From lattice BF gauge theory to area-angle Regge calculus

TL;DR

The paper builds a direct link from lattice BF theory to area-angle Regge calculus by enforcing Plebanski simplicity constraints on bivectors and carefully solving parallel transport (gluing) relations. The resulting formalism identifies area data $A_f$ and 3d/4d dihedral angles as the fundamental geometric variables, recasting the action in a compactified Regge form that includes an Immirzi parameter $\gamma$; in nondegenerate configurations the Immirzi parameter drops from the equations of motion, yielding a Regge-like dynamics. A mixed framework is developed that preserves lattice BF variables ($g_{vt}$, bivectors) while reproducing area-angle RC constraints, including a $\mathrm{U}(1)\times\mathrm{U}(1)$ connection for the angular data. The authors also show how classic BF spin foams emerge in the unconstrained limit and how local observables (areas and 3d angles) translate into spin-foam insertions (Casimirs and 6j-symbols), providing a concrete path to derive spin foam models from discrete path integrals. Overall, the work clarifies the geometric content of area-angle RC within a BF-based discretization and offers a practical route to connect lattice gravity, Regge calculus, and spin foam models.

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 à 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.

Figures (2)

• Figure 1: In the dual picture, the three tetrahedra $t$, $t'$ and $t"$ of the 4-simplex $v$ become edges meeting at the vertex $v$. Each triangle being shared by two tetrahedra in $v$, the boundaries of the dual faces in the neibourhood of $v$ are made of two dual edges. For a pair of tetrahedra meeting at $f$, one can equivalently write the dihedral angle $\theta_{tt"}$ between them in terms of the 3d angles using three different intermediate tetrahedra (the boundary of a 4-simplex being made of five tetrahedra). This leads to the constraints \ref{['consistency 3d angles']} between the 3d angles, as proposed in area-angleRC
• Figure 2: The left figure shows the graph weighting the tetrahedron $t$ and resulting from the insertion of the 3d angle between $f_1$ and $f_2$ in $t$. In contrast with pure BF theory, it does not simply consist of the contraction of the intertwiners coming from the integrations of $g_1$ and $g_2$ because of the insertion of a link in the spin 1 representation between the links of $f_1$ and $f_2$. This graph reduces to the evaluation of a 6j-symbol if the pairing is conveniently chosen, displaying the role of the intertwiners in the quantization of the 3d angle.