Knots and Quantum Gravity: Progress and Prospects

Abstract

Recent work on the loop representation of quantum gravity has revealed previously unsuspected connections between knot theory and quantum gravity, or more generally, 3-dimensional topology and 4-dimensional generally covariant physics. We review how some of these relationships arise from a `ladder of field theories' including quantum gravity and BF theory in 4 dimensions, Chern-Simons theory in 3 dimensions, and the G/G gauged WZW model in 2 dimensions. We also describe the relation between link (or multiloop) invariants and generalized measures on the space of connections. In addition, we pose some research problems and describe some new results, including a proof (due to Sawin) that the Chern-Simons path integral is not given by a generalized measure.

22 pages in LaTeX

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Cited by in corpus (13)

• Spin Foam Models
• A Short Review of Loop Quantum Gravity
• Spin Network States in Gauge Theory
• 4-Dimensional BF Theory as a Topological Quantum Field Theory
• Ashtekar's formulation for $N=1,2$ supergravities as "constrained" BF theories
• Towards a Gravitational Analog to S-duality in Non-abelian Gauge Theories
• Quantum Gravity Vacuum and Invariants of Embedded Spin Networks
• Quantum Gravity Hamiltonian for Manifolds with Boundary
• Local symmetries of the non-Abelian two-form
• Flat Spacetime Vacuum in Loop Quantum Gravity
• Dimension of the moduli space and Hamiltonian analysis of BF field theories
• A semiclassical interpretation of the topological solutions for canonical quantum gravity
• Magnetic monopole is photon