4-dimensional BF Gravity on the Lattice

TL;DR

This work constructs a 4D lattice gravity model from SU(2) BF theory by placing the 2-form field $B$ on triangles and dual-link holonomies $U$ on the dual lattice, with curvature defined via holonomies around dual plaquettes. A vanishing holonomy constraint links $B$ to the curvature, and the partition function is shown to reduce to a product of generalized 15-$j$ symbols, with discretized triangle areas arising from the logarithmic lattice action. The authors provide a detailed graphical Pachner move analysis proving topological invariance under 4D moves, establishing triangulation independence and enabling a continuum limit that recovers the continuum $BF$ action. The framework unifies lattice gauge theory with Regge calculus concepts, suggests a natural discretization of geometric data, and offers avenues for extending to higher dimensions, coupling to matter, and possible $q$-deformations.

Abstract

We propose the lattice version of $BF$ gravity action whose partition function leads to the product of a particular form of 15-$j$ symbol which corresponds to a 4-simplex. The action is explicitly constructed by lattice $B$ field defined on triangles and link variables defined on dual links and is shown to be invariant under lattice local Lorentz transformation and Kalb-Ramond gauge transformation. We explicitly show that the partition function is Pachner move invariant and thus topological. The action includes the vanishing holonomy constraint which can be interpreted as a gauge fixing condition. This formulation of lattice $BF$ theory can be generalized into arbitrary dimensions.

Figures (4)

• Figure 1: $\tilde{P}$ is the dual plaquette dual to the triangle $t$ associated to the 2-form $B(t)$. Dual link variable $U = e^{i \omega}$ is located on the dual link which constructs the boundary of the dual plaquette $\tilde{P}$.
• Figure 2: A simple setup to show the integrated lattice Bianchi identity. $A$, $B$, $C$, $D$, $E$ and $F$ are sites and $ACDEF$, $ABCEF$, $ABCDF$, $BCDEF$ are the centers of 4-simplex. Thick lines are the dual links and thin lines are original links.
• Figure 3: Graphical presentation of geometrical structure of a 4-simplex $ABCDE$
• Figure 4: Pachner moves in 4 dimensions. (1) 1-5 move : $(ABCDE) \rightarrow (BCDEF)(ACDEF)(ABDEF)(ABCEF)(ABCDF)$, (2) 2-4 move : $(ACDEF)(BCDEF) \rightarrow (ABDEF)(ABCEF)(ABCDF)(ABCDE)$ , and (3) 3-3 move : $(BCDEF)(ACDEF)(ABDEF) \rightarrow (ABCDE)(ABCEF)(ABCDF)$.