Let C be a smooth projective curve of genus at least 2, and let N be the moduli space of semistable rank-two vector bundles of odd degree on C. We construct a semi-orthogonal decomposition in the derived category of N conjectured by Belmans, Galkin and Mukhopadhyay and by Narasimhan. It has blocks of the form D(C_d) where C_d are d-th symmetric powers of C, and the semi-orthogonal complement to these blocks is conjecturally trivial.

In order to prove our result, we use the moduli spaces of stable pairs over C. Such spaces are related to each other via GIT wall crossing, and the method of windows allows us to understand the relationship between the derived categories on either side of a given wall.

This is joint work with J. Tevelev.