A realization of certain modules for the N=4N=4 superconformal algebra and the affine Lie algebra A2(1)A_2 ^{(1)}

We shall first present an explicit realization of the simple $N=4$ superconformal vertex algebra $L_{c} ^{N=4}$ with central charge $c=-9$. This vertex superalgebra is realized inside of the $b c \beta \gamma$ system and contains a subalgebra isomorphic to the simple affine vertex algebra $L_{A_1} (- \tfrac{3}{2} \Lambda_0)$. Then we construct a functor from the category of $L_{c} ^{N=4}$--modules with $c=-9$ to the category of modules for the admissible affine vertex algebra $L_{A_{2} } (-\tfrac{3}{2} \Lambda_0)$. By using this construction we construct a family of weight and logarithmic modules for $L_{c} ^{N=4}$ and $L_{A_{2} } (-\tfrac{3}{2} \Lambda_0)$. We also show that a coset subalgebra of $L_{A_{2} } (-\tfrac{3}{2} \Lambda_0)$ is an logarithmic extension of the $W(2,3)$--algebra with $c=-10$. We discuss some generalizations of our construction based on the extension of affine vertex algebra $L_{A_1} (k \Lambda_0)$ such that $k+2 = 1/p$ and $p$ is a positive integer.Comment: 27 page

A construction of some ideals in affine vertex algebras

Let N_{k} (\g) be a vertex operator algebra (VOA) associated to the generalized Verma module for affine Lie algebra of type $A_{\ell -1} ^{(1)}$ or $C_{\ell} ^{(1)}$. We construct a family of ideals J_{m,n} (\g) in N_{k} (\g), and a family V_{m,n} (\g) of quotient VOAs. These families include VOAs associated to the integrable representations, and VOAs associated to admissible representations at half-integer levels investigated in q-alg/9502015. We also explicitly identify the Zhu's algebras A(V_{m,n} (\g)) and find a connection between these Zhu's algebras and Weyl algebras.Comment: 10 pages, Latex, minor change

Free field realization of the twisted Heisenberg-Virasoro algebra at level zero and its applications

We investigate the free fields realization of the twisted Heisenberg-Virasoro algebra $\mathcal{H}$ at level zero. We completely describe the structure of the associated Fock representations. Using vertex-algebraic methods and screening operators we construct singular vectors in certain Verma modules as Schur polynomials. We completely solve the irreducibility problem for tensor product of irreducible highest weight modules with intermediate series. We also determine the fusion rules for an interesting subcategory of $\mathcal{H}$-modules. Finally, as an application we present a free field realization of the $W(2,2)$-algebra and interpret the $W(2,2)$-singular vectors as $\mathcal{H}$-singular vectors in Verma modules.Comment: To appear in Journal of Pure and Applied Algebra, 24 page

Self-dual and logarithmic representations of the twisted Heisenberg--Virasoro algebra at level zero

This paper is a continuation of arXiv:1405.1707. We present certain new applications and generalizations of the free field realization of the twisted Heisenberg-Virasoro algebra ${\mathcal H}$ at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for ${\mathcal H}$ is presented by combining a bosonic construction of Whittaker modules from arXiv:1409.5354 with a construction of logarithmic modules for vertex algebras. As an application, we prove that there exists a non-split self-extension of irreducible self-dual module which is a logarithmic module of rank two. We construct a large family of logarithmic modules containing different types of highest weight modules as subquotients. We believe that these logarithmic modules are related with projective covers of irreducible modules in a suitable category of ${\mathcal H}$-modules.Comment: 22 pages, 6 figure