arXiv subjects

Drazen Adamovic

Publications and source records attributed to Drazen Adamovic.

At least 55 records · Page 3Linked to original sources

arXiv2016

Finite vs infinite decompositions in conformal embeddings

Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras $V_{\mathbf{k}}(\mathfrak g^0)\subset V_{k}(\mathfrak g)$, corresponding to an embedding of a maximal equal rank reductive subalgebra $\mathfrak g^0$ into a simple Lie algebra $\mathfrak g$, is conformal if and only if the corresponding central charges are equal. We classify the equal rank conformal embeddings. Furthermore we describe, in almost all cases, when $V_{k}(\mathfrak g)$ decomposes finitely as a $V_{\mathbf{k}}(\mathfrak g^0)$-module.

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arXiv2015

On principal realization of modules for the affine Lie algebra $A_1 ^{(1)}$ at the critical level

We present complete realization of irreducible $A_1 ^{(1)}$-modules at the critical level in the principal gradation. Our construction uses vertex algebraic techniques, the theory of twisted modules and representations of Lie conformal superalgebras. We also provide an alternative Z-algebra approach to this construction. All irreducible highest weight $A_1 ^{(1)}$-modules at the critical level are realized on the vector space $M_{\tfrac{1}{2} + \Bbb Z} (1) ^{\otimes 2}$ where $M_{\tfrac{1}{2} + \Bbb Z} (1) $ is the polynomial ring ${\Bbb C}[α(-1/2), α(-3/2), ...]$. Explicit combinatorial bases for these modules are also given.

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arXiv2015

Whittaker modules for the affine Lie algebra $A_1 ^{(1)}$

We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra $\widehat{sl_2}$ of type $A_1^{(1)}$ with noncritical level which are also irreducible Whittaker modules over $\widetilde{sl_2} =\widehat{sl_2} + {\Bbb C} d $ with the same Whittaker function and central charge. We have to modulo a central character for ${sl_2}$ to obtain irreducible degenerate Whittaker $\widehat{sl_2} $-modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker $\widehat{sl_2}$--module by a submodule generated by a scalar action of central elements of the vertex algebra $V_{-2}(sl_2)$ is irreducible as $\widehat{sl_2}$--module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for $\widetilde{sl_2}$. Quite surprisingly, with the same Whittaker function and the same central character of $V_{-2}(sl_2)$, some irreducible $\widetilde{sl_2}$ Whittaker modules can have semisimple or free action of $d$. At last, by using vertex algebraic techniques we present a Wakimoto type construction of a family of generalized Whittaker irreducible modules for $\widehat{sl_2}$ at the critical level. This family includes all classical Whittaker modules at critical level. We also have Wakimoto type realization for irreducible degenrate Whittaker modules for $\widehat{sl_2}$ at noncritical level.

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arXiv2015

Vertex Algebras $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$ and Constant Term Identities

We consider $AD$-type orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ extending the well-known $c=1$ orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras $A(\mathcal{W}(p)^{A_m})$ and $A(\mathcal{W}(p)^{D_m})$, where $A_m$ and $D_m$ are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible $\mathcal{W}(p)^Γ$-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of $m$ and $p$ with a computer software. As a result, we argue that if certain constant term properties hold, the irreducible modules constructed in [Commun. Contemp. Math. 15 (2013), 1350028, 30 pages, arXiv:1212.5453; Internat. J. Math. 25 (2014), 1450001, 34 pages, arXiv:1304.5711] provide a complete list of irreducible $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$-modules. This paper is a continuation of our previous work on the $ADE$ subalgebras of the triplet vertex algebra $\mathcal{W}(p)$.

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arXiv2015

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.

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arXiv2014

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

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 βγ$ system and contains a subalgebra isomorphic to the simple affine vertex algebra $L_{A_1} (- \tfrac{3}{2} Λ_0)$. 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 $L_{A_{2} } (-\tfrac{3}{2} Λ_0)$. By using this construction we construct a family of weight and logarithmic modules for $L_{c} ^{N=4}$ and $L_{A_{2} } (-\tfrac{3}{2} Λ_0)$. We also show that a coset subalgebra of $L_{A_{2} } (-\tfrac{3}{2} Λ_0)$ 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 $L_{A_1} (k Λ_0)$ such that $k+2 = 1/p$ and $p$ is a positive integer.

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arXiv2014

A classification of irreducible Wakimoto modules for the affine Lie algebra $A_1 ^{(1)}$

By using methods developed in arXiv:math/0602181 we study the irreducibility of certain Wakimoto modules for $\widehat{sl_2}$ at the critical level. We classify all $χ\in {\Bbb C}((z))$ such that the corresponding Wakimoto module $W_χ$ is irreducible. It turns out that zeros of Schur polynomials play important rule in the classification result.

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arXiv2013

ADE subalgebras of the triplet vertex algebra W(p): A-series

Motivated by \cite{am1}, for every finite subgroup $Γ\subset PSL(2,\mathbb{C})$ we investigate the fixed point subalgebra $\triplet^Γ$ of the triplet vertex $\mathcal {W}(p)$, of central charge $1-\frac{6(p-1)^{2}}{p}$, $p\geq2$. This part deals with the $A$-series in the ADE classification of finite subgroups of $PSL(2,\mathbb{C})$. First, we prove the $C_2$-cofiniteness of the $A_m$-fixed subalgebra $\triplet^{A_m}$. Then we construct a family of $\am$-modules, which are expected to form a complete set of irreps. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for $m=2$. We also present a rigorous proof of the fact that the full automorphism group of $\triplet$ is $PSL(2,\mathbb{C})$.

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arXiv2013

ADE subalgebras of the triplet vertex algebra W(p): D_m-series

We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra W(p). This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra $\bar{M(1)} ^+$, the $\Z_2$--orbifold of the singlet vertex algebra $\bar{M(1)}$. Then we classify irreducible modules and determine Zhu's and $C_2$--algebra for the vertex algebra $\triplet ^{D_2}$. A general method for construction of twisted $\triplet$--modules is also introduced. We also discuss classification of twisted $\bar{M(1)}$--modules including the twisted Zhu's algebra $A_Ψ (\bar{M(1)})$, which is of independent interest. The category of admissible $Ψ$-twisted $\bar{M(1)}$-modules is expected to be semisimple. We also prove $C_2$-cofiniteness of $\triplet^{D_m}$ for all $m$, and give a conjectural list of irreducible $\triplet^{D_m}$-modules. Finally, we compute characters of the relevant irreducible modules and describe their modular closure.

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arXiv2013

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 Adamović and O. Perše (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 Perše (2012) has the isomorphic fusion algebra. This enables us to decompose certain $E_6 ^{(1)}$ and $F_4 ^{(1)}$--modules at negative levels.

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arXiv2012

C_2-cofinite W-algebras and their logarithmic representations

We discuss our recent results on the representation theory of $\mathcal{W}$--algebras relevant to Logarithmic Conformal Field Theory. First we explain some general constructions of $\mathcal{W}$-algebras coming from screening operators. Then we review the results on $C_2$--cofiniteness, the structure of Zhu's algebras, and the existence of logarithmic modules for triplet vertex algebras. We propose some conjectures and open problems which put the theory of triplet vertex algebras into a broader context. New realizations of logarithmic modules for $\mathcal{W}$-algebras defined via screenings are also presented.

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arXiv2012

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.

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arXiv2011

Some general results on conformal embeddings of affine vertex operator algebras

We give a general criterion for conformal embeddings of vertex operator algebras associated to affine Lie algebras at arbitrary levels. Using that criterion, we construct new conformal embeddings at admissible rational and negative integer levels. In particular, we construct all remaining conformal embeddings associated to automorphisms of Dynkin diagrams of simple Lie algebras. The semisimplicity of the corresponding decompositions is obtained by using the concept of fusion rules for vertex operator algebras.

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arXiv2011

The structure of Zhu's algebras for certain W-algebras

We introduce a new approach that allows us to determine the structure of Zhu's algebra for certain vertex operator (super)algebras which admit horizontal $\mathbb{Z} $-grading. By using this method and an earlier description of Zhu's algebra for the singlet W-algebra, we completely describe the structure of Zhu's algebra for the triplet vertex algebra W(p). As a consequence, we prove that Zhu's algebra A(W(p)) and the related Poisson algebra P(W(p)) have the same dimension. We also completely describe Zhu's algebras for the N=1 triplet vertex operator superalgebra SW(m). Moreover, we obtain similar results for the c=0 triplet vertex algebra W_{2,3} important in logarithmic conformal field theory. Because our approach is "internal" we had to employ several constant term identities for purposes of getting right upper bounds on dimension of Zhu's algebras.

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arXiv2011

On W-algebra extensions of (2,p) minimal models: p > 3

This is a continuation of arXiv:0908.4053, where, among other things, we classified irreducible representations of the triplet vertex algebra W_{2,3}. In this part we extend the classification to W_{2,p}, for all odd p>3. We also determine the structure of the center of the Zhu algebra A(W_{2,p}) which implies the existence of a family of logarithmic modules having L(0)-nilpotent ranks 2 and 3. A logarithmic version of Macdonald-Morris constant term identity plays a key role in the paper.

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arXiv2009

On W-algebras associated to (2,p) minimal models and their representations

For every odd p \geq 3, we investigate representation theory of the vertex algebra WW_{2,p} associated to (2,p) minimal models for the Virasoro algebras. We demonstrate that vertex algebras WW_{2,p} are C_2--cofinite and irrational. Complete classification of irreducible representations for WW_{2,3} is obtained, while the classification for p \geq 5 is subject to certain constant term identities. These identities can be viewed as "logarithmic deformations" of Dyson and Selberg constant term identities, and are of independent interest.

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arXiv2009

Lattice construction of logarithmic modules for certain vertex algebras

A general method for constructing logarithmic modules in vertex operator algebra theory is presented. By utilizing this approach, we give explicit vertex operator construction of certain indecomposable and logarithmic modules for the triplet vertex algebra W(p) and for other subalgebras of lattice vertex algebras and their N=1 super extensions. We analyze in detail indecomposable modules obtained in this way, giving further evidence for the conjectural equivalence between the category of W(p)-modules and the category of modules for the restricted quantum group $\bar{\mathcal{U}}_q(sl_2)$ at root of unity. We also construct logarithmic representations for a certain affine vertex operator algebra at admissible level realized in \cite{A-2005}. In this way we prove the existence of the logarithmic representations predicted in \cite{G}. Our approach enlightens related logarithmic intertwining operators among indecomposable modules, which we also construct in the paper.

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arXiv2008

The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector

We classify irreducible $σ$-twisted modules for the N=1 super triplet vertex operator superalgebra $\mathcal{SW}(m)$ introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of $σ$-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the $SL(2,\mathbb{Z})$-closure of the space spanned by irreducible characters, irreducible supercharacters and $σ$-twisted irreducible characters is $(9m+3)$-dimensional. We present strong evidence that this is also the (full) space of generalized characters for $\mathcal{SW}(m)$. We are also able to relate irreducible $\mathcal{SW}(m)$ characters to characters for the triplet vertex algebra $\mathcal{W}(2m+1)$, studied in [Adamovic D., Milas A., Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857].

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