Posted Content•
Bershadsky-Polyakov vertex algebras at positive integer levels and duality
TL;DR: In this article, the authors study the simple Bershadsky-polyakov algebra W_k = \mathcal{W}_k(sl_3,f_{\theta})$ at positive integer levels and classify their irreducible modules.
Abstract: We study the simple Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_{\theta})$ 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 $\mathcal W_k$.
Citations
More filters
TL;DR: In this article, Adamovic and Kontrec showed that the simple Bershadsky-polyakov algebras with admissible non-integral weights are always rational in the category of highest-weight modules.
Abstract: The Bershadsky–Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $$\mathfrak {sl}_{3}$$
and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications (Arakawa in Commun Math Phys 323:627–633, 2013, Adamovic and Kontrec in Classification of irreducible modules for Bershadsky–Polyakov algebra at certain levels). In particular, we prove that the simple Bershadsky–Polyakov algebras with admissible nonintegral $$\mathsf {k}$$
are always rational in category $$\mathscr {O}$$
, whilst they always admit nonsemisimple relaxed highest-weight modules unless $$\mathsf {k}+\frac{3}{2} \in \mathbb {Z}_{\geqslant 0}$$
.
18 citations
07 Mar 2023
TL;DR: In this paper , it was shown that a natural construction of relaxed highest-weight Bershadsky polyakov modules gives a set of irreducible weight modules whose weight spaces are finite-dimensional.
Abstract: The Bershadsky--Polyakov algebras are the subregular quantum hamiltonian reductions of the affine vertex operator algebras associated with $\mathfrak{sl}_3$. In arXiv:2007.00396 [math.QA], we realised these algebras in terms of the regular reduction, Zamolodchikov's W$_3$-algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highest-weight Bershadsky--Polyakov modules gives modules that are generically irreducible. Here, we prove that this construction, when combined with spectral flow twists, gives a complete set of irreducible weight modules whose weight spaces are finite-dimensional. This gives a simple independent proof of the main classification theorem of arXiv:2007.03917 [math.RT] for nondegenerate admissible levels and extends this classification to a category of weight modules. We also deduce the classification for the nonadmissible level $\mathsf{k}=-\frac{7}{3}$, which is new.
1 citations
Posted Content•
TL;DR: In this paper, a subalgebra of the Clifford vertex superalgebra was introduced which is completely reducible as a $L^{Vir} (-2,0)$-cofinite, but it is not conformal and is not isomorphic to the symplectic fermion algebra.
Abstract: We introduce a subalgebra $\overline F$ of the Clifford vertex superalgebra ($bc$ system) which is completely reducible as a $L^{Vir} (-2,0)$-module, $C_2$-cofinite, but it is not conformal and it is not isomorphic to the symplectic fermion algebra $\mathcal{SF}(1)$. We show that $\mathcal{SF}(1)$ and $\overline{F}$ are in an interesting duality, since $\overline{F}$ can be equipped with the structure of a $\mathcal{SF}(1)$-module and vice versa.
Using the decomposition of $\overline F$ and a free-field realization from arXiv:1711.11342, we decompose $L_k(\mathfrak{osp}(1\vert 2))$ at the critical level $k=-3/2$ as a module for $L_k(\mathfrak{sl}(2))$. The decomposition of $L_k(\mathfrak{osp}(1\vert 2))$ is exactly the same as of the $N=4$ superconformal vertex algebra with central charge $c=-9$, denoted by $\mathcal V^{(2)}$. Using the duality between $\overline{F}$ and $\mathcal{SF}(1)$, we prove that $L_k(\mathfrak{osp}(1\vert 2))$ and $\mathcal V^{(2)}$ are in the duality of the same type. As an application, we construct and classify all irreducible $L_k(\mathfrak{osp}(1\vert 2))$-modules in the category $\mathcal O$ and the category $\mathcal R$ which includes relaxed highest weight modules.
We also describe the structure of the parafermion algebra $N_{-3/2}(\mathfrak{osp}(1\vert 2))$ as a $N_{-3/2}(\mathfrak{sl}(2))$-module.
We extend this example, and for each $p \ge 2$, we introduce a non-conformal vertex algebra $\mathcal A^{(p)}_{new}$ and show that $\mathcal A^{(p)}_{new} $ is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra $ \mathcal V^{(p)} _{new}$ which is isomorphic to the logarithmic vertex algebra $\mathcal V^{(p)}$ as a module for $\widehat{\mathfrak{sl}}(2)$.
TL;DR: In this article , a certain subalgebra F of the Clifford vertex superalgebra (bc system) was introduced, which is completely reducible (as a LVir(−2,0) -module) and C2-cofinite, but not conformal (and not isomorphic to the symplectic fermion algebra SF(1)).
References
More filters
Book•
01 Jan 1997TL;DR: In this paper, a formal distribution a(z,w) = 2 QFT and chiral algebras is defined and the Virasoro algebra is defined, which is a generalization of the Wightman axioms.
Abstract: Preface. 1: Wightman axioms and vertex algebras. 1.1: Wightman axioms of a QFT. 1.2: d = 2 QFT and chiral algebras. 1.3: Definition of a vertex algebra. 1.4: Holomorphic vertex algebras. 2: Calculus of formal distributions. 2.1: Formal delta-function. 2.2: An expansion of a formal distribution a(z,w). 2.3: Locality. 2.4: Taylor's formula. 2.5: Current algebras. 2.6: Conformal weight and the Virasoro algebra. 2.7: Lie superalgebras of formal distributions and conformal superalgebras. 3: Local fields. 3.1: Normally ordered product. 3.2: Dong's lemma. 3.3: Wick's theorem and a "non-commutative" generalization. 3.4: Restricted and field representations of Lie superalgebras of formal distributions. 3.5: Free (super)bosoms. 3.5: Free (super)fermions. 4: Structure theory of vertex algebras. 4.1: Consequences of translation covariance. 4.2: Quasisymmetry. 4.3: Superalgebras, ideals, and tensor products. 4.4: Uniqueness theorem. 4.5: Existence theorem. 4.6: Borcherds OPE formula. 4.7: Vertex algebras associated to Lie superalgebras of formal distributions. 4.8: Borcherds identity. 4.9: Graded and Mobius conformal vertex algebras. 4.10: Conformal vertex algebras. 4.11: Field algebras. 5: Examples of vertex algebras and their applications. 5.1: Charged free fermions. 5.2: Boson-fermion correspondence and KP hierarchy. 5.3: gl and W. 5.4: Lattice vertex algebras. 5.5: Simple lattice vertex algebras. 5.6: Root lattice vertex algebras and affine vertex algebras. 5.7: Conformal structure for affine vertex algebras. 5.8: Superconformal vertex algebras. 5.9: On classification of conformal superalgebras. Bibliography. Index
1,373 citations
"Bershadsky-Polyakov vertex algebras..." refers background in this paper
...Li construction from [34] (see also [26]) we get that...
[...]
TL;DR: In this article, it was shown that the characters of the integrable highest weight modules of affine Lie algebras and the minimal series of the Virasoro algebra give rise to conformal field theories.
Abstract: In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual action of the modular group SL2(Z). The similar result for the minimal series of the Virasoro algebra is observed in [Ca] and [IZ]. In both cases one uses the explicit character formulas to prove the modular invariance. The character formula for the affine Lie algebra is computed in [K], and the character formula for the Virasoro algebra is essentially contained in [FF]; see [R] for an explicit computation. This mysterious connection between the infinite dimensional Lie algebras and the modular group can be explained by the two dimensional conformal field theory. The highest weight modules of affine Lie algebras and the Virasoro algebra give rise to conformal field theories. In particular, the conformal field theories associated to the integrable highest modules and minimal series are rational. The characters of these modules are understood to be the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. From this point of view, the role of the modular group SL2(Z) is manifest. In the study of conformal field theory, physicists arrived at the notion of chiral algebras (see e.g. [MS]). Independently, in the attempt to realize the Monster sporadic group as a symmetry group of certain algebraic structure, an infinite dimensional graded representation of the Monster sporadic group, the so called Moonshine module, was constructed in [FLM1]. This algebraic structure was later found in [Bo] and called the vertex algebra; the first axioms of vertex operator algebras were formulated in that paper. The proof that the Moonshine module is a vertex operator algebra and the Monster group acts as its automorphism group was given in [FLM2]. Notably the character of the Moonshine module is also a modular function, namely j(τ) − 744. It turns out that the vertex operator algebra can be regarded as a rigorous mathematical definition of the chiral algebras in the physical literature. And it is expected that a pair of isomorphic vertex operator algebras and their representations (corresponding to the holomorphic and antiholomorphic sectors) are the basic objects needed to build a conformal field theory of a certain type.
1,122 citations
"Bershadsky-Polyakov vertex algebras..." refers background in this paper
...a highest weight vector vx,y such that J0vx,y = xvx,y, Jnvx,y = 0 forn>0, L0vx,y = yvx,y, Lnvx,y = 0 forn>0, G± n vx,y = 0 forn≥ 1. Let Aω(V) denote the Zhu algebra associated to the VOA V (cf. [41]) with the Virasorovectorω, and let [v] be the image of v∈ Vunder the mapping V → Aω(V). For the Zhu algebra Aω(Wk) it holds that: Proposition 3.2 ([9], Proposition3.2.). There exists a homomorphism Φ...
[...]
TL;DR: In this article, additional symmetries in two-dimensional conformal field theory generated by spin s = 1/2, 1,...,3 currents were investigated and the generators of the symmetry form associative algebras with quadratic determining relations.
Abstract: This paper investigates additional symmetries in two-dimensional conformal field theory generated by spin s = 1/2, 1,...,3 currents. For spins s = 5/2 and s = 3, the generators of the symmetry form associative algebras with quadratic determining relations. ''Minimal models'' of conforma field theory with such additional symmetries are considered. The space of local fields occurring in a conformal field theory with additional symmetry corresponds to a certain (in general, reducible) representation of the corresponding algebra of the symmetry.
910 citations
"Bershadsky-Polyakov vertex algebras..." refers methods in this paper
...In [8], the Bershadsky-Polyakov algebraWk is realized as a vertex subalgebra of Zk⊗Π(0) (where Zk = Wk(sl(3), fprinc) is the Zamolodchikov W -algebra [40]), for 2k + 3 / ∈ Z≥0....
[...]
Book•
02 Oct 2012
TL;DR: In this paper, the notion of vertex operator algebra was introduced and a formal series and the formal delta function were derived from the axiomatic definition of a vertex operator and its application in the formal calculus.
Abstract: 1 Introduction.- 1.1 Motivation.- 1.2 Example of a vertex operator.- 1.3 The notion of vertex operator algebra.- 1.4 Simplification of the definition.- 1.5 Representations and modules.- 1.6 Construction of families of examples.- 1.7 Some further developments.- 2 Formal Calculus.- 2.1 Formal series and the formal delta function.- 2.2 Derivations and the formal Taylor Theorem.- 2.3 Expansions of zero and applications.- 3 Vertex Operator Algebras: The Axiomatic Basics.- 3.1 Definitions and some fundamental properties.- 3.2 Commutativity properties.- 3.3 Associativity properties.- 3.4 The Jacobi identity from commutativity and associativity.- 3.5 The Jacobi identity from commutativity.- 3.6 The Jacobi identity from skew symmetry and associativity.- 3.7 S3-symmetry of the Jacobi identity.- 3.8 The iterate formula and normal-ordered products.- 3.9 Further elementary notions.- 3.10 Weak nilpotence and nilpotence.- 3.11 Centralizers and the center.- 3.12 Direct product and tensor product vertex algebras.- 4 Modules.- 4.1 Definition and some consequences.- 4.2 Commutativity properties.- 4.3 Associativity properties.- 4.4 The Jacobi identity as a consequence of associativity and commutativity properties.- 4.5 Further elementary notions.- 4.6 Tensor product modules for tensor product vertex algebras.- 4.7 Vacuum-like vectors.- 4.8 Adjoining a module to a vertex algebra.- 5 Representations of Vertex Algebras and the Construction of Vertex Algebras and Modules.- 5.1 Weak vertex operators.- 5.2 The action of weak vertex operators on the space of weak vertex operators.- 5.3 The canonical weak vertex algebra ?(W) and the equivalence between modules and representations.- 5.4 Subalgebras of ?(W).- 5.5 Local subalgebras and vertex subalgebras of ?(W).- 5.6 Vertex subalgebras of ?(W) associated with the Virasoro algebra.- 5.7 General construction theorems for vertex algebras and modules.- 6 Construction of Families of Vertex Operator Algebras and Modules.- 6.1 Vertex operator algebras and modules associated to the Virasoro algebra.- 6.2 Vertex operator algebras and modules associated to affine Lie algebras.- 6.3 Vertex operator algebras and modules associated to Heisenberg algebras.- 6.4 Vertex operator algebras and modules associated to even lattices-the setting.- 6.5 Vertex operator algebras and modules associated to even lattices-the main results.- 6.6 Classification of the irreducible L?(?, O)-modules for g finite-dimensional simple and ? a positive integer.- References.
892 citations
Additional excerpts
...[35]): Com(S, V ) := {v ∈ V | anv = 0, ∀a ∈ S, ∀n ∈ Z≥0}....
[...]
Book•
01 Jan 2000TL;DR: Vertex algebra bundles are associated with Lie algebras and operator product expansion (OPE) as mentioned in this paper, and vertex algebra bundles can be used to represent internal symmetries of vertex algebra.
Abstract: Introduction Definition of vertex algebras Vertex algebras associated to Lie algebras Associativity and operator product expansion Applications of the operator product expansion Modules over vertex algebras and more examples Vertex algebra bundles Action of internal symmetries Vertex algebra bundles: Examples Conformal blocks I Conformal blocks II Free field realization I Free field realization II The Knizhnik-Zamolodchikov equations Solving the KZ equations Quantum Drinfeld-Sokolov reduction and $\mathcal{W}$-algebras Vertex Lie algebras and classical limits Vertex algebras and moduli spaces I Vertex algebras and moduli spaces II Chiral algebras Factorization Appendix Bibliography Index List of frequently used notation.
721 citations
Additional excerpts
...Φ(z) = X n∈Z Φ(n+ 1 2)z −n−1 generate on F1/2 = ^ (Φ(−n−1/2)| n∈ Z≥0) a unique structure of a vertex superalgebra with conformal vector ωF1/2 = 1 2Φ(− 3 2)Φ(−2) 1 , of central charge cF1/2 = 1/2 (cf. [23], [26]). Note also that the field Φ(z) is usually called neutral fermion field, and F1/2 is called free-fermion theory in physics literature. A basis of F1/2 is given by Φ(−n1 − 1 2)...Φ(−nr − 1 2), whe...
[...]