Gopakumar–Vafa invariant

In theoretical physics, Rajesh Gopakumar and Cumrun Vafa introduced in a series of papers^[1][2][3][4] 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:

${\displaystyle \sum _{g=0}^{\infty }~\sum _{\beta \in H_{2}(M,\mathbb {Z} )}{\text{GW}}(g,\beta )q^{\beta }\lambda ^{2g-2}=\sum _{g=0}^{\infty }~\sum _{k=1}^{\infty }~\sum _{\beta \in H_{2}(M,\mathbb {Z} )}{\text{BPS}}(g,\beta ){\frac {1}{k}}\left(2\sin \left({\frac {k\lambda }{2}}\right)\right)^{2g-2}q^{k\beta }}$ ,

where

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

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:

${\displaystyle Z_{top}=\exp \left[\sum _{g=0}^{\infty }~\sum _{k=1}^{\infty }~\sum _{\beta \in H_{2}(M,\mathbb {Z} )}{\text{BPS}}(g,\beta ){\frac {1}{k}}\left(2\sin \left({\frac {k\lambda }{2}}\right)\right)^{2g-2}q^{k\beta }\right]\ .}$ 

Notes

1. Gopakumar & Vafa 1998a
2. Gopakumar & Vafa 1998b
3. Gopakumar & Vafa 1999
4. Gopakumar & Vafa 1998d

References

• Gopakumar, Rajesh; Vafa, Cumrun (1998a), M-Theory and Topological strings-I, arXiv:hep-th/9809187, Bibcode:1998hep.th....9187G
• Gopakumar, Rajesh; Vafa, Cumrun (1998b), M-Theory and Topological strings-II, arXiv:hep-th/9812127, Bibcode:1998hep.th...12127G
• Gopakumar, Rajesh; Vafa, Cumrun (1999), "On the Gauge Theory/Geometry Correspondence", Adv. Theor. Math. Phys., 3 (5): 1415–1443, arXiv:hep-th/9811131, Bibcode:1998hep.th...11131G, doi:10.4310/ATMP.1999.v3.n5.a5, S2CID 13824856
• Gopakumar, Rajesh; Vafa, Cumrun (1998d), "Topological Gravity as Large N Topological Gauge Theory", Adv. Theor. Math. Phys., 2 (2): 413–442, arXiv:hep-th/9802016, Bibcode:1998hep.th....2016G, doi:10.4310/ATMP.1998.v2.n2.a8, S2CID 16676561
• Ionel, Eleny-Nicoleta; Parker, Thomas H. (2018), "The Gopakumar–Vafa formula for symplectic manifolds", Annals of Mathematics, Second Series, 187 (1): 1–64, arXiv:1306.1516, doi:10.4007/annals.2018.187.1.1, MR 3739228, S2CID 7070264