From lattice BF gauge theory to area-angle Regge calculus

TL;DR

This paper develops a framework connecting lattice BF gauge theory with area-angle Regge calculus in 4D, incorporating simplicity constraints, geometric variables, and the Immirzi parameter, facilitating the derivation of spin foam models.

Contribution

It provides a precise method to derive area-angle Regge calculus from constrained BF theory, unifying variables and formulating an action including the Immirzi parameter.

Findings

Identifies geometric quantities within BF lattice gauge theory.

Shows how simplicity constraints encode Regge calculus constraints.

Recovers standard BF spin foam model without simplicity constraints.

Abstract

We consider Riemannian 4d BF lattice gauge theory, on a triangulation of spacetime. Introducing the simplicity constraints which turn BF theory into simplicial gravity, some geometric quantities of Regge calculus, areas, and 3d and 4d dihedral angles, are identified. The parallel transport conditions are taken care of to ensure a consistent gluing of simplices. We show that these gluing relations, together with the simplicity constraints, contain the constraints of area-angle Regge calculus in a simple way, via the group structure of the underlying BF gauge theory. This provides a precise road from constrained BF theory to area-angle Regge calculus. Doing so, a framework combining variables of lattice BF theory and Regge calculus is built. The action takes a form {\it \`a la Regge} and includes the contribution of the Immirzi parameter. In the absence of simplicity constraints, the standard spin foam model for BF theory is recovered. Insertions of local observables are investigated, leading to Casimir insertions for areas and 6j-symbols for 3d angles. The present formulation is argued to be suitable for deriving spin foam models from discrete path integrals.