  Let $C$ be a smooth projective curve of genus $g\ge2$ and let $N$ be the
moduli space of stable rank $2$ vector bundles on $C$ of odd degree. We
construct a semi-orthogonal decomposition of the bounded derived category of
$N$ conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. It has
two blocks for each $i$-th symmetric power of $C$ for $i=0,\ldots,g-2$ and one
block for the $(g-1)$-st symmetric power. We conjecture that the subcategory
generated by our blocks has a trivial semi-orthogonal complement, proving the
full BGMN conjecture. Our proof is based on an analysis of wall-crossing
between moduli spaces of stable pairs, combining classical vector bundles
techniques with the method of windows.
