Abstract:
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.
