Let $M$ be a closed $3$-manifold and $\rho : \pi_1(M) \to \operatorname{SL}_2(\mathbb C)$ a representation. (Feel free to replace $\rho$ with a flat $\mathfrak{sl}_2$ connection with holonomy $\rho$.) It is quite natural to think of the (hyperbolic) volume and Chern-Simons invariant as real and imaginary parts of a complex volume $$ \operatorname{V}(M,\rho) = \operatorname{Vol}(M, \rho) + i \operatorname{CS}(M,\rho) \pmod{2\pi^2 i} $$ (For example, methods for computing $\operatorname{CS}$ using dilogarithms actually compute $\operatorname{V}$.) When $M$ is hyperbolic we can let $\rho$ be the canonical hyperbolic structure and these become topological invariants of $M$.
It is relatively clear how to understand $\operatorname{Vol}(M)$ in this case: it's the volume of the metric, and it serves as a measure of complexity of $M$. However, I'm less clear on what $\operatorname{CS}(M)$ means. Some facts I know:
- Because $\operatorname{CS}(\bar{M}) = -\operatorname{CS}(M)$ we can use it as an obstruction to homotopy equivalences $M \to \bar M$ between $M$ and its mirror image.
- In their $\eta$-invariant papers Atiyah-Patodi-Singer mention that $\operatorname{CS}$ is an obstruction to conformal embeddings of $M$ in $\mathbb R^n$.
- I've seen it mentioned that $\operatorname{CS}$ behaves well under (branched?) covers but never seen the details.
- The original definition of $\operatorname{CS}$ has it appearing as a boundary term in Chern-Weil theory, which presumably has a geometric interpretation, although maybe one mostly focusing on $4$-manifolds.
What other geometric/topological interpretations of $\operatorname{CS}$ are there? I'm particularly interested in hyperbolic $3$-manifolds.
