Group-Theoretical Approach to Extended Conformal Supersymmetry: Function Space Realizations and Invariant Differential Operators

We give function space realizations of all representations of the conformal superalgebra su(2,2/N) and of the supergroup SU(2, 2 /N) induced from irreducible finite-dimensional Lorentz and SU(N) representations realized without spin and isospin indices. We use the lowest weight module structure of our su(2,2/N) representations to present a general procedure (adapted from the semisimple Lie algebra case) for the canonical construction of invariant differential operators closely related to the reducible (indecomposable) representations. All conformal supercovariant derivatives are obtained in this way. Examples of higher order invariant differential operators are given.     

On the classification of diagonal coset modular invariants

We relate in a novel way the modular matrices of GKO diagonal cosets without fixed points to those of WZNW tensor products. Using this we classify all modular invariant partition functions ofsu(3) _k su(3)₁/su(3) _k+1 for all positive integer levelk, andsu(2) _k su(2) _l /su(2) _k+1 for allk and infinitely manyl (in fact, for eachk a positive density ofl). Of all these classifications, only that forsu(2) _k su(2)₁/su(2) _k+1 had been known. Our lists include many new invariants.Supported in part by NSERC.     

Fundamental Fermions Fit Inside One <Emphasis Type="Italic">su</Emphasis>(1|5) Irreducible Representation

The Lie superalgebra su(1|5) has irreducible representations of dimension 32, in which the 32 fundamental fermions of one generation (leptons and quarks, of left and right chirality, and their antiparticles) can be accommodated. The branching of these su(1|5) representations with respect to its subalgebra su(3)× su(2)× u(1) reproduces precisely the classification of these fundamental fermions according to the gauge group su(3)^c× su(2)^w× u(1)^w of the Standard Model. Furthermore, a simple construction of the relevant representations is given, and some consequences are discussed.     

Dynamical and structural symmetries for the highest Landau levels on the <Emphasis Type="Italic">AdS</Emphasis><Subscript>2</Subscript>

The aim of the paper is to use the recurrence relations with respect to both indices of the associated Legendre functions for the extraction of the Dirac quantization condition and dynamical symmetry group U(1, 1) corresponding to the highest Landau levels on the hyperbolic plane with uniform magnetic field B. Irreducible representations of the su(2) algebra are obtained by the ladder differential operators which change B by 1/2 unit and mode number by one unit. Two different classes of the irreducible representations of SU(1, 1) with the even and odd boson numbers 2B − 1/2 are extracted for the Bargmann indices 1/4 and 3/4, respectively. Finally, we show that shape invariance symmetry is realized by the ladder operators which shift only the magnetic field B by 1/2 unit.     

Graded Hopf maps and fuzzy superspheres

We argue supersymmetric generalizations of fuzzy two- and four-spheres based on the unitary-orthosymplectic algebras, uosp(N|2) and uosp(N|4), respectively. Supersymmetric version of Schwinger construction is applied to derive graded fully symmetric representation for fuzzy superspheres. As a classical counterpart of fuzzy superspheres, graded versions of 1st and 2nd Hopf maps are introduced, and their basic geometrical structures are studied. It is shown that fuzzy superspheres are represented as a “superposition” of fuzzy superspheres with lower supersymmetries. We also investigate algebraic structures of fuzzy two- and four-superspheres to identify su(2|N) and su(4|N) as their enhanced algebraic structures, respectively. Evaluation of correlation functions manifests such enhanced structure as quantum fluctuations of fuzzy supersphere.     

N=2 affine superalgebras and Hamiltonian reduction inN=2 superspace

We constructN=2 affine current algebras for the superalgebrassl(n/n-1)⁽¹⁾ in terms ofN=2 supercurrents subjected to nonlinear constraints and discuss the general procedure of the hamiltonian reduction inN=2 superspace at the classical level. We consider in detail the simplest case ofN=2sl(2/1)⁽¹⁾ and show howN=2 superconformal algebra inN=2 superspace follows via the hamiltonian reduction. Applying the hamiltonian reduction to the case ofN=2sl(3/2)⁽¹⁾, we find two new extendedN=2 superconformal algebras in a manifestly supersymmetricN=2 superfield form. Decoupling of four component currents of dimension 1/2 in them yields, respectively,u(2/1) andu(3) Knizhnik-Bershadsky superconformal algebras. We also discuss how theN=2 superfield formulations ofN=2W ₃ andN=2W ₃ ⁽²⁾ superconformal algebras come out in this framework, as well as some unusual extendedN=2 superconformal algebras containing constrainedN=2 stress tensor and/or spin 0 supercurrents.     

Classification of irreducible super-unitary representations ofsu(p,q/n)

In this paper we classify all the irreducible super-unitary representations ofsu(p,q/n), which can be integrated up to a unitary representation ofS(U(p,q)×U(n)), a Lie group corresponding to the even part ofsu(p,q/n). Note that a real form of the Lie superalgebrasl(m/n;) which has non-trivial superunitary representations is of the formsu(p,q/n)(p+q=m) orsu(m/r,s)(r+s=n). Moreover, we give an explicit realization for each irreducible super-unitary representation, using the oscillator representation of an orthosymplectic Lie superalgebra.     

Non-unitary minimal models,Bailey's Lemma and <Emphasis Type="Italic">N</Emphasis>=1,2 Superconformal algebras

Using the Bailey flow construction, we derive character identities for the N=1 superconformal models SM(p′,2p+p′) and SM(p′,3p′−2p), and the N=2 superconformal model with central charge c=3 from the nonunitary minimal models M(p,p′). A new Ramond sector character formula for representations of N=2 superconformal algebras with central element c=3 is given. Supported in part by NSF grant DMS-0200774.     

AdS superalgebras with brane charges

We consider the inclusion of brane charges in AdS₅ superalgebras that contain the maximal central extension of the super-Poincaré algebra on ∂AdS₅. For theories with N supersymmetries on the boundary, the maximal extension is OSp(1/8N,R), which contains the group Sp(8N,R)⊃U(2N,2N)⊃SU(2,2)×U(N) as extension of the conformal group. An “intermediate” extension to U(2N,2N/1) is also discussed, as well as the inclusion of brane charges in AdS₇ and AdS₄ superalgebras. BPS conditions in the presence of brane charges are studied in some details.     

Random Matrix Theory and L-Functions at s= 1/2

Recent results of Katz and Sarnak [8, 9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the L-functions within these families at the central point s= 1/2 and those of the characteristic polynomials Z(U,θ) of matrices U with respect to averages over SO(2N) and USp(2N) at the corresponding point θ= 0, using techniques previously developed for U(N) in [10]. For any matrix size N we find exact expressions for the moments of Z(U,0) for each ensemble, and hence calculate the asymptotic (large N) value distributions for Z(U,0) and log Z(U,0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for L-functions. The value distributions suggest consequences for the non-vanishing of L-functions at the central point. Received: 1 February 2000 / Accepted: 24 March 2000     

Indecomposable representations of the algebra su(2,1)

Representations of the Lie algebra sl(3) with highest weight are analyzed. Invariant subspaces of indecomposable representations are determined. We study the decomposition of these representations with respect to the subalgebras su(2) and su(1,1) (in their obvious imbedding in su(2,1)).For special cases this decomposition gives indecomposable non multiplicity free representations (indecomposable pairs) with highest weight. These were discussed in [1] and appear also in the decomposition so(3,2) su(1,1) of the Rac representation, [7].     

Decompositions of superfields inD=4 conformal SUSY

We use character formulae for the positive-energy unitary irreducible representations of theN-extendedD=4 conformal superalgebras su(2, 2/N) that were derived earlier to give decompositions of long superfields as they descend to the unitarity threshold.     

On the Unitarity of <Emphasis Type="Italic">D</Emphasis> = 9, 10, 11 Conformal Supersymmetry

We consider the unitarity of D = 9, 10, 11 conformal supersymmetry using the recently established classification of the UIRs of the superalgebras osp(12n, ).     

Singlet Deformation of <Emphasis Type="Italic">N</Emphasis> = (1,1) Superspace

We discuss the SO(4) × SU(2) invariant deformation of the Euclidean N = (1,1) supersymmetric theories in the framework of the harmonic superspace.     

核类似铜酸盐超导模型的sus(1,1)⊕sud(1,1)代数结构

由Iachello提出的核类似铜酸盐超导模型具有su(3)代数结构,其平均场近似下的Hamilton量可以写成su(3)生成元的线性组合.通过代数生成元的实现,该模型约化的Hamilton量具有su_s(1,1)⊕su_d(1,1)代数结构,利用相干算子U(θ,φ)的幺正变换,可得到系统约化Hamilton量的能隙方程及本征值,发现应用不同代数结构求解会得到不同侧重点的分析结果.     

Hermitean Oscillator-like Realizations of Classical Algebras and Superalgebras in Hilbert Space with Positive Definite Metric

The hermitean oscillator-like realizations of classical algebras in terms of bosonic and fermionic creation and annihilation operators are given. The hermitean realizations of classical superalgebras using boson-fermion oscillators are explicitely described. The assumption of positive definite metric in a Hilbert space of the oscillators states is exploited. Due to this fact, the realizations of superalgebras in the Hilbert space can be constructed only for: the real orthosymplectic superalgebra osp (N; 2M; R); the unitary compact superalgebra su (N; M); the unitary noncompact one SU(N; K, M); and the quaternionic unitary superalgebra uu_α(N; M; H).     

ISO(3,1¦N) andOSp(N¦4) supergravities

We advance anISO(3,1¦N) extended Poincaré supergravity and anOSp(N¦4) de Sitter supergravity by using the supergauge action mechanisms of supergroups on the superspaces and by treating the gravitational parts of these two supergravities as the gauge theories of gravity, give a new matrix representation ofISO(3,1¦N) generators and a new one ofOSp(N¦4) ones, obtain the commutation and anticommutation relations ofiso(3, 1¦N) andosp(N¦4) superalgebras, construct the actions of these Supergravities and discuss some other problems. A particle multiplets method based on the supersymmetry transformation is used and the probable numbers of particles of different helicities in the two supergravities are given.     

Dual properties of orthogonal polynomials of discrete variables associated with the quantum algebra <Emphasis Type="Italic">U</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(<Emphasis Type="Italic">su</Emphasis>(2))

We show that for every set of discrete polynomials y _n (x(s)) on the lattice x(s), defined on a finite interval (a, b), it is possible to construct two sets of dual polynomials z _k (ξ(t)) of degrees k = s-a and k = b-s-1. Here we do this for the classical and alternative Hahn and Racah polynomials as well as for their q-analogs. Also we establish the connection between classical and alternative families. This allows us to obtain new expressions for the Clerbsch-Gordan and Racah coefficients of the quantum algebra U _q (su(2)) in terms of various Hahn and Racah q-polynomials. Dedicated to the memory of our teacher and friend Arnold F. Nikiforov (18.11.1930–27.12.2005).     

The superconformal algebra in higher dimensions

The superconformal algebra for 4/4N-dimensional super-Minkowski space (d=4) can be identified with the simple superalgebra su (2,2/N). For even-dimension d=5,6 the superconformal algebra can be identified with a real form of the simple superalgebras F(4), D(4,1) respectively in Kac's classification. For even-dimension d>-7 it is impossible to define a superconformal algebra satisfying three natural conditions: (1) it acts as infinitesimal automorphisms on super-Minkowski space; (2) this action extends the natural action of the super-Poincaré algebra; (3) when the action of the even part of the superconformal algebra is reduced to an infinitesimal action on ordinary Minkowski space, it extends the natural action of the conformal algebra so (2, d).