Chern-Simons theory, Stokes' Theorem, and the Duflo map

Abstract

We consider a novel derivation of the expectation values of holonomies in Chern-Simons theory, based on Stokes' Theorem and the functional properties of the Chern-Simons action. It involves replacing the connection by certain functional derivatives under the path integral integral. It turns out that ordering choices have to be made in the process, and we demonstrate that, quite surprisingly, the Duflo isomorphism gives the right ordering, at least in the simple cases that we consider. In this way, we determine the expectation values of unknotted, but possibly linked, holonomy loops for SU(2) and SU(3), and sketch how the method may be applied to more complicated cases. Our manipulations of the path integral are formal but well motivated by a rigorous calculus of integration on spaces of generalized connections which has been developed in the context of loop quantum gravity.

26 pages, 8 figures

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• Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity
• Gravitational edge modes: From Kac-Moody charges to Poincaré networks
• Black Holes as Quantum Gravity Condensates
• Turaev-Viro amplitudes from 2+1 Loop Quantum Gravity
• Black hole horizons from within loop quantum gravity
• Horizon entropy with loop quantum gravity methods
• Chern-Simons expectation values and quantum horizons from LQG and the Duflo map
• Quantum Mechanics on SO(3) via Non-commutative Dual Variables
• Entropy of Quantum Black Holes
• Non-commutative holonomies in 2+1 LQG and Kauffman's brackets
• Non-commutative Fourier transform for the Lorentz group via the Duflo map
• 2+1 gravity with positive cosmological constant in LQG: a proposal for the physical state
• Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data
• Abelian Chern-Simons theory, Stokes' theorem, and generalized connections
• Extensions of the Duflo map and Chern-Simons expectation values
• Quantum surface holonomies for loop quantum gravity and their application to black hole horizons
• New insights in quantum geometry