arXiv subjects

Drazen Adamovic

Publications and source records attributed to Drazen Adamovic.

At least 37 records · Page 2Linked to original sources

arXiv2020

The $N=1$ super Heisenberg-Virasoro vertex algebra at level zero

We study the representation theory of the N=1 super Heisenberg-Virasoro vertex algebra at level zero, which extends the previous work on the Heisenberg-Virasoro vertex algebra arXiv:math/0201314, arXiv:1405.1707 and arXiv:1703.00531 to the super case. We calculated all characters of irreducible highest weight representations by investigating certain Fock space representations. Quite surprisingly, we found that the maximal submodules of certain Verma modules are generated by subsingular vectors. The formulas for singular and subsingular vectors are obtained using screening operators appearing in a study of certain logarithmic vertex algebras in arXiv:0908.4053.

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arXiv2020

Bershadsky-Polyakov vertex algebras at positive integer levels and duality

We study the simple Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_θ)$ at positive integer levels and classify their irreducible modules. In this way we confirm the conjecture from arXiv:1910.13781. Next, we study the case $k=1$. We discover that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple afine vertex superalgebra $L_{k'} (osp(1 \vert 2))$ for $k'=-5/4$. Using the free-field realization of $L_{k'} (osp(1 \vert 2))$ from arXiv:1711.11342, we get a free-field realization of $\mathcal W_k$ and their highest weight modules. In a sequel, we plan to study fusion rules for $\mathcal W_k$.

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arXiv2020

On Zhu's algebra and $C_2$--algebra for symplectic fermion vertex algebra $SF(d)^+$

In this paper, we study the family of vertex operator algebras $SF(d)^+$, known as symplectic fermions. This family is of a particular interest because these VOAs are irrational and $C_2$-cofinite. We determine the Zhu's algebra $A(SF(d)^+)$ and show that the equality of dimensions of $A(SF(d)^+)$ and the $C_2$--algebra $\mathcal P(SF(d)^+)$ holds for $d \geq 2$ (the case of $d=1$ was treated by T. Abe). We use these results to prove a conjecture by Y. Arike and K. Nagatomo on the dimension of the space of one-point functions on $SF(d)^+$.

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arXiv2020

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

We study parafermion vertex algebras $N_{-3/2}(\frak{sl}(2))$ and $N_{-3/2}(\frak{sl}(3))$. Using the isomorphism between $N_{-3/2}(\frak{sl}(3))$ 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 $N_{-3/2}(\frak{sl}(3))$ is a direct sum of irreducible $N_{-3/2}(\frak{sl}(2))$-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 $c=-10$.

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arXiv2020

The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$

The vertex algebras $V^{(p)}$ and $R^{(p)}$ introduced in [2] are very interesting relatives of the famous triplet algebras of logarithmic CFT. The algebra $V^{(p)}$ (respectively, $R^{(p)}$) is a large extension of the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ (respectively, $L_k(\mathfrak{sl}_2)$ times a Heisenberg algebra), at level $k=-2+1/p$ for positive integer $p$. In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on $V^{(p)}$ and we decompose $V^{(p)}$ as an $L_k(\mathfrak{sl}_2)$-module and $R^{(p)}$ as an $L_k(\mathfrak{gl}_2)$-module. The decomposition of $V^{(p)}$ shows that $V^{(p)}$ is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of $V^{(p)}$ is the logarithmic doublet algebra $A^{(p)}$ introduced in [12], while the reduction of $R^{(p)}$ yields the $B^{(p)}$-algebra of [39]. Conversely, we realize $V^{(p)}$ and $R^{(p)}$ from $A^{(p)}$ and $B^{(p)}$ via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category $KL_{k}$ of ordinary $L_k(\mathfrak{sl}_2)$-modules at level $k=-2+1/p$ is a rigid vertex tensor category equivalent to a twist of the category Rep$(SU(2))$. This finally completes rigid braided tensor category structures for $L_k(\mathfrak{sl}_2)$ at all levels $k$. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both $R^{(p)}$ and $B^{(p)}$ are certain non-principal W-algebras of type $A$ at boundary admissible levels. The same uniqueness result also shows that $R^{(p)}$ and $B^{(p)}$ are the chiral algebras of Argyres-Douglas theories of type $(A_1, D_{2p})$ and $(A_1, A_{2p-3})$.

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arXiv2019

Classification of irreducible modules for Bershadsky-Polyakov algebra at certain levels

We study the representation theory of the Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_θ)$. In particular, Zhu algebra of $\mathcal W_k$ is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector. We classify all modules in the category $\mathcal{O}$ for the Bershadsky-Polyakov algebra $\mathcal W_k$ when $k=-5/3, -9/4, -1,0$. In the case $k=0$ we show that the Zhu algebra $A(\mathcal W_k)$ has $2$--dimensional indecomposable modules.

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arXiv2019

On irreducibility of modules of Whittaker type for cyclic orbifold vertex algebra

We extend the Dong-Mason theorem on the irreducibility of modules for orbifold vertex algebras from [C. Dong, G. Mason, Duke Math. J. 86 (1997)] 305-321] for the category of weak modules. Let $V$ be a vertex operator algebra, $g$ an automorphism of order $p$. Let $W$ be an irreducible weak $V$--module such that $W,W\circ g,\dots,W\circ g^{p-1}$ are inequivalent irreducible modules. We prove that $W$ is an irreducible weak $V^{\left\langle g\right\rangle }$-module. This result can be applied on irreducible modules of certain Lie algebra $\mathfrak L$ such that $W,W\circ g,\dots,W\circ g^{p-1}$ are Whittaker modules having different Whittaker functions. We present certain applications in the cases of the Heisenberg and Weyl vertex operator algebras.

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arXiv2019

On fusion rules and intertwining operators for the Weyl vertex algebra

In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way we confirm the conjecture on fusion rules based on the Verlinde algebra. We explicitly construct intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra $\widehat{gl(1 \vert 1)}$.

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arXiv2019

Realizations of simple affine vertex algebras and their modules: the cases $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$

We study embeddings of the simple admissible affine vertex algebras $V_k(sl(2))$ and $V_k(osp(1,2))$, $k \notin {\Bbb Z}_{\ge 0}$, into the tensor product of rational Virasoro and $N=1$ Neveu-Schwarz vertex algebra with lattice vertex algebras. We prove that the admissible affine vertex algebra $V_k(sl(2))$ can be embedded into vertex algebra $L^{Vir} (c_{p,p'}, 0) \otimes Π(0)$ where $L^{Vir} (c_{p,p'}, 0) $ is suitable minimal Virasoro vertex algebra and $Π(0)$ is a vertex algebra of lattice type. By using these realizations we construct a family of weight, logarithmic and Whittaker $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$--modules. As an application, we construct all irreducible degenerate Whittaker modules for $V_k(sl(2))$.

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arXiv2018

On some vertex algebras related to $V_{-1}(\frak{sl} (n) )$ and their characters

We consider several vertex operator (super)algebras closely related to $V_{-1}(\frak{sl} (n) )$, $n \ge 3$ : (a) the parafermionic subalgebra $K(\frak{sl}(n),-1)$ for which we completely describe its inner structure, (b) the vacuum algebra $Ω(V_{-1}(\frak{sl} (n) ) )$, and (c) an infinite extension $\mathcal U$ of $V_{-1}(\frak{sl} (n) )$ constructed by combining certain irreducible ordinary modules with integral weights. It turns out that $\mathcal U$ is isomorphic to the coset vertex algebra $\frak{psl}(n|n) _1 / \frak{sl}(n)_1$, $n \ge 3$. We show that $V_{-1}(\frak{sl}(n))$ admits precisely $n$ ordinary irreducible modules, up to isomorphism. This leads to the conjecture that ${\mathcal U}$ is {\em quasi-lisse}. We present evidence in support of this conjecture: we prove that the (super)character of $\mathcal U$ is quasi-modular of weight one by virtue of being the constant term of a meromorphic Jacobi form of index zero. Explicit formulas and MLDE for characters and supercharacters are given for $\frak{g}=\frak{sl}(3)$ and outlined for general $n$. We present a conjectural family of 2nd order MLDEs for characters of vertex algebras $\frak{psl}(n|n) _1$, $n \geq 2$. We finish with a theorem pertaining to characters of $\frak{psl}(n|n)_1$ and $\mathcal U$-modules.

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arXiv2018

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.

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arXiv2018

Kostant's pair of Lie type and conformal embeddings

We deal with some aspects of the theory of conformal embeddings of affine vertex algebras, providing a new proof of the Symmetric Space Theorem and a criterion for conformal embeddings of equal rank subalgebras. We finally study some examples of embeddings at the critical level. We prove a criterion for embeddings at the critical level which enables us to prove equality of certain central elements.

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arXiv2017

On classification of non-equal rank affine conformal embeddings and applications

We complete the classification of conformal embeddings of a maximally reductive subalgebra $\mathfrak k$ into a simple Lie algebra $\mathfrak g$ at non-integrable non-critical levels $k$ by dealing with the case when $\mathfrak k$ has rank less than that of $\mathfrak g$. We describe some remarkable instances of decomposition of the vertex algebra $V_{k}(\mathfrak g)$ as a module for the vertex subalgebra generated by $\mathfrak k$. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings $A_1 \times A_1 \hookrightarrow C_3$ at level $k=-1/2$, and obtain explicit branching rules by applying certain $q$-series identity. In the analysis of conformal embedding $A_1 \times D_4 \hookrightarrow C_8$ at level $k=-1/2$ we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.

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arXiv2017

A note on the affine vertex algebra associated to $\frak{gl}(1 \vert 1)$ at the critical level and its generalizations

In this note we present an explicit realization of the affine vertex algebra $V^{cri}(\frak{gl}(1 \vert 1)) $ inside of the tensor product $F\otimes M$ where $F$ is a fermionic verex algebra and $M$ is a commutative vertex algebra. This immediately gives an alternative description of the center of $V^{cri}(\frak{gl}(1 \vert 1) ) )$ as a subalgebra $M _ 0$ of $M$. We reconstruct the Molev-Mukhin formula for the Hilbert-Poincare series of the center of $V^ {cri}(\frak{gl}(1 \vert 1) )$. Moreover, we construct a family of irreducible $V^{cri}(\frak{gl}(1 \vert 1))$ -modules realized on $F$ and parameterized by $χ^+, χ^- \in {\Bbb C}((z)). $ We propose a generalization of $V^ {cri}(\frak{gl}(1 \vert 1))$ as a critical level version of the super $\mathcal W_{1+\infty}$ vertex algebra.

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arXiv2017

Conformal embeddings of affine vertex algebras in minimal $W$-algebras II: decompositions

We present methods for computing the explicit decomposition of the minimal simple affine $W$-algebra $W_k(\mathfrak g, θ)$ at a conformal level $k$ as a module for its maximal affine subalgebra $\mathcal V_k(\mathfrak g^{\natural})$. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when $\mathfrak g^{\natural}$ is a semisimple Lie algebra, we show that, for a suitable conformal level $k$, $W_k(\mathfrak g, θ)$ is isomorphic to an extension of $\mathcal V_k(\mathfrak g^{\natural})$ by its simple module. We are able to prove that in certain cases $W_k(\mathfrak g, θ)$ is a simple current extension of $\mathcal V_k(\mathfrak g^{\natural})$. In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple $W$-algebra $W_{k}(sl(4), θ)$ at $k=-8/3$. We prove, as conjectured in arXiv:1407.1527, that $W_{k}(sl(4), θ)$ is isomorphic to the vertex algebra $\mathcal R^{(3)}$, and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra $V_k (sl(n))$ at certain admissible levels and for $V_k (sl(m | n)), m\ne n, m,n\geq 1$ at arbitrary levels.

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arXiv2017

On Free Field Realizations of $W(2,2)$-Modules

The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra $\mathcal H$ at level zero as modules for the $W(2,2)$-algebra by using construction from [J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:1405.1707]. We prove that the irreducible highest weight ${\mathcal H}$-module is irreducible as $W(2,2)$-module if and only if it has a typical highest weight. Finally, we construct a screening operator acting on the Heisenberg-Virasoro vertex algebra whose kernel is exactly $W(2,2)$ vertex algebra.

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arXiv2016

Some applications and constructions of intertwining operators in LCFT

We discuss some applications of fusion rules and intertwining operators in the representation theory of cyclic orbifolds of the triplet vertex operator algebra. We prove that the classification of irreducible modules for the orbifold vertex algebra W(p)^{A_m} follows from a conjectural fusion rules formula for the singlet vertex algebra modules. In the p=2 case, we computed fusion rules for the irreducible singlet vertex algebra modules by using intertwining operators. This result implies the classification of irreducible modules for W(2)^{A_m}, conjectured previously in [4]. The main technical tool is a new deformed realization of the triplet and singlet vertex algebras, which is used to construct certain intertwining operators that can not be detected by using standard free field realizations.

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arXiv2016

Conformal embeddings of affine vertex algebras in minimal $W$-algebras I: structural results

We find all values of $k\in \mathbb C$, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra $W_k(\mathfrak g,θ)$ is conformal, where $\mathfrak g$ is a basic simple Lie superalgebra and $-θ$ its minimal root. In particular, it turns out that if $W_k(\mathfrak g,θ)$ does not collapse to its affine part, then the possible values of these $k$ are either $-\frac{2}{3} h^\vee$ or $-\frac{h^\vee-1}{2}$, where $h^\vee$ is the dual Coxeter number of $\mathfrak g$ for the normalization $(θ,θ)=2$. As an application of our results, we present a realization of simple affine vertex algebra $V_{-\tfrac{n+1}{2} } (sl(n+1))$ inside of the tensor product of the vertex algebra $W_{\tfrac{n-1}{2}} (sl(2| n), θ)$ (also called the Bershadsky-Knizhnik algebra) with a lattice vertex algebra.

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