The BGMN conjecture via stable pairs

Let C be a smooth projective curve of genus g≥2, and let N be the moduli space of stable vector bundles on C of rank 2 and fixed determinant of odd degree. We construct a semiorthogonal decomposition of Db(N) conjectured by Narasimhan and by Belmans, Galkin, and Mukhopadhyay. It has two blocks for each ith symmetric power of C for i=0,…,g−2 and one block for the (g−1)st symmetric power. We conjecture that the subcategory generated by our blocks has a trivial semiorthogonal 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.