A canonical connection on bundles on Riemann surfaces and Quillen connection on the theta bundle

We investigate the symplectic geometric and differential geometric aspects of the moduli space of connections on a compact Riemann surface $X$. Fix a theta characteristic $K_X^{1\mathrm{/}2}$ on $X$; it defines a theta divisor on the moduli space $\mathcal{M}$ of stable vector bundles on $X$ of rank $r$ degree zero. Given a vector bundle $E\in \mathcal{M}$ lying outside the theta divisor, we construct a natural holomorphic connection on $E$ that depends holomorphically on $E$. Using this holomorphic connection, we construct a canonical holomorphic isomorphism between the following two: \begin{enumerate} \item the moduli space $\mathcal{C}$ of pairs $(E,D)$, where $E\in \mathcal{M}$ and $D$ is a holomorphic connection on $E$, and \item the space $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{n}(\mathrm{\Theta })$ given by the sheaf of holomorphic connections on the line bundle on $\mathcal{M}$ associated to the theta divisor. \end{enumerate} The above isomorphism between $\mathcal{C}$ and ${\rm Conn}(\Theta)$ is symplectic structure preserving, and it moves holomorphically as $X$ runs over a holomorphic family of Riemann surfaces.