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

Hanno Sahlmann, Thomas Thiemann

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.

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.

Paper Structure

This paper contains 8 sections, 1 theorem, 88 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Surface ordering, factor ordering, regularization
  3. The use of the Duflo isomorphism
  4. Unknots
  5. Hopf link
  6. Discussion
  7. Chern-Simons theory expectation values
  8. Lie-algebra results

Key Result

Proposition 2.1

The holonomy $h_{\partial S}$ around the surface (i.e. along the path $S(0,0)$ to $S(1,0)$ to $S(1,1)$ to $S(0,1)$ and back to $S(0,0)$) can be expressed as

Figures (8)

  • Figure 1: A simple Wilson loop $\alpha$ and a surface $S$ bounded by it
  • Figure 2: Examples for Seifert surfaces of knots and links: For the Hopf rings (left) and the trefoil knot (right). Graphics here and in some of the following figures created with SeifertViewSV)
  • Figure 3: The simple situation, with two holonomies meeting the surface transversally in one point.
  • Figure 4: The Hopf link $H_+$, already drawn with surface $S$
  • Figure 5: An example for the 'key chain link': $L_3$ with its Seifert surface
  • ...and 3 more figures

Theorems & Definitions (1)

  • Proposition 2.1: nastokes