Logarithmic intertwining operators and W(2,2p-1)-algebras

For every $p \geq 2$, we obtained an explicit construction of a family of $\mathcal{W}(2,2p-1)$-modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual $\mathcal{W}(2,2p-1)$-modules, we described their internal structure, and computed their graded dimensions. In addition, we constructed certain hidden logarithmic intertwining operators among two ordinary and one logarithmic $\mathcal{W}(2,2p-1)$-modules. This work, in particular, gives a mathematically precise formulation and interpretation of what physicists have been referring to as "logarithmic conformal field theory" of central charge $c_{p,1}=1-\frac{6(p-1)^2}{p}, p \geq 2$. Our explicit construction can be easily applied for computations of correlation functions. Techniques from this paper can be used to study the triplet vertex operator algebra $\mathcal{W}(2,(2p-1)^3)$ and other logarithmic models.Comment: 22 pages; v2: misprints corrected, other minor changes. Final version to appear in Journal of Math. Phy

An explicit realization of logarithmic modules for the vertex operator algebra W_{p,p'}

By extending the methods used in our earlier work, in this paper, we present an explicit realization of logarithmic \mathcal{W}_{p,p'}-modules that have L(0) nilpotent rank three. This was achieved by combining the techniques developed in \cite{AdM-2009} with the theory of local systems of vertex operators \cite{LL}. In addition, we also construct a new type of extension of $\mathcal{W}_{p,p'}$, denoted by $\mathcal{V}$. Our results confirm several claims in the physics literature regarding the structure of projective covers of certain irreducible representations in the principal block. This approach can be applied to other models defined via a pair screenings.Comment: 18 pages, v2: one reference added, other minor change

Logarithmic vertex algebras related to sp(4)

We present several results and conjectures pertaining to parafermion vertex algebra and related logarithmic vertex algebras. Starting from the tensor product of two copies of the singlet vertex algebra M(2), we consider various subalgebras that appear in its decomposition including N-1(sl(2)) and its Z2-fixed point algebra, and the S2-symmetric orbifold of the singlet vertex algebra M(2). In particular, we show that N-1(sl(2)) has an extension to a W-algebra of type (2, 3, 4, 5, 6, 7, 8). Finally we state some conjectures about singlet and triplet type W-algebras of type sp(4) and their characters

On coset vertex algebras with central charge 1

We present a coset realization of the vertex operator algebra $V_L ^+$ with central charge $1$. We investigate the vertex operator algebra $V_{Z sqrt{2n}} ^+$ (resp. $V_{2Z sqrt{2n+1}} ^+$) as a vertex subalgebra of $L_{D_n ^{(1)}}(Lambda _0) otimes L_{D_n ^{(1)}}(Lambda _0)$ (resp. $L_{B_n ^{(1)}}(Lambda _0) otimes L_{B_n ^{(1)}}(Lambda _0)$). Our construction is based on the boson-fermion correspondence and certain conformal embeddings

Representations of certain non-rational vertex operator algebras of affine type

In this paper we study a series of vertex operator algebras of integer level associated to the affine Lie algebra $A_{\ell}^{(1)}$. These vertex operator algebras are constructed by using the explicit construction of certain singular vectors in the universal affine vertex operator algebra $N(n-2,0)$ at the integer level. In the case $n=1$ or $l=2$, we explicitly determine Zhu's algebras and classify all irreducible modules in the category $\mathcal{O}$. In the case $l=2$, we show that the vertex operator algebra $N(n-2,0)$ contains two linearly independent singular vectors of the same conformal weight.Comment: 15 pages, LaTeX; final version, to appear in J. Algebr

Conformal embeddings in affine vertex superalgebras

This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra $V_k(\mathfrak g)$ where $\mathfrak g=\mathfrak g_{\bar 0}\oplus \mathfrak g_{\bar 1}$ is a basic classical simple Lie superalgebras. Let $\mathcal V_k (\mathfrak g_{\bar 0})$ be the subalgebra of $V_k(\mathfrak g)$ generated by $\mathfrak g_{\bar 0}$. We first classify all levels $k$ for which the embedding $\mathcal V_k (\mathfrak g_{\bar 0})$ in $V_k(\mathfrak g)$ is conformal. Next we prove that, for a large family of such conformal levels, $V_k(\mathfrak g)$ is a completely reducible $\mathcal V_k (\mathfrak g_{\bar 0})$--module and obtain decomposition rules. Proofs are based on fusion rules arguments and on the representation theory of certain affine vertex algebras. The most interesting case is the decomposition of $V_{-2} (osp(2n +8 \vert 2n))$ as a finite, non simple current extension of $V_{-2} (D_{n+4}) \otimes V_1 (C_n)$. This decomposition uses our previous work [10] on the representation theory of $V_{-2} (D_{n+4})$.Comment: Latex file, 45 pages, to appear in Advances in Mathematic

Fusion rules and complete reducibility of certain modules for affine Lie algebras

We develop a new method for obtaining branching rules for affine Kac-Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in Adamovi\'c and O. Per\v{s}e (2008) is closed under fusion. Then we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type $A_{\ell-1}^{(1)}$, obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for $\ell \ge 3$. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type $C_{\ell}^{(1)}$. Next we notice that the category of $D_{2 \ell -1}^{(1)}$ modules at level $- 2 \ell +3$ obtained in Per\v{s}e (2012) has the isomorphic fusion algebra. This enables us to decompose certain $E_6 ^{(1)}$ and $F_4 ^{(1)}$--modules at negative levels.Comment: 18 pages; final version, to appear in Journal of Algebra and Its Application

Defining relations for minimal unitary quantum affine W-algebras

We prove that any unitary highest weight module over a universal minimal quantum affine $W$-algebra at non-critical level descends to its simple quotient. We find the defining relations of the unitary simple minimal quantum affine $W$-algebras and the list of all their irreducible positive energy modules. We also classify all irreducible highest weight modules for the simple affine vertex algebras in the cases when the associated simple minimal $W$-algebra is unitary.Comment: Latex file, 24 pages, revised versio

Lie superalgebras and irreducibility of A_1^(1)-modules at the critical level

We introduce the infinite-dimensional Lie superalgebra ${\mathcal A}$ and construct a family of mappings from certain category of ${\mathcal A}$-modules to the category of A_1^(1)-modules of critical level. Using this approach, we prove the irreducibility of a family of A_1^(1)-modules at the critical level. As a consequence, we present a new proof of irreducibility of certain Wakimoto modules. We also give a natural realizations of irreducible quotients of relaxed Verma modules and calculate characters of these representations.Comment: 21 pages, Late

The N=1 triplet vertex operator superalgebras

We introduce a new family of C_2-cofinite N=1 vertex operator superalgebras SW(m), $m \geq 1$, which are natural super analogs of the triplet vertex algebra family W(p), $p \geq 2$, important in logarithmic conformal field theory. We classify irreducible SW(m)-modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible SW(m) characters. Finally, we contemplate possible connections between the category of SW(m)-modules and the category of modules for the quantum group U^{small}_q(sl_2), q=e^{\frac{2 \pi i}{2m+1}}, by focusing primarily on properties of characters and the Zhu's algebra A(SW(m)). This paper is a continuation of arXiv:0707.1857.Comment: 53 pages; v2: references added; v3: a few changes; v4: final version, to appear in CM