Noncommutative resolution of $SU_C(2)$

We study the derived category of the moduli space $SU_C(2)$ of rank $2$ vector bundles on a smooth projective curve $C$ of genus $g\ge 2$ with trivial determinant. This generalizes the recent work by Tevelev and Torres on the case with fixed odd determinant. Since $SU_C(2)$ is singular, we work with its resolution of singularities, specifically with the noncommutative resolution constructed by P\u{a}durariu and \v{S}penko--Van den Bergh (in the more general setting of symmetric stacks). We show that this noncommutative resolution admits a semiorthogonal decomposition into derived categories of symmetric powers $Sym^{2k}C$ for $2k\le g-1$. In the case of even genus, each block appears four times. This is also true in the case of odd genus, except that the top symmetric power $Sym^{g-1}C$ appears twice. In the case of even genus, the noncommutative resolution is strongly crepant in the sense of Kuznetsov and categorifies the intersection cohomology of $SU_C(2)$. Since all of its components are "geometric," our semiorthogonal decomposition provides evidence for the expectation, which dates back to the work of Newstead and Tyurin, that $SU_C(2)$ is a rational variety. Finally, we study mutations of semiorthogonal decompositions on the Hecke correspondence, answering a question of P\u{a}durariu and Toda.