  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 S^3 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.
