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Gopakumar–Vafa invariant

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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:

∑ g = 0 ∞   ∑ β ∈ H 2 ( M , Z ) GW ( g , β ) q β λ 2 g − 2 = ∑ g = 0 ∞   ∑ k = 1 ∞   ∑ β ∈ H 2 ( M , Z ) BPS ( g , β ) 1 k ( 2 sin ⁡ ( k λ 2 ) ) 2 g − 2 q k β {\displaystyle \sum _{g=0}^{\infty }~\sum _{\beta \in H_{2}(M,\mathbb {Z} )}{\text{GW}}(g,\beta )q^{\beta }\lambda ^{2g-2}=\sum _{g=0}^{\infty }~\sum _{k=1}^{\infty }~\sum _{\beta \in H_{2}(M,\mathbb {Z} )}{\text{BPS}}(g,\beta ){\frac {1}{k}}\left(2\sin \left({\frac {k\lambda }{2}}\right)\right)^{2g-2}q^{k\beta }} ,

where

  • β {\displaystyle \beta } is the class of pseudoholomorphic curves with genus g,
  • λ {\displaystyle \lambda } is the topological string coupling,
  • q β = exp ⁡ ( 2 π i t β ) {\displaystyle q^{\beta }=\exp(2\pi it_{\beta })} with t β {\displaystyle t_{\beta }} the Kähler parameter of the curve class β {\displaystyle \beta } ,
  • GW ( g , β ) {\displaystyle {\text{GW}}(g,\beta )} are the Gromov–Witten invariants of curve class β {\displaystyle \beta } at genus g {\displaystyle g} ,
  • BPS ( g , β ) {\displaystyle {\text{BPS}}(g,\beta )} are the number of BPS states (the Gopakumar–Vafa invariants) of curve class β {\displaystyle \beta } at genus g {\displaystyle 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 t o p = exp ⁡ [ ∑ g = 0 ∞   ∑ k = 1 ∞   ∑ β ∈ H 2 ( M , Z ) BPS ( g , β ) 1 k ( 2 sin ⁡ ( k λ 2 ) ) 2 g − 2 q k β ]   . {\displaystyle Z_{top}=\exp \left\ .}

Notes

  1. Gopakumar & Vafa 1998a
  2. Gopakumar & Vafa 1998b
  3. Gopakumar & Vafa 1999
  4. Gopakumar & Vafa 1998d

References


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