nLab Gopakumar-Vafa invariant

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String theory

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Idea

Gopakumar-Vafa invariants are supposed to count the number of pseudoholomorphic curves and represent the number of BPS states on a three-dimensional projective Calabi-Yau manifold, hence are of interest in string theory.

Rajesh Gopakumar and Cumrun Vafa first described the invariant in (GV 98a, GV 98b, GV 98c, GV 99). Instead of directly defining the invariant, they instead described properties and connections to other invariants. Gromov-Witten invariants and Pandharipande-Thomas invariants also count the numbers of holomorphic curves, so equivalences between the three invariants known as GW/PT/GV correspondence are conjectured. But all of them use additional data: Gromov-Witten invariants additionally use maps into the manifold, Pandharipande-Thomas invariants additionally use points on the curve (D0-D2-D6 bound states) and Gopakumar-Vafa invariants additionally use vector bundles on the curve (D2-branes). As a result, the Gromov-Witten invariant is rational, hence not really counting a number of curves. On the other hand, Pandharipande-Thomas invariants and Gopakumar-Vafa invariants are integers.

GW/PT/GV correspondence

Let MM be a three-dimensional Calabi-Yau manifold, β∈H 2(M,ℤ)\beta\in H_2(M,\mathbb{Z}) the homology class and gg the genus of a pseudoholomorphic curve. Let GW(g,β)∈ℚ\operatorname{GW}(g,\beta)\in\mathbb{Q} be the Gromov-Witten invariant, PT(g,β)∈ℤ\operatorname{PT}(g,\beta)\in\mathbb{Z} be the Pandharipande-Thomas invariant and GV(n,β)∈ℤ\operatorname{GV}(n,\beta)\in\mathbb{Z} be the Gopakumar-Vafa invariant, then:

∑ g=0 ∞∑ β∈H 2(M,ℤ)GW(g,β)q βλ 2g−2=∑ k=1 ∞∑ g=0 ∞∑ β∈H 2(M,ℤ)GV(g,β)k(2sin(kλ2)) 2g−2q kβ; \sum_{g=0}^\infty\sum_{\beta\in H_2(M,\mathbb{Z})}\operatorname{GW}(g,\beta)q^{\beta}\lambda^{2g-2} =\sum_{k=1}^\infty\sum_{g=0}^\infty\sum_{\beta\in H_2(M,\mathbb{Z})}\frac{\operatorname{GV}(g,\beta)}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}q^{k\beta};
ln(1+∑ n=−∞ ∞∑ β∈H 2(M,ℤ)PT(n,β)q βλ n)=∑ k=1 ∞∑ n=−∞ ∞∑ β∈H 2(M,ℤ)GV(g,β)k(−1) g−1((−λ) k2−(−λ) k2) 2g−2q kβ. \ln\left( 1+\sum_{n=-\infty}^\infty\sum_{\beta\in H_2(M,\mathbb{Z})}\text{PT}(n,\beta)q^\beta\lambda^n \right) =\sum_{k=1}^\infty\sum_{n=-\infty}^\infty\sum_{\beta\in H_2(M,\mathbb{Z})}\frac{\operatorname{GV}(g,\beta)}{k}(-1)^{g-1}\left( (-\lambda)^{\frac{k}{2}}-(-\lambda)^{\frac{k}{2}} \right)^{2g-2}q^{k\beta}.

(Ionel & Parker 99, Eq. (0.1), Toda 21, Eq. (7.2) & Thrm. 7.1)

References

Original introduction:

Further references:

See also:

Last revised on December 7, 2024 at 23:52:38. See the history of this page for a list of all contributions to it.

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