Graduate Study Chern-Simons Invariant: Understanding 3-Manifold Measurement

[nateHI]

I've been studying the Witten-Reshetikhin-Turaev (WRT) invariant of 3-manifolds but have almost zero background in physics. The WRT of a 3-manifold is closely related to the Chern-Simons (CS) invariant via the volume conjecture. My question is, what does the CS invariant of a 3-manifold measure? I mean, if it's an invariant then it must give you some information about the manifold, right?

 

[azntoon]

The Chern-Simons (CS) invariant of a 3-manifold is an important topological invariant that measures the extent to which the curvature of a given 3-manifold deviates from being constant across the manifold. Specifically, it measures the integral of the "Chern-Simons 3-form" over the 3-manifold. This 3-form is related to the curvature of the 3-manifold and is defined using the connection of a principal G-bundle on the 3-manifold. The CS invariant is interesting in that it is a topological invariant of the 3-manifold, meaning that it is independent of the metric or any other choice of coordinates. This makes it useful for studying the topology of 3-manifolds.