Gopakumar–Vafa invariant

In theoretical physics Rajesh Gopakumar and Cumrun Vafa introduced new topological invariants, which named Gopakumar–Vafa invariant, that represent the number of BPS states on Calabi–Yau 3-fold, in a series of papers. (see Gopakumar & Vafa (1998a),Gopakumar & Vafa (1998b) and also see Gopakumar & Vafa (1998c), Gopakumar & Vafa (1998d).)　They lead the following formula generating function for the Gromov–Witten invariant on Calabi–Yau 3-fold M.

where ${\displaystyle GW(g,\beta )}$ is Gromov–Witten invariant, ${\displaystyle \beta }$ the number of pseudoholomorphic curves with genus g and ${\displaystyle BPS(g,\beta )}$ the number of the BPS states.

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