Counting perverse coherent systems on Calabi-Yau 4-folds

TL;DR

This paper extends the concept of counting invariants of stable perverse coherent systems from Calabi-Yau 3-folds to Calabi-Yau 4-folds, providing explicit computations and conjecturing a wall-crossing formula.

Contribution

It introduces new counting invariants for Calabi-Yau 4-folds and computes them across all stability chambers, generalizing previous 3-fold results.

Findings

Computed invariants in all stability chambers

Studied invariants of local resolved conifold using localization

Conjectured a wall-crossing formula that generalizes previous results

Abstract

Nagao-Nakajima introduced counting invariants of stable perverse coherent systems on small resolutions of Calabi-Yau 3-folds and determined them on the resolved conifold. Their invariants recover DT/PT invariants and Szendr\"oi's non-commutative invariants in some chambers of stability conditions. In this paper, we study an analogue of their work on Calabi-Yau 4-folds. We define counting invariants for stable perverse coherent systems using primary insertions and compute them in all chambers of stability conditions. We also study counting invariants of local resolved conifold $\mathcal{O}_{\mathbb{P}^1}(-1,-1,0)$ defined using torus localization and tautological insertions. We conjecture a wall-crossing formula for them, which upon dimensional reduction recovers Nagao-Nakajima's wall-crossing formula on resolved conifold.