Gopakumar–Vafa invariant - Alchetron, the free social encyclopedia

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.

∑ g ≥ 0 , n ≥ 1 , β ∈ H 2 ( M , Z ) G W ( g , β ) q β λ 2 g − 2 = ∑ k > 0 , r ≥ 0 , β ∈ H 2 ( M , Z ) B P S ( r , β ) 1 k ( 2 sin ⁡ ( k λ 2 ) ) 2 r − 2 q k β

where G W ( g , β ) is Gromov–Witten invariant, β the number of pseudoholomorphic curves with genus g and B P S ( r , β ) 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 t o p = exp ⁡ [ ∑ k > 0 , r ≥ 0 , β ∈ H 2 ( M , Z ) B P S ( r , β ) 1 k ( 2 sin ⁡ ( k λ 2 ) ) 2 r − 2 q k β ⋅ t ] .

References

Gopakumar–Vafa invariant Wikipedia

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