Topological Operators and Completeness of Spectrum in Discrete Gauge Theories

TL;DR

This paper investigates the relationship between spectrum completeness and topological operators in discrete gauge theories, revealing that spectrum completeness correlates with the absence of specific topological operators, and discusses implications for quantum gravity.

Contribution

It refines the understanding of spectrum completeness by linking it to the absence of certain topological operators beyond traditional global symmetries.

Findings

Completeness of the spectrum is equivalent to the absence of Gukov-Witten operators in 3D discrete gauge theories.

The analysis extends to higher spacetime dimensions, establishing a broader framework.

Evidence suggests topological operators are absent in consistent quantum gravity theories.

Abstract

In many gauge theories, the existence of particles in every representation of the gauge group (also known as completeness of the spectrum) is equivalent to the absence of one-form global symmetries. However, this relation does not hold, for example, in the gauge theory of non-abelian finite groups. We refine this statement by considering topological operators that are not necessarily associated with any global symmetry. For discrete gauge theory in three spacetime dimensions, we show that completeness of the spectrum is equivalent to the absence of certain Gukov-Witten topological operators. We further extend our analysis to four and higher spacetime dimensions. Since topological operators are natural generalizations of global symmetries, we discuss evidence for their absence in a consistent theory of quantum gravity.