Fusion basis for lattice gauge theory and loop quantum gravity

TL;DR

The paper introduces a fusion basis for the gauge-invariant Hilbert space in lattice gauge theory and loop quantum gravity, emphasizing magnetic and electric excitations and their hierarchical structure for improved analysis.

Contribution

It presents a novel fusion basis that simplifies the study of excitations and coarse-graining in lattice gauge theories and loop quantum gravity, contrasting with traditional spin-network approaches.

Findings

Fusion basis classifies excitations via Drinfel'd double representations.

It facilitates the study of large-scale structure and coarse-graining.

Provides a hierarchical framework for multi-scale state construction.

Abstract

We introduce a new basis for the gauge-invariant Hilbert space of lattice gauge theory and loop quantum gravity in $(2+1)$ dimensions, the fusion basis. In doing so, we shift the focus from the original lattice (or spin-network) structure directly to that of the magnetic (curvature) and electric (torsion) excitations themselves. These excitations are classified by the irreducible representations of the Drinfel'd double of the gauge group, and can be readily "fused" together by studying the tensor product of such representations. We will also describe in detail the ribbon operators that create and measure these excitations and make the quasi-local structure of the observable algebra explicit. Since the fusion basis allows for both magnetic and electric excitations from the onset, it turns out to be a precious tool for studying the large scale structure and coarse-graining flow of lattice gauge theories and loop quantum gravity. This is in neat contrast with the widely used spin-network basis, in which it is much more complicated to account for electric excitations, i.e. for Gau\ss~constraint violations, emerging at larger scales. Moreover, since the fusion basis comes equipped with a hierarchical structure, it readily provides the language to design states with sophisticated multi-scale structures. Another way to employ this hierarchical structure is to encode a notion of subsystems for lattice gauge theories and $(2+1)$ gravity coupled to point particles. In a follow-up work, we will exploit this notion to provide a new definition of entanglement entropy for these theories.