In the paper ``Analytic Continuation Of Chern-Simons Theory'' (arXiv:1001.2933) Witten postulates that hyperbolic volume of 3-dimensional manifold coincides with the value of the Chern-Simons functional of the hyperbolic connection (see section 5.3.4). 


Let me state this more precisely.


Let $M$ be a three dimensional spin manifold. Consider a Riemannian metric $\rho$ on $M$ with constant negative curvature $-1$. The universal cover ($\tilde{M},\tilde{\rho}$) is isometric to the hyperbolic space (${H^3},\rho^{st}$). The fundamental group of $M$ acts on $H^3$ by isometries. It therefore defines a homomorphism $g:\pi\_1(M)\to \text {Isom}(H^3)=\text {PSL}(2,{\mathbb{C}})$. 


**Def** The *hyperbolic connection* $A\_{\rho}$ on the trivial $PSL(2,\mathbb{C})$-bundle $E$ on $M$ is a flat connection with monodromy representation $g$.


**Rem** The inclusion $\text{SO}(3)\subset \text{PSL}(2,\mathbb{C})$ is a homotopy equivalence. Since $M$ has a spin structure we can lift $A\_{\rho}$ to an $SL(2,\mathbb{C})$-connection.


**Def** The value of the *Chern-Simons functional* on an $\text{SL}(2,\mathbb{C})$-connection $A$ in the trivial bundle on $M$ is given by
\begin{equation}
CS(A):=\int\_{M}tr[A,dA]+\frac{2}{3}tr[A,A\wedge A].
\end{equation}
Here $tr[\cdot,\cdot]$ is defined as follows:
\begin{equation}
tr[\cdot,\cdot]: \Omega^n(M, g)\otimes\Omega^m(M, g) \xrightarrow{\wedge} \Omega^{m+n} (M, g\otimes g) \xrightarrow{tr} \Omega^{m+n} (M,\mathbb{C}). 
\end{equation}
Here $g$ is a simple Lie algebra and the trace over the last arrow is the standard non-degenerate invariant symmetric bilinear form on $g$.


**Rem** A gauge transformation $s \in \Omega^0(M,E)$ changes CS by an integer: $CS(A)-CS(s^\*A)\in 2\pi \mathbb{Z}$.


Finally, what I'm seeking for is a reference for the formula (which seems to be well known)
\begin{equation}
2\pi \text{ Im } \text{CS}(A\_{\rho})=\text{Vol}\_{\rho}.
\end{equation}



Hyperbolic Volume and Chern-Simons
This is a Theorem of Yoshida, the reference is 


* Yoshida, Tomoyoshi: ''The η-invariant of hyperbolic 3-manifolds.'' Invent. Math. 81, 473-514 (1985). <http://mathlab.snu.ac.kr/~top/articles/Yoshida.pdf>


The proof is by explicit computation and comparison of the Chern-Simons form and the volume form. 


There is an alternative proof using the Extended Bloch Group, the reference is


* Neumann, Walter D.: ''Extended Bloch group and the Cheeger-Chern-Simons class.'' Geom. Topol. 8, 413-474 (2004). <http://arxiv.org/abs/math/0307092>


with some more details in


* Goette, Sebastian; Zickert, Christian K.: ''The extended Bloch group and the Cheeger-Chern-Simons class.'' Geom. Topol. 11, 1623-1635 (2007). <http://www2.math.umd.edu/~zickert/The%20extended%20Bloch%20group%20and%20the%20Cheeger-Chern-Simons%20class%20(Goette,%20Zickert).pdf>


