Physics:Gopakumar–Vafa invariant

Short description: Topological invariants concerning BPS states

In theoretical physics, Rajesh Gopakumar and Cumrun Vafa introduced in a series of papers 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:

$\sum _{g=0}^{\mathrm{\infty }}\sum _{\beta \in H_2(M,\mathbb{\mathbb{Z}})}\text{GW}(g,\beta )q^{\beta }{\lambda }^{2g-2}=\sum _{g=0}^{\mathrm{\infty }}\sum _{k=1}^{\mathrm{\infty }}\sum _{\beta \in H_2(M,\mathbb{\mathbb{Z}})}\text{BPS}(g,\beta )\frac{1}{k}{(2\sin \left(\frac{k\lambda }{2}\right))}^{2g-2}{q}^{k\beta }$ ,

where

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

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:

$Z_{top}=\exp [\sum _{g=0}^{\mathrm{\infty }}\sum _{k=1}^{\mathrm{\infty }}\sum _{\beta \in H_2(M,\mathbb{\mathbb{Z}})}\text{BPS}(g,\beta )\frac{1}{k}{(2\sin \left(\frac{k\lambda }{2}\right))}^{2g-2}{q}^{k\beta }].$

References

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