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

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 ${O}^{\oplus r}_{X}(-n)\rightarrow F$ where $F$ is a pure coherent sheaf with one dimensional support, $r>1$ and $n\gg0$ 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.