International Journal of Geometric Methods in Modern Physics

We regard classical phase space as a generalized complex manifold and analyze the B-transformation properties of the ⋆-product of functions. The C^⋆-algebra of smooth functions transforms in the expected way, while the C^⋆-algebra of holomorphic functions (when it exists) transforms nontrivially. The B-transformed ⋆-product encodes all the properties of phase-space quantum mechanics in the presence of a background magnetic field.

A novel Chern–Simons E₈ gauge theory of gravity in D = 15 based on an octicE₈ invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is developed. A grand unification model of gravity with the other forces is very plausible within the framework of a supersymmetric extension (to incorporate spacetime fermions) of this Chern–Simons E₈ gauge theory. We review the construction showing why the ordinary 11D Chern–Simons gravity theory (based on the Anti de Sitter group) can be embedded into a Clifford-algebra valued gauge theory and that an E₈ Yang–Mills field theory is a small sector of a Clifford (16) algebra gauge theory. An E₈ gauge bundle formulation was instrumental in understanding the topological part of the 11-dim M-theory partition function. The nature of this 11-dim E₈ gauge theory remains unknown. We hope that the Chern–Simons E₈ gauge theory of gravity in D = 15 advanced in this work may shed some light into solving this problem after a dimensional reduction.

The braneworld theory appear with the purpose of solving the problem of the hierarchy of the fundamental interactions. The perspectives of the theory emerge as a new physics, for example, deviation of the law of Newton's gravity. One of the principles of the theory is to suppose that the braneworld is local submanifold in a space of high dimension, the bulk, solution of Einstein's equations in high dimension. In this paper we approach the mathematical consistency of this theory with a new proof of the fundamental theorem of submanifolds for the case of semi-Riemannian manifolds. This theorem consists of an essential mathematical support for this new theory. We find the integrability conditions for the existence of space–time submanifolds in a pseudo-Euclidean space.

We give a justification of the perturbation theory for contributions of Morse critical points to the spectral decomposition in deformation quantization.

A generalized geometric method is developed for constructing exact solutions of gravitational field equations in Einstein theory and generalizations. First, we apply the formalism of nonholonomic frame deformations (formally considered for nonholonomic manifolds and Finsler spaces) when the gravitational field equations transform into systems of nonlinear partial differential equations which can be integrated in general form. The new classes of solutions are defined by generic off-diagonal metrics depending on integration functions on one, two and three (or three and four) variables if we consider four (or five) dimensional spacetimes. Second, we use a general scheme when one (two) parameter families of exact solutions are defined by any source-free solutions of Einstein's equations with one (two) Killing vector field(s). A successive iteration procedure results in new classes of solutions characterized by an infinite number of parameters for a non-Abelian group involving arbitrary functions on one variable. Five classes of exact off-diagonal solutions are constructed in vacuum Einstein and in string gravity describing solitonic pp-wave interactions. We explore possible physical consequences of such solutions derived from primary Schwarzschild or pp-wave metrics.

The relevant material on differential calculus on graded infinite order jet manifolds and its cohomology is summarized. This mathematics provides the adequate formulation of Lagrangian theories of even and odd variables on smooth manifolds in terms of the Grassmann-graded variational bicomplex.

We present, in a pedagogical style, many instances of reduction procedures appearing in a variety of physical situations, both classical and quantum. We concentrate on the essential aspects of any reduction procedure, both in the algebraic and geometrical setting, elucidating the analogies and the differences between the classical and the quantum situations.