Formulae of one-partition and two-partition Hodge integrals

In [46], Witten related topological string theory on the cotangent bundle T M of a three manifold M to the Chern–Simons gauge theory on M . Gopakumar and Vafa [11] related the topological string theory on the deformed conifold (the cotangent bundle T S of the 3–sphere) to that on the resolved conifold (the total space of OP1. 1/ ̊OP1. 1/! P ). These works lead to a duality between the topological string theory on the resolved conifold and the Chern–Simons theory on S . A mathematical consequence of this duality is a surprising relationship between Gromov–Witten invariants, which arise in the topological string theory, and knot invariants, which arise in the Chern–Simons theory. Ooguri and Vafa [43] proposed that Chern–Simons knot invariants of a knot K in S should be related to open Gromov–Witten invariants of .X;LK /, where X is the resolved conifold and LK is a Lagrangian submanifold of X canonically associated to K .