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:

,

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 (1998c), On the Gauge Theory/Geometry Correspondence, arXiv:hep-th/9811131, Bibcode:1998hep.th...11131G
  • Gopakumar, Rajesh; Vafa, Cumrun (1998d), Topological Gravity as Large N Topological Gauge Theory, arXiv:hep-th/9802016, Bibcode:1998hep.th....2016G
  • 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


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