Vertex operator superalgebras and their representations (original) (raw)

On Exceptional Vertex Operator (Super) Algebras

Developments in Mathematics, 2014

We consider exceptional vertex operator algebras and vertex operator superalgebras with the property that particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendents of the vacuum. We show that the genus one partition function and characters for simple ordinary modules must satisfy modular linear differential equations. We show the rationality of the central charge and module lowest weights, modularity of solutions, the dimension of each graded space is a rational function of the central charge and that the lowest weight primaries generate the algebra. We also discuss conditions on the reducibility of the lowest weight primary vectors as a module for the automorphism group. Finally we analyse solutions for exceptional vertex operator algebras with primary vectors of lowest weight up to 9 and for vertex operator superalgebras with primary vectors of lowest weight up to 17/2. Most solutions can be identified with simple ordinary modules for known algebras but there are also four conjectured algebras generated by weight two primaries and three conjectured extremal vertex operator algebras generated by primaries of weight 3, 4 and 6 respectively.

On a family of vertex operator superalgebras

2021

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight3 2 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V(2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V( 3 2 ) is naturally a V(2)-module equipped with a V(2)-valued symmetric bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A-module equipped with a symmetric A-valued bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L(A,U) and a simple vertex operator superalgebra LL(A,U)(l, 0) for every nonzero number l such...

Vertex operators for algebras and superalgebras

Nuclear Physics B - Proceedings Supplements, 1988

We emphasize the role of the boson-ferrnion correspondence in two dimensionnal conformal field theory for the realization of level one representations of affine untwisted and twisted Kac-Moody algebras via vertex operators. Using also the boson-boson correspondence, vertex operators for contragredient affine superalgebras can be constructed.

Representations of vertex operator algebras

2013

In this thesis we study the representation theory of vertex operator algebras. The thesis consists of two parts. The first part deals with the connection among rationality, regularity and C2C_2C2-cofiniteness of vertex operator algebras. It is proved that if any ZZZ-graded weak module for a vertex operator algebra V is completely reducible, then V is rational and C2C_2C2-cofinite. That is, V is regular. This gives a natural characterization of regular vertex operator algebras. Motivated by classification of rational vertex operator algebras with central charge c = 1. We compute the quantum dimensions of irreducible modules of the rational and C2-cofinite vertex operator algebra VL2A4V_{L_2}^{A_4}VL_2A_4 in the other part.This result will be used to determine the fusion rules for this algebra.

Vertex operator superalgebras and the 16-fold way

Transactions of the American Mathematical Society

Let V V be a vertex operator superalgebra with the natural order 2 automorphism σ \sigma . Under suitable conditions on V V , the σ \sigma -fixed subspace V 0 ¯ V_{\bar 0} is a vertex operator algebra and the V 0 ¯ V_{\bar 0} -module category C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a modular tensor category. In this paper, we prove that C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a fermionic modular tensor category and the Müger centralizer C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 of the fermion in C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is generated by the irreducible V 0 ¯ V_{\bar 0} -submodules of the V V -modules. In particular, C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 is a super-modular tensor category and C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a minimal modular extension of C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 . We provide a construction of a vertex operator superalgebra V l V^l for each positive integer l l such that C ( V l ) 0 ¯ \mathcal {C}_{{(V^l)_{\bar 0}}} is a minimal modular extension of C ...

Sugawara and Vertex Operator Constructions for Deformed Virasoro Algebras

Annales Henri Poincaré, 2006

From the defining exchange relations of the A q,p (gl N) elliptic quantum algebra, we construct subalgebras which can be characterized as q-deformed W N algebras. The consistency conditions relating the parameters p, q, N and the central charge c are shown to be related to the singularity structure of the functional coefficients defining the exchange relations of specific vertex operators representations of A q,p (gl N) available when N = 2.

Vertex operator representations of quantum affine superalgebras

arXiv: Quantum Algebra, 2017

Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1 irreducible integrable highest weight representations and classify the finite dimensional irreducible representations of Uq(g). This makes essential use of the Drinfeld realisation for Uq(g), and quantum correspondences between affine Kac-Moody superalgebras, developed in earlier papers.

Spectral Triples and the Super-Virasoro Algebra

Communications in Mathematical Physics, 2010

We construct infinite dimensional spectral triples associated with representations of the super-Virasoro algebra. In particular the irreducible, unitary positive energy representation of the Ramond algebra with central charge c and minimal lowest weight h = c/24 is graded and gives rise to a net of even θ-summable spectral triples with non-zero Fredholm index. The irreducible unitary positive energy representations of the Neveu-Schwarz algebra give rise to nets of even θ-summable generalised spectral triples where there is no Dirac operator but only a superderivation.

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

Journal of Algebra and Its Applications, 2021

We study the representation theory of the [Formula: see text] super Heisenberg–Virasoro vertex algebra at level zero, which extends the previous work on the Heisenberg–Virasoro vertex algebra [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342; D. Adamović and G. Radobolja, Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero, Commun. Contemp. Math. 21(2) (2019) 1850008; Y. Billig, Representations of the twisted Heisenberg–Virasoro algebra at level zero, Can. Math. Bull. 46(4) (2003) 529–537] 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 obta...

Classification of some three-dimensional vertex operator algebras

arXiv (Cornell University), 2019

We discuss the classification of strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy. Our Main Theorem 1 provides a classification of all such VOAs in the form of one infinite family of affine VOAs, one individual affine algebra and two Virasoro algebras, together with a family of eleven exceptional character vectors and associated data that we call the U-series. We prove that there are at least 15 VOAs in the U-series occurring as commutants in a Schellekens list holomorphic VOA. These include the affine algebra E 8,2 and Höhn's Baby Monster VOA VB (0) but the other 13 seem to be new. The idea in the proof of our Main Theorem is to exploit properties of a family of vector-valued modular forms with rational functions as Fourier coefficients, which solves a family of modular linear differential equations in terms of generalized hypergeometric series. CONTENTS 14 5. Positivity restrictions 20 6. The remaining fibers 22 7. Trimming down to Theorem 1 29 8. Solutions with y = −1/2 31 9. The U-series 41 Appendix A. Primes in progressions 46 Appendix B. Affine algebras 47 References 51