Symmetry, Integrability and Geometry: Methods and Applications, 2008, том 4

A Unified Model of Phantom Energy and Dark Matter

2008-01-01T00:00:00Z

A Unified Model of Phantom Energy and Dark Matter Chaves, M.; Singleton, D. To explain the acceleration of the cosmological expansion researchers have considered an unusual form of mass-energy generically called dark energy. Dark energy has a ratio of pressure over mass density which obeys w = p/ρ < −1/3. This form of mass-energy leads to accelerated expansion. An extreme form of dark energy, called phantom energy, has been proposed which has w = p/ρ < −1. This possibility is favored by the observational data. The simplest model for phantom energy involves the introduction of a scalar field with a negative kinetic energy term. Here we show that theories based on graded Lie algebras naturally have such a negative kinetic energy and thus give a model for phantom energy in a less ad hoc manner. We find that the model also contains ordinary scalar fields and anti-commuting (Grassmann) vector fields which act as a form of two component dark matter. Thus from a gauge theory based on a graded algebra we naturally obtained both phantom energy and dark matter.

Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems

2008-01-01T00:00:00Z

Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems Beffa, G.M. In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The paper discusses the close connection between different types of geometries and the type of equations they realize. In particular, we describe the direct relation between symmetric spaces and equations of KdV-type, and the possible geometric origins of this connection.

Relative differential K-characters

2008-01-01T00:00:00Z

Relative differential K-characters Maghfoul, M. We define a group of relative differential K-characters associated with a smooth map between two smooth compact manifolds. We show that this group fits into a short exact sequence as in the non-relative case. Some secondary geometric invariants are expressed in this theory.

A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)

2008-01-01T00:00:00Z

A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary) Paneitz, S.M. This is the original manuscript dated March 9th 1983, typeset by the Editors for the Proceedings of the Midwest Geometry Conference 2007 held in memory of Thomas Branson. Stephen Paneitz passed away on September 1st 1983 while attending a conference in Clausthal and the manuscript was never published. For more than 20 years these few pages were circulated informally. In November 2004, as a service to the mathematical community, Tom Branson added a scan of the manuscript to his website. Here we make it available more formally. It is surely one of the most cited unpublished articles. The differential operator defined in this article plays a key rôle in conformal differential geometry in dimension 4 and is now known as the Paneitz operator.