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

TL;DR

This paper computes Wilson loop expectation values in $ ext{U}(1)^n$ Chern-Simons theory on closed 3-manifolds, demonstrating their topological invariance and exploring duality and zero modes.

Contribution

It provides explicit calculations of observables, analyzes topological sectors, confirms invariance, and discusses duality and zero modes in $ ext{U}(1)^n$ Chern-Simons theory.

Findings

Expectation values depend on topological sectors.

Expectation values are topological invariants.

The paper confirms a form of CS duality.

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.