Gopakumar–Vafa Large N Duality is a correspondence between
Chern–Simons invariants of a link in a 3–manifold and relative
Gromov–Witten invariants of a 6–dimensional symplectic manifold
relative to a Lagrangian submanifold. We address the correspondence
between the Chern–Simons free energy of S3 with no link and the
Gromov–Witten invariant of the resolved conifold in great detail.
This case avoids mathematical difficulties in formulating a
definition of relative Gromov–Witten invariants, but includes all of
the important ideas.
There is a vast amount of background material related to this
duality. We make a point of collecting all of the background
material required to check this duality in the case of the
3–sphere, and we have tried to present the material in a way
complementary to the existing literature. This paper contains a
large section on Gromov–Witten theory and a large section on quantum
invariants of 3–manifolds. It also includes some physical
motivation, but for the most part it avoids physical terminology.
Keywords
Gromov–Witten invariants,
Chern–Simons invariants, Reshetikhin–Turaev
invariants, Gopakumar–Vafa conjecture, Large N Duality,
3–manifold, symplectic manifold, quantum invariants
