An Analysis of the Representations of the Mapping Class Group of a Multi-Geon Three-Manifold

TL;DR

This paper analyzes the unitary irreducible representations of the mapping class group of certain 3-manifolds, revealing implications for quantum gravity sectors and the behavior of topological geons.

Contribution

It provides a group-theoretic analysis of the UIRs of the mapping class group for manifolds formed by connected sums, linking representation theory to quantum gravity implications.

Findings

UIRs form theta sectors in quantum gravity theories

Analysis of the structure of UIRs for semi-direct product groups

Qualitative insights into geon behavior in different quantum sectors

Abstract

It is well known that the inequivalent unitary irreducible representations (UIR's) of the mapping class group $G$ of a 3-manifold give rise to ``theta sectors'' in theories of quantum gravity with fixed spatial topology. In this paper, we study several families of UIR's of $G$ and attempt to understand the physical implications of the resulting quantum sectors. The mapping class group of a three-manifold which is the connected sum of $\R^3$ with a finite number of identical irreducible primes is a semi-direct product group. Following Mackey's theory of induced representations, we provide an analysis of the structure of the general finite dimensional UIR of such a group. In the picture of quantized primes as particles (topological geons), this general group-theoretic analysis enables one to draw several interesting qualitative conclusions about the geons' behavior in different quantum sectors, without requiring an explicit knowledge of the UIR's corresponding to the individual primes.