Unitary Weight Research Articles

UNITARY REPRESENTATIONS OF W INFINITY ALGEBRAS

We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of W infinity algebras [Formula: see text] with central charges (2, 1, 3, 2M, N, 2M+N). The characters of these representations are computed. We construct a new extended superalgebra [Formula: see text], whose bosonic sector is [Formula: see text]. Its representations obtained from a free field realization with central charge 2M+N, are classified into two classes: continuous series and discrete series. For the former there exists a supersymmetry, but for the latter a supersymmetry exists only for M=N.

AN EXPLICIT FORMULA FOR SOME SINGULAR VECTORS OF THE NEVEU–SCHWARZ ALGEBRA

Similarly to the Virasoro algebra, the Neveu–Schwarz algebra has a discrete series of unitary irreducible highest weight representations. These are labeled by the values of [Formula: see text] (the central charge) and of the highest weight hpq = [(p (m + 2) − qm)2 − 4]/(8m (m + 2)) where m, p, q are some integers. The Verma modules constructed with these values (c, h) are not irreducible, however, as they contain two Verma submodules, each generated by a singular vector ψp,q (of weight hpq + pq/2) and ψm−p, m+2−q (of weight hpq + (m−p)(m+2−q)/2), respectively. We give an explicit expression for these singular vectors whenever one of its indices is 1.

Annihilators and associated varieties of unitary highest weight modules

Annihilators and associated varieties of unitary highest weight modules

Derived Functors of Unitary Highest Weight Modules at Reduction Points

The derived functors introduced by Zuckerman are applied to the unitary highest weight modules of the Hermitian symmetric pairs of classical type. The construction yields "small" unitary representations which do not have a highest weight. The infinitesimal character parameter of the modules we consider is such that their derived functors are nontrivial in more than one degree; at the extreme degrees where the cohomology is nonvanishing, it is possible to determine the ${\mathbf {K}}$-spectrum of the resulting representations completely. Using this information it is shown that, in most cases, the derived functor modules are unitary, irreducible, and not of highest weight type. Their infinitesimal character and lowest ${\mathbf {K}}$-type are also easily computed.

UNITARY REPRESENTATIONS OF TWISTED N = 4 SUPERCONFORMAL ALGEBRAS

We discuss a class of twisted N = 4 superconformal algebras, containing an SU (2) ⊕ SU (2) ⊕ U (1) Kac–Moody subalgebra. We construct the spectral flow which relates different boundary conditions corresponding to this twisted algebras. A general formula for the unitary highest weight representations is derived. For the central charge of the Virasoro c = 3, we verify this formula using an explicit realization.

Unitary lowest weight representations of the noncompact supergroup OSp(2m*/2n)

The oscillator construction of the unitary lowest weight representations of the noncompact supergroup OSp(2m*/2n) with the even subgroup SO*(2m)×USp(2n) is given. Simple rules for determining the SO*(2m)×USp(2n) decomposition of the lowest weight representations of OSp(2m*/2n) are derived. The particular examples of OSp(4*/2) with the even subgroup SU(1,1)×SU(2)×SU(2) and OSp(4*/6) with the even subgroup SU(1,1)×SU(2)×Usp(6) are studied in greater detail.

Derived functors of unitary highest weight modules at reduction points

The derived functors introduced by Zuckerman are applied to the unitary highest weight modules of the Hermitian symmetric pairs of classical type. The construction yields "small" unitary representations which do not have a highest weight. The infinitesimal character parameter of the modules we consider is such that their derived functors are nontrivial in more than one degree; at the extreme degrees where the cohomology is nonvanishing, it is possible to determine the K {\mathbf {K}} -spectrum of the resulting representations completely. Using this information it is shown that, in most cases, the derived functor modules are unitary, irreducible, and not of highest weight type. Their infinitesimal character and lowest K {\mathbf {K}} -type are also easily computed.

An 𝐿²-cohomology construction of unitary highest weight modules for 𝑈(𝑝,𝑞)

In this paper a geometric construction is given of all unitary highest weight modules of G = U ⁡ ( p , q ) G = \operatorname {U} (p,q) . The construction is based on the unitary model of the k k th tensor power of the metaplectic representation in a Bargmann-Segal-Fock space of square-integrable differential forms. The representations are constructed as holomorphic sections of certain vector bundles over G / K G/K , and the construction is implemented via an integral transform analogous to the Penrose transform of mathematical physics.

Super loop groups, Hamiltonian actions and super Virasoro algebras

The quotient $${{\widetilde{LG}} \mathord{\left/ {\vphantom {{\widetilde{LG}} G}} \right. \kern-\nulldelimiterspace} G}$$ of a super loop group $$\widetilde{LG}$$ by the subgroup of constant loops is given a supersymplectic structure and identified through a moment map embedding with a coadjoint orbit of the centrally extended super loop algebra . The algebra $$\widetilde{diff}^c S^1 $$ of super-conformal vector fields on the circle is shown to have a natural representation as Hamiltonian vector fields on $${{\widetilde{LG}} \mathord{\left/ {\vphantom {{\widetilde{LG}} G}} \right. \kern-\nulldelimiterspace} G}$$ generated by an equivariant moment map . This map is obtained by composition of315-8 with a super Poisson map defining a supersymmetric extension of the classical Sugawara formula. Upon quantization, this yields the corresponding formula of Kac and Todorov on unitary highest weight representations. For any homomorphism ρ:u(1)→G, an associated “twisted” moment map is also derived, generating a super Poisson bracket realization of a super Virasoro subalgebra $$\widetilde{Vir}$$ of the semi-direct sum . The corresponding super Poisson map is interpreted as a nonabelian generalization of the super Miura map and applied to two super KdV hierarchies to derive corresponding integrable generalized super MKdV hierarchies in .

Tensor Products of Unitary Super-Virasoro Modules With Central Charge 7/10

AbstractThe two Virasoro superalgebras, known as the Neveu- Schwarz algebra and the Ramond algebra, each have two unitary irreducible lowest weight modules with central charge . In this paper, I show how tensor products of these modules decompose into finite direct sums of irreducible modules with central charge .

Classification of unitary highest weight representations for noncompact real forms

Using Jakobsen theorems, unitarizability in Hermitian symmetric spaces is discussed. The set of all missing highest weights is explicitly calculated and the construction of their corresponding highest weights vectors is studied.

Lowest weight representations of some infinite dimensional groups on fock spaces

The results of Kashiwara and Vergne on the decomposition of the tensor products of the ‘Segal-Shale-Weil representation’ are extended to the infinite dimensional case and give all unitary lowest weight representations. Our methods are basically algebraic. When restricted to the finite dimensional case, they yield a new proof.

On the unitary representations of a class of N = 4 superconformal algebras

We derive properties of unitary highest weight representations of the N = 4 superconformal algebras in 2 dimensions, characterized by two SU(2) Kač-Moody subalgebras. For the range 3 ⩽ c < 6 we can prove that we have constructed all unitary highest weight representations. These reduce in N = 2 to the discrete series. For other values of the parameters we derive bounds on the values of h, and we conjecture that these are complete. This is motivated by explicit constructions of representations for various values of the parameters. For the so-called massless representations we do indeed prove that our treatment is complete.

IMPROVED FORMULATION OF GNO FERMIONIZATION THEOREM

It is pointed out that in the Kac-Moody algebras fulfilling the fermionization criterion of Goddard, Nahm and Olive and having a non-minimal value of the central charge k, only a proper subset of the allowed unitary highest weight representations can actually be encoded in a free fermion theory. These truly fermionizable representations are selected by a very specific non-regular embedding of the fermionizable Kac-Moody algebra into the lowest level SO (N F ) Kac-Moody algebra, N F being both the number of fermions and the dimension of the GNO symmetric space. This embedding is a particular case of the embeddings considered by Bais and Bouwknegt and by Schellekens and Warner, for which the Virasoro central charge of the subgroup is equal to that of the group. Furthermore, these fermionizable representations span an orbit of the modular group always leading to a non trivial modular invariant partition function.

Affine superalgebras, unitary representations and the sugawara construction

Unitary highest weight representations of the untwisted, affine, basic classical Lie superalgebras are considered. Unitary constraints (for both even and odd generators) relate the value of the central extension to the vacuum quantum numbers. The only consistent possibilities are for certain compact type I superalgebras, or for those type I and II superalgebras yielding spinor gradings of rigid two-dimensional conformal transformations. The Sugawara construction for these is presented and the value of the Virasoro central charge listed.

Analogues of Kostant's -cohomology formulas for unitary highest weight modules.

Article Analogues of Kostant's -cohomology formulas for unitary highest weight modules. was published on January 1, 1988 in the journal Journal für die reine und angewandte Mathematik (volume 1988, issue 392).

Gradient-type differential operators and unitary highest weight representations of SU( p, q)

Let C p + q be equipped with a hermitian form of signature ( p, q) and let SU( p, q) denote the subgroup of the corresponding invariance group U( p, q) consisting of matrices with determinant 1. To certain highest weights λ, we associate a first-order group invariant linear differential operator D λ; whose kernel contains a unitary highest weight representation with highest weight λ. The Fock model realization of unitary highest weight representations of U( p, q) is the fundamental tool used to implement this construction. The operator D λ is shown to be equivalent to an operator D ̃ λ which acts on Hol( G K , H λ) , the space of holomorphic vector valued functions defined on G K . We identify a set Λ 1 of highest weights such that Ker( D λ) is a proper subspace of Hol( G K , H λ) and show that those λ in Λ 1 correspond to points occurring at the far right of the discrete set in the classification scheme of Enright, Howe, and Wallach. First-order differential equations arising from this proper containment are explicitly derived from the operator D̃ λ. We illustrate the fundamental nature of these first-order equations by deriving from them a system which completely determines the irreducible spaces for ladder representations of SU( p, q).

Unitary lowest weight representations of the noncompact supergroup OSp(2n/2m,R)

The oscillator construction of the unitary irreducible lowest (highest) weight representations of the noncompact supergroup OSp(2n/2m,R) with even subgroup SO(2n)×Sp(2m,R) is given. In particular, simple rules for determining the SO(2n)×Sp(2m,R) decomposition of the unitary lowest weight representations of OSp(2n/2m,R) are derived.

Branching rules for unitary representations of Virasoro and super-Virasoro algebras at c=1

The branching rules for unitary highest weight representations of Virasoro and super-Virasoro algebras that contain a subalgebra with central charge c=1 are presented. These rules are useful for identifying points of higher symmetry in two-dimensional critical systems with c=1 and for the analysis of a defected Ising chain.

Unitary highest weight representations of noncompact supergroups

The oscillator method for the construction of unitary highest (or lowest) weight representations of noncompact groups and supergroups is generalized. Within this generalization, the method yields unitary highest weight representations of all simple supergroups whose even subgroups are in the form of a direct product of a compact group with a simple noncompact group. The method is illustrated by studying in detail the unitary highest weight representations of the supergroup OSp (2n+1/2m,R). The generalized supercoherent states associated with these unitary representations are also defined.