On some natural torsors over moduli spaces of parabolic bundles

A moduli space $\mathcal{N}$ of stable parabolic vector bundles, of rank $r$ and parabolic degree zero, on a $n$-pointed curve has two naturally occurring holomorphic $T^{\ast }\mathcal{N}$–torsors over it. One of them is given by the moduli space of pairs of the form $(E_{\ast },D)$, where $E_{\ast }\in \mathcal{N}$ and $D$ is a connection on $E_{\ast }$. This $T^{\ast }\mathcal{N}$–torsor has a $C^{\mathrm{\infty }}$ section that sends any $E_{\ast }\in \mathcal{N}$ to the unique connection on it with unitary monodromy. The other $T^{\ast }\mathcal{N}$–torsor is given by the sheaf of holomorphic connections on a theta line bundle over $\mathcal{N}$. This $T^{\ast }\mathcal{N}$–torsor also has a $C^{\mathrm{\infty }}$ section given by the Hermitian structure on the theta bundle constructed by Quillen. We prove that these two $T^{\ast }\mathcal{N}$–torsors are isomorphic by a canonical holomorphic map. This holomorphic isomorphism interchanges the above two $C^{\mathrm{\infty }}$ sections.