However, M ′′ = 0 implies that J −1 ⊠ M ′′ is a non-zero quotient of M, by (1), so we must have J −1 ⊠ M ′′ ∼ = M, as M is simple. Fusing with J now gives J ⊠ M ∼ = M ′′ , so we conclude that M ′ = 0 and that J ⊠ M is simple. The simplicity of J ∼ = J ⊠ V now follows from that of V, completing the proof of (3).To prove (4), note that applying right-exactness to the short exact sequence 0where f is the induced map from J ⊠ M ′ to J ⊠ M that might not be an inclusion. Fusing with J −1 and applying (2.4), we arrive at 5)

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On C2-cofiniteness of parafermion vertex operator algebras

Cited by 28 publications

References 25 publications

Singular Vectors and Zhu’s Poisson Algebra of Parafermion Vertex Operator Algebras

Singular Vectors and Zhu’s Poisson Algebra of Parafermion Vertex Operator Algebras

Cosets of Bershadsky–Polyakov algebras and rational $${\mathcal W}$$ W -algebras of type A

Schur–weyl Duality for Heisenberg Cosets

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