Gravity in non-commutative geometry

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.Dedicated to H. ArakiSupported in part by the Swiss National Foundation (SNF)     

Two-dimensional Lorentz-Weyl anomaly and gravitational Chern-Simons theory

Two-dimensional chiral fermions and bosons, more generally conformal blocks of two-dimensional conformal field theories, exhibit Weyl-, Lorentz- and mixed Lorentz-Weyl anomalies. A novel way of computing these anomalies for a system of chiral bosons of arbitrary conformal spinj is sketched. It is shown that the Lorentz- and mixed Lorentz-Weyl anomalies of these theories can be cancelled by the anomalies of a three-dimensional classical Chern-Simons action for the spin connection, expressed in terms of the dreibein field. Some tentative applications of this result to string theory are indicated.     

Massive supergravity from spontaneously breaking orthosymplectic gauge symmetry

We construct a Lagrangian density which is manifestly invariant under the orthosymplectic gauge group OSP(1; 4) and under general coordinate transformations. This is done by the use of two multiplets, the symmetric and antisymmetric representations of OSP(1; 4). We present the general features of OSP(m; 2n) and, in particular, its irreducible representations. The absence of OSP(1; 4) symmetry from the ground state indicates that one of the scalar fields, which is an element of the symmetric multiplet, has a nonvanishing vacuum expectation value. A shift in the fields reveals the physical spectrum of our Lagrangian. Two Goldstone fields are present, a vector and a spinor, corresponding to the breakdown of OSP(1; 4) to the Lorentzian group. The full Lagrangian contains a graviton, a massive $\text{spin}\text{-}\text{3}\text{2}$ field, and two massive scalar fields. The generalization to OSP(2; 4) is immediate.     

N = 4 supergravity coupled to N = 4 matter and hidden symmetries

We present the N = 1 supergravity in 10 dimensions obtained by truncating the reduced N = 1 supergravity from 11 dimensions. This is further reduced to 4 dimensions to give SU(4) supergravity coupled to six SO(4) vector multiplets. As the reduction is from 10 dimensions, the theory is expected to have the symmetry SL(6R)_global×SO(6)_local, but we give a theoretical argument that this can be extended to SO(6,6)×SU(1,1)_global and SO(6)×SO(6)×U(1)_local.     

Rapid access to novel 2-alkylthiopyrimidine derivatives and attempt of their Tacrine analogs synthesis

A variety of novel 2-alkylthiopyrimidines were synthesized through simple condensation of arylidenemalononitriles with different 2-alkylthiouronium halide derivatives catalyzed by anhydrous potassium carbonate (K₂CO₃). The reactions have been carried out under mild conditions in i-PrOH, and the products were obtained in moderate to good yields with a simple work-up method. Subsequently, some examples of these compounds have been converted into Tacrine analogs by applying the Friedländer reaction.     

Supersymmetric electro-weak effects ong μ−2

A model independent analysis of the super-symmetric electroweak contribution tog _μ?2 is discussed within the framework ofN=1 Supergravity unified theory. A detailed comparison with existing experiment of two models (R.G. and T.B.) is carried out. The supersymmetric electro-weak contributions are found to be characteristically different and generally larger than the electro-weak contributions of the standard theory, and in many cases significantly larger. Effects of the hidden sector and the photino mass dependence ofg _μ?2 are also investigated. Present data already eliminates some choices of parameters. Reduction of existing experimental errors by a factor of 3 will make contact with most R.G. models and by a factor of 10 with most T.B. models.     

Hermitian Geometry and Complex Space-Time

We consider a complex Hermitian manifold of complex dimensions four with a Hermitian metric and a Chern connection. It is shown that the action that determines the dynamics of the metric is unique, provided that the linearized Einstein action coupled to an antisymmetric tensor is obtained, in the limit when the imaginary coordinates vanish. The unique action is of the Chern-Simons type when expressed in terms of the Kähler form. The antisymmetric tensor field has gauge transformations coming from diffeomorphism invariance in the complex directions. The equations of motion must be supplemented by boundary conditions imposed on the Hermitian metric to give, in the limit of vanishing imaginary coordinates, the low-energy effective action for a curved metric coupled to an antisymmetric tensor.