On parafermion vertex algebras of $\mathfrak{sl}(2{)}_{-3/2}$ and $\mathfrak{sl}(3{)}_{-3/2}$

Abstract

We study parafermion vertex algebras $N_{-3/2}(\mathfrak{sl}(2))$ and $N_{-3/2}(\mathfrak{sl}(3))$. Using the isomorphism between $N_{-3/2}(\mathfrak{sl}(3))$ and the logarithmic vertex algebra ${\mathcal{W}}^0(2{)}_{A_2}$ from [2], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra $\mathcal{W}(2,3)$ of central charge $c=-10$, and that $N_{-3/2}(\mathfrak{sl}(3))$ is a direct sum of irreducible $N_{-3/2}(\mathfrak{sl}(2))$-modules. As a byproduct, we prove certain conjectures about the vertex algebra ${\mathcal{W}}^0(p{)}_{A_2}$. We also obtain a vertex-algebraic proof of the irreducibility of a family of $\mathcal{W}(2,3{)}_c$ modules at $c=-10$.

22 pages

References in corpus (4)

• Logarithmic CFTs connected with simple Lie algebras
• On the Feigin-Tipunin conjecture
• Semi-infinite cohomology and the linkage principle for $W$-algebras
• A note on the "logarithmic-W_3" octuplet algebra and its Nichols algebra

Cited by in corpus (1)

• A duality between vertex superalgebras $L_{-3/2}(\mathfrak{osp}(1|2))$ and ${\mathcal{V}}^{(2)}$ and generalizations to logarithmic vertex algebras