Noncommutative resolution of SUC (2)

We study the derived category of the moduli space SUC (2) of rank 2 vector bundles on a smooth projective curve C of genus g 2 with trivial determinant. This generalizes the recent work by Tevelev and Torres on the case with fixed odd determinant. Since SUC (2) is singular, we work with its resolution of singularities, specifically with the noncommutative resolution constructed by Padurariu and Spenko-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^2kC for 2k 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-1C 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 SUC (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 SUC (2) is a rational variety.