Extended geometric Supergravity on Group Manifolds with Spontaneous Fibration

We review and develop geometrical gauging involving the sequence: Lie group/Principal Bundle, for an Internal symmetry group/Soft Group Manifold, for Non-Internal groups. In the Internal case, we rederive the Ghost-fields and the BRS transformations and equations geometrically. In the case of a Soft Group Manifold, we study Spontaneous Fibration, rederive Gravity and Supergravity and then present Extended geometric Supergravity theories.     

Leptonic flavor and CP violation

Recent neutrino oscillation data teach us that the neutrinos have masses and that they mix. We discuss two ways that can be used to probe other non-standard leptonic physics. We show that non-standard neutrino interaction can be probed in neutrino oscillation experiments and discuss sneutrino-antisneutrino mixing.     

Magnetic correlations on fractals

The critical behavior of magnetic spin models on various fractal structures is reviewed, with emphasis on branching and nonbranching Koch curves and Sierpiriski gaskets and carpets. The spin correlation function is shown to have unusual exponential decays, e.g., of the form exp[-(r/gx) ^x ], and to crossover to other forms at larger distancesr. The various fractals are related to existing models for the backbone of the infinite incipient cluster at the percolation threshold, and conclusions are drawn regarding the behavior of spin correlations on these models.     

Duality,crossing and MacLane’s coherence

It is shown that MacLane’ rectangle, pentagon and hexagon identities in category theory, when applied in particle physics to duality diagrams or to rational conformal field theories in two dimensions, yield the necessary physical algebraic constraints. Supported in part by DOE grant no. DE-FG05-85ER40200 Supported by US-Israel BSF grant 87-00009/1.     

Irreducible gauge theory of a consolidated Salam-Weinberg model

We derive the Salam-Weinberg model by gauging an internal simple supergroup $\text{SU}\text{(}\text{2}\text{1}\text{)}$. The theory uniquely assigns the correct SU(2)_L ? U(1) eigenvalues for all leptons, fixes θ_W = 30°, generates the W^±_σ, Z⁰_σ and A_σ together with the Higgs-Goldstone $\text{I}{}_{\mathrm{<mtext>L</mtext>}}\text{=}\text{1}\text{2}$ scalar multiplets as gauge fields, and imposes the standard spontaneous breakdown of SU(2)_L ? U(1). The masses of intermediate bosons and fermions are directly generated by $\text{SU}\text{(}\text{2}\text{1}\text{)}$ universality, which also fixes the Higgs field coupling.     

Noncommutative Geometry, Superconnections and Riemannian Gravity as a Low-Energy Theory

A superconnection is a supermatrix whose evenpart contains the gaugepotential one-forms of a localgauge group, while the odd parts contain the (zero-form)Higgs fields breaking the local symmetry spontaneously. The combined grading is thus odd everywhere andthe superconnection can be directly derived from aformulation of Noncommutative Geometry, as theappropriate one-form in the relevant form calculus. The simple supergroup (4, ) (rank = 3) in Kac' classification (evensubgroup (4,)) provides themost economical spontaneous breaking of (4,) as gauge group leaving just local (1,3) unbroken. Post-Riemannian SKY gravity thereby yields Einstein's theory asa low-energy (longer range) effective theory. The theoryis renormalizable and may be unitary.     

Inhibition mode of a bisubstrate inhibitor of KDO8P synthase: a frequency-selective REDOR solid-state and solution NMR characterization

In this report the mode of inhibition of mechanism-based inhibitor (2, K(i) = 0.4 microM) of 3-deoxy-d-manno-2-octulosonate-8-phosphate synthase (KDO8PS), which was designed to mimic the combined key features of its natural substrates arabinose-5-phosphate (A5P) and phoshoenolpyruvate (PEP) into a single molecule, was investigated. Our earlier solid-state NMR observations identified the inhibitor to bind in a way that partly mimics A5P, while the phosphonate moiety of its PEP-mimicking part exhibits no interactions with enzyme residues. This result was apparently in disagreement with the competitive inhibition of 2 against PEP and with the later solved crystal structure of KDO8PS-2 binary complex identifying the interactions of its PEP-mimicking part with the enzyme residues that were not detected by solid-state NMR. To solve this discrepancy, further solid-state REDOR NMR and (31)P solution NMR experiments were applied to a variety of enzyme complexes with the substrates and inhibitor. In particular, a novel frequency-selective REDOR experiment was developed and applied. Integration of the solution and solid-state NMR data clearly demonstrates that under conditions of stoichiometric enzyme-ligand ratio at thermodynamic equilibrium (a) PEP binding is unperturbed by the presence of 2 and (b) both PEP and 2 can bind simultaneously to the synthase, i.e., form a ternary complex with PEP occupying its own subsite and 2 occupying A5P's subsite. The latter observation suggests that under the conditions used in our NMR measurements, the inhibition pattern of 2 against PEP should have a mixed type character. Furthermore, the NMR data directly demonstrate the distinction between the relative binding strength of the two moieties of 2: enzyme interactions with PEP-mimicking moiety are much weaker than those with the A5P moiety. This observation is in agreement with KDO8PS-2 crystal structure showing only remote contacts of the phosphonate due to large structural changes of binding site residues. It is concluded that these phosphonate-enzyme interactions evidenced by both (31)P solution NMR and X-ray are too weak to be preserved under the lyophilization of KDO8PS-2 binary complex and therefore are not evidenced by the solid-state REDOR spectra.     

Packing dimension and Cartesian products

We show that for any analytic set in , its packing dimension can be represented as , where the supremum is over all compact sets in , and denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if . In contrast, we show that the dual quantity , is at least the ``lower packing dimension' of , but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)

     

Some double-valued representations of the linear groups

We review the mathematical theory ofSL(n, R) and its double-covering group , especially forn = 2, 3, 4. After discussing a variety of physical applications, we show that provides holonomic curved space (world) spinors with an infinite number of components. We construct the relevant holonomic manifield and discuss the gravitational interaction of a proton as an example.Invited paper, dedicated to Eugene Paul Wigner on the occasion of his eightieth birthday.Supported in part by a U.S. Department of Energy Research Grant.     

Intersecting random translates of invariant Cantor sets

Summary Given two Cantor setsX andY in [0, 1), invariant under the mapxb x mod 1, the Hausdorff dimension of (X+t)Y is constant almost everywhere. WhenX,Y are defined by admissible digits in baseb, and more generally by sofic systems, we compute this dimension in terms of the largest Lyapunov exponent of a random product of matrices. The results are extended to higher dimensions and multiple intersections.Oblatum 17-VIII-1990Support was provided by an IBM fellowship.Partially supported by a grant from the Landau Centre for Mathematical Analysis