Quantum geometry from 2+1 AdS quantum gravity on the torus

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.

19 pages, 11 figures, revised with more explanations, improved figures and extra figures. To appear GERG

References in corpus (8)

• Comparative Quantizations of (2+1)-Dimensional Gravity
• Equivalent Quantisations of (2+1)-Dimensional Gravity
• Classical and Quantum Evolutions of the de Sitter and the anti-de Sitter Universes in 2+1 dimensions
• Wilson Loops and Black Holes in 2+1 Dimensions
• Quantum Holonomies in (2+1)-Dimensional Gravity
• Parametrization of the moduli space of flat SL(2,R) connections on the torus
• Reduced Phase Space of the first order Einstein Gravity on $\mathbf{R}\times T^2$
• A quantum Goldman bracket in 2+1 quantum gravity

Cited by in corpus (3)

• 2-Group Actions and Moduli Spaces of Higher Gauge Theory
• Quantum Holonomies and the Heisenberg Group
• Theory of intersecting loops on a torus