Geometry of Brane-Worlds

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.