Tautological algebra of the moduli stack of semistable bundles of rank 2 on a general curve

Our aim is to determine the tautological algebra generated by the cohomology classes of the Brill-Noether loci in the rational cohomology of the moduli stack ${\mathcal{U}}_C(n,d)$ of semistable bundles of rank $n$ and degree $d$. We show that for a general smooth projective curve $C$ of genus $g\ge 2$, $d=2g-2$, the tautological algebra of ${\mathcal{U}}_C(2,2g-2)$ (resp. the moduli stack ${\mathcal{S}\mathcal{U}}_C(2,\mathcal{L})$ of semistable bundles of rank $2$ and determinant $\mathcal{L}$ with $\mathrm{deg}(\mathcal{L})=2g-2$) is generated by the divisor classes (resp. the class of the Theta divisor $\mathrm{\Theta }$). This is previously known in rank one situation, called the (classical) Porteous formula.