Volume 8, issue 2 (2004)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Hodge integrals and invariants of the unknot

Andrei Okounkov and Rahul Pandharipande

Geometry & Topology 8 (2004) 675–699

arXiv: math.AG/0307209

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.

Keywords
Hodge integrals, unknot, Gopakumar–Mariño–Vafa formula
Mathematical Subject Classification 2000
Primary: 14H10
Secondary: 57M27
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Publication
Received: 30 September 2003
Revised: 22 April 2004
Published: 24 April 2004
Proposed: Robion Kirby
Seconded: Shigeyuki Morita, Ralph Cohen
Authors
Andrei Okounkov
Department of Mathematics
Princeton University
Princeton
New Jersey 08544
USA
Rahul Pandharipande
Department of Mathematics
Princeton University
Princeton
New Jersey 08544
USA