Fusion basis for lattice gauge theory and loop quantum gravity

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 have exploited this notion to provide a new definition of entanglement entropy for these theories.

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…One can understand lattice gauge theory configurations in terms of defect excitations [8]. The notion of these excitations depends on a choice of vacuum.…”

“…From a lattice gauge theory or loop quantum gravity perspective these ribbon operators unify flux and holonomy operators [8,65]. Open ribbon operators require the presence of punctures, we will therefore consider only closed ribbons.…”

“…The fusion basis in (2+1) dimensions is perfectly suited for coarse graining [8], as it encodes excitations on different coarse graining scales. The B 1 and B 2 bases describe, however, excitations on the lattice scale (with respect to two different vacua).…”

“…The coarse graining of non-Abelian gauge theories leads to the emergence of torsion degrees of freedom, which on the lattice level are described by violations of the Gauß constraints [8,[15][16][17]80]. This brings as to the question of how to accommodate such torsion degrees of freedom, see also the paragraph on the introduction of boundaries.…”

“…These bases will allow a wide range of further applications, e.g. for background independent definitions of entanglement entropy [7] or for the construction of coarse graining schemes [8]. Let us shortly discuss these various points:…”