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Drazen Adamovic

Publications and source records attributed to Drazen Adamovic.

68 records · Page 4Linked to original sources
arXiv2008

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 πi}{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.

math.QA↗
arXiv2007

On the triplet vertex algebra W(p)

We study the triplet vertex operator algebra $\mathcal{W}(p)$ of central charge $1-\frac{6(p-1)^2}{p}$, $p \geq 2$. We show that $\trip$ is $C_2$-cofinite but irrational since it admits indecomposable and logarithmic modules. Furthermore, we prove that $\trip$ is of finite-representation type and we provide an explicit construction and classification of all irreducible $\mathcal{W}(p)$-modules and describe block decomposition of the category of ordinary $\trip$-modules. All this is done through an extensive use of Zhu's associative algebra together with explicit methods based on vertex operators and the theory of automorphic forms. Moreover, we obtain an upper bound for ${\rm dim}(A(\mathcal{W}(p)))$. Finally, for $p$ prime, we completely describe the structure of $A(\trip)$. The methods of this paper are easily extendable to other $\mathcal{W}$-algebras and superalgebras.

math.QA↗
arXiv2007

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.

math.QA↗
arXiv2007

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.

math.QA↗
arXiv2006

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.

math.QA↗
arXiv2004

A construction of admissible $A_1^{(1)}$-modules of level $-{4/3}$

By using generalized vertex algebras associated to rational lattices, we construct explicitly the admissible modules for the affine Lie algebra $A_1 ^{(1)}$ of level $-{4/3}$. As an application, we show that the W(2,5) algebra with central charge c=-7 investigated in math.QA/0207155 is a subalgebra of the simple affine vertex operator algebra $L(-{4/3}Λ_0)$.

math.QA↗
arXiv2002

Classification of irreducible modules of certain subalgebras of free boson vertex algebra

Let M(1) be the vertex algebra for a single free boson. We classify irreducible modules of certain vertex subalgebras of M(1) generated by two generators. These subalgebras correspond to the W(2, 2p-1)--algebras with central charge $1- 6 \frac{(p - 1) ^{2}}{p}$ where p is a positive integer, $p \ge 2$. We also determine the associated Zhu's algebras.

math.QA↗
arXiv2001

Regularity of certain vertex operator algebras with two generators

For every $m \in {\C} \setminus \{0, -2\}$ and every nonnegative integer $k$ we define the vertex operator (super)algebra $D_{m,k}$ having two generators and rank $ \frac{3 m}{m + 2}$. If $m$ is a positive integer then $D_{m,k}$ can be realized as a subalgebra of a lattice vertex algebra. In this case, we prove that $D_{m,k}$ is a regular vertex operator (super)algebra and find the number of inequivalent irreducible modules.

math.QA↗
arXiv2001

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.

math.QA↗
arXiv1999

Representations of the vertex algebra $W_{1+\infty}$ with a negative integer central charge

Let $\D$ be the Lie algebra of regular differentialoperators on ${\C} \setminus \{0\}$, and ${\hD}= {\D} + {\C} C$ be the central extension of ${\D}$. Let $W_{1+\infty,-N}$ be the vertex algebra associated to the irreducible vacuum $\hD$-module with the central charge $c=-N$. We show that $W_{1+\infty,-N}$ is a subalgebra of the Heisenberg vertex algebra M(1) with $2 N$ generators, and construct 2N-dimensional family of irreducible $W_{1+\infty,-N}$-modules. Considering these modules as $\hD$-modules, we identify the corresponding highest weights.

math.QA↗
arXiv1998

Representations of N=2 superconformal vertex algebra

Let $L_c$ be simple vertex operator superalgebra(SVOA) associated to the vacuum representation of N=2 superconformal algebra with the central charge $c$. Let $c_m = {3m}/{m+2}$. We classify all irreducible modules for the SVOA $L_{c_m}$. When $m$ is an integer we prove that the set of all unitary representations of N=2 superconformal algebra with the central charge $c_m$ provides all irreducible $L_{c_m}$-modules. When $m \notin {\N} $ and $m$ is an admissible rational number we show that irreducible $L_{c_m}$-modules are parameterized with the union of one finite set and union of finitely many rational curves.

math.QA↗
arXiv1998

Some Rational Vertex Algebras

Let $L((n-\tfrac 3 2)Λ_0)$, $n \in \Bbb N$, be a vertex operator algebra associated to the irreducible highest weight module $L((n-\tfrac 3 2)Λ_0)$ for a symplectic affine Lie algebra. We find a complete set of irreducible modules for $L((n-\tfrac 3 2)Λ_0)$ and show that every module for $L((n-\tfrac 3 2)Λ_0)$ from the category $\Cal O$ is completely reducible.

q-alg↗