Ponzano-Regge Model on Manifold with Torsion

TL;DR

This paper extends the Ponzano-Regge model to manifolds with torsion, linking quantum angular momentum symbols to discrete gravity actions, and explores their asymptotic behavior in a torsional setting.

Contribution

It introduces a novel approach to incorporate torsion into the Ponzano-Regge model, relating angular momentum, torsion, and discrete gravity actions on simplicial manifolds.

Findings

Relation between 6j symbols and Regge action with torsion established

Asymptotic formula for the partition function derived in the torsional case

Connection to Feynman path integral without cosmological constant discussed

Abstract

The connection between angular momentum in quantum mechanics and geometric objects is extended to manifold with torsion. First, we notice the relation between the $6j$ symbol and Regge's discrete version of the action functional of Euclidean three dimensional gravity with torsion, then consider the Ponzano and Regge asymptotic formula for the Wigner $6j$ symbol on this simplicial manifold with torsion. In this approach, a three dimensional manifold $M$ is decomposed into a collection of tetrahedra, and it is assumed that each tetrahedron is filled in with flat space and the torsion of $M$ is concentrated on the edges of the tetrahedron, the length of the edge is chosen to be proportional to the length of the angular momentum vector in semiclassical limit. The Einstein-Hilbert action is then a function of the angular momentum and the Burgers vector of dislocation, and it is given by summing the Regge action over all tetrahedra in $M$. We also discuss the asymptotic approximation of the partition function and their relation to the Feynmann path integral for simplicial manifold with torsion without cosmological constant.