Torsors over moduli spaces of vector bundles over curves of fixed determinant

Let $\mathcal{M}$ be a moduli space of stable vector bundles of rank $r$ and determinant $\xi$ on a compact Riemann surface $X$. Fix a semistable holomorphic vector bundle $F$ on $X$ such that $\chi (E\otimes F)=0$ for $E\in \mathcal{M}$. Then any $E\in \mathcal{M}$ with $H^0(X,E\otimes F)=0=H^1(X,E\otimes F)$ has a natural holomorphic projective connection. The moduli space of pairs $(E,\mathrm{\nabla })$, where $E\in \mathcal{M}$ and $\mathrm{\nabla }$ is a holomorphic projective connection on $E$, is an algebraic $T^{\ast }\mathcal{M}$–torsor on $\mathcal{M}$. We identify this $T^{\ast }\mathcal{M}$–torsor on $\mathcal{M}$ with the $T^{\ast }\mathcal{M}$–torsor given by the sheaf of connections on an ample line bundle over $\mathcal{M}$.