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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:
- ,
where
- is the class of pseudoholomorphic curves with genus g,
- is the topological string coupling,
- with the Kähler parameter of the curve class ,
- are the Gromov–Witten invariants of curve class at genus ,
- are the number of BPS states (the Gopakumar–Vafa invariants) of curve class at genus .
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:
Notes
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
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