The BPS invariants of Calabi-Yau threefolds were introduced by Gopakumar and Vafa as an index of M2 branes wrapping holomorphic curves. In joint work with Klemm and Vafa, we introduced methods of both string theory and algebraic geometry for computing the invariants in examples. These computations have since been made mathematically rigorous using the stable pairs of Pandharipande and Thomas.

In this talk, I employ refinements of the stable pair invariants with a motivic flavor to identify the underlying SU(2) x SU(2) representations of Gopakumar and Vafa. Results are in agreement with recent work of Huang, Kashani-Poor, and Klemm, where these SU(2) x SU(2) representations were found for local P2 by B-model methods.