Subsingular Vectors Research Articles

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

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 obtained using screening operators appearing in a study of certain logarithmic vertex algebras [D. Adamović and A. Milas, On W-algebras associated to (2, [Formula: see text]) minimal models and their representations, Int. Math. Res. Notices 2010(20) (2010) 3896–3934].

Galilean W3 algebra

Galilean W3 vertex operator algebra (VOA) GW3(cL,cM) is constructed as a universal enveloping vertex algebra of certain non-linear Lie conformal algebra. It is proved that this algebra is simple by using the determinant formula of the vacuum module. The reducibility criterion for Verma modules is given, and the existence of subsingular vectors is demonstrated. Free field realization of GW3(cL,cM) and its highest weight modules are obtained within a rank 4 lattice VOA.

On the 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.

Verma modules over the W(2,2) algebras

We use a new method to study the singular and subsingular vectors of Verma modules over the W ( 2 , 2 ) algebras and show that a subsingular vector exists in the Verma module under some conditions.

Subsingular vectors in Verma modules, and tensor product of weight modules over the twisted Heisenberg-Virasoro algebra and W(2, 2) algebra

We show that subsingular vectors may exist in Verma modules over W(2, 2), and present the subquotient structure of these modules. We prove conditions for irreducibility of the tensor product of intermediate series module with a highest weight module. Relation to intertwining operators over vertex operator algebra associated with W(2, 2) is discussed. Also, we study the tensor product of intermediate series and a highest weight module over the twisted Heisenberg-Virasoro algebra, and present series of irreducible modules with infinite-dimensional weight spaces.

The Adapted Ordering Method for Lie algebras and superalgebras and their generalizations

In 1998 the Adapted Ordering Method was developed for the representation theory of the superconformal algebras in two dimensions. It allows us to determine maximal dimensions for a given type of space of singular vectors, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this paper we present the Adapted Ordering Method for general Lie algebras and superalgebras and their generalizations, provided they can be triangulated. We also review briefly the results obtained for the Virasoro algebra and for the N = 2 and Ramond N = 1 superconformal algebras.

EMBEDDING DIAGRAMS OF THE N=2 SUPERCONFORMAL ALGEBRA UNDER SPECTRAL FLOW

The embedding diagrams of representations of the N=2 superconformal algebra with central charge c=3 are given. Some non-unitary representations possess subsingular vectors that are systematically described. The structure of the embedding diagrams is largely defined by the spectral flow symmetry. As an additional consistency check the action of the spectral flow on the characters is calculated.

CONSTRUCTION FORMULAE FOR SINGULAR VECTORS OF THE TOPOLOGICAL AND OF THE RAMOND N = 2 SUPERCONFORMAL ALGEBRAS

We write down one-to-one mappings between the singular vectors of the Neveu–Schwarz N = 2 superconformal algebra and 16 + 16 types of singular vectors of the topological and of the Ramond N = 2 superconformal algebras. As a result one obtains the construction formulae for the latter using Dörrzapf's construction formulae for the Neveu–Schwarz singular vectors. The indecomposable singular vectors of the topological and of the Ramond N = 2 algebras ("no-label" and "no-helicity" singular vectors) cannot be mapped to singular vectors of the Neveu–Schwarz N = 2 algebra, but to subsingular vectors, for which no construction formulae exist.

Transmutations between singular and subsingular vectors of the N = 2 superconformal algebras

We present subsingular vectors of the N = 2 superconformal algebras other than the ones which become singular in chiral Verma modules, reported recently by Gato-Rivera and Rosado. We show that two large classes of singular vectors of the topological algebra become subsingular vectors of the antiperiodic NS algebra under the topological untwistings. These classes consist of BRST-invariant singular vectors with relative charges q = −2, −1 and zero conformal weight, and nolabel singular vectors with q = 0, −1. In turn the resulting NS subsingular vectors are transformed by the spectral flows into subsingular and singular vectors of the periodic R algebra. We write down these singular and subsingular vectors starting from the topological singular vectors at levels 1 and 2.

Singular Dimensions¶of the N= 2 Superconformal Algebras II:¶The Twisted N= 2 Algebra

We introduce a suitable adapted ordering for the twisted N=2 superconformal algebra (i.e. with mixed boundary conditions for the fermionic fields). We show that the ordering kernels for complete Verma modules have two elements and the ordering kernels for G-closed Verma modules just one. Therefore, spaces of singular vectors may be two-dimensional for complete Verma modules whilst for G-closed Verma modules they can only be one-dimensional. We give all singular vectors for the levels 1/2, 1, and 3/2 for both complete Verma modules and G-closed Verma modules. We also give explicit examples of degenerate cases with two-dimensional singular vector spaces in complete Verma modules. General expressions are conjectured for the relevant terms of all (primitive) singular vectors, i.e. for the coefficients with respect to the ordering kernel. These expressions allow to identify all degenerate cases as well as all G-closed singular vectors. They also lead to the discovery of subsingular vectors for the twisted N=2 superconformal algebra. Explicit examples of these subsingular vectors are given for the levels 1/2, 1, and 3/2. Finally, the multiplication rules for singular vector operators are derived using the ordering kernel coefficients. This sets the basis for the analysis of the twisted N=2 embedding diagrams.

Families of singular and subsingular vectors of the topological N = 2 superconformal algebra

We analyze several issues concerning the singular vectors of the topological N = 2 superconformal algebra. First we investigate which types of singular vectors exist, regarding the relative U(1) charge and the BRST-invariance properties, finding four different types in chiral Verma modules and twenty-nine different types in complete Verma modules. Then we study the family structure of the singular vectors, every member of a family being mapped to any other member by a chain of simple transformations involving the spectral flows. The families of singular vectors in chiral Verma modules follow a unique pattern (four vectors) and contain subsingular vectors. We write down these families until level 3, identifying the subsingular vectors. The families of singular vectors in complete Verma modules follow infinitely many different patterns, grouped roughly in five main kinds. We present a particularly interesting thirty-eight-member family at levels 3, 4, 5, and 6, as well as the complete set of singular vectors at level 1 (twenty-eight different types). Finally we analyze the Dörrzapf conditions leading to two linearly independent singular vectors of the same type, at the same level in the same Verma module, and we write down four examples of those pairs of singular vectors, which belong to the same thirty-eight-member family.

The Structure of Verma Modules over the N \equals; 2 Superconformal Algebra

We classify degeneration patterns of Verma modules over the N=2 superconformal algebra in two dimensions. Explicit formulae are given for singular vectors that generate maximal submodules in each of the degenerate cases. The mappings between Verma modules defined by these singular vectors are embeddings; in particular, their compositions never vanish. As a by-product, we also obtain general formulae for N=2 subsingular vectors.

Chiral determinant formulae and subsingular vectors for the N = 2 superconformal algebras

We derive conjectures for the N = 2 “chiral” determinant formulae of the topological algebra, the antiperiodic NS algebra, and the periodic R-algebra, corresponding to incomplete Verma modules built on chiral topological primaries, chiral and antichiral NS primaries, and Ramond ground states, respectively. Our method is based on the analysis of the singular vectors in chiral Verma modules and their spectral flow symmetries, together with some computer exploration and some consistency checks. In addition, and as a consequence, we uncover the existence of subsingular vectors in these algebras, giving examples (subsingular vectors are non-highest-weight null vectors which are not descendants of any highest-weight singular vectors).

Conditionally Invariant Equations from Subsingular Vectors

We discuss the relation between subsingular vectors of Verma modules over semisimple Lie algebras {\cal G} and differential equations which are conditionally {\cal G}-invariant. We treat the example o the conformal algebra su(2,2) and its complexification sl(4). The conditionally invariant equations are the d'Alembert equation and a new equation arising from a subsingular vector proposed by Bernstein-Gel'fand-Gel'fand.

Probing integrable perturbations of conformal theories using singular vectors

It has been known for some time that the (1,3) perturbations of the (2 k + 1,2) Virasoro minimal models have conserved currents which are also singular vectors of the Virasoro algebra. This also turns out to hold for the (1,2) perturbation of the (3 k ± 1,3) models. In this paper we investigate the requirement that a perturbation of an extended conformal field theory has conserved currents which are also singular vectors. We consider conformal field theories with W 3 and (bosonic) WBC 2 = W(2,4) extended symmetries. Our analysis relies heavily on the general conjecture of de Vos and van Driel relating the multiplicities of W-algebra irreducible modules to the Kazhdan-Lusztig polynomials of a certain double coset. Granting this conjecture, the singular-vector argument provides a direct way of recovering all known integrable perturbations. However, W models bring a slight complication in that the conserved densities of some (1, 2)-type perturbations are actually subsingular vectors, that is, they become singular vectors only in a quotient module.

Subsingular vectors and conditionally invariant (q-deformed) equations

We give a systematic discussion of the relation between subsingular vectors of Verma modules over semisimple Lie algebras G and differential equations which are conditionally G-invariant. This is extended to the Drinfeld-Jimbo q-deformation Uq(G) of G. We treat in detail the conformal algebra su(2,2), its complexification sl(4) and their q-deformations. The conditionally invariant equations are the d`Alembert equation and a new equation arising from a subsingular vector proposed by Bernstein-Gel`fand-Gel`fand (1971). We also give the q-difference analogues of these equations.

The Kazhdan-Lusztig conjecture for finiteW-algebras

We study the representation theory of finiteW-algebras. After introducing parabolic subalgebras to describe the structure ofW-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finiteW-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors and find that, forW-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a ‘dual’ analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infiniteW-algebras.