Knot Invariants and Topological Strings

TL;DR

This paper provides evidence for a deep connection between large N Chern-Simons theory and topological string theory, demonstrating that knot invariants match across these theories and revealing new integral invariants related to M-brane spectra.

Contribution

It introduces a reformulation of knot invariants in terms of new integral invariants and explores their relation to M-brane spectra and superpotential terms in supersymmetric theories.

Findings

Wilson loop observables match to all orders in N between theories

New integral invariants encode M2 brane spectrum and spin

Link established between knot invariants and superpotential terms

Abstract

We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of a simple knot on S^3 (for any representation) agrees to all orders in N with the corresponding quantity on the topological string side. For a general knot, we find a reformulation of the knot invariant in terms of new integral invariants, which capture the spectrum (and spin) of M2 branes ending on M5 branes embedded in the resolved conifold geometry. We also find an intriguing link between knot invariants and superpotential terms generated by worldsheet instantons in N=1 supersymmetric theories in 4 dimensions.