B^F Theory and Flat Spacetimes

Abstract

We propose a reduced constrained Hamiltonian formalism for the exactly soluble $B \wedge F$ theory of flat connections and closed two-forms over manifolds with topology $Σ^3 \times (0,1)$. The reduced phase space variables are the holonomies of a flat connection for loops which form a basis of the first homotopy group $π_1(Σ^3)$, and elements of the second cohomology group of $Σ^3$ with value in the Lie algebra $L(G)$. When $G=SO(3,1)$, and if the two-form can be expressed as $B= e\wedge e$, 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.