B^F Theory and Flat Spacetimes

Details B^F Theory and Flat Spacetimes B^F Theory and Flat Spacetimes

1993-11-26

arxiv.org/abs/gr-qc/9311033v1

Commun.Math.Phys. 170 (1995) 63-78

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

21 pp., Mexico Preprint ICN-UNAM-93-12

Scientific paper

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 $\Sigma^3 \times (0,1)$. The reduced phase space variables are the holonomies of a flat connection for loops which form a basis of the first homotopy group $\pi_1(\Sigma^3)$, and elements of the second cohomology group of $\Sigma^3$ 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.

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