Relation between Regge calculus and BF theory on manifolds with defects. (English) Zbl 1416.83087

Kisielowski, Marcin

Ann. Henri Poincaré 20, No. 5, 1403-1437 (2019).

Summary: In Regge calculus, the space-time manifold is approximated by certain abstract simplicial complex, called a pseudomanifold, 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 consider an action functional which coincides with the standard BF action for suitably regular manifolds with defects and fields. We show that 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, the degrees of freedom of Regge calculus are traded for discrete degrees of freedom of topological BF theory.

MSC:

83D05	Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83C27	Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory
53Z05	Applications of differential geometry to physics
81T45	Topological field theories in quantum mechanics

Keywords:

Regge calculus; manifolds with defects; pseudomanifold; topological BF theory

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