A simple proof of Gopakumar-Vafa conjecture for local toric Calabi-Yau manifolds
Let \(X\) be a Calabi--Yau threefold. Its Gromov--Witten invariants are the expected numbers of curves of given genera lying on it counted with particular rational coefficients. So, they are rational numbers. Gopakumar and Vafa, based on the compactification of M-theory on Calabi--Yau threefold defined Gopakumar--Vafa invariants, which are integral numbers. The Gopakumar--Vafa conjecture states the particular relation between Gromov--Witten and Gopakumar--Vafa invariants. This conjecture gives the recursive procedure of calculation of Gopakumar--Vafa invariants in terms of Gromov--Witten ones; in fact, this procedure is the mathematical definition of Gopakumar--Vafa invariants, so the conjecture states that they are integral. Let \(X\) be a local toric Calabi--Yau variety. Then one can define the local version of Gopakumar--Vafa conjecture in the similar way as in the usual one. The author proves the conjecture in this case (Theorem 1). The main result implies also that local Gopakumar--Vafa invariants vanish at large genera.
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