Curve counting via stable objects in derived categories of Calabi-Yau 4-folds

TL;DR

This paper interprets a conjectural Gopakumar-Vafa type formula for Calabi-Yau 4-folds through wall-crossing phenomena in derived categories, introducing new invariants and providing evidence via examples.

Contribution

It introduces invariants counting stable pairs on CY 4-folds and proposes a wall-crossing formula that recovers the GV type conjecture, linking stable objects to enumerative invariants.

Findings

Wall-crossing formula recovers GV type conjecture

Examples support the proposed invariants and formula

Stable objects correspond to D0-D2-D8 bound states

Abstract

In our previous paper with Maulik, we proposed a conjectural Gopakumar-Vafa (GV) type formula for the generating series of stable pair invariants on Calabi-Yau (CY) 4-folds. The purpose of this paper is to give an interpretation of the above GV type formula in terms of wall-crossing phenomena in the derived category. We introduce invariants counting LePotier's stable pairs on CY 4-folds, and show that they count certain stable objects in D0-D2-D8 bound states in the derived category. We propose a conjectural wall-crossing formula for the generating series of our invariants, which recovers the conjectural GV type formula. Examples are computed for both compact and toric cases to support our conjecture.