Gopakumar–Vafa invariant
Topological invariants concerning BPS states
In theoretical physics , Rajesh Gopakumar and Cumrun Vafa introduced in a series of papers[ 1] [ 2] [ 3] [ 4] 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 β -->
∈ ∈ -->
H
2
(
M
,
Z
)
GW
(
g
,
β β -->
)
q
β β -->
λ λ -->
2
g
− − -->
2
=
∑ ∑ -->
g
=
0
∞ ∞ -->
∑ ∑ -->
k
=
1
∞ ∞ -->
∑ ∑ -->
β β -->
∈ ∈ -->
H
2
(
M
,
Z
)
BPS
(
g
,
β β -->
)
1
k
(
2
sin
-->
(
k
λ λ -->
2
)
)
2
g
− − -->
2
q
k
β 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 β -->
{\displaystyle \beta }
pseudoholomorphic curves with genus g ,
λ λ -->
{\displaystyle \lambda }
q
β β -->
=
exp
-->
(
2
π π -->
i
t
β β -->
)
{\displaystyle q^{\beta }=\exp(2\pi it_{\beta })}
t
β β -->
{\displaystyle t_{\beta }}
β β -->
{\displaystyle \beta }
GW
(
g
,
β β -->
)
{\displaystyle {\text{GW}}(g,\beta )}
β β -->
{\displaystyle \beta }
g
{\displaystyle g}
BPS
(
g
,
β β -->
)
{\displaystyle {\text{BPS}}(g,\beta )}
β β -->
{\displaystyle \beta }
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 β -->
∈ ∈ -->
H
2
(
M
,
Z
)
BPS
(
g
,
β β -->
)
1
k
(
2
sin
-->
(
k
λ λ -->
2
)
)
2
g
− − -->
2
q
k
β β -->
]
.
{\displaystyle Z_{top}=\exp \left[\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 }\right]\ .}
Notes
References
Gopakumar, Rajesh; Vafa, Cumrun (1998a), M-Theory and Topological strings-I , arXiv :hep-th/9809187 Bibcode :1998hep.th....9187G Gopakumar, Rajesh; Vafa, Cumrun (1998b), M-Theory and Topological strings-II , arXiv :hep-th/9812127 Bibcode :1998hep.th...12127G Gopakumar, Rajesh; Vafa, Cumrun (1999), "On the Gauge Theory/Geometry Correspondence", Adv. Theor. Math. Phys. , 3 (5): 1415–1443, arXiv :hep-th/9811131 Bibcode :1998hep.th...11131G , doi :10.4310/ATMP.1999.v3.n5.a5 , S2CID 13824856 Gopakumar, Rajesh; Vafa, Cumrun (1998d), "Topological Gravity as Large N Topological Gauge Theory", Adv. Theor. Math. Phys. , 2 (2): 413–442, arXiv :hep-th/9802016 Bibcode :1998hep.th....2016G , doi :10.4310/ATMP.1998.v2.n2.a8 , S2CID 16676561 Ionel, Eleny-Nicoleta ; Parker, Thomas H. (2018), "The Gopakumar–Vafa formula for symplectic manifolds", Annals of Mathematics 187 (1): 1–64, arXiv :1306.1516 doi :10.4007/annals.2018.187.1.1 , MR 3739228 , S2CID 7070264
![{\displaystyle Z_{top}=\exp \left[\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 }\right]\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb2d64366ad33fcdf01aaab7918abafe3e1ebb1)
