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Observables in $\mathrm{U}(1)^n$ Chern-Simons theory

Michail Tagaris, Frank Thuillier

Abstract

In this article, we will compute the expectation value of observables (which appear as Wilson loops) in $\mathrm{U}(1)^n$ Chern-Simons theory for closed oriented $3$-manifolds. We will show how the various topological sectors of the observable affect the expectation value and confirm that it is a topological invariant. We will also exhibit in this case as well a form of the CS duality introduced in previous works. Finally, to complete the treatment of this theory, we will compute its zero modes and the equations of motion.

Observables in $\mathrm{U}(1)^n$ Chern-Simons theory

Abstract

In this article, we will compute the expectation value of observables (which appear as Wilson loops) in Chern-Simons theory for closed oriented -manifolds. We will show how the various topological sectors of the observable affect the expectation value and confirm that it is a topological invariant. We will also exhibit in this case as well a form of the CS duality introduced in previous works. Finally, to complete the treatment of this theory, we will compute its zero modes and the equations of motion.
Paper Structure (8 sections, 77 equations, 3 figures)

This paper contains 8 sections, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Expectation value of $\mathrm{U}(1)^n$ observables
  3. Computing the expectation value
  4. Reciprocity and degenerate case
  5. CS duality
  6. Examples
  7. Conclusion
  8. Equations of motion and zero modes

Figures (3)

  • Figure 1: Extracting the framing and the interlinking from the torsion parts of the observable.
  • Figure 2: Observable of example 1, we have split the figure into three parts. The first two each showcase the torsion component of the observable (in red) corresponding to a different copy of the gauge group, along with the surgery link (in black). The third part showcases the trivial component.
  • Figure 3: Representation of $H^{1}_{\mathrm{DB}}(M)$ as a fibration with base space $H^2(M)$ and fibers $\Omega^1(M)/\Omega^1_\mathbb{Z}(M)$

Theorems & Definitions (2)

  • Example 1
  • Example 2