Refined Chern-Simons Theory and Topological String

TL;DR

This paper connects refined Chern-Simons theory with topological string theory, deriving the refined topological vertex from link invariants, and explores dualities and mirror symmetry in refined invariants for toric Calabi-Yau manifolds.

Contribution

It derives the refined topological vertex from refined Chern-Simons theory and proposes a formalism for arbitrary toric Calabi-Yau manifolds using a pair of vertices and Morse flow.

Findings

Derived the refined topological vertex from link invariants.

Established dual descriptions of toric Calabi-Yau manifolds with branes.

Linked large N duality to mirror symmetry of refined knot invariants.

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.