Hodge integrals and invariants of the unknot

Abstract

We prove the Gopakumar-Mariño-Vafa formula for special cubic Hodge integrals. The GMV formula arises from Chern-Simons/string duality applied to the unknot in the three sphere. The GMV formula is a q -analog of the ELSV formula for linear Hodge integrals. We find a system of bilinear localization equations relating linear and special cubic Hodge integrals. The GMV formula then follows easily from the ELSV formula. An operator form of the GMV formula is presented in the last section of the paper.

A mathematical theory of the topological vertex

“…The main result of [22] is the following formula conjectured by Mariño and Vafa [25] (see Okounkov and Pandharipande [29] for another proof):…”

A Proof of a Conjecture of Mariño-Vafa on Hodge Integrals

“…The purpose of this paper is to supply various computational details, and to present some related results. See [25] for another approach to the Mariño-Vafa formula.…”

Localization and string duality

“…See [14] for the details of the derivation in this case. The second is by OkounkovPandhrapande [26], they gave a different approach by using the ELSV formula as initial value, and as well as the λ g conjecture and other recursion relations from localization on the moduli spaces of stable maps to P 1 .…”