DIMENSIONAL REDUCTION BY A TWO - FORM ( another alternative to compactification )

It is shown that the local coupling of a higher dimensional graviton to a closed degenerate two-form produces dimensional reduction by spontaneous breakdown of extra-dimensional translational symmetry. Four dimensional Poincaré invariance emerges as residual symmetry. As a specific example, a six dimensional geometry coupled to a closed rank 2 two-form yields the 'ground state' ds 2 = e −|ξ| 2 /4l 2 η µν dx µ dx ν + δ ij dξ i dξ j with l a fundamental length scale. At low energies, space-time reduces to four observable dimensions and general relativistic gravity is reproduced. There has been recently a renewed interest in higher dimensional geometry as an intermediate step in between today's high energies field theories and a foretold final theory. Besides the search for phenomenological implications, one of the hottest problems remains that of reproducing the effective four dimensional nature of space-time at low energies scales. To one side the traditional hypothesis of compact extra dimensions has been fully reconsidered [1]. To the other, a remarkable alternative has been put forward by showing that dimensional reduction take place without compactification in a setup with a 3-brane embedded in the higher dimensional space [2]. In both approaches dimensional reduction is produced by global topological assumptions –compactness of extra-dimensions in the first case, 3-brane boundary conditions in the second one. In this letter we explore an alternative based on the local interaction with a closed degenerate two-form. The specific model we consider is a six dimensional space-time (M 6 , g IJ) on which a closed rank 2 two-form B IJ is defined 1. No physical hypothesis are made on the nature of B IJ. In solving generalized Einstein's equations it is found that translational invariance is spontaneously broken and four dimensional Poincaré invariance emerges as the residual symmetry of the geometric 'ground state'. The corresponding metric tensor reproduces –up to the functional form of the wrap factor and two instead of one extra dimensions– the one recently investigated by L. Randall and R. Sundrum [2]. On low energies scales space-time dynamically reduces to four effective dimensions and the ordinary general relativistic picture is recovered.