Genus-zero and genus-one GV/DT₄ correspondence for general Calabi–Yau four-folds

Establish, for arbitrary Calabi–Yau 4-folds and a suitable choice of orientation, the equality between primary DT₄ invariants of one-dimensional stable sheaves and genus-zero Gopakumar–Vafa type invariants, together with the stated descendant DT₄ formula in terms of genus-zero and genus-one Gopakumar–Vafa invariants and meeting invariants.

Background

The paper states a general conjectural correspondence relating DT₄ invariants of one-dimensional stable sheaves to the Gopakumar–Vafa type invariants introduced by Klemm and Pandharipande. The conjecture includes both a primary genus-zero identity and a descendant genus-one identity involving meeting invariants.

For total spaces of canonical bundles of Fano three-folds, the conjecture additionally specifies an orientation expected to make both identities hold. The paper proves the primary identity for the local projective plane and verifies the descendant identity only in finite degree, so the general conjecture remains unresolved.