Polynomial invariants for torus knots and topological strings

TL;DR

This paper tests a conjecture linking Chern-Simons gauge theory and topological string theory by calculating Wilson loop vevs for torus knots and confirming their agreement with string theory predictions, supporting knot invariants as enumerative generating functions.

Contribution

It develops a systematic method to derive string amplitudes from Chern-Simons Wilson loop vevs and confirms their agreement for torus knots, strengthening the gauge-string correspondence.

Findings

Complete agreement between gauge theory calculations and string theory predictions.

New systematic procedure for extracting string amplitudes from Wilson loop vevs.

Evidence that knot invariants serve as generating functions for enumerative problems.

Abstract

We make a precision test of a recently proposed conjecture relating Chern-Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern-Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU(N) for torus knots. We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes. We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern-Simons vevs. Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems.