Quantum geometry from 2+1 AdS quantum gravity on the torus
J. E. Nelson, R. F. Picken
TL;DR
This paper explores the quantum geometry of 2+1 AdS quantum gravity on a torus, revealing novel features of Wilson observables, including signed area phases and a q-deformed Goldman bracket, with a geometric interpretation via a quantum Pick's formula.
Contribution
It introduces a quantum version of Pick's formula and analyzes the properties of Wilson observables related to loop intersections in 2+1 AdS quantum gravity.
Findings
Wilson observables exhibit signed area phases
The commutator describes loop intersections
A geometric interpretation via quantum Pick's formula
Abstract
Wilson observables for 2+1 quantum gravity with negative cosmological constant, when the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our study of this bracket, which can be interpreted as a q-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick's formula for the area of a polygon with integer vertices.
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