Knots and Quantum Gravity: Progress and Prospects

TL;DR

Baez surveys the interplay between knot theory and quantum gravity through a ladder of field theories: $4$-D general relativity and BF theory, $3$-D Chern-Simons theory, and the $2$-D $G/G$ gauged WZW model. Loop states connect to diffeomorphism invariance, enabling knot invariants to solve the Hamiltonian constraint. The paper analyzes the relation between link invariants and generalized measures on the space of connections, including the Ashtekar–Lewandowski measure, and shows how skein invariants arise as loop transforms while noting Sawin’s result that the Chern-Simons path integral is not a generalized measure. It ends with open problems and directions toward a rigorous, gauge-invariant formulation of quantum gravity grounded in these topological insights.

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.