Thurston Geometries from Eleven Dimensions

In three dimensions, a `master theory' for all Thurston geometries requires imaginary flux. However, these geometries can be obtained from physical three-dimensional theories with various additional scalar fields, which can be interpreted as moduli in various compactifications of a higher-dimensional `master theory'. Three Thurston geometries are of the form N_2 x S^1, where N_2 denotes a two-dimensional Riemannian space of constant curvature. This enables us to twist these spaces, via T-duality, into other Thurston geometries as a U(1) bundle over N_2. In this way, Hopf T-duality relates all but one of the geometries in the higher-dimensional M-theoretic framework. The exception is the `Sol geometry,' which results from the dimensional reduction of the decoupling limit of the D3-brane in a background B-field.