Explicit formulae for rank zero DT invariants and the OSV conjecture

TL;DR

This paper derives explicit formulas connecting different types of Donaldson-Thomas invariants on Calabi-Yau 3-folds, providing new insights into the OSV conjecture and extending to rank two invariants in certain cases.

Contribution

It introduces two distinct wall-crossing methods to explicitly relate rank 0, rank 1, and rank 2 DT invariants, and refines the OSV conjecture for Calabi-Yau 3-folds.

Findings

Derived explicit formulas for rank 0 DT invariants in terms of rank 1 invariants.

Proved a modified version of Toda's OSV conjecture for specific Calabi-Yau 3-folds.

Provided a formula for rank two DT invariants when the Picard rank is one.

Abstract

Fix a Calabi-Yau 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as the quintic 3-fold. By two different wall-crossing arguments we prove two different explicit formulae relating rank 0 Donaldson-Thomas invariants (counting torsion sheaves on $X$ supported on ample divisors) in terms of rank 1 Donaldson-Thomas invariants (counting ideal sheaves of curves) and Pandharipande-Thomas invariants. In particular, we prove a slight modification of Toda's formulation of OSV conjecture for $X$. When $X$ is of Picard rank one, we also give an explicit formula for rank two DT invariants in terms of rank zero and rank one DT invariants.