  In this paper, we present an investigation of the Gopakumar-Vafa (GV)
invariant, a curve-counting integral invariant associated with Calabi-Yau
threefolds, as proposed by physicists. Building upon the conjectural definition
of the GV invariant in terms of perverse sheaves, as formulated by Maulik-Toda
in 2016, we focus on the total space of the canonical bundle of $\mathbb{P}^2$
and compute the relevant invariants. We establish a conjectural correspondence
between the Gopakumar-Vafa and Pandharipande-Thomas invariants at the level of
perverse sheaves, drawing inspiration from the work of Migliorini, Shende, and
Viviani. This work serves as a significant step towards validating the
conjecture and deepening our understanding of the GV invariant and its
connections to algebraic geometry and physics.
