TITLE: Is there a non-abelian version of the Torelli map?
QUESTION [11 upvotes]: Let $C$ be a connected compact oriented real surface of genus $g$, let $G$ be a connected compact Lie group and let $G_\mathbb{C}$ be the complexification of $G$. One considers the moduli space $M (C,G)$ of isomorphism classes of representations of the fundamental group $\pi_1 (C)$ in $G$. It is a compact real algebraic variety (in general singular) with a natural symplectic structure (see Atiyah-Bott, Goldman,...)
If one fixes a complex structure on $C$, one can consider the moduli space $M_{hol} (C,G)$ of degree zero semistable holomorphic $G_\mathbb{C}$-bundles on $C$. It is a projective complex algebraic variety (in general singular) with a natural polarization. The theorem of Narasimhan-Seshadri (for $G=U(n)$, extended by Ramanathan for general $G$) gives an isomorphism between the underlying real analytic structures of $M(C,G)$ and $M_{hol}(C,G)$, with identification of the natural symplectic form on $M(C,G)$ with the natural Kähler form on $M_{hol}(C,G)$. In other words, a choice of complex structure on $C$ gives a natural polarized complex algebraic structure on $M(C,G)$ compatible with the symplectic structure. In vague terms, my question is: how does this structure changes when the complex structure
on $C$ changes?
More precisely, I would like to consider the moduli space $\mathcal{M}$ of polarized complex algebraic structures on $M(C,G)$ compatible with the symplectic structure. I don't know if such moduli space exists.
Q1) Is there a way to make sense of the moduli space $\mathcal{M}$ ? Locally, globally?
If $\mathcal{M}$ makes sense then the Narasimhan-Seshadri theorem gives a map $f: M_g \rightarrow \mathcal{M}$, $C \mapsto M_{hol}(C, G)$, where $M_g$ is the moduli space of complex structures on $C$.
Q2) What can be said on the map $f$ ? Is it injective ? What is the dimension of its image?...
When $G=U(1)$, we simply have $\mathcal{M} = A_g$, the moduli space of principally polarized abelian varieties of dimension $g$ and $f: M_g \rightarrow A_g$ is simply the Torelli map $C \mapsto Jac(C)$, which is injective by the Torelli theorem. So my questions could be reformulated as: is there some non-abelian version of the Torelli map and of the Torelli theorem (where "non-abelian" refers to the replacement of $G=U(1)$ by a general $G$) ?

REPLY [5 votes]: In the analytic category, this question is addressed  in N.Hitchin's paper Flat connections and geometric quantization.
I will write $M$ for your $M_{hol}(C,G)$, and will pretend that $M$ is smooth 
(you can replace $G$ by a central extension, or use the normality of $M$ plus Hartogs-type arguments). Then, if  $P$ be a stable $G_{\mathbb{C}}$-bundle (with no extra automorphisms), $T_{M,[P]}= H^1(C, ad P)$.
The infinitesimal deformation map
$$
H^1(C,T_C) \longrightarrow H^1(M, T_M)
$$
is injective. This is due to Narasimhan and Ramanan when $M$ is smooth (they treat unitary bundles with coprime rank and degree), and Hitchin gives independent proof in the singular case.
Next, cup product gives a map
$$
H^0(T_M^2)\otimes H^1(\Omega^1_M)\to H^1(T_M),
$$
and under multiplication with the Atiyah-Bott symplectic form, the polarisation-preserving deformations correspond to sections of $H^0(M,Sym^2 T_M)\subset H^0(M, T_M^2)$. We can pull these to (the total space of) $\pi:T^\vee_M\to M$:
$$
H^0(M, Sym^2 T_M)\subset H^0(T^\vee_M,  Sym^2 \pi^\ast T_M).
$$
Proposition 2.16 in Hitchin's paper says that the cup product pairing
$$
H^1(C,T_C)\otimes H^0(C, ad P\otimes K_C)\to H^1(C,ad P)
$$
gives, for any class in $H^1(C,T_C)$, the corresponding deformation of the Kaehler polarisation of the moduli space. If you're working with a classical group, the section of $Sym^2\pi^\ast T_M$, corresponding to $[\kappa]\in H^1(C, T_C)$ is 
$$
G(\alpha,\alpha)=\int_C Tr(\alpha^2)\wedge \kappa,\quad   \alpha\in T^\vee_{M,P}=H^0(C, ad P\otimes K_C).
$$
In general, you take the quadratic part of the Hitchin map.
There is indeed an analogy with the abelian case. Hitchin considers a line bundle over Teichmueller space, whose fibres are the ``non-abelian theta-functions'' (of sufficiently high level), and constructs a projectively flat connection on it. The (projective) flatness is the analogue of the heat equation for theta-functions. The connection is known as the Hitchin connection, WZW-connection, or KZB-connection.