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}


2 Answers 2


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


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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