Physics:Gopakumar–Vafa invariant

Short description: Topological invariants concerning BPS states

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

where

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

• Gopakumar, Rajesh; Vafa, Cumrun (1998a), M-Theory and Topological strings-I, Bibcode: 1998hep.th....9187G 
• Gopakumar, Rajesh; Vafa, Cumrun (1998b), M-Theory and Topological strings-II, Bibcode: 1998hep.th...12127G 
• Gopakumar, Rajesh; Vafa, Cumrun (1999), "On the Gauge Theory/Geometry Correspondence", Adv. Theor. Math. Phys. 3 (5): 1415–1443, doi:10.4310/ATMP.1999.v3.n5.a5, Bibcode: 1998hep.th...11131G 
• Gopakumar, Rajesh; Vafa, Cumrun (1998d), "Topological Gravity as Large N Topological Gauge Theory", Adv. Theor. Math. Phys. 2 (2): 413–442, doi:10.4310/ATMP.1998.v2.n2.a8, Bibcode: 1998hep.th....2016G 
• "The Gopakumar–Vafa formula for symplectic manifolds", Annals of Mathematics, Second Series 187 (1): 1–64, 2018, doi:10.4007/annals.2018.187.1.1