Some Results of Marino-Vafa Formula.pdf

1、Based on the string duality Marifio and Vafa [MV] made a conjecture between Chem-Simons and string theory which generalizes the well-known Witten COnjecture/Kontsevich theorem([w]，[Ko1]，[CLL]，[KarL]，[KiL]，[LX4]，[Mil，Mi2]

2、，[S]，[MS]，[OP1]，[LX4]).In 2003，C.-C.Liu，K.Liu，and J.Zhou [LLZ1]gave a proof of Mari(n)o-Vafa conjecture fanother proof see [OP1])using the localization technique on the moduli space of relative morphisms (M)g,0(P1，μ)cons

3、tructed by J.Li [Li1，Li2].In this paper，we use the Mari(n)oVafa formula to give a simple proof ofλ9 conjecture([FabP3]，[GJV2]，[LiY]，[LLZ2])and Witten con]ecture，also find a closed formula of a certain type of Hodge integ