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.

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

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

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

References

• Gopakumar, Rajesh; Vafa, Cumrun (1998a), M-Theory and Topological strings-I
• Gopakumar, Rajesh; Vafa, Cumrun (1998b), M-Theory and Topological strings-II
• Gopakumar, Rajesh; Vafa, Cumrun (1998c), On the Gauge Theory/Geometry Correspondence
• Gopakumar, Rajesh; Vafa, Cumrun (1998d), Topological Gravity as Large N Topological Gauge Theory