B^F Theory and Flat Spacetimes

TL;DR

This paper develops a Hamiltonian formalism for the B∧F theory of flat connections, characterizing the phase space and solutions as flat spacetimes with possible global torsion, and classifies such spacetimes based on geometric structures.

Contribution

It introduces a reduced Hamiltonian approach for B∧F theory on manifolds with specific topology, linking solutions to flat spacetimes with torsion and classifying their geometric structures.

Findings

Phase space variables are holonomies and cohomology elements.

Solutions generally represent spacetimes with global torsion.

Classification of flat spacetimes with homogeneous foliation.

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 $\Sigma^3 \times (0,1)$. 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.