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

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