The BGMN conjecture via stable pairs

TL;DR

This paper constructs a semi-orthogonal decomposition of the derived category of a moduli space of stable rank 2 bundles on a curve, providing evidence for the BGMN conjecture through wall-crossing analysis.

Contribution

It offers a new semi-orthogonal decomposition of the derived category of the moduli space, advancing the proof of the BGMN conjecture using wall-crossing and window techniques.

Findings

Constructed a semi-orthogonal decomposition matching the conjecture

Identified blocks corresponding to symmetric powers of the curve

Provided evidence that the subcategory generates the entire derived category

Abstract

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.