Geometry of Brane-Worlds

TL;DR

This paper explores the most general geometric framework for brane-world models, demonstrating compatibility with various Lagrangians, deriving a canonical structure, and analyzing gauge fields and extra dimension sizes.

Contribution

It introduces a broad geometric scenario for brane-worlds, including a canonical quantization approach and insights into gauge field confinement and extra dimension sizes.

Findings

Embedding equations are compatible with a wide class of Lagrangians.

A non-trivial canonical structure suggests a path to quantum brane-world geometry.

Confined gauge fields exist in six or more dimensions, with calculated extra dimension sizes.

Abstract

The most general geometrical scenario in which the brane-world program can be implemented is investigated. The basic requirement is that it should be consistent with the confinement of gauge interaction, the existence of quantum states and the embedding in a bulk with arbitrary dimensions, signature and topology. It is found that the embedding equations are compatible with a wide class of Lagrangians, starting with a modified Einstein-Hilbert Lagrangian as the simplest one, provided minimal boundaries are added to the bulk. A non-trivial canonical structure is derived, suggesting a canonical quantization of the brane-world geometry relative to the extra dimensions, where the quantum states are set in correspondence with high frequency gravitational waves. It is shown that in the cases of at least six dimensions, there exists a confined gauge field included in the embedding structure. The size of extra dimensions compatible with the embedding is calculated and found to be different from the one derived with product topology.