Английская Википедия:Gopakumar–Vafa invariant

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Шаблон:Short description 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:

<math>\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{BPS}(g,\beta)\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}q^{k\beta}</math> ,

where

• <math>\beta</math> is the class of pseudoholomorphic curves with genus g,
• <math>\lambda</math> is the topological string coupling,
• <math>q^\beta=\exp(2\pi i t_\beta)</math> with <math>t_\beta</math> the Kähler parameter of the curve class <math>\beta</math>,
• <math>\text{GW}(g,\beta)</math> are the Gromov–Witten invariants of curve class <math>\beta</math> at genus <math>g</math>,
• <math>\text{BPS}(g,\beta)</math> are the number of BPS states (the Gopakumar–Vafa invariants) of curve class <math>\beta</math> at genus <math>g</math>.

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:

<math>Z_{top}=\exp\left[\sum_{g=0}^\infty~\sum_{k=1}^\infty~\sum_{\beta\in H_2(M,\mathbb{Z})}\text{BPS}(g,\beta)\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}q^{k\beta}\right]\ .</math>

References

• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation

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