Moduli of flat connections vs Teichmuller space on surfaces The dimension of moduli of flat $SU(2)$-connection is the same as the dimension of Teichmuller space, both on a surface of genus $g$; namely both dimensions are equal to $6g-6$. Is this a coincidence or is there a way to see a correspondence?
 A: As far as I know, those spaces have the same dimension simply because $SL_2 (\mathbb{R})$ and $SU(2)$ have the same dimension.
If $M$ is a manifold and $G$ is a Lie group, then the holonomy map sends a flat $G-$connection $\nabla$ to a representation $\pi^\nabla: \pi_1 (M) \rightarrow G$. This map sends pairs of gauge equivalent connections to conjugate representations. Furthermore, one can prove that this defines a bijection between flat $G-$connections up to gauge equivalence and representations up to conjugation (I believe this is called Riemann-Hilbert correspondence).
In your case, the space of flat $SU(2)-$connections on a hyperbolic surface $S$ (modulo gauge equivalence) is identified with $Hom(\pi_1 (S), SU(2) )/G$ (the equivalence comes from identifying conjugate representations). A presentation for $\pi_1 (S)$ is given by $\langle a_1,\ldots,a_{2g} \vert \prod_{i=1}^g [a_i,a_{i+g}] \rangle$ (where $g$ is the genus of $S$), therefore $Hom(\pi_1(S), SU(2))$ has dimension $(2g).3 - 3$ because $\dim SU(2) = 3$. If you also quotient by the action of $SU(2)$ by conjugation, you get the desired $6g - 3 - 3$.
Notice that this computation only relied on the fact that $\dim SU(2) = 3$, so if you repeat the argument with $SL_2 (\mathbb{R})$, the dimension of $Hom (\pi_1 (S), SL_2 (\mathbb{R}))/G$ (the Teichmuller space of $S$ *) is also $6g-6$.
I'm no expert, so maybe there's a deeper relationship between those spaces that I'm missing.
*I should add "discrete and faithful representations" (or irreducible), but forgive my laziness.
