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 Poincar\'e 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' $$ds^2={\rm e}^{-|\xi|^2/4l^2}\eta_{\mu\nu}dx^\mu dx^\nu+\delta_{ij} d\xi^i d\xi^j$$ with $l$ a fundamental length scale. At low energies, space-time reduces to four observable dimensions and general relativistic gravity is reproduced.