Discrete Series Of Unitary Representations Research Articles

Gupta-Bleuler quantization for minimally coupled scalar fields in de Sitter space

We present a fully covariant quantization of the minimally coupled massless field on de Sitter space, thanks to a new representation of the canonical commutation relations. We thus obtain a formalism free of any infrared divergence. Our method is based on a rigorous group-theoretical approach combined with a suitable adaptation (Krein spaces) of the Wightman-Gärding axiomatic for massless fields (Gupta-Bleuler scheme). We make explicit the correspondence between unitary irreducible representations of the de Sitter group and the field theory on de Sitter spacetime. The minimally coupled massless field is associated with a representation which is the lowest term of the discrete series of unitary representations of the de Sitter group. In spite of the presence of negative-norm modes in the theory, no negative energy can be measured: expressions such as nk1nk2 ... |T00|nk1nk2 ... are always positive.

Realization of the discrete series of unitary representations of Sl(2,R) in terms of creation and annihilation operators

A basis for the Hilbert space of the discrete series of unitary representations of Sl(2,R) is constructed as a complete set of coherent states of pairs of particle–antiparticle creation operators. The integer q specifying an irreducible representation is the eigenvalue of the charge operator. Representations with charges ∓q are equivalent.

Characters of super-Kac-Moody algebras

The partition functions for super-Wess-Zumino-Witten models can be expressed in terms of characters of super-Kac-Moody algebras. These characters are examined with the emphasis on maintaining supersymmetry explicitly. It is shown that an analogoue of Borel-Weil theory is at least formally relevant to the representation theory of super-Kac-Moody algebras, and that the characters have an interpretation in terms of fixed points of the action of the corresponding group on a homogeneous superspace. Characters with nontrivial dependence on the supermodular parameters of superconformal and supersymmetric Yang-Mills backgrounds on the torus with (++) spin structure are computed, and for the case of SU(2), they are used to extend the conventional GKO construction for the characters of the discrete series of unitary representations of the superconformal algebra with c< 3 2 to accomodate the odd superconformal supercharacters of Cohn and Friedan. This extension of the GKO construction requires the incorporation of a “spectator” space of free fermions in the standard GKO construction of superconformal characters relevant to the (+−), (−+) and (−−) spin structures.

Unitary construction of extended conformal algebras

Particular examples and the general structure of extended conformal symmetries in coset conformal field theories are discussed. Discrete series of unitary representations, whose existence had been previously conjectured, are constructed for a class of extended conformal algebras introduced by Fateev and Zamolodchikov (FZ). The construction is a generalisation of the coset construction of the discrete series for the superconformal algebra using the coset spaces so(N) ⊕ su(N)/so(N), for fixed N. The N = 3 series is the FZ S3 algebra and the N = 4 series consists of two commuting copies of the superconformal algebra. A general method for analysing the extended conformal symmetries present in a particular coset theory and of constructing discrete series of representations of extended symmetry algebras is outlined.

A connection between the twisted N = 2 superconformal and SU(2) Kac-moody algebras

We prove that there is a one-to-one correspondence between the unitary highest weight representations of the (twisted) SU(2) Kac-Moody algebra and the discrete series of unitary representations of the twisted N = 2 superconformal algebra. The algebras are represented on the same Hilbert space having the same Virasoro algebra.

N=1 superconformal operator product expansions and superfield fusion rules

The N=1 superfield OPEs are constructed in an explicit form. A new type of superfield fusion rules (even, odd and even-odd) are found in the case of the discrete series of unitary representations of the N=1 Neveu-Schwarz algebra. Examination of the N=1 superfield OPE algebras shows and that the N=2 superconformal algebra arises as a subalgebra in the c=1 superconformal ( N=1) minimal model, interpreted earlier as the Kosterlitz-Thouless point of the critical O(2) gaussian model.

Discrete series of unitary representations of the Lie algebra of the group O(p, q)

Discrete series of unitary representations of the Lie algebra of the group O(p, q)