Geometry & Topology Monographs 8 (2006) 105–128

DOI: 10.2140/gtm.2006.8.105

arXiv: math.AG/0502430

Abstract

Prompted by the duality between open string theory on noncompact Calabi–Yau threefolds and Chern–Simons theory on three-manifolds, M Mariño and C Vafa conjectured a formula of one-partition Hodge integrals in term of invariants of the unknot. Many Hodge integral identities, including the λ_g conjecture and the ELSV formula, can be obtained by taking limits of the Mariño–Vafa formula.

Motivated by the Mariño–Vafa formula and formula of Gromov–Witten invariants of local toric Calabi–Yau threefolds predicted by physicists, J Zhou conjectured a formula of two-partition Hodge integrals in terms of invariants of the Hopf link and used it to justify the physicists’ predictions.

In this expository article, we describe proofs and applications of these two formulae of Hodge integrals based on joint works of K Liu, J Zhou and the author. This is an expansion of the author’s talk of the same title at the BIRS workshop The Interaction of Finite Type and Gromov–Witten Invariants, November 15th–20th 2003.

Keywords

Hodge integrals, Mariño–Vafa formula, knot invariants, HOMFLY polynomials, Gromov–Witten theory, Chern–Simons theory

Mathematical Subject Classification

Primary: 14N35, 53D45

Secondary: 57M25

References

Publication

Received: 1 January 2005
Revised: 26 January 2005
Accepted: 1 February 2005
Published: 22 April 2006

Authors