Wall-crossing and invariants of higher rank Joyce-Song stable pairs

TL;DR

This paper develops a higher rank version of Joyce-Song stable pairs on Calabi-Yau threefolds, introducing new invariants and analyzing their wall-crossing behavior using advanced categorical techniques.

Contribution

It introduces higher rank Joyce-Song pairs and defines their stability, extending the original theory to a broader class of sheaves on Calabi-Yau threefolds.

Findings

Defined higher rank Joyce-Song pairs for $r>1$

Established stability conditions for these pairs

Computed invariants via wall-crossing techniques

Abstract

We introduce a higher rank analog of the Joyce-Song theory of stable pairs. Given a nonsingular projective Calabi-Yau threefold $X$, we define the higher rank Joyce-Song pairs given by ${O}^{r}_{X}(-n)\rightarrow F$ where $F$ is a pure coherent sheaf with one dimensional support, $r>1$ and $n\gg 0$ is a fixed integer. We equip the higher rank pairs with a Joyce-Song stability condition and compute their associated invariants using the wallcrossing techniques in the category of weakly semistable objects.