Refined Chern-Simons Theory and Topological String

TL;DR

This work develops a refined version of the gauge–string duality by linking refined Chern-Simons theory to refined topological strings on toric Calabi–Yau manifolds. It derives the refined topological vertex from refined CS in the large $N$ limit (AK vertex) and connects it to the CIV vertex via a change of basis, including a second conjugate vertex, thereby embedding refined knot observables into a vertex framework. The authors extend the duality to more general geometries through quiver CS theories and demonstrate a detailed case study on a local ${f P}^2$ geometry, obtaining a vertex-based decomposition and explicit gluing rules. They propose a refined vertex formalism for arbitrary toric Calabi–Yau manifolds using a Morse flow on the toric graph, aligning with emerging ideas on refinement and chamber structure, and they explain a refined large $N$ mirror symmetry for knot invariants. Overall, the paper provides a comprehensive ladder from refined CS theory to a prospective, geometry-wide refined vertex calculus with broad implications for topological strings and knot homologies.

Abstract

We show that refined Chern-Simons theory and large N duality can be used to study the refined topological string with and without branes. We derive the refined topological vertex of hep-th/0701156 and hep-th/0502061 from a link invariant of the refined SU(N) Chern-Simons theory on S^3, at infinite N. Quiver-like Chern-Simons theories, arising from Calabi-Yau manifolds with branes wrapped on several minimal S^3's, give a dual description of a large class of toric Calabi-Yau. We use this to derive the refined topological string amplitudes on a toric Calabi-Yau containing a shrinking P^2 surface. The result is suggestive of the refined topological vertex formalism for arbitrary toric Calabi-Yau manifolds in terms of a pair of vertices and a choice of a Morse flow on the toric graph, determining the vertex decomposition. The dependence on the flow is reminiscent of the approach to the refined topological string in upcoming work of Nekrasov and Okounkov. As a byproduct, we show that large N duality of the refined topological string explains the ``mirror symmetry`` of the refined colored HOMFLY invariants of knots.

Figures (17)

• Figure 1: The geometric transition relating $T^*S^3$ and $Y={\cal O}(-1)\oplus{\cal O}(-1)\rightarrow {\mathbb P^1}$.
• Figure 2: ${\mathbb C}^3$ with three stacks of branes on Lagrangians $L_1$, $L_2$, $L_3$.
• Figure 3: The $T^*S^3$ with three brane stacks. The M2 branes wrapping with holomorphic annuli are schematically shown.
• Figure 4: Branes on the conifold in the large $N$ limit.
• Figure 5: A "double" Hopf link, the starting point for derivation of topological vertex.