Poisson fiber bundles and coupling Dirac structures

Poisson fiber bundles are studied. We give sufficient conditions for the existence of a Dirac structure on the total space of a Poisson fiber bundle endowed with a compatible connection. We also provide some examples.      

Cartan geometries and multiplicative forms

In this paper we show that Cartan geometries can be studied via transitive Lie groupoids endowed with a special kind of vector-valued multiplicative 1-forms. This viewpoint leads us to a more general notion, that of Cartan bundle, which encompasses both Cartan geometries and G-structures.     

The action of the group of bundle-automorphisms on the space of connections and the geometry of gauge theories

Some aspects of the geometry of gauge theories are sketched in this review. We deal essentially with Yang-Mills theory, discussing the structure of the space of gauge orbits and the geometrical interpretation of ghosts and anomalies. Occasionally we deal also with classical gauge theories of gravitation and in particular we study the action of the group of diffeomorphisms on the space of linear connections. Finally we comment on the mathematical interpretation of anomalies in field theories.     

Background Independent Quantum Mechanics, Classical Geometric Forms and Geometric Quantum Mechanics—I

The geometry of the symplectic structures and Fubini-Study metric is discussed. Discussion in the paper addresses geometry of Quantum Mechanics in the classical phase space. Also, geometry of Quantum Mechanics in the projective Hilbert space has been discussed for the chosen Quantum states. Since the theory of classical gravity is basically geometric in nature and Quantum Mechanics is in no way devoid of geometry, the explorations pertaining to more and more geometry in Quantum Mechanics could prove to be valuable for larger objectives such as understanding of gravity.     

Chern–Simons classes for a superconnection

In this note we define the Chern–Simons classes of a flat superconnection, D+L$D+L$, on a complex Z/2Z$\mathbb{Z}/2\mathbb{Z}$-graded vector bundle E$E$ on a manifold such that D$D$ preserves the grading and L$L$ is an odd endomorphism of E$E$. As an application, we obtain a definition of Chern–Simons classes of a (not necessarily flat) morphism between flat vector bundles on a smooth manifold. An application of Reznikov's theorem shows the triviality of these classes when the manifold is a compact Kähler manifold or a smooth complex quasi-projective variety in degrees >1$>1$.     

Effect of organic salts on micellar growth and structure studied by rheology

We report a rheological study on the effect of adding organic salts [sodium tosylate (NaTos) and benzoic acid potassium salt (BaPs)] on the micellar growth and structure of aqueous solutions of cethyltrimethylammonium chloride (CTAC) at a constant molar concentration ratio [salt]/[CTAC]. The rheological data show two well-defined domains of growth characterized by scaling laws for the surfactant concentration. The addition of NaTos leads to an unusual maximum in the viscosity-surfactant concentration curve. Before the maximum (domain 1), the analysis of the data (η₀, τ_R and G ₀) suggests the presence of branched micelles (connections). After the maximum (domain 2), however, the exponents of the scaling laws do not reflect either the relaxation of this branched structure or that of an entangled transient network structure. A faster mechanism of relaxation, not yet elucidated governs their dynamics. The exponents of the power laws in the presence of the BaPs are found, however, to be in accordance with the theory of equilibrium polymers. Received: 15 April 1998 Accepted in revised form: 20 October 1998     

A kinematic model for continuous distributions of dislocations

In continuum theory of defects the notion of a flat connection is employed. This paper gives a characterisation of these connections via injective ³-valued differential forms. For material structures with continuous distributions of dislocations, a configuration space in the sense of global analysis is introduced and analysed. A kinematics for these dislocations is formulated which generalises from elasticity.     

On the quantum Hall effect

We consider a particle in a 2 dimensional plane in a periodic potential and a homogeneous magnetic field perpendicular to the plane. Kubo's expression for conductivity of the Hall current is an integer.

This result of Thouless, Kohomoto, Nightingale and den Nijs is interpreted geometrically.