Itzhak Bars

Dimension and character formulas for Lie supergroups

A character formula is derived for Lie supergroups. The basic technique is that of symmetrization and antisymmetrization associated with Young tableaux generalized to supergroups. We rewrite the characters of the ordinary Lie groups U(N), O(N), and Sp(2N) in terms of traces in the fundamental representation. It is then shown that by simply replacing traces with supertraces the characters of certain representations for U(N/M) and OSP(N/2M) are obtained. Dimension formulas are derived by calculating the characters of a special diagonal supergroup element with (+1) and (−1) eigenvalues. Formulas for the eigenvalues of the quadratic Casimir operators are given. As applications, the decomposition of a representation into representations of subgroups is discussed. Examples are given for the Lie supergroup SU(6/4) which has physical applications as a dynamical supersymmetry in nuclei.

U(6/4) supersymmetry in nuclei

We suggest that supergroups and superalgebras may be useful in classifying the spectra of certain even-even and even-odd nuclei. We show, in particular, that properties of many states in several nuclei, including excitation energies, electromagnetic transition rates and transfer reaction intensities, can be described by a U(6/4) supersymmetry. Our analysis provides the first evidence for the occurrence of supersymmetry in nature.

Representations of Supergroups

An explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations. The rules of supersymmetrization are characterized by extended Young supertableaux. This constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all Casimir operators can be calculated. We have applied this approach so far to the supergroups SU(N/M), OSP(N/2M), P(N), for which we have obtained all the representations constructible as direct products of the fundamental (defining) representations. An argument is presented toward the irreducibility of all these representations.

Branching rules for the supergroup SU(N/M) from those of SU(N+M)

The decomposition of representations of supergroups into representations of subgroups is needed in practical applications. In this paper we set up and exploit a fruitful one‐to‐one correspondence between the Lie group branching SU (N+M)⊇SU(N)⊗SU(M)⊗U(1) and the supergroup branchings SU(N/M)⊇SU(N)⊗SU(M)⊗U(1) and SU(N1+N2/M1+M2)⊇SU(N1/M1) ⊗SU(N2/M2)⊗U(1). A simple and useful prescription is discovered for obtaining the SU(N/M) branching rules from those of SU(N+M) for any representation. A large class of examples, sufficient for many physical applications we can foresee, are explicitly worked out and tabulated.

Kac–Dynkin diagrams and supertableaux

We show the relation between Kac–Dynkin diagrams and supertableaux. We show the relation between Kac–Dynkin diagrams and supertableaux.

U(N) integral for the generating functional in lattice gauge theory

The one link integral or equivalently the generating functional of U(N) integrals in the lattice gauge theory is explicitly evaluated in terms of a character expansion.

Theoretical and phenomenological constraints on preon models and roles of supergroups

Abstract Models of massless composite quarks and leptons are proposed on the basis of the following requirements: 1. (1) Confining hypercolor gauge forces which are asymptotically free with a scale Λ H in the TeV range. 2. (2) Anomaly free gauge sector. 3. (3) Anomaly matching between preons and composite fermions for consistency of an unbroken chiral symmetry. 4. (4) Decoupling of heavy preons via a parity doubling of the composite fermions that contain them. 5. (5) Massless composite fermions contain the minimal number of valence preons that make a hypercolor singlet. Exotics are excluded or assumed to be massive. 6. (6) Indices of composites do not exceed 1. 7. (7) Pauli principle is satisfied with a completely symmetric spatial wave function. 8. (8) The preonic chiral symmetries contain SU(3) × SU(2) × U(1). When the symmetry is broken down to this group, the only remaining massless fermions fall just into the patterns of observed families. 9. (9) Unobserved processes such as proton decay, e + e − → μ ± e ∓ , eN→ μ N, υ →e γ , K L → μ ± e −+ , K + → π + ± μ ∓ , Δm (K L −K S ), and anomalous magnetic moments of e and μ are suppressed or do not occur. 10. (10) A mechanism for SU(2) w breaking that generates masses for quarks and leptons exists. We find that requirement (4) always imposes the structure of representations of supergroups SU( N / M ). Unlike supersymmetric models, the grading here is between left-handed and right-handed fermions rather than between bosons and fermions. Only the even subgroup of the supergroup is a symmetry, but preons and composite fermions must be classified in irreducible representations of SU( N / M ) as if the full supergroup were a symmetry. Using supertableaux techniques, we find and classify all models containing 3 preons in a composite fermion. We study models in which the preons fall into one of these structures with respect to hypercolor: (i) A single irreducible representation R of any hypercolor, group (including direct products) with R × R × R ∼ 1, (ii) Two irreducible representations R 1 , R 2 of hypercolor with R 1 R 2 ∗ R 2 ∗ ∼ 1, and (iii) Three irreducible representations satisfying R 1 R 2 R 3 ∼ 1. We treat the hypercolor group G H and the representations R i as unknowns and follow a strategy for finding constraints on G H and R i which lead to all possible models consistent with conditions (1)–(5), then (6)–(7) and finally (8)–(10). Explicitly we construct many models that satisfy conditions (1)–(6). One model of type (i) satisfies requirement (7) but no model of type (i) satisfies the additional requirements (8)–(10). Only two classes of models of type (ii) and one class of type (iii) are found to pass tests (1)–(7). Among then we find an SU(3) × SU(2) × U(1) embedding which satisfies the remaining physical requirements (8)–(10). Δm (K L − K S ) provides the most severe bounds on the hypercolor scale Λ H .

A quantum string theory of hadrons and its relation to quantum chromodynamics in two dimensions

Abstract We propose an interacting quantum string theory of hadrons which has a close relation with quantum chromodynamics (QCD). Hadrons are constructed by attaching, massive, spin - 1 2 quarks, with internal symmetry, to the ends of strings. The string has independent longitudinal modes of oscillation which play a major role in the agreement with QCD. The string-string interactions are formulated in a new Hamiltonian formalism which allows strings to join or split only by quark-antiquark annihilation. Weak-electromagnetic gauge bosons are coupled to the flavor group of the quarks. By specializing to two dimensions, it is shown that quantum chromodynamics is equivalent to this string theory, in the sense that the mesonic and baryonic spectra, and the strong, weak and electromagnetic vertices for hadrons are identical in these two theories.

Gravity with extra gauge symmetry

Abstract In addition to the tangent space Lorentz gauge invariance, we introduce an extra gauge principle in the gauge formulation of gravity, in any dimension. The connection ω ij μ , which starts out as an elementary field, undergoes a gauge transformation which mixes tangent space indices ( i , j ) with base indices ( μ ). The equations of motion determine ω ij μ as the sum of usual riemannian connection and a completely antisymmetric torsion piece. In 11 dimensions our theory reproduces the bosonic sector of supergravity if we identify torsion with the 3-index gauge potential A μυλ . This generalizes Englerts recent observation, since in our formulation torsion = A μυλ for any field configuration, not just a solution.

Family structure with composite quarks and leptons

Abstract A theory of composite quarks and leptons as composites of three spin - 1 2 preons is proposed. It is argued that the persistent mass condition is a natural requirement in such models. A specific realistic model in constructed that predicts up to 8 SO(10) families in the spinor representation, for which masses are generated by explicit small breaking of the underlying chiral symmetries. The small masses of the families are explained as being due only to SU(3)C⊗SU(2)W⊗U(1)W symmetry.