Semi-orthogonal decompositions of moduli spaces

Let $C$ be a smooth projective curve of genus at least $2$ and let $N$ be the moduli space of stable rank $2$ vector bundles on $C$ with fixed odd determinant. 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 $i = 0,...,g-2$ and one block for the $(g - 1)$-st symmetric power.
The proof contains two parts. Semi-orthogonality, proved jointly with Sebastian Torres, relies on hard vanishing theorems for vector bundles on the moduli space of stable pairs. The second part, elimination of the phantom, requires analysis of weaving patterns in derived categories.