Gopakumar–Vafa invariant

In theoretical physics, Rajesh Gopakumar and Cumrun Vafa introduced in a series of papers new topological invariants, called Gopakumar–Vafa invariants, that represent the number of BPS states on a Calabi–Yau 3-fold. They lead to the following generating function for the Gromov–Witten invariants on a Calabi–Yau 3-fold ''M'':

:$\sum_^\infty~\sum_ \text(g,\beta)q^\lambda^=\sum_^\infty~\sum_^\infty~\sum_\text(g,\beta)\frac\left(2\sin\left(\frac\right)\right)^q^$ ,

where
* $\beta$ is the class of pseudoholomorphic curves with genus ''g'',
* $\lambda$ is the topological string coupling,
* $q^\beta=\exp(2\pi i t_\beta)$ with $t_\beta$ the Kähler parameter of the curve class $\beta$,
* $\text(g,\beta)$ are the Gromov–Witten invariants of curve class $\beta$ at genus $g$,
* $\text(g,\beta)$ are the number of BPS states (the Gopakumar–Vafa invariants) of curve class $\beta$ at genus $g$.

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:

:Z_=\exp\left sum_^\infty~\sum_^\infty~\sum_\text(g,\beta)\frac\left(2\sin\left(\frac\right)\right)^q^\right .

Notes

References

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Quantum field theory
Algebraic geometry
String theory

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