Quantum K-invariants and Gopakumar-Vafa invariants II. Calabi-Yau threefolds at genus zero

This is the second part of our ongoing project on the relations between Gopakumar-Vafa BPS invariants (GV) and quantum K-theory (QK) on the Calabi–Yau threefolds (CY3). We show that on CY3 a genus zero quantum K-invariant can be written as a linear combination of a finite number of Gopakumar–Vafa invariants with coefficients from an explicit multiple cover formula''. Conversely, GV can be determined by QK in a similar manner. The technical heart is a proof of a remarkable conjecture by Hans Jockers and Peter Mayr. This result is consistent with the virtual Clemens conjecture’’ for the Calabi–Yau threefolds. A heuristic derivation of the relation between QK and GV via the virtual Clemens conjecture and the multiple cover formula is also given.