Freund–Rubin compactification
Freund–Rubin compactification is a form of dimensional reduction in which a field theory in d-dimensional spacetime, containing gravity and some field whose field strength is a rank s antisymmetric tensor, 'prefers' to be reduced down to a spacetime with a dimension of either s or d-s.
Consider General Relativity in d spacetime dimensions. In the presence of an antisymmetric tensor field (without external sources), the Einstein Field equations, and the equations of motion for the antisymmetric tensor are
Where the stress-energy tensor takes the form
Being a rank s antisymmetric tensor, the field strength has a natural ansatz for its solution, proportional to the Levi-Civita tensor on some s-dimensional manifold.
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