In theoretical physics, Rajesh Gopakumar and Cumrun Vafa introduced in a series of papers numerical invariants of Calabi-Yau threefolds, later referred to as the Gopakumar–Vafa invariants. These physically defined invariants represent the number of BPS states on a Calabi–Yau threefold. In the same papers, the authors also derived the following formula which relates the Gromov–Witten invariants and the Gopakumar-Vafa invariants.
∑ g = 0 ∞ ∑ β ∈ H 2 ( M , Z ) GW ( g , β ) q β λ 2 g − 2 = ∑ g = 0 ∞ ∑ k = 1 ∞ ∑ β ∈ H 2 ( M , Z ) GV ( g , β ) 1 k ( 2 sin ( k λ 2 ) ) 2 g − 2 q k β {\displaystyle \sum _{g=0}^{\infty }~\sum _{\beta \in H_{2}(M,\mathbb {Z} )}{\text{GW}}(g,\beta )q^{\beta }\lambda ^{2g-2}=\sum _{g=0}^{\infty }~\sum _{k=1}^{\infty }~\sum _{\beta \in H_{2}(M,\mathbb {Z} )}{\text{GV}}(g,\beta ){\frac {1}{k}}\left(2\sin \left({\frac {k\lambda }{2}}\right)\right)^{2g-2}q^{k\beta }} , where
β {\displaystyle \beta } is the class of holomorphic curves with genus g,
λ {\displaystyle \lambda } is the topological string coupling, mathematically a formal variable,
q β = exp ( 2 π i t β ) {\displaystyle q^{\beta }=\exp(2\pi it_{\beta })} with t β {\displaystyle t_{\beta }} the Kähler parameter of the curve class β {\displaystyle \beta } ,
GW ( g , β ) {\displaystyle {\text{GW}}(g,\beta )} are the Gromov–Witten invariants of curve class β {\displaystyle \beta } at genus g {\displaystyle g} ,
GV ( g , β ) {\displaystyle {\text{GV}}(g,\beta )} are the Gopakumar–Vafa invariants of curve class β {\displaystyle \beta } at genus g {\displaystyle g} . Notably, Gromov-Witten invariants are generally rational numbers while Gopakumar-Vafa invariants are always integers.
As a partition function in topological quantum field theory Gopakumar–Vafa invariants can be viewed as a partition function in topological quantum field theory. They are proposed to be the partition function in Gopakumar–Vafa form:
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