Fusion basis for lattice gauge theory and loop quantum gravity

TL;DR

This work introduces a fusion basis for the gauge-invariant Hilbert space of (2+1)D lattice gauge theories and loop quantum gravity, recasting excitations in terms of irreducible representations of the Drinfel'd double $\mathcal{D}(\mathcal{G})$ and organizing their fusion via tensor products. By pairing a lattice-based BF vacuum with ribbon operators, the authors build a framework that diagonalizes both magnetic and electric observables and provides a natural multi-scale/coarse-graining structure. The fusion basis, together with open and closed ribbon operators, yields a robust set of gauge-invariant Dirac observables and supports a coarse-graining scheme that remains well-behaved under non-Abelian coarse-graining, addressing limitations of the spin-network basis. The paper also develops a lattice-independent description using punctured surfaces and Ocneanu's tube algebra, linking to the representation theory of the Drinfel'd double and setting the stage for future work on entanglement entropy and higher-dimensional generalizations.

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 Gauß~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.

Figures (10)

• Figure 1: Examples of minimal graphs on ${\mathbb S}_2$ and ${\mathbb S}_3$, respectively.
• Figure 2: Action of the open ribbon operator.
• Figure 3: The construction of a closed ribbon operator. The left panel shows the auxiliary puncture and an auxiliary link with holonomy $k$ going to this puncture. This holonomy plays no role in the final action of the closed ribbon operator as described in (\ref{['pcr1']}).
• Figure 4: The closed ribbon operator applied to a state on a cylinder.
• Figure 5: Two braided closed ribbon operators. These can be constructed by gluing open ribbon operators, which act in a certain order, e.g. under--crossing pieces of ribbons act before the over--crossing pieces of ribbons.