4-dimensional BF Gravity on the Lattice

TL;DR

This paper develops a lattice formulation of 4-dimensional BF gravity, demonstrating its topological invariance and gauge symmetries, with potential for generalization to higher dimensions.

Contribution

It introduces a lattice BF gravity action with explicit construction, invariance properties, and a connection to 15-j symbols, advancing discrete quantum gravity models.

Findings

Partition function expressed via 15-j symbol for 4-simplex

Proven invariance under Pachner moves, confirming topological nature

Inclusion of gauge fixing via holonomy constraint

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.