Instituto de Ciencias Nucleares, Circuito Exterior, C.U.

Abstract: We propose a reduced constrained Hamiltonian formalism for the exactly soluble B ∧ F theory of flat connections and closed two-forms over manifolds with topology Σ× (0, 1). The reduced phase space variables are the holonomies of a flat connection for loops which form a basis of the first homotopy group π1(Σ ), and elements of the second cohomology group of Σ with value in the Lie algebra L(G). When G = SO(3, 1), and if the two-form can be expressed as B = e ∧ e, for some vierbein field e, then the variables represent a flat spacetime. This is not always possible: We show that the solutions of the theory generally represent spacetimes with “global torsion”. We describe the dynamical evolution of spacetimes with and without global torsion, and classify the flat spacetimes which admit a locally homogeneous foliation, following Thurston’s classification of geometric structures. PACS 04.20, 02.20.+b, 03.50.Kk, 04.50.+h