Geometry & Topology, Vol. 8 (2004) Paper no. 17, pages 675--699.

Hodge integrals and invariants of the unknot

A Okounkov, R Pandharipande

Abstract. We prove the Gopakumar-Marino-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-Marino-Vafa formula

AMS subject classification. Primary: 14H10. Secondary: 57M27.

DOI: 10.2140/gt.2004.8.675

E-print: arXiv:math.AG/0307209

Submitted to GT on 30 September 2003. (Revised 22 April 2004.) Paper accepted 13 February 2004. Paper published 24 April 2004.

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A Okounkov, R Pandharipande
Department of Mathematics, Princeton University
Princeton, NJ 08544, USA

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