Genus expansion of HOMFLY polynomials

TL;DR

This paper explores higher genus corrections to HOMFLY polynomials in Chern-Simons theory, expressing expansion coefficients via symmetric group characters and connecting to Hurwitz theory and integrability.

Contribution

It introduces a genus expansion framework for HOMFLY polynomials using cut-and-join operators, linking knot invariants to Hurwitz theory and integrability properties.

Findings

Expansion coefficients are expressed through eigenvalues of cut-and-join operators.

The z-expansion is naturally exponentiated, revealing integrability structures.

The genus expansion differs from the weak coupling expansion at finite N.

Abstract

In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus corrections to this formula in the form of expansion in powers of z = q-q^{-1}. Expansion coefficients are expressed through the eigenvalues of the cut-and-join operators, i.e. symmetric group characters. Moreover, the z-expansion is naturally exponentiated. Representation through cut-and-join operators makes contact with Hurwitz theory and its sophisticated integrability properties. Our formulas describe the shape of genus expansion for the HOMFLY polynomials, which for their matrix model counterparts is usually controlled by Virasoro like constraints and AMM/EO topological recursion. The genus expansion differs from the better studied weak coupling expansion at finite number of colors N, which is described in terms of the Vassiliev invariants and Kontsevich integral.