On a Proof of a Conjecture of Marino-vafa on Hodge Integrals

where ψi = c1(Li) is the first Chern class of Li, and λj = cj(E) is the j-th Chern class of the Hodge bundle. Hodge integrals arise in the calculations of Gromov-Witten invariants by localization techniques [14, 7]. The explicit evaluation of Hodge integrals is a difficult problem. The Hodge integrals involving only ψ classes can be computed recursively by Witten’s conjecture [26] proven by Kontsevich [13]. Algorithms of computing Hodge integrals are described in [2]. In [23], M. Mariño and C. Vafa obtained a closed formula for a generating function of certain open Gromov-Witten invariants, some of which has been reduced to Hodge integrals by localization techniques which are not fully clarified mathematically. This leads to a conjectural formula of Hodge integrals. To state this formula, we introduce some notation, following [28]. Let