A proof of GW/DT4 conjecture on local projective plane

We prove a closed formula for DT4 invariants of one dimensional stable sheaves on the total space of ${\mathcal{O}}_{{\mathbb{P}}^2}(-1,-2)$ for all degree curve classes. During the proof, orientations of moduli spaces are chosen in a canonical way. Combining with Gopakumar-Vafa type invariants proposed and computed by Klemm-Pandharipande, we verify a conjecture proposed by the author, Maulik and Toda.