Gopakumar–Vafa invariant

In theoretical physics Rajesh Gopakumar and Cumrun Vafa introduced new topological invariants, which named Gopakumar–Vafa invariant, that represent the number of BPS states on Calabi–Yau 3-fold, in a series of papers. (see Gopakumar & Vafa (1998a),Gopakumar & Vafa (1998b) and also see Gopakumar & Vafa (1998c), Gopakumar & Vafa (1998d).)　They lead the following formula generating function for the Gromov–Witten invariant on Calabi–Yau 3-fold M.

\sum _{g\geq 0,n\geq 1,\beta \in H_{2}(M,\mathbb {Z} )}GW(g,\beta )q^{\beta }\lambda ^{2g-2}=\sum _{k>0,g\geq 0,\beta \in H^{2}(M,\mathbb {Z} )}BPS(g,\beta ){\frac {1}{k}}\left(2\sin \left({\frac {k\lambda }{2}}\right)\right)^{2g-2}q^{k\beta }

where $GW(g,\beta )$ is Gromov–Witten invariant, $\beta$ the number of pseudoholomorphic curves with genus g and $BPS(g,\beta )$ the number of the BPS states.

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:

Z_{top}=\exp \left[\sum _{\begin{smallmatrix}k>0,\ g\geq 0,\\\beta \in H_{2}(M,\mathbb {Z} )\end{smallmatrix}}BPS(g,\beta ){\frac {1}{k}}\left(2\sin \left({\frac {k\lambda }{2}}\right)\right)^{2g-2}q^{k\beta \cdot t}\right]\ .

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

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