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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}

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2 Answers 2

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This is a Theorem of Yoshida, the reference is

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

with some more details in

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The first reference known to me is

Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982). ZBL0496.57005.

However, I highly recommend various papers by Walter Neumann for much more.

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