# Relation between Regge calculus and BF theory on manifolds with defects

## TL;DR

This paper establishes a connection between Regge calculus and BF theory by constructing manifolds with defects from simplicial complexes, enabling a reformulation of Regge geometry in terms of topological BF theory.

## Contribution

It introduces a method to encode Regge geometry within BF theory on manifolds with defects, bridging discrete geometry and topological field theory.

## Key findings

- The constructed action matches the standard BF action for regular manifolds.
- Solutions of the field equations reproduce Regge actions at boundary data.
- Degrees of freedom are transferred from Regge calculus to BF theory.

## Abstract

In Regge calculus the space-time manifold is approximated by certain abstract simplicial complex, called a pseudo-manifold, and the metric is approximated by an assignment of a length to each 1-simplex. In this paper for each pseudomanifold we construct a smooth manifold which we call a manifold with defects. This manifold emerges from the purely combinatorial simplicial complex as a result of gluing geometric realizations of its n-simplices followed by removing the simplices of dimension n-2. The Regge geometry is encoded in a boundary data of a BF-theory on this manifold. We construct an action functional which coincides with the standard BF action for suitably regular manifolds with defects and fields. On the other hand, the action evaluated at solutions of the field equations satisfying certain boundary conditions coincides with an evaluation of the Regge action at Regge geometries defined by the boundary data. As a result we trade the degrees of freedom of Regge calculus for discrete degrees of freedom of topological BF theory.