Abstract:

Gromov-Witten invariants give a virtual count of the number of curves on a smooth projective variety with given conditions. In general, Gromov-Witten invariants are rational numbers due to multiple cover contributions. To isolate contributions not involving multiple covers, people define Gopakumar-Vafa type invariants (particularly on certain projective varieties) and conjecture their integrality.

In this talk, I will review the genus zero multiple cover formula on semi-positive varieties and define the genus zero Gopakumar--Vafa type invariants. Finally, I will outline the proof of the integrality of Gopakumar--Vafa type invariants in this case. The main technique is to relate Gopakumar--Vafa type invariants to quantum $K$-invariants and to utilize the integrality of the latter.